A new approach to velocity averaging lemmas in Besov spaces

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J. Math. Pures Appl. 101 (2014) 495–551 www.elsevier.com/locate/matpur

A new approach to velocity averaging lemmas in Besov spaces Diogo Arsénio a,∗,1 , Nader Masmoudi b,1 a Institut de Mathématiques de Jussieu – Paris Rive Gauche, Campus des Grands Moulins, Bâtiment Sophie Germain, 75013 Paris, France b Courant Institute of Mathematical Sciences, 251 Mercer St, New York, NY 10012, United States

Received 27 July 2012 Available online 3 July 2013

Abstract We develop a new approach to velocity averaging lemmas based on the dispersive properties of the kinetic transport operator. This method yields unprecedented sharp results in critical Besov spaces, which display, in some cases, a gain of one full derivative. p Moreover, the study of dispersion allows to treat the case of Lrx Lv integrability with r  p. We also establish results on the control 1 of concentrations in the degenerate Lx,v case, which is fundamental in the study of hydrodynamic limits of the Boltzmann equation. © 2013 Elsevier Masson SAS. All rights reserved. Résumé On développe une nouvelle approche des lemmes de moyenne utilisant les propriétés dispersives de l’opérateur de transport cinétique. Cette méthode fournit des résultats précis dans des espaces de Besov critiques donnant, dans certain cas, un gain d’une p dérivée entière. De plus, l’étude de la dispersion permet de traiter le cas d’intégrabilité Lrx Lv avec r  p. On présente également des 1 résultats de contrôle des concentrations dans le cas dégénéré Lx,v , ce qui est fondamental dans l’étude des limites hydrodynamiques de l’équation de Boltzmann. © 2013 Elsevier Masson SAS. All rights reserved. Keywords: Kinetic transport equation; Velocity averaging; Dispersion; Besov spaces

1. Introduction The regularizing properties of the kinetic transport equation were first established in [14] for the basic Hilbertian D case. Essentially, the main result therein states that if f (x, v), g(x, v) ∈ L2 (RD x × Rv ), where D  1 is the dimension, satisfy the stationary transport relation v · ∇x f (x, v) = g(x, v), * Corresponding author.

E-mail addresses: [email protected] (D. Arsénio), [email protected] (N. Masmoudi). 1 Both authors were partially supported by the NSF grant DMS-0703145.

0021-7824/$ – see front matter © 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.matpur.2013.06.012

(1.1)

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in the sense of distributions, then, for any given φ(v) ∈ C0∞ (RD ), the velocity average verifies that   1 f (x, v)φ(v) dv ∈ H 2 RD x .

(1.2)

RD

This kind of smoothing property was then further investigated in [13], where the Hilbertian case was refined and D some extensions to the non-Hilbertian case f, g ∈ Lp (RD x × Rv ), with p = 2, were obtained, but these were not optimal. The first general results in the non-Hilbertian case establishing an optimal gain of regularity were obtained in [12] using Besov spaces and interpolation theory. The methods employed therein were not optimal in all aspects, though, but they were robust and thus allowed to further extend the averaging lemmas to settings bearing much more generality and still exhibiting an optimal gain of regularity. To be precise, the results from [12] were able to treat the case v · ∇x f (x, v) = (1 − x )α (1 − v )β g(x, v),

(1.3)

D q D D where 0  α < 1, β  0, f ∈ Lp (RD x × Rv ) and g ∈ L (Rx × Rv ), with 1 < p, q < ∞. Another interesting q D p generalization from [12] concerned the case where f, g ∈ L (Rv ; L (RD x )) satisfy the transport equation (1.1) with 1 < q  p < ∞. The corresponding results in standard Sobolev spaces were then obtained in [7], but the methods of proof were complicated and relied on harmonic analysis on product spaces. Unfortunately, as pointed out in [28], the proofs in D q D D [7] were flawed in the non-homogeneous cases, i.e. in the case f ∈ Lp (RD x × Rv ) and g ∈ L (Rx × Rv ), with q D p D 1 < p, q < ∞, p = q, and in the case f, g ∈ L (Rv ; L (Rx )), with 1 < q < p < ∞. Note that another general approach, which yielded similar results in abstract interpolation spaces resulting from the real interpolation of Besov spaces, was developed in [18]. The interesting case of velocity averaging where f (x, v) and g(x, v) have more local integrability in v than q D in x, i.e. f, g ∈ Lp (RD x ; L (Rv )) with 1 < p < q < ∞, was not addressed until [28]. To be precise, the simple question raised therein was: is it possible to improve, at least locally, the properties of the velocity averages in the q D p D D case f, g ∈ Lp (RD x ; L (Rv )), with p  q, with respect to the case f, g ∈ L (Rx × Rv )? The answer from [28] was definitely affirmative, even though it did not provide a general approach to this setting. Some other similar but very specific cases were treated in [18]. In the present work, whose main results are exposed in Section 4, we provide a very general, robust and unified method for establishing velocity averaging lemmas. In particular, our approach yields sharp results in critical Besov spaces. Indeed, previous methods (cf. [12,18], for instance) only provided optimal gains of regularity on velocity averages in some non-critical larger Besov spaces, for they were based on abstract interpolation procedures. It also allows us to treat some cases where the local integrability in velocity is improved and, thus, to extend the discussion initiated in [28] on the improved gain of regularity with respect to the velocity integrability. Our approach has its own limitations, though, and doesn’t apply to the whole range of parameters 1  p  q  ∞. However, as discussed later on, it exhibits some optimality in the range of parameters where it is valid. Finally, it is to be emphasized that the methods presented here can be used to establish corresponding results in critical Sobolev spaces. The main idea in our analysis, presented in Section 3, consists in utilizing the so-called dispersive properties of the kinetic transport equation, which are best expressed in the interpolation formula

t f (x, v) = f (x − tv, v) +

g(x − sv, v) ds,

(1.4)

0

formally satisfied by solutions of the transport equation (1.1) (cf. Section 3 for its derivation), together with standard averaging methods. Moreover, this interpolation formula will be well-suited for the application of the so-called dispersive estimates first used in [5] and further developed in [8,21]. It is at first uncertain whether these dispersive properties are even related to velocity averaging lemmas. However, our work clearly establishes a rather strong link between these two properties. Most of the above-mentioned developments came to include more cases that were imposed by the underlying applications. In particular, one of the last improvements of averaging lemmas from [15] was motivated by the hydrodynamic limit of the Boltzmann equation [16], where it was necessary to have a refined averaging lemma in

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D L1 (RD x × Rv ). This was maybe the first time that some dispersive estimates were used in the proof of an averaging lemma. This paper is also somewhat motivated by the hydrodynamic limit of the Boltzmann equation but in the case of D long-range interactions [1,2]. Thus, our work allows to obtain an extension of the averaging lemma in L1 (RD x × Rv ) from [15], which is crucially employed in [1,2]. The main novelty in the L1 averaging lemmas compared to [15] is that we are able to include derivatives in the right-hand side of the transport equation, i.e. we consider the relation (1.3) rather than (1.1). This is actually crucial when dealing with the Boltzmann equation in the presence of long-range interactions, since the Boltzmann collision operator is only defined (after renormalization) in the sense of distributions as a singular integral operator. We present this application of our main results to the theory of hydrodynamic limits in Section 5. See also [4] for a different approach to the same problem based on hypoellipticity. For the sake of simplicity, we will not treat the time dependent transport equation

(∂t + v · ∇x )f (t, x, v) = g(t, x, v).

(1.5)

However, it is to be emphasized that similar results can be obtained in this setting and that the main results presented in Section 4 can be adapted to include a time variable. Still, for completeness, without stating precise results, we detail a time dependent strategy in Appendix C. The results in this article rely on the theory of function spaces and interpolation. Thus, prior to stating the main theorems in Section 4, we present in the preliminary Section 2 the main tools and notations on the theory of Besov spaces that will be utilized throughout this work, and in Section 3 we explain the key ideas behind our approach to velocity averaging lemmas leading to the main results from Section 4. Regarding interpolation theory, we provide in Appendix A, for later reference, a very brief introduction. We refer the reader to [6,27] for more details on the subject. In Section 5, we apply our main results to establish an endpoint result on the control of concentrations in L1 , which has fundamental consequences on the study of hydrodynamic limits of the Boltzmann equation. Finally, Section 6 contains the proofs of the main results from Section 4. 2. Littlewood–Paley decompositions and Besov spaces We will denote the Fourier transform fˆ(ξ ) = Ff (ξ ) =



e−iξ ·v f (v) dv

(2.1)

RD

and its inverse g(v) ˜ =F

−1

1 g(v) = (2π)D

 eiv·ξ g(ξ ) dξ.

(2.2)

RD

We introduce now a standard Littlewood–Paley decomposition of the frequency space into dyadic blocks. To this end, let ψ(ξ ), ϕ(ξ ) ∈ C0∞ (RD ) be such that     1 ψ, ϕ  0 are radial, supp ψ ⊂ |ξ |  1 , supp ϕ ⊂  |ξ |  2 2 ∞   and 1 = ψ(ξ ) + ϕ 2−k ξ , for all ξ ∈ RD . (2.3) k=0

Defining the scaled functions

ψδ (ξ ) = ψ( ξδ )

and ϕδ (ξ ) = ϕ( ξδ ), for any δ > 0, one has then that   δ supp ψδ ⊂ {|ξ |  δ}, supp ϕδ ⊂  |ξ |  2δ 2 ∞ and 1 ≡ ψδ + ϕδ2k . k=0

(2.4)

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Furthermore, we will use the Fourier multiplier operators     Sδ , δ : S RD → S RD

(2.5)

(here S denotes the space of tempered distributions) defined by     Sδ f = F −1 ψδ ∗ f and δ f = F −1 ϕδ ∗ f,

(2.6)

so that Sδ f +

∞

δ2k f = f,

(2.7)

k=0

where the series is convergent in S . For notational convenience, we also introduce, for every 0 < δ1 < δ2 , the operators 0 = S 1

and [δ1 ,δ2 ] = S2δ2 − Sδ1 ,

(2.8)

so that F[δ1 ,δ2 ] f coincides with Ff on {δ1  ξ  δ2 } and is supported on the domain { δ21  ξ  2δ2 }. s (RD ), for any s ∈ R and 1  p, q  ∞, as the subspaces of Now, we may define the standard Besov spaces Bp,q tempered distributions endowed with the norm

f Bp,q s (RD ) =

q 0 f Lp (RD )

+

∞

1 q

2

ksq

q 2k f Lp (RD )

(2.9)

,

k=0

if q < ∞, and with the obvious modifications in case q = ∞. s (RD ), for any s ∈ R and 1  p, q  ∞, as the subspaces We also introduce the homogeneous Besov spaces B˙ p,q of tempered distributions endowed with the semi-norm

f B˙ s

D p,q (R )

=

∞

1 q

q 2ksq 2k f Lp (RD )

(2.10)

,

k=−∞

if q < ∞, and with the obvious modifications in case q = ∞. s (RD ) is an analog of the standard Sobolev space W s,p (RD ), In terms of dimensional analysis, the Besov space Bp,q in the sense that both spaces enjoy similar scaling properties, no matter what the value of 1  q  ∞ is. The same is s (RD ) and W ˙ s,p (RD ). Moreover, one can show that true regarding their homogeneous counterparts, B˙ p,q  D    D s s R ⊂ W s,p RD ⊂ Bp,2 R , (2.11) Bp,p for any s ∈ R and 1 < p  2, and that

 D    D s s R ⊂ W s,p RD ⊂ Bp,p R , Bp,2

(2.12)

for any s ∈ R and 2  p < ∞. For functions depending on two variables x ∈ RD and v ∈ RD , we will use the Littlewood–Paley decomposition on each variable. That is, denoting D = {0} ∪ {2k ∈ N: k ∈ N}, we can write f (x, v) = xi vj f (x, v), (2.13) i,j ∈D x v D for any f (x, v) ∈ S (RD x × Rv ), where we employ the superscripts to emphasize that the multipliers i and j t,s D D solely act on the respective variables x and v. Thus, we define now the mixed Besov spaces Br,p,q (Rx × Rv ), for any s, t ∈ R and 1  p, q, r  ∞, as the subspaces of tempered distributions endowed with the norm t,s f Br,p,q (RD ×RD ) = x

v

i,j ∈D

q xi vj f Lr (RD ;Lp (RD )) x v

sq

(1 ∨ i) (1 ∨ j ) tq

1 q

,

(2.14)

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499

if q < ∞, and with the obvious modifications in case q = ∞, where the symbol a ∨ b, for any a, b ∈ R, stands for the maximum between a and b. t,s D We also introduce the homogeneous mixed Besov spaces B˙ r,p,q (RD x × Rv ), for any s, t ∈ R and 1  p, q, r  ∞, as the subspaces of tempered distributions endowed with the norm 1 q

ktq lsq x v q t,s f B˙ r,p,q = 2 2  f , (2.15)  (RD ×RD ) Lr (RD ;Lp (RD )) 2k 2l x

v

x

k,l∈Z

v

if q < ∞, and with the obvious modifications in case q = ∞. t,s D In terms of dimensional analysis, the Besov space Br,p,q (RD x × Rv ) is an analog of the two variable Sobolev t s p D space composed of functions f (x, v) such that (1 − x ) 2 (1 − v ) 2 f ∈ Lr (RD x ; L (Rv )), in the sense that both spaces enjoy similar scaling properties, no matter what the value of 1  q  ∞ is. The same is true regarding their homogeneous counterparts. s D r Next, in addition to the spaces Lr (RD x ; Bp,q (Rv )), for any s ∈ R and 1  p, q, r  ∞, which are defined as L s r D s D  spaces with values in the Banach spaces Bp,q , we further define the spaces L (Rx ; Bp,q (Rv )) as the subspaces of tempered distributions endowed with the norm

1 ∞ q

v q

q

 f r D p D + f r D s 2ksq vk f r D p D , (2.16) D = L (Rx ;Bp,q (Rv ))

0

L (Rx ;L (Rv ))

2

k=0

L (Rx ;L (Rv ))

if q < ∞, and with the obvious modifications in case q = ∞. This kind of spaces were first introduced by Chemin and Lerner in [9] with (x, v) replaced by (t, x) (time and space) and were used in many problems related to the Navier–Stokes equations (cf. [11] for instance). One can check easily that, if q  r, then   D    D  s s r RD ⊂L (2.17) Lr R D x ; Bp,q Rv x ; Bp,q Rv , and that, if q  r, then Furthermore, it holds that

  D    D  s s r RD ⊂ L r RD L x ; Bp,q Rv x ; Bp,q Rv .

(2.18)

   D   D  0,s  D 0,s r D s Rx × R D ⊂ Br,p,∞ Rx × R D Br,p,1 v ⊂ L Rx ; Bp,q Rv v ,

(2.19)

for all s ∈ R and 1  p, q, r  ∞. D ˙s r (RD Finally, we introduce the homogeneous Besov spaces L x ; Bp,q (Rv )), for any s ∈ R and 1  p, q, r  ∞, as the subspaces of tempered distributions endowed with the norm

∞ 1 q

ksq v q f L 2 (2.20) 2k f Lr (RD ;Lp (RD )) , r (RD ;B˙ s (RD )) = x

p,q

v

k=−∞

x

v

if q < ∞, and with the obvious modifications in case q = ∞. s D r (RD In terms of dimensional analysis, the Besov space L x ; Bp,q (Rv )) is an analog of the two variable Sobolev s p D space composed of functions f (x, v) such that (1 − v ) 2 f ∈ Lr (RD x ; L (Rv )), in the sense that both spaces enjoy similar scaling properties, no matter what the value of 1  q  ∞ is. The same is true regarding their homogeneous counterparts. 3. Ellipticity, dispersion and averaging Here, we explain the concepts which will lead to the proofs of the main results in this work. The classical theory of velocity averaging lemmas in L2x,v , first developed in [13,14], is based on a simple but D ingenious microlocal decomposition. More precisely, if f (x, v), g(x, v) ∈ L2 (RD x × Rv ) satisfy the transport relation (1.1) then, considering the Fourier transforms fˆ(η, v) and g(η, ˆ v) in the space variable only, it holds that ˆ iv · ηf (η, v) = g(η, ˆ v). (3.1)

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Therefore, it is possible to exploit some ellipticity of the transport operator as long as one remains on an appropriate microlocal domain. In other words, we may invert the transport operator as long as the quantity |v · η| remains uniformly bounded away from zero: fˆ(η, v) =

  on |v · η| > 1 , say.

1 g(η, ˆ v), iv · η

(3.2)

Thus, introducing some cutoff function ρ ∈ S(R) (S denotes the Schwartz space of rapidly decaying functions) such that ρ(0) = 1 and an interpolation parameter t > 0, we may decompose 1 − ρ(tv · η) g(η, ˆ v). fˆ(η, v) = ρ(tv · η)fˆ(η, v) + iv · η

(3.3)

It follows that each term in the right-hand side may then be estimated locally in L2v and the remainder of the proof simply consists in choosing the optimal value for the interpolation parameter t (which will depend on η). The  conclusion of this method yields that, for every test function φ(v) ∈ C0∞ , the velocity average f (x, v)φ(v) dv 1

belongs to Hx2 . This approach yields optimal results and exhibits the crucial regularizing properties of the transport operator, which, we insist, is based on exploiting some partial ellipticity. p As mentioned before, several extensions of this method are possible (cf. [7,12]), in particular, the Lx,v case of 2 velocity averaging lemmas is obtained by interpolating the preceding Lx,v result with the degenerate case in L1x,v . Indeed, if f (x, v), g(x, v) ∈ L1x,v , then absolutely no regularity may be gained on the velocity averages from the transport equation, which is unfortunately optimal as far as the gain of regularity is concerned. In this work, we obtain refined velocity averaging results by further exploiting the dispersive properties of the transport operator discovered by Castella and Perthame in [8]. This requires the development of a suitable interpolation formula, more refined than (3.3), and the study of its properties. Thus, introducing an interpolation parameter t > 0, it trivially holds, from (1.1), that  (∂t + v · ∇x )f = g, (3.4) f (t = 0) = f. Hence the interpolation formula, t f (x, v) = f (x − tv, v) +

g(x − sv, v) ds,

(3.5)

0

which is in fact dual to the interpolation formula employed in [15] and is merely Duhamel’s representation formula for the time dependent transport equation (3.4). At this point, it is already possible to obtain an elementary dispersive result based on (3.5), which is, in fact, a quantification of the equi-integrability result from [15]. We formulate this basic result in the following proposition: 1 1 p D Proposition 3.1. Let f (x, v), g(x, v) ∈ Lr (RD x ; L (Rv )), where 1  r  p  ∞ and D( r − p ) < 1, be such that

Then,

v · ∇x f = g.

(3.6)

  D  r f ∈ Lp RD x ; L Rv ,

(3.7)

and the following estimate holds 1−D( 1 − p1 )

f Lpx Lr  Cr,p f Lr Lp r v

x

v

D( 1 − p1 )

gLr Lr p x

v

,

(3.8)

where Cr,p > 0 only depends on r and p. Proof. This simple bound will follow from the dispersive properties of the transport operator, which were first established by Castella and Perthame in [8]. More precisely, we will utilize the following dispersive estimate, valid for any 1  r  p  ∞, which is a simple consequence of the change of variables v → y = x − tv,

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h(x − tv, v)

=t

p

Lx Lrv

=t



x −y

h y,

t

− Dr

−D( 1r − p1 )



h(x, v)

t

p



x −y

h y,

t

− Dr

Lx Lry p

Lrx Lv

501

p

Lry Lx

(3.9)

.

Thus, applying (3.9) to the interpolation formula (3.5), we easily obtain that f 

p Lx Lrv

t

−D( 1r − p1 )

t f 

p Lrx Lv

+

s

−D( 1r − p1 )

ds gLr Lpv ,

(3.10)

x

0

where the above integral is finite if and only if D( 1r − p1 ) < 1. Then, optimizing the preceding estimate in t > 0 yields (3.8), which concludes the proof of the proposition. 2 Note that, whereas the elementary dispersive estimate (3.9) on the transport flow is valid for the full range of parameters 1  r  p  ∞, the conclusion of Proposition 3.1 requires the condition D( 1r − p1 ) < 1, which may seem, at first, wrongly suboptimal. However, it is readily seen from the simplicity of the preceding proof that this restriction is natural and, in fact, it is optimal in the sense that the transport equation (3.6) cannot yield a gain in regularity greater than one full derivative, since the transport operator is a differential operator of order one. Indeed, reasoning in the p space variable x only, the integrability of f in x is improved by Proposition 3.1 from Lrx to Lx , which corresponds −D( 1 − 1 ),p

precisely to a gain of D( 1r − p1 ) derivatives when compared to the Sobolev injection Lrx ⊂ Wx r p . This kind of restriction on the parameters will be thoroughly discussed in Section 4 in connection to our main results. 1 · D D Next, considering any f ∈ S(RD x × Rv ), χ ∈ S(R ) and denoting χλ (·) = λD χ( λ ), where λ > 0, one easily verifies the following rule of action of convolutions on velocity averages,    1 x −y χλ ∗x f (x − tv, v) dv = χ f (y − tv, v) dv dy λD λ RD

RD ×RD



1

= RD ×RD



=

( λt )D

χ

 f (x − tw, v) dw dv

w−v λ t

(χ λ ∗v f )(x − tw, w) dw, t

(3.11)

RD

where we used the change of variables (v, y) → (v, w = x−y t + v). In the notation of Section 2 and in particular (2.6) and (2.8), one then checks employing the above rule of action of convolutions that the dyadic frequency blocks act on velocity averages according to the identities, where δ > 0,    v  St f (x − tv, v) dv, x0 f (x − tv, v) dv =    v  tδ f (x − tv, v) dv. (3.12) xδ f (x − tv, v) dv = Next, we apply this identity to the velocity averages of the above interpolation formula (3.5) to deduce 





f (x, v) dv =

x0 RD

 f (x, v) dv =

RD

t 



(x − tv, v) dv +

RD

 x2k

Stv f

RD



 vt2k f (x − tv, v) dv +

0 RD t 



 Ssv g (x − sv, v) dv ds,



 vs2k g (x − sv, v) dv ds.

(3.13)

0 RD

The above representation formula can be used explicitly to establish the simplest forms of our main results. It is the first key idea in our approach. Indeed, it shows that the space frequencies of the averages of f (x, v) may be

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controlled with the velocity frequencies of f (x, v) and g(x, v). This property of transfer of frequencies is linked to the hypoellipticity of the transport operator and we refer to [4] for recent developments on this matter.  Moreover, it can be used in some case to give a sense to the velocity average RD f (x, v) dv even when f (x, v) is a priori not globally integrablein velocity. Indeed, supposing that we can show (using dispersive estimates for instance) t that (vt2k f )(x − tv, v) and 0 (vs2k g)(x − sv, v) ds actually are globally integrable in velocity for each given t > 0,  then we may nevertheless define the velocity average x2k RD f (x, v) dv by the identities (3.13), and similarly for the low frequencies component. This principle is used implicitly in the statements of Theorems 4.1, 4.2, 4.3 and 4.4, but we will never render this argument explicit for the sake of simplicity. A very simple but useful refinement of the above formulas (3.13) follows from the fact that S2x x0 = x0 and x [2k−1 ,2k+1 ] x2k = x2k . Thus, we obtain 

 f (x, v) dv =

x0 RD



RD



 (x − tv, v) dv +

S2x Stv f 

RD



 S2x Ssv g (x − sv, v) dv ds,

0 RD



 x[2k−1 ,2k+1 ] vt2k f (x − tv, v) dv

f (x, v) dv =

x2k

t 

RD

t  +



 x[2k−1 ,2k+1 ] vs2k g (x − sv, v) dv ds,

(3.14)

0 RD

which considerably decreases the set of frequencies of f (x, v) and g(x, v) required to control the frequencies of the velocity averages. p p The formulas (3.13) and (3.14) will be used to treat the velocity averages in the L1x Lv and the L1v Lx settings, which are endpoint cases. As usual, the more general cases will then be obtained by interpolation with the classical L2x,v case of velocity averaging. However, a significant obstruction to this interpolating strategy lies in that the representation formulas (3.3), which exploits the elliptic properties of the transport operator, and (3.5), which is based on the dispersion of the transport operator, are of different nature and thus seem at first to be incompatible. There is however an elementary but crucial link between them which we establish now. Similar ideas used in [4].  are 1 irs ρ(s) To this end, we first notice, recalling the Fourier inversion formula ρ(r) = 2π e ˆ ds, that it trivially holds, R for any t ∈ R, that  1 ˆ ds, (3.15) fˆ(η, v)ρ(tη · v) = fˆ(η, v)eistη·v ρ(s) 2π and similarly, since 1 = ρ(0) =

1 2π



R

ˆ ds R ρ(s)

and further noticing st

−

eiη·vσ dσ = 0

1 − eistη·v , iη · v

(3.16)

that 1 − ρ(tη · v) 1 g(η, ˆ v) = iη · v 2π

 R

1 − eistη·v 1 g(η, ˆ v) ρ(s) ˆ ds = − iη · v 2π

 st g(η, ˆ v)eiη·vσ dσ ρ(s) ˆ ds. R 0

Then, simply taking the inverse Fourier transform of the above identities, we obtain   1 1 Fx−1 ρ(tη · v)Fx f (x, v) = Fx−1 eistη·v Fx f (x, v)ρ(s) ˆ ds = f (x + stv, v)ρ(s) ˆ ds, 2π 2π R

and

(3.17)

R

(3.18)

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1 − ρ(tη · v) Fx−1 Fx g(x, v) = − iη · v

=−

1 2π 1 2π

 st

503

Fx−1 eiσ η·v Fx g(x, v) dσ ρ(s) ˆ ds

R 0  st

g(x + σ v, v) dσ ρ(s) ˆ ds.

(3.19)

R 0

Therefore, incorporating identities (3.18) and (3.19) into the interpolation formulas (3.3) or (3.5), we obtain, for each t ∈ R, the following refined decomposition f (x, v) = TAt f (x, v) + tTBt g(x, v), where TAt f (x, v) = 1 TBt g(x, v) = −

1 t 2π

 s

1 2π

 st R 0

(3.20)



 f (x + stv, v)ρ(s) ˆ ds = R

f (x − stv, v)ρ(s) ˜ ds, R

1 g(x + σ v, v) dσ ρ(s) ˆ ds = − 2π

 s g(x + σ tv, v) dσ ρ(s) ˆ ds R 0

g(x − σ tv, v) dσ ρ(s) ˜ ds,

=

(3.21)

R 0

and Fx TAt f (η, v) = ρ(tη · v)Fx f (η, v), 1 − ρ(tη · v) Fx g(η, v), Fx TBt g(η, v) = itη · v

(3.22)

which shows that formulas (3.3) and (3.5) are in fact equivalent. Indeed, it is readily seen that (3.3) follows from (3.20) by use of the Fourier transform, while (3.5) is deduced from (3.20) by setting ρ(s) ˜ equal to an arbitrary approximation of the Dirac mass at s = 1, which makes sense since we have merely imposed on the cutoff that   1 ˆ ds = R ρ(s) ˜ ds = 1. Equivalently, formula (3.20) can be derived from (3.5) by replacing t by st ρ(0) = 2π R ρ(s) and then integrating against ρ(s) ˜ ds. Notice also that, by setting ρ(s) ˜ = e−s 1{s0} in (3.20), one arrives at the standard representation formula, for any t > 0, ∞ f (x, v) =

f (x − stv, v)e 0

∞ =

−s

s +t

g(x − σ tv, v)dσ e−s ds

0

 s 1 f (x − sv, v) + g(x − sv, v) e− t ds, t

(3.23)

0

which is usually obtained by directly solving the equivalent transport equation   1 1 + v · ∇x f = f + g, t t

(3.24)

but we will not make any use of this decomposition (cf. [15,18] for uses of this formula). Essentially, this particular choice of cutoff is not appropriate because its frequencies are not well localized, even though its Fourier transform decays exponentially. The importance of having a strong frequencies cutoff is made fully explicit in the localization identities (3.30) of Proposition 3.2 below. Much more importantly, this refined decomposition (3.20) shows that the elliptic and dispersive properties of the transport operator may be exploited through the same interpolation formula. Thus, employing formula (3.12) on the

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transfer of frequencies for velocity averages with the above decomposition, we obtain the following crucial proposition, which will systematically be the starting point of the proofs for the averaging lemmas presented in this work. Proposition 3.2. Let f (x, v), g(x, v) ∈ S(RD × RD ) be such that v · ∇x f = g. For all t > 0, δ  0 and for every cutoff function ρ ∈ S(R) such that ρ(0) = consider the decomposition  xδ f (x, v) dv = Atδ f (x) + tBδt g(x),

1 2π



ˆ ds R ρ(s)

=



˜ ds R ρ(s)

(3.25) = 1, we

(3.26)

RD

where the operators

Atδ

and

Bδt

are defined by



Atδ f (x) = xδ

TAt f (x, v) dv, RD



Bδt g(x) = xδ

TBt g(x, v) dv.

(3.27)

RD

Then it holds that



Fx−1 ρ(tη · v)Fx xδ f (x, v) dv,

Atδ f (x) = RD



Fx−1 τ (tη · v)Fx xδ g(x, v) dv,

Bδt g(x) =

(3.28)

RD

where τ (s) =

1−ρ(s) is

is smooth, and   Atδ f (x) =

xδ f (x R

RD



  1  Bδt g(x) =

xδ g(x R

0

 − stv, v) dv ρ(s) ˜ ds,

− σ stv, v) dv dσ s ρ(s) ˜ ds.

(3.29)

RD

Furthermore, if the cutoff ρ ∈ S(R) is such that ρ˜ is compactly supported inside [1, 2], then, for every δ > 0,     vf , vg , At0 f = At0 S2x S4t B0t g = B0t S2x S4t     v g . Atδ f = Atδ xδ vtδ f , Bδt g = Bδt xδ S8tδ (3.30) [ 2 ,2δ]

[ 2 ,2δ]

[ 2 ,4tδ]

Proof. It is readily seen that identities (3.28) and (3.29) are a mere transcription in terms of velocity averages of properties (3.21) and (3.22) for the operators TAt and TBt . Therefore, we only have to justify the crucial property (3.30). To this end, notice first that property (3.12) on the transfer of spatial to velocity frequencies implies, according to (3.29), that, when δ > 0,     v  Sst f (x − stv, v) dv ρ(s) At0 f (x) = ˜ ds, R

  1 

RD

B0t g(x) =



Sσv st g

R

0 RD





(x − σ stv, v) dv dσ s ρ(s) ˜ ds,

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  vstδ f (x − stv, v) dv ρ(s) ˜ ds,

 



Atδ f (x) = R

  1 

RD

Bδt g(x) =



vσ stδ g

R

505





(x − σ stv, v) dv dσ s ρ(s) ˜ ds.

(3.31)

0 RD

Then, since t  st  2t and σ st  2t on the support of ρ(s), ˜ it holds that v, Sstv = Sstv S4t

v, Sσv st = Sσv st S4t

vstδ = vstδ vtδ

[ 2 ,4tδ]

,

v . vσ stδ = vσ stδ S8tδ

(3.32)

Hence, employing identities (3.12) again, we find that     v v  t Sst S4t f (x − stv, v) dv ρ(s) A0 f (x) = ˜ ds R

RD

 

=



v x0 S4t f

R

RD

  1  B0t g(x) =

v g Sσv st S4t

R

0 RD

  1 

= R

Atδ f (x) =

  v   v x0 S4t g (x − σ stv, v) dv dσ s ρ(s) ˜ ds = B0t S4t g (x),



 (x − stv, v) dv ρ(s) ˜ ds





vstδ v[ tδ ,4tδ] f 2

R

RD

 

=



xδ v[ tδ ,4tδ] f

R



2

RD  1



Bδt g(x) = R

=

2

  v vσ stδ S8tδ g (x − σ stv, v) dv dσ s ρ(s) ˜ ds



v xδ S8tδ g

R

   (x − stv, v) dv ρ(s) ˜ ds = Atδ v[ tδ ,4tδ] f (x),





0 RD

  1 

Finally, utilizing that

  (x − σ stv, v) dv dσ s ρ(s) ˜ ds

0 RD

 

S2x x0



  v  (x − stv, v) dv ρ(s) ˜ ds = At0 S4t f (x),



  v   (x − σ stv, v) dv dσ s ρ(s) ˜ ds = Bδt S8tδ g (x).

(3.33)

0 RD

= x0

and xδ

[ 2 ,2δ]

xδ = xδ concludes the proof of the proposition.

2

4. Main results In this section, we present our main results. p

4.1. Velocity averaging in L1x Lv , inhomogeneous case p

Our first result concerns the endpoint case L1x Lv . 1 α D 1 (RD Theorem 4.1. Let f (x, v) ∈ L x ; Bp,q (Rv )), where 1  p, q  ∞ and α > −D(1 − p ) > −1, be such that

v · ∇x f = g β D 1 (RD for some g(x, v) ∈ L x ; Bp,q (Rv )), where β ∈ R.

(4.1)

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If β + D(1 − p1 ) < 1, then



 D s Rx , f (x, v) dv ∈ Bp,q

(4.2)

RD

where s =

α+D(1− p1 ) 1+α−β

− D(1 − p1 ), and the following estimate holds





f (x, v) dv  C f L β α (dv)) + g1 1 (dx;Bp,q

L (dx;B s (dx) Bp,q

RD



p,q (dv))

(4.3)

,

where the constant C > 0 only depends on fixed parameters. If β + D(1 − p1 ) > 1 or, if β + D(1 − p1 ) = 1 and q = 1, then   D s Rx , f (x, v) dv ∈ Bp,∞

(4.4)

RD

where s = 1 − D(1 − p1 ), and the following estimate holds





f (x, v) dv  C x,v 1 (dx;B β 0 f L1 (dx;Lp (dv)) + gL

p,q (dv))

s Bp,∞ (dx)

RD



(4.5)

,

where the constant C > 0 only depends on fixed parameters. α D 1 (RD Notice that, with the sole bound f (x, v) ∈ L x ; Bp,q (Rv )), and in particular without any information on the transport equation (4.1), it is only possible to deduce, by Sobolev embedding, that the velocity average satisfies, for any φ(v) ∈ C0∞ (RD ),    −D(1− p1 )  D  Rx . (4.6) f (x, v)φ(v) dv ∈ L1 RD x ⊂ Bp,∞ RD

Therefore, in the above theorem, the gain on the velocity average can be measured by the difference between the regularity index s in (4.3) and (4.5), and the regularity index −D(1 − p1 ) obtained by Sobolev embedding. Thus, as α+D(1− p1 ) 1+α−β . 1 − D(1 − p1 ), or as

long as β + D(1 − p1 ) < 1, the above theorem yields a gain of regularity of

Note that this gain approaches one full derivative as β tends to α tends to infinity. However, since the transport operator is a differential operator of order one, it can never yield a net gain of regularity greater than one. This is precisely the reason why, when β + D(1 − p1 )  1, the averaging lemma saturates and only yields a maximal gain of one full derivative, independently of α. Very loosely speaking, this result shows that the transport operator v · ∇x is fully invertible when g is very regular in velocity. Thus, it becomes an elliptic operator through velocity averaging. This is also the reason why, quite remarkably, only the low frequencies of f are involved in this case. α+D(1− 1 )

Notice also that the condition α > −D(1 − p1 ) above is very natural, since otherwise the gain 1+α−βp becomes negative and thus the averaging lemma turns out to be weaker than the Sobolev embedding. However, the condition D(1 − p1 ) < 1 seems less natural. Actually, its necessity comes from the handling of the low velocity frequencies of g(x, v) (it can be interpreted as β + D(1 − p1 ) < 1 with β = 0 for those low frequencies). Thus, it is possible to remove this condition by considering a corresponding version of Theorem 4.1 for homogeneous Besov spaces, which is the content of Theorem 4.3 below. The next result extends the preceding theorem to the case including spatial derivatives in the right-hand side. a,α 1 D Theorem 4.2. Let f (x, v) ∈ B1,p,q (RD x × Rv ), where 1  p, q  ∞, a ∈ R and α > −D(1 − p ) > −1, be such that

v · ∇x f = g for some

b,β g(x, v) ∈ B1,p,q (RD x

× RD v ),

where β ∈ R and b  a − 1.

(4.7)

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

If β + D(1 − p1 ) < 1, then



507

 D s Rx , f (x, v) dv ∈ Bp,q

(4.8)

RD

where s = (1 + b − a)

α+D(1− p1 )

+ a − D(1 − p1 ), and the following estimate holds



 

f (x, v) dv  C f B a,α (RDx ×RDv ) + gB b,β (RD ×RD ) ,

s 1,p,q x v 1,p,q 1+α−β

RD

(4.9)

Bp,q (dx)

where the constant C > 0 only depends on fixed parameters. If β + D(1 − p1 ) > 1 or, if β + D(1 − p1 ) = 1 and q = 1, then   D s Rx , f (x, v) dv ∈ Bp,q

(4.10)

RD

where s = (1 + b − a) + a − D(1 − p1 ), and the following estimate holds





f (x, v) dv  C x,v 0 f L1 (dx;Lp (dv)) + gB b,β

RD



D D 1,p,q (Rx ×Rv )

s (dx) Bp,q

,

where the constant C > 0 only depends on fixed parameters. If β + D(1 − p1 ) = 1 and q = 1, then, for every  > 0,   D s− Rx , f (x, v) dv ∈ Bp,q

(4.11)

(4.12)

RD

where s = (1 + b − a) + a − D(1 − p1 ), and the following estimate holds





f (x, v) dv  C f B a,α (RDx ×RDv ) + gB b,β

RD

s− Bp,q (dx)

1,p,q

D D 1,p,q (Rx ×Rv )



,

(4.13)

where the constant C > 0 only depends on fixed parameters, in particular on  > 0. The remarks formulated above about Theorem 4.1 are still valid here regarding Theorem 4.2. α+D(1− 1 )

Thus, as long as β + D(1 − p1 ) < 1, the above theorem yields a gain of regularity of (1 + b − a) 1+α−βp (compared to the Sobolev embedding). Notice that this gain approaches a derivative of order (1 + b − a) as β tends to 1 − D(1 − p1 ), or as α tends to infinity, which is optimal for a differential operator of order one. Therefore, the averaging lemma saturates beyond the value β + D(1 − p1 ) = 1 and only yields at most a gain of 1 + b − a derivatives. We do not know whether it is possible to achieve a full gain of 1 + b − a derivatives in the case β + D(1 − p1 ) = 1 and q = 1. Nevertheless, the cases β + D(1 − p1 ) > 1 or β + D(1 − p1 ) = 1 and q = 1 do yield an exact full gain of 1 + b − a derivatives, independently of α, which, again, is largely optimal. Quite remarkably, this is the first time that a velocity averaging result achieves exactly the maximal gain of regularity, i.e. one full derivative in the case a = b = 0, say. Moreover, it is worth noting that only the low frequencies of f are involved in this case, which, very loosely speaking, shows that the transport operator v · ∇x is fully invertible when g is very regular in velocity. Notice also that the conditions α > −D(1 − p1 ) and b  a − 1 above are very natural, since otherwise the gain α+D(1− 1 )

(1 + b − a) 1+α−βp becomes negative and thus the averaging lemma turns out to be weaker than the Sobolev embedding. Finally, as for Theorem 4.1, the condition D(1 − p1 ) < 1 seems less natural. Actually, its necessity comes from the handling of the low velocity frequencies of g(x, v) (it can be interpreted as β + D(1 − p1 ) < 1 with β = 0 for those

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low frequencies). Thus, it is possible to remove this condition by considering a corresponding version of Theorem 4.2 for homogeneous Besov spaces, which is the content of Theorem 4.4 below. p

4.2. Velocity averaging in L1x Lv , homogeneous case p

This section contains the results in the endpoint case L1x Lv formulated with homogeneous Besov spaces and corresponding to Theorems 4.1 and 4.2. 1 D ˙α 1 (RD Theorem 4.3. Let f (x, v) ∈ L x ; Bp,q (Rv )), where 1  p, q  ∞ and α > −D(1 − p ), be such that

v · ∇x f = g

(4.14)

1 D ˙β 1 (RD for some g(x, v) ∈ L x ; Bp,q (Rv )), where β < 1 − D(1 − p ). Then,   D s Rx , f (x, v) dv ∈ B˙ p,q

(4.15)

RD

where s =

α+D(1− p1 ) 1+α−β

− D(1 − p1 ), and the following estimate holds





f (x, v) dv

˙s

 Cf 

Bp,q (dx)

RD

1) 1−β−D(1− p 1+α−β α (dv)) 1 (dx;B˙ p,q L

× g

1) α+D(1− p 1+α−β β 1 (dx;B˙ p,q L (dv))

,

(4.16)

where the constant C > 0 only depends on fixed parameters. a,α 1 D Theorem 4.4. Let f (x, v) ∈ B˙ 1,p,q (RD x × Rv ), where 1  p, q  ∞, a ∈ R and α > −D(1 − p ), be such that

v · ∇x f = g for some Then,

b,β g(x, v) ∈ B˙ 1,p,q (RD x

(4.17)

1 × RD v ), where b  a − 1 and β < 1 − D(1 − p ).



 D s Rx , f (x, v) dv ∈ B˙ p,q

(4.18)

RD

where s = (1 + b − a)

α+D(1− p1 ) 1+α−β

+ a − D(1 − p1 ), and the following estimate holds





f (x, v) dv

˙s RD

Bp,q (dx)

 Cf 

1) 1−β−D(1− p 1+α−β a,α D B˙ 1,p,q (RD x ×Rv )

× g

1) α+D(1− p 1+α−β b,β D B˙ 1,p,q (RD x ×Rv )

,

(4.19)

where the constant C > 0 only depends on fixed parameters. Furthermore, if β + D(1 − p1 ) = 1 and q = 1, then the above estimate remains valid. 4.3. The classical L2x L2v case revisited We give now a new very general version of the classical velocity averaging lemma in L2x L2v in terms of Besov p spaces. The proofs of this formulation have the advantage of employing the same principles and ideas as in the L1x Lv cases. In particular, they are based on the same decompositions and operators (provided by Proposition 3.2 below), p which will be crucial in order to carry out interpolation arguments between the L1x Lv and L2x L2v cases later on. Note that, as usual, exploiting the Hilbertian structure of Besov spaces (i.e. choosing q = 2), the theorem below can readily be reformulated in terms of more standard Sobolev spaces.

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509

a,α 1 D Theorem 4.5. Let f ∈ B2,2,q (RD x × Rv ), where a ∈ R, α > − 2 and 1  q  ∞, be such that

v · ∇x f = g

(4.20)

b,β

D for some g ∈ B2,2,q (RD x × Rv ), where b  a − 1 and β ∈ R.

If β < 12 , then, for any φ ∈ C0∞ (RD ),



 D s Rx , f (x, v)φ(v) dv ∈ B2,q

(4.21)

RD α+ 1

2 where s = (1 + b − a) 1+α−β + a, and the following estimate holds





f (x, v)φ(v) dv

RD

s (RD ) B2,q x

  Cφ f B a,α

D D 2,2,q (Rx ×Rv )

+ gB b,β



D D 2,2,q (Rx ×Rv )

(4.22)

,

where the constant Cφ > 0 only depends on φ and other fixed parameters. If β > 12 or, if β = 12 and q = 1, then, for any φ ∈ C0∞ (RD ),   D s Rx , f (x, v)φ(v) dv ∈ B2,q

(4.23)

RD

where s = 1 + b, and the following estimate holds







f (x, v)φ(v) dv

 Cφ x,v 0 f L2 (RD ×RD ) + gB b,β

RD

x

s (RD ) B2,q x

v



,

D D 2,2,q (Rx ×Rv )

where the constant Cφ > 0 only depends on φ and other fixed parameters. If β = 12 and q = 1, then, for any φ ∈ C0∞ (RD ) and every  > 0,  s−  D  Rx , f (x, v)φ(v) dv ∈ B2,q

(4.24)

(4.25)

RD

where s = 1 + b, and the following estimate holds





f (x, v)φ(v) dv  Cφ f B a,α

RD

s− B2,q (RD x )

D D 2,2,q (Rx ×Rv )

+ gB b,β



D D 2,2,q (Rx ×Rv )

,

(4.26)

where the constant Cφ > 0 only depends on φ and other fixed parameters, in particular on  > 0. α+ 1

2 Notice that the theorem above provides a net gain of regularity of (1 + b − a) 1+α−γ derivatives, where

γ = min{β, 12 }. Therefore, the restrictions α > − 12 and 1 + b − a  0 on the parameters are, in fact, quite natural since the gain of regularity would possibly be negative otherwise. Furthermore, the threshold at the value β = 12 stems from the fact that, since the transport operator is a differential operator of order one, it cannot yield, in the case a = b = 0 say, a gain of regularity which would be superior to one α+ 1

2  1, which implies γ  12 . full derivative. In other words, necessarily 1+α−γ Quite remarkably, as for Theorem 4.2, the above theorem does achieve the maximal gain of regularity of 1 + b − a derivatives in the cases β > 12 or β = 12 and q = 1, independently of α, which is unprecedented. Moreover, it is worth noting that only the low frequencies of f are involved in this case, which, very loosely speaking, shows that the transport operator v · ∇x is fully invertible when g is very regular in velocity.

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4.4. The L1x Lv and L2x L2v cases reconciled The following theorem results from a simple interpolation between Theorems 4.2 and 4.5. A more general, but far more complicated, interpolation procedure will yield the more general Theorem 4.7 below. a,α D Theorem 4.6. Let f (x, v) ∈ Br,p,q (RD x × Rv ), where 1  r  p  ∞, 1  q < ∞, a ∈ R and 1 1 1 1 α > r − 1 − D( r − p ) > − r , be such that

v · ∇x f = g b,β D for some g(x, v) ∈ Br,p,q (RD x × Rv ), where β ∈ R and 1 1 1 If β < r − D( r − p ), then, for any φ ∈ C0∞ (RD ),



(4.27)

b  a − 1.

 D s Rx , f (x, v)φ(v) dv ∈ Bp,q

(4.28)

RD 1+α−( 1 −D( 1 − 1 ))

r r p where s = (1 + b − a) + a − D( 1r − p1 ), and the following estimate holds 1+α−β



 

f (x, v)φ(v) dv a,α  Cφ f Br,p,q D + g b,β D , (RD

s

Br,p,q (RD x ×Rv ) x ×Rv )

(4.29)

Bp,q (dx)

RD

where the constant Cφ > 0 only depends on φ and other fixed parameters. If β > 1r − D( 1r − p1 ) or, if β = 1r − D( 1r − p1 ) and q = 1, then, for any φ ∈ C0∞ (RD ),   D s Rx , f (x, v)φ(v) dv ∈ Bp,q

(4.30)

RD

where s

= 1 + b − D( 1r

− and the following estimate holds



 

a,α D D + g b,β D D ,

f (x, v)φ(v) dv  Cφ x,v 0 f Br,p,q

s

(Rx ×Rv ) Br,p,q (Rx ×Rv ) 1 p ),

(4.31)

Bp,q (dx)

RD

where the constant Cφ > 0 only depends on φ and other fixed parameters. If β = 1r − D( 1r − p1 ) and q = 1, then, for any φ ∈ C0∞ (RD ) and every  > 0,   D s− Rx , f (x, v)φ(v) dv ∈ Bp,q

(4.32)

RD

where s

= 1 + b − D( 1r

− and the following estimate holds



 

f (x, v)φ(v) dv a,α  Cφ f Br,p,q D + g b,β D) , (RD

s− Br,p,q (RD ×R x ×Rv ) x v RD

1 p ),

(4.33)

Bp,q (dx)

where the constant Cφ > 0 only depends on φ and other fixed parameters, in particular on  > 0. Notice that the theorem above corresponds exactly to Theorems 4.2 and 4.5 in the limiting cases r = 1 and r = 2, respectively. It provides a net gain of regularity, compared to the Sobolev embedding, of (1 + b − a)

1+α−( 1r −D( 1r − p1 )) 1+α−γ

derivatives, where γ = min{β, 1r − D( 1r − p1 )}. Therefore, the restrictions α > 1r − 1 − D( 1r − p1 ) and 1 + b − a  0 on the parameters are, in fact, quite natural since the gain of regularity would possibly be negative otherwise. Furthermore, the threshold at the value β = 1r − D( 1r − p1 ) stems from the fact that, since the transport operator is a differential operator of order one, it cannot yield, in the case a = b = 0 say, a gain of regularity which would be superior to one full derivative. In other words, necessarily

1+α−( 1r −D( 1r − p1 )) 1+α−γ

 1, which implies γ 

1 r

− D( 1r − p1 ).

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

511

Quite remarkably, as for Theorems 4.2 and 4.5, the above theorem does achieve the maximal gain of regularity of 1 + b − a derivatives in the cases β > 1r − D( 1r − p1 ) or β = 1r − D( 1r − p1 ) and q = 1, independently of α, which is unprecedented. Moreover, it is worth noting that only the low frequencies of f are involved in this case, which, very loosely speaking, shows that the transport operator v · ∇x is fully invertible when g is very regular in velocity. The following theorem is the most general result presented in this work. However, it does not contain all the previous theorems. It follows from a general abstract interpolation procedure of the preceding results. It is to be emphasized that, whereas all the results presented so far were set in the simpler case where the integrability and the summability indices defining the Besov spaces are identical for f (x, v) and for g(x, v), the result below does not assume any a priori relationship between the indices defining the function spaces for f (x, v) and g(x, v) (except for the rather nonrestrictive condition (4.36) below). D D Theorem 4.7. Let f (x, v) ∈ Bra,α 0 ,p0 ,q0 (Rx × Rv ), where 1  r0  p0  r0  ∞, 1  q0 < ∞, a ∈ R and 1 1 1 α > r0 − 1 − D( r0 − p0 ), be such that

v · ∇x f = g

(4.34)

D for some g(x, v) ∈ Br1 ,p1 ,q1 (RD x × Rv ), where 1  r1  p1  r1  ∞, 1  q1 < ∞, b ∈ R and β < satisfy  2 1 1 −1−D − >0 or p1 = r1 = 2 r1 r1 p1 b,β

1 r1

− D( r11 −

1 p1 )

(4.35)

and (1 − θ ) where

1 1 1 1 +θ = (1 − θ ) + θ , p0 p1 q0 q1

(4.36)



θ=

α+1−

Then, for any χ, φ ∈ C0∞ (RD ),

1 r0

  α + 1 − r10 + D r10 − p10     + D r10 − p10 + −β + r11 − D r11 − 

1 p1

 ∈ (0, 1).

(4.37)

 D s f (x, v)χ(x)φ(v) dv ∈ Bp,p Rx ,

(4.38)

RD

where

  1 1 1 1 s = (1 − θ ) a − D − − +θ b−D + θ, r0 p0 r1 p1 1 1 1 1 1 = (1 − θ ) + θ , = (1 − θ ) + θ p p0 p1 q0 q1

and the following estimate holds



f (x, v)χ(x)φ(v) dv

RD

s (dx) Bp,p

  Cφ f Bra,α,p 0

0 ,q0

D (RD x ×Rv )

+ gB b,β

D D r1 ,p1 ,q1 (Rx ×Rv )

(4.39)



,

(4.40)

where the constant Cφ > 0 only depends on φ and other fixed parameters. Furthermore, if p  r0 , the space localization through a cutoff χ(x) is not necessary and it is possible to take χ(x) ≡1 in the above statements. The interpretation of net gain of regularity is not as straightforward as it is for the preceding theorems. Thus, we provide now a somewhat alternative analysis of the regularity index s.

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The above theorem essentially establishes an estimate on the velocity average which stems from an interpolation of order θ between the controls  a−D( r1 − p1 )     0 0 RD (4.41) f (x, v)φ(v) dv ∈ Bra0 ,q0 RD x ⊂ Bp0 ,q0 x RD

and



b−D( r1 − p1 )     1 1 RD g(x, v)φ(v) dv ∈ Brb1 ,q1 RD x ⊂ Bp1 ,q1 x ,

(4.42)

RD

which follow from standard Sobolev embeddings. If the functions f (x, v) and g(x, v) were linked by a relation f = T g, where T is some bounded and invertible operator of differential order r ∈ R acting only on x, then it would be natural to expect, by interpolation of order θ , a control on the velocity average in the Besov space (1−θ)(a−D( r1 − p1 ))+θ(b−D( r1 − p1 )+r)  0

Bp,p

0

1

1

 RD x ,

(4.43)

where = (1 − + = (1 − + This formal reasoning shows that the net gain of regularity given by a differential relation f = T g of order r through an interpolation of order θ is at most θ r, when compared to a differential operator of zero (e.g. the identity). From that viewpoint, the above theorem asserts that, the transport operator T = v · ∇x being a differential operator of order one, it is possible to obtain a maximal net gain of regularity θ through velocity averaging. We insist that here the net gain of regularity is found by comparing the actual regularity index s with the interpolation of the indices obtained by Sobolev embeddings. Therefore, the restriction α > r10 − 1 − D( r10 − p10 ) on the parameters is, in fact, quite natural since the gain of regularity θ would possibly be negative otherwise. Furthermore, the constraint β < r11 − D( r11 − p11 ) stems from the fact that, since the transport operator is a differential operator of order one, it cannot yield a gain of regularity θ which would be superior to one full derivative. In other words, necessarily θ < 1, which implies β < r11 − D( r11 − p11 ). It is quite interesting to note that some kind of space localization is definitely necessary in the case p < r0 of the above theorem. It is in fact explicit from the proofs that this restriction comes from the control of low frequencies. Indeed, let us suppose that the estimate (4.40) holds with χ ≡ 1 for some given choice of parameters in the one-dimensional case D = 1. Further consider f (x, v) ∈ S(Rx × Rv ) such that its space and velocity frequencies are localized in a bounded domain. In other words, we suppose that x2k f = v2k f = 0, for every k  0, say. Therefore, in virtue of estimate (4.40), it holds that



 

f (x, v)φ(v) dv  Cφ f  r0 p0 + v∂x f  r1 p1 . (4.44)

p

Lx Lv Lx Lv 1 p

θ ) p10

θ p11

θ ) q10

θ q11 .

Lx

RD

In particular, since the transformation fR (x, v) = f ( Rx , v) preserves de localization of low frequencies for any R > 1, we deduce that it must also hold that







1  

p R f (x, v)φ(v) dv = fR (x, v)φ(v) dv

 Cφ fR Lr0 Lp0 + v∂x fR Lr1 Lpv 1 p

Lx

RD

−

v

1  1  −1 = Cφ R r0 f Lr0 Lp0 + R r1 v∂x f Lr1 Lpv 1 . x

1

x

p

Lx

RD

1

v

x

x

(4.45)

It follows that R p r0 must remain bounded as R tends towards infinity, which forces r0  p. Finally, we would like to emphasize that we have chosen to present, in this work, cases of velocity averaging p lemmas dealing with Lrx Lv integrability only, where r  p, principally because our analysis of dispersion allowed to handle these previously unsettled cases. But we insist that this is by no means a restriction of our method, which p is, in fact, very robust and enables to also treat the actually easier setting of Lv Lrx integrability, where p  r, and thus, to recover most of previously known results in sharper Besov spaces. Our precise interpolation techniques can r p even reach settings where the left-hand side f enjoys Lx0 Lv 0 integrability, where r0  p0 , while the right-hand side g p1 r1 displays Lv Lx integrability, where p1  r1 , and vice versa.

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513

5. Control of concentrations in L1 The methods developed in Section 3, which eventually led to the main results from Section 4, were initially motivated by an important application to the viscous incompressible hydrodynamic limit of the Boltzmann equation with long-range interactions in [1,2]. We present in this section some technical lemmas which were crucially employed therein and are contained in our main results. Essentially, we show here how the methods from Section 3 can be used to gain control over the concentrations in kinetic transport equations through velocity averaging. This application relies on a refined regularity estimate for the Boltzmann equation without cutoff established in [3], wherein the reader will also find a complete discussion of this application to the hydrodynamic limit. An alternative method has been developed in [4]. The difficult problem of concentrations in the viscous incompressible hydrodynamic limit of the Boltzmann equation was first successfully handled in [16] for bounded collision kernels with cutoff and then, using the same approach, extended in [17,19] to more general cross-sections with cutoff. In any case, the method consisted in an application of a simple, but subtle, velocity averaging lemma in L1 established in [15], which we briefly present now. Recall first the following definition from [15]. 1 D D Definition. A bounded sequence {fn (x, v)}∞ n=0 ⊂ Lloc (Rx × Rv ) is said to be equi-integrable in v if and only if for D D every η > 0 and each compact K ⊂ R × R , there exists δ > 0 such that for any measurable set Ω ⊂ RD × RD with supx∈RD 1Ω (x, v) dv < δ, we have that    fn (x, v) dx dv < η for every n. (5.1) Ω∩K

The crucial compactness lemma in L1 obtained in [15] and relevant to the rigorous derivations of the hydrodynamic limit in the cutoff case [16,17,19] is contained in the following theorem. 1 D D Theorem 5.1. (See [15].) Let the sequence {fn (x, v)}∞ n=0 be bounded in Lloc (Rx × Rv ), equi-integrable in v, and such that

v · ∇x fn = gn

(5.2)

1 D D for some sequence {gn }∞ n=0 bounded in Lloc (Rx × Rv ). Then,

1. {fn }∞ n=0 is equi-integrable (in all variables x and v) and, 2. for each ψ ∈ L∞ (RD ) with compact support, the family of velocity averages    fn (x, v)ψ(v) dv is relatively compact in L1loc RD .

(5.3)

RD

Unfortunately, the above result is not directly applicable to the viscous incompressible hydrodynamic limit of the Boltzmann equation with long-range interactions. Indeed, its use is prevented by the particular structure of the Boltzmann collision operator, which behaves as a nonlinear differential operator for collision kernels without cutoff (cf. [1–3]). Consequently, the strategy from [16,17,19] would require a generalization of the preceding theorem to kinetic transport equations (5.2) with velocity derivatives in its right-hand side. However, a simple counterexample shows that a straight generalization won’t be possible. Indeed, consider the locally integrable functions fn (x, v) = nϕ(nx1 ) cos(nv1 ), gn (x, v) = v1 nϕ (nx1 ) sin(nv1 ) and

hn (x, v) = nϕ (nx1 ) sin(nv1 ),

(5.4)

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where ϕ ∈ C0∞ (R) has non-trivial mass. Then, one easily checks that v · ∇x fn = ∂v1 gn − hn

and {fn }∞ n=0 is equi-integrable in v.

(5.5)

However, {fn }∞ n=0 is not equi-integrable in all variables, which shows that the first assertion of Theorem 5.1 doesn’t hold if one has derivatives on the right-hand side of the transport equation (5.2). Nevertheless, note that the velocity averages of fn do converge strongly to zero and thus, it is unclear whether there holds a direct extension of the second assertion of Theorem 5.1 or not. Furthermore, this counterexample uses crucially the fact that fn is allowed to alternate sign and so, it might still be possible that preventing this oscillatory behavior by simply imposing a non-negativity condition on the fn ’s would allow for a generalization of the result. We wish now to extend the above Theorem 5.1 in order to allow the use of derivatives in the right-hand side of the transport equation, which, again, is necessary for the rigorous derivation of the hydrodynamic limit in the non-cutoff case. It turns out that a slightly better control on the high velocity frequencies of the fn ’s will suffice to provide a relevant generalization. This is the content of the coming theorem. 0 1 D D Theorem 5.2. Let {fn (x, v)}∞ n=0 be a bounded sequence in L (Rx ; B1,1 (Rv )), such that fn  0 and γ

v · ∇x fn = (1 − v ) 2 gn

(5.6)

1 D D for some sequence {gn (x, v)}∞ n=0 bounded in L (Rx × Rv ) and some γ ∈ R. Then,

{fn }∞ n=0

is equi-integrable (in all variables x and v).

(5.7)

0 are in order. First of all, it terms of scaling, it is similar to L1 , which is Some remarks on the Besov space B1,1 0 is strictly why Theorem 5.2 can be considered an appropriate substitute to Theorem 5.1. In fact, the Besov space B1,1 0 can be interpreted as a function in L1 whose high frequencies smaller than L1 . Loosely speaking, an element of B1,1 are slightly better behaved than a merely integrable function. This behavior of high frequencies is expressed in (2.9) 0 and, therefore, it cannot be interpreted by the mere summability of dyadic frequency blocks defining the norm of B1,1 0 is only as a regularity condition since the control of the frequencies is not quantified. Thus, the Besov space B1,1 slightly better than L1 . 0 necessarily enjoys an L log L bound. As explained in the proof of Theorem 5.2, a non-negative function in B1,1 The converse is not true and it does not seem to be true either, if we drop the non-negativity condition. In fact, we are 0 which would easily guarantee the membership in B 0 of a given not aware of any explicit characterization of B1,1 1,1 0 , such as the subspaces B s , for any 1  q  ∞ function. It is, however, easy to identify convenient subsets of B1,1 1,q and s > 0. In particular, in the application of the results of the present section to the study of the hydrodynamic limit 0 on an appropriate sequence of of the Boltzmann equation with long-range interactions in [2], a uniform bound in B1,1 functions is provided by a weak regularity estimate (cf. [3]) valid for the non-cutoff Boltzmann equation. The above theorem will be obtained as a corollary of the following proposition, which is, in fact, a particular case of Theorem 4.1. Nevertheless, we do provide below a complete justification for this result, because its proof is simpler and contains some of the essential ideas presented in Section 3. Thus, we hope that it will also serve as a primer to the more convoluted demonstrations of Section 6. 0 D Proposition 5.3. Let f (x, v) ∈ L1 (RD x ; B1,1 (Rv )) be such that

v · ∇x f = g

(5.8)

β

D for some g(x, v) ∈ L1 (RD x ; B1,1 (Rv )), where β ∈ R. Then,   D 0 Rx , f (x, v) dv ∈ B1,1 RD

(5.9)

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

and the following estimate holds





f (x, v) dv

RD

0 (dx) B1,1

  C f L1 (dx;B 0

1,1 (dv))

515



+ gL1 (dx;B β

1,1 (dv))

(5.10)

,

where the constant C > 0 only depends on fixed parameters. 1 B 1 ⊂ L 1 B 2 whenever β1  β2 , we may assume without any Proof. Notice first that, in virtue of the inclusions L 1,1 1,1 1 B β = L1 B β , for any β ∈ R. loss of generality that β < 1. Furthermore, it is to be emphasized that L 1,1 1,1 Integrating the dyadic interpolation formula (3.13) in space yields β



x

 k

f (x, v) dv

2

t

 v k f

L1x,v

t2

L1x

RD

β

+

k=0

0 β k 1−β

. Thus, summing the above estimate over k

∞ ∞ 

v

v

 k f 1 +

 k g 1 ds.  L L t 2 s2 tk

L1x

RD

(5.11)

x,v

For each value of k, we fix now the interpolation parameter t as tk = 2 yields

 ∞

x

 k f (x, v) dv

2

v

 k g 1 ds. L s2

k

x,v

k=0

(5.12)

x,v

k=0 0

The last term above is then handled through the following calculation, noticing that

tk−1 2

< tk ,

tk−1

tk ∞ tk ∞ 2

v

v

v

 k g 1 ds =

 k g 1 ds +

 k g 1 ds L L L s2 s2 s2 x,v

k=0 0

x,v

k=0 0

x,v

tk−1 2

tk−1 tk ∞

v

v 1

 k g 1 ds s2k−1 g L1 ds + = Lx,v s2 x,v 2 k=0

tk−1 2

0

∞ tk ∞ tk

v

1 v

 k g 1 ds,

s2k g L1 ds + = Lx,v s2 x,v 2 k=−1 0

k=0

(5.13)

tk−1 2

from which we deduce t(−1) ∞ tk ∞ tk

v

v

v

 k g 1 ds =

 s g 1 ds + 2

 k g 1 ds. L L L s2 s2 x,v

k=0 0

2

x,v

k=0

0

(5.14)

x,v

tk−1 2

Consequently, combining (5.12) with (5.14), we infer

 ∞

x

 k f (x, v) dv

2 k=0

RD

L1x

∞ ∞ tk

v

v

 k g 1 ds + Cg 1

t 2k f L1 + 2   (dx;B β L s2 L k

k=0

x,v

k=0

x,v

1,1 (dv))

,

(5.15)

tk−1 2

where C > 0 is a fixed constant that only depends on fixed parameters. Now, let us recall a basic principle from the theory of Littlewood and Paley. Consider any δ > 0. Then, there lδ lδ +1 . In the notation of Section 2 and specially (2.6), it then holds that exists 2 a unique integer lδ such that 2  δ < 2 D j =−1 ϕ2lδ +j ≡ 1 on the support of ϕδ , which therefore implies for any f ∈ S (R ) and any 1  p  ∞ that

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δ f Lp

2



= 2lδ +j δ f

j =−1

C

2

2lδ +j f Lp ,

(5.16)

j =−1

Lp

for some fixed constant C > 0 independent of δ. We wish now to apply this principle to our situation. Thus, denoting by [·] the integer part of a number and noticing 1 [k 1−β ]

that 2

 t k 2k = 2

1 k 1−β

<2

1 [k 1−β ]+1

, we obtain, on the one hand, that

∞ ∞ 2

v

v

 k f 1  C

 L t 2 [k k

k=0

x,v

2

j =−1 k=0

1 1−β ]+j

f L1  Cf L 1 (dx;B 0

1,1 (dv))

x,v

.

(5.17)

And, on the other hand, we have that ∞ tk

v

 k g 1 ds  L s2 k=0

x,v

2

∞ tk

v



1 j =−[ 1−β ]−2 k=0

tk−1 2

C

2

1 j =−[ 1−β ]−2 k=0

C

2

∞ 



tk−1

v 1 g L1 tk − [k 1−β ]+j x,v 2 2

2

1 [k 1−β ]+j β

v[k 2

1 j =−[ 1−β ]−2 k=0

 CgL 1 (dx;B β

1,1 (dv))

Finally, combining (5.15) with (5.17) and (5.18) yields





f (x, v) dv  C f L 1 (dx;B 0

RD

1 1−β ]+j

g L1

x,v

(5.18)

.

1,1 (dv))

0 (dx) B1,1

x,v

2

tk−1 2

∞

vs2k g L1 ds

1 ]+j [k 1−β



+ gL 1 (dx;B β

1,1 (dv))

,

(5.19)

2

where the constant C > 0 only depends on fixed parameters, which concludes our proof. Proof of Theorem 5.2. Notice first that there exists β < 1 so that  γ β  D  (1 − v ) 2 gn ∈ L1 RD x ; B1,1 Rv .

(5.20)

We may therefore straightforwardly apply Proposition 5.3 to infer that ∞   D 0 Rx . fn (x, v) dv is bounded in B1,1

(5.21)

n=0

RD

Let us recall now a few basic facts from the theory of functions spaces. We refer the reader to [22] or [27] for more s (RD ), for any 1  p, q < ∞ and s ∈ R, details on the subject. The norm that defines the Triebel–Lizorkin spaces Fp,q is given by (in the notation of Section 2)



s = f Fp,q

RD

∞   q  0 f (x)q + 2ksq 2k f (x)

1

p q

p

dx

,

(5.22)

k=0

0 = F 0 and therefore B 0 is continuously embedded in F 0 , which is where f ∈ S (RD ). Thus, it is clear that B1,1 1,1 1,1 1,2 nothing but the local Hardy space h1 (cf. [27, Section 2.3.5]).

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517

Therefore, it holds that ∞



  is bounded in h1 RD x

fn (x, v) dv n=0

RD

and



∞ fn (x, v) n=0

  D  is bounded in L1 RD x ; h1 Rv .

(5.23)

We recall now Stein’s L log L result (cf. [24], [25, Chapter 1, §5.2] and [26, Chapter I, §8.14, Chapter III, §5.3]), which states that a sequence of positive functions uniformly bounded in h1 necessarily satisfies locally a uniform L log L bound. Here, for a given compact subset K ⊂ RD , the Orlicz space L log L(K) is defined as the Banach space endowed with the norm     |f (x)| f L log L(K) = inf λ > 0: h dx  1 , (5.24) λ K

where h(z) = (1 + z) log(1 + z) − z is a convex function. Thus, we conclude that, for any compact subsets Kx , Kv ⊂ RD , ∞  fn (x, v) dv is bounded in L log L(Kx ) n=0

RD

 ∞   and fn (x, v) n=0 is bounded in L1 RD x ; L log L(Kv ) . It then follows that

∞



is equi-integrable

fn (x, v) dv RD

and

(5.25)

n=0



∞ fn (x, v) n=0

is equi-integrable in v,

(5.26)

and this is, together with the positiveness of the sequences, exactly what is required by Lemma 5.2 of [15] in order to deduce that ∞  (5.27) fn (x, v) n=0 is equi-integrable (in all variables x and v), which concludes the proof of the theorem.

2

We provide now simple extensions of Theorem 5.2 and Proposition 5.3 to the time dependent transport equation with a vanishing temporal derivative. These variants of the preceding results are precisely the versions needed for their application to the viscous incompressible hydrodynamic limit of the Boltzmann equation without cutoff in [2]. Moreover, the proofs provided below show how to smoothly adapt the general methods developed in this work to the time dependent kinetic transport equation. 0 ∞ 1 D D 1 D D Theorem 5.4. Let {fn (t, x, v)}∞ n=0 be a bounded sequence in L (Rt ; L (Rx × Rv )) and in L (Rt × Rx ; B1,1 (Rv )), such that fn  0 and that γ

(δn ∂t + v · ∇x )fn = (1 − v ) 2 gn

(5.28)

1 D D for some sequence {gn (t, x, v)}∞ n=0 bounded in L (Rt × Rx × Rv ), some γ ∈ R and where the sequence δn > 0 vanishes as n → ∞. Then,

{fn }∞ n=0

is equi-integrable (in all variables t, x and v).

As before, the above theorem will be obtained as a corollary of the following proposition:

(5.29)

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0 1 D D Proposition 5.5. Let {fn (t, x, v)}∞ n=0 be a bounded sequence in L (Rt × Rx ; B1,1 (Rv )) such that

(δn ∂t + v · ∇x )fn = gn

(5.30)

1 D D for some bounded sequence {gn (x, v)}∞ n=0 in L (Rt × Rx ; B1,1 (Rv )), where β ∈ R, and the sequence δn > 0 vanishes as n → ∞. Then, ∞    D  0 Rx . (5.31) fn (t, x, v) dv is bounded in L1 Rt ; B1,1 β

n=0

RD

Proof. We give here a short demonstration of this lemma by following closely the proof of Proposition 5.3 and emphasizing the necessary changes. Thus, we first introduce an interpolation parameter s ∈ R. Then, it trivially holds that  (∂s + δn ∂t + v · ∇x )fn = gn , (5.32) fn (s = 0) = fn . Hence the interpolation formula s fn (t, x, v) = fn (t − δn s, x − sv, v) +

gn (t − δn σ, x − σ v, v) dσ.

(5.33)

0

It follows that, in virtue of (3.12),    v  x 2k fn (t, x, v) dv = s2k fn (t − δn s, x − sv, v) dv RD

RD

s 

+



 vσ 2k gn (t − δn σ, x − σ v, v) dv dσ,

(5.34)

0 RD

which substitutes the interpolation formula (3.13). Then, integrating in time and space yields



x

 k fn (t, x, v) dv

2

RD

L1t,x

 vs2k fn L1

t,x,v

s +

k=0

L1t,x

RD

t,x,v

(5.35)

dσ.

0

For each value of k, we fix now the interpolation parameter s as sk = 2

 ∞

x

 k fn (t, x, v) dv

2

v

 k gn 1 L σ2

∞

v

 

, which yields, summing over k,

∞ 

v

 k gn 1 + L σ2 sk

k fn

sk 2

β k 1−β

L1t,x,v

k=0

t,x,v

dσ.

(5.36)

k=0 0

The remainder of the demonstration only consists in an analysis of the velocity frequencies and it is thus strictly identical to the proof of Proposition 5.3. And so, we infer that



 

fn (t, x, v) dv (5.37)  C fn L1 (dt dx;B 0 (dv)) + gn L1 (dt dx;B β (dv)) ,

0 (dx)) L1 (dt;B1,1

RD

which concludes our proof.

1,1

1,1

2

Proof of Theorem 5.4. Notice first that there exists β < 1 so that  γ β  D  (1 − v ) 2 gn ∈ L1 Rt × RD x ; B1,1 Rv .

(5.38)

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519

We may therefore straightforwardly apply Proposition 5.5 to infer that ∞    D  0 Rx . fn (t, x, v) dv is bounded in L1 Rt ; B1,1

(5.39)

n=0

RD

Then, just as in the proof of Theorem 5.2, using Stein’s L log L result, we conclude that, for any compact subsets Kx , Kv ⊂ RD , ∞    fn (t, x, v) dv is bounded in L1 Rt ; L log L(Kx ) n=0

RD

∞  and fn (t, x, v) n=0

  is bounded in L1 Rt × RD x ; L log L(Kv ) .

(5.40)

∞ 1 D D Since {fn (t, x, v)}∞ n=0 is also bounded in L (Rt ; L (Rx × Rv )), it then follows that ∞   fn (t, x, v) dx dv is equi-integrable in t, n=0

RD ×RD

∞



is equi-integrable in x

fn (t, x, v) dv RD

and

n=0

∞  fn (t, x, v) n=0

is equi-integrable in v.

(5.41)

By the positiveness of the sequences, it is then possible to apply here Lemma 5.2 of [15] iteratively (one may also consult the proof of Proposition 3.3.5 of [23]) in order to conclude that  ∞ fn (t, x, v) n=0 is equi-integrable (in all variables t, x and v), (5.42) which concludes the proof of the theorem.

2

6. Proofs of the main results We proceed now to the demonstrations of the main results from Section 4. Since all our proofs are based on the crucial decomposition (3.26) given by Proposition 3.2, we will systematically start the proofs by establishing sharp dispersive estimates on the operators Atδ and Bδt appearing in the right-hand side of (3.26). In accordance with the hypotheses of Proposition 3.2, we will always consider the operators Atδ and Bδt as defined for some given cutoff   1 ρ ∈ S(R) such that ρ(0) = 2π ˆ ds = R ρ(s) ˜ ds = 1 and that ρ˜ is compactly supported inside [1, 2]. R ρ(s) For convenience of the reader, we have consistently labeled the coming sections so that the proofs of the theorems from Sections 4.1, 4.2, 4.3 and 4.4 are respectively contained in Section 6.1, 6.2, 6.3 and 6.4. p

6.1. Velocity averaging in L1x Lv , inhomogeneous case p

In this section, we show the proofs of the averaging lemmas in the endpoint case L1x Lv , i.e. of Theorems 4.1 and 4.2. For the sake of simplicity, we will only consider here the operators Atδ and Bδt as defined for the specific cutoff ρ(s) ˜ = δ{s=1} , so that  t Aδ f (x) = xδ f (x − tv, v) dv, RD

1  Bδt g(x) =

xδ g(x − σ tv, v) dv dσ. 0 RD

(6.1)

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This simplification is performed at absolutely no cost of generality, since it is then possible to easily extend the results to the case of a general smooth cutoff ρ ∈ S(R) by considering    st  st   Aδ f (x) ρ(s) Bδ g(x) s ρ(s) ˜ ds and ˜ ds. (6.2) R

R

It is to be emphasized that the above simplification is possible only because the identities (3.28) from Proposition 3.2 will not be employed here. Thus, we start by establishing, in Lemma 6.1 below, sharp estimates on the operators Atδ and Bδt based on dispersion. We then proceed to the proofs of Theorems 4.1 and 4.2, which are essentially an interpolation argument based on the estimates from Lemma 6.1. Lemma 6.1. For every 1  p, q  ∞, α ∈ R, k ∈ N and t  2−k , it holds that

1

A f p  C v f 1 p  Cf 1 D α D 0 0 L (Rx ;Bp,q (Rv )) L L L x

x

v

p



(6.3)

and

t

A k f 2

p

Lx

C

 t

D(1− p1 )

v

 k f t2

L1x Lv

C α+D(1− p1 ) kα t 2

f L α D , 1 (RD x ;Bp,q (Rv ))

(6.4)

where C > 0 is independent of t and 2k . Furthermore, if  1 D 1− < 1, p then, for any β ∈ R, it holds that

1

B g 0

p

Lx

(6.5)

 C S2v g L1 Lp  CgL 1 (RD ;B β x

v

x

(6.6)

D p,q (Rv ))

and, if further β + D(1 − p1 ) = 1,

t

B k g 2

p

Lx

C

 t

β+D(1− p1 ) kβ 2

C

 t

γ +D(1− p1 ) kγ 2

2

−jk (1−β−D(1− p1 ))



v0 g L1 Lp x v

gL 1 (RD ;B β x

D p,q (Rv ))

+

j k +1 j =0

2

−(jk +1−j )(1−β−D(1− p1 )) jβ

2



v2j g L1 Lp x v

,

(6.7)

where γ = min{β, 1 − D(1 − p1 )}, jk  1 is the largest integer such that 2jk −1  t2k and C > 0 is independent of t and 2k . Finally, if β + D(1 − p1 ) = 1, then

t

B k g 2

p

Lx

 j k +1

v

 1 C C v jβ  kβ 0 g L1 Lp + 2 2j g L1 Lp  kβ log 1 + t2k q gL β D , 1 (RD x v x v x ;Bp,q (Rv )) t2 t2

where C > 0 is independent of t and

(6.8)

j =0

2k .

Proof. These bounds will follow from the dispersive properties of the transport operator, which were first established by Castella and Perthame in [8]. More precisely, we will utilize the following dispersive estimate, valid for any 1  p  ∞, which is a simple consequence of the change of variables v → y = x − tv,



 



x−y −D(1− p1 ) −D

h(x − tv, v) p 1 = t −D h y, x − y

h(x, v) 1 p . (6.9)

 t =t h y,



Lx Lv Lx Lv p p t t Lx L1y L1y Lx

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

521

Thus, we easily deduce, utilizing identities (3.12), that

1





A f (x) p  C v f (x, v) 1 p  C f (x, v) 1 D α 0 0 L L L L (R ;B x

v

x

(6.10)

D p,q (Rv ))

x

and

t

A k f (x) 2

p

Lx

 Ct

 −α−D(1− p1 ) −kα 2 f (x, v) L vt2k f (x, v) L1 Lp  Ct 1 (RD ;B α

 −D(1− p1 )

x

v

x

D p,q (Rv ))

,

(6.11)

which concludes the estimate for A10 and At2k . As for B01 and B2t k , we first obtain 1

1

B g(x)

p Lx

0

C

s

  −D(1− p1 )

S v g (x, v) 1 p s Lx Lv

(6.12)

ds

0

and

t

B k g(x) 2

1 p

Lx

C

 vst2k g (x, v) L1 Lp ds.

 −D(1− p1 )

(st)

(6.13)

v

x

0

Then, since D(1 −

1 p) < 1

and

Ssv

= Ssv S2v ,

1

B g(x) 0

p

Lx

we deduce that



  C S2v g (x, v) L1 Lp  C g(x, v) L 1 (RD ;B β x

v

D p,q (Rv ))

x

(6.14)

,

which concludes the estimate on the low frequency component. Regarding the high frequencies, we further split the integral in (6.13) into small dyadic intervals    j −1 j    1 1 2 2 2 I0 = 0, k , , k , , k I1 = ..., Ij = k k t2 t2 t2 t2 t2  jk −1  2 ,1 , and finally Ijk = t2k

(6.15)

where jk  1 is the largest integer such that 2jk −1  t2k . Thus, on each dyadic interval Ij , where 1  j  jk , the frequency st2k is between 2j −1 and 2j . Therefore, we deduce that, for any 1  j  jk ,  

−D(1− p1 ) −D(1− p1 ) v

v j −2 j +1 v k g 1 p ds (st) st2k g L1 Lp ds = (st) L L [2 ,2 ] st2 v

x

Ij

x

Ij

 C|Ij |2

v

v[2j −2 ,2j +1 ] g L1 Lp

−(j −k)D(1− p1 )

x

v

C (j −k)(1−D(1− p1 ))

v j −2 j +1 g 1 p . = 2 Lx Lv [2 ,2 ] t

(6.16)

Furthermore, when j = 0, it is readily seen, since D(1 − p1 ) < 1, that  

1 −D(1− p1 )

v k g 1 p ds = (st)−D(1− p ) S v v k g 1 p ds (st) 4 st2 L L L L st2 x

I0

v

x

I0



C

−D(1− p1 )

(st)

ds S4v g L1 Lp x

I0

=C Thus, on the whole, we have obtained that

1 t (1 − D(1 −

v

1 p ))

2

v

−k(1−D(1− p1 ))

S v g 1 p . 4 Lx Lv

(6.17)

522

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

t

B k g 2

1 p

Lx

C

vst2k g L1 Lp ds

−D(1− p1 )

(st)

x

v

0

=C

jk  j =0 I

vst2k g L1 Lp ds

−D(1− p1 )

(st)

x

v

j

 j k +1

C −k(1−D(1− p1 )) (j −k)(1−D(1− p1 )) v v

2 0 g L1 Lp + 2j g L1 Lp 2  x v x v t j =0

−β−D(1− p1 ) −kβ −j (1−β−D(1− p1 ))

v g 1 p  Ct 2 2 k 0

+

j k +1

2

−(jk +1−j )(1−β−D(1− p1 )) jβ

2

j =0

Lx Lv



v2j g L1 Lp x v

(6.18)

.

It follows that, if 1 > β + D(1 − p1 ), a further application of Hölder’s inequality to the preceding estimate yields

t

B k g(x) 2

 Ct

p

Lx

 −j (1−β−D(1− 1 )) ∞ −β−D(1− p1 ) −kβ

g(x, v) 1 D β D . p 2 2  (Rx ;Bp,q (Rv )) j =0 q L

(6.19)

And in case 1 < β + D(1 − p1 ), we obtain

t

B k g(x) 2

p Lx





1   C k(D(1− p1 )−1)

2j (1−β−D(1− p )) ∞ q g(x, v) 1 D β D . 2  (Rx ;Bp,q (Rv )) j =0  L t

Finally, if 1 = β + D(1 − p1 ), since jk + 3 

t

B k g 2

p

Lx

log(t2k+4 ) log 2 ,

(6.20)

then

 j k +1

C  kβ v0 g L1 Lp + 2jβ v2j g L1 Lp x v x v t2 j =0

1 C  kβ (jk + 3) q gL β D 1 (RD x ;Bp,q (Rv )) t2 1  C  kβ log 1 + t2k q gL β D , 1 (RD x ;Bp,q (Rv )) t2 which concludes the proof of the lemma. 2

(6.21)

We proceed now to the proof of Theorem 4.1, which is the simplest of our main results. Proof of Theorem 4.1. According to Proposition 3.2, we begin with the following dyadic interpolation formulas, for each k ∈ N,  x0 f (x, v) dv = A10 f (x) + B01 g(x), RD



f (x, v) dv = At2k f (x) + tB2t k g(x).

x2k

(6.22)

RD

Then, a direct application of Lemma 6.1 yields the estimates, for every t  2−k ,



x

 β α D + Cg1 D 1 (RD

0 f (x, v) dv  Cf L L (R ;B x ;Bp,q (Rv )) p

RD

Lx

x

D p,q (Rv ))

,

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551



x

 k f (x, v) dv

2

p

t

Lx

RD

α+D(1− p1 )

× 2

α t2k vt2k f L1 Lp +



C



x

2kα

−jk (1−β−D(1− p1 ))

v

v0 g L1 Lp x v

+

523

C t

β−1+D(1− p1 ) kβ 2

j k +1

2



−(jk +1−j )(1−β−D(1− p1 )) jβ

v2j g L1 Lp x v

2

j =0

, (6.23)

2jk −1

 t2k

where jk  1 is the largest integer such that and C > 0 is independent of t and which concludes the estimate on the low frequencies of the velocity averages x0 RD f (x, v) dv. Note that the above estimate remains valid for the value β + D(1 − p1 ) = 1. Next, in order to treat the high frequencies, for each value of k, we have to select the value of the interpolation parameter t which will optimize (6.23), i.e. minimize its right-hand side. The heuristic argument yielding the optimal value for t goes as follows. In virtue of Lemma 6.1, the estimate on the high frequencies in (6.23) essentially amounts to

 

x 1 1

 k f (x, v) dv  C + , (6.24)

2 p α+D(1− p1 ) kα γ −1+D(1− p1 ) kγ Lx t 2 t 2 D 2k ,

R

where γ = min{β, 1 − D(1 − p1 )}. Therefore, up to multiplicative constants, the right-hand side above will be minimized when both terms are equal, which leads to an optimal value of the interpolation parameter t of tk = 2 where we have used the hypothesis α > −D(1 −

1 p)

α−γ −k 1+α−γ

 2−k ,

(6.25)

to deduce that 1 + α − γ > 0.

Note that, in the case β + D(1 − p1 )  1, we can choose t = ∞, which is, in fact, more optimal than t2k = 2 since it eliminates the first term in the right-hand side of the above estimates. This case is discussed later on. Thus, in the case β + D(1 − p1 ) < 1 so that γ = β, setting t = tk in (6.23), denoting s =

α+D(1− p1 ) 1+α−γ

1 k 1+α−γ

,

− D(1 − p1 ),

k ] and summing over k, yields that noticing that jk = [1 + 1+α−γ

∞

 

ks x

2  k

f (x, v) dv

2

p

RD

k=0 q

Lx



 −k 1 (1−β−D(1− 1 )) ∞ v α ∞

 g 1 p p  C tk 2k vt 2k f L1 Lp k=0 q + C 2 1+α−β 0 k=0 q Lx Lv k x v

j +1 ∞ k





−(j +1−j )(1−β−D(1− p1 )) jβ + C 2 k 2 v2j g L1 Lp

. x v

q j =0

(6.26)

k=0 

Then, it is readily seen that the first term in the right-hand side above is controlled by f L α D , for 1 (RD x ;Bp,q (Rv )) k

−j (1−β−D(1− p1 ))

1

tk 2k = 2 1+α−β → ∞ as k → ∞. Furthermore, writing aj = 2jβ v2j gL1 Lpv 1{j 0} and bj = 2 x for all j ∈ Z, we see that a = {aj }j ∈Z ∈ q , b = {bj }j ∈Z ∈ 1 and that

j +1 ∞ k



 ∞

−(jk +1−j )(1−β−D(1− p1 )) jβ v 2 2 2j g L1 Lp = (a ∗ b)jk +1 k=0 q

x v

q j =0

1{j 0} ,

k=0 

 aq b1  CgL 1 (RD ;B β

D p,q (Rv ))

x

Thus, on the whole, combining the preceding estimates with (6.26), we deduce that

∞

 

ks x

2  k

 Cf 1 D α D + Cg 1 D β f (x, v) dv  (R ;B L (Rx ;Bp,q (Rv ))

2

L p

RD

Lx

k=0 q

x

(6.27)

.

D p,q (Rv ))

,

(6.28)

524

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

which concludes the proof in the case β + D(1 − p1 ) < 1. Regarding the case β + D(1 − p1 )  1 so that γ = 1 − D(1 − p1 ), we note that the estimate (6.23) on the high frequencies implies that, recalling jk satisfies t2k  2jk  t2k+1 ,



x C

 k f (x, v) dv α D 1 (RD

p  α+D(1− 1 ) kα f L

2 x ;Bp,q (Rv )) p 2 Lx t D R

 j k +1

v

v 1 C j (1−β−D(1− )) jβ

 g 1 p + p 2  g p . + 2 (6.29) 0 Lx Lv L1x Lv 2j k(1−D(1− p1 )) 2 j =0 α+D(1− 1 )

Therefore, letting t tend to infinity, denoting s = 1+α−γp − D(1 − p1 ) = 1 − D(1 − p1 ) and noticing that limt→∞ jk = ∞, we find



 ∞



1 j (1−β−D(1− )) x

v p 2jβ v g p . 2ks f (x, v) dv 2 (6.30)

 C 0 g L1 Lp +

2k L1 L 2j p

x

Lx

RD

It then follows that

 

ks x

2  k f (x, v) dv

2

and, in case β + D(1 − p1 ) > 1,

 

ks x

2  k f (x, v) dv

2 RD

x

j =0

v

∞

p

 CgL 1 (RD ;B β

,

(6.31)

∞

p

 CgL 1 (RD ;B β

,

(6.32)

k=0 ∞

Lx

RD

v

k=0 ∞

Lx

x

x

D p,1 (Rv ))

D p,q (Rv ))

2

which concludes the proof of the theorem.

Next is the demonstration of Theorem 4.2, which builds upon the previous proof of Theorem 4.1. Proof of Theorem 4.2. As in the proof of Theorem 4.1, we begin with the dyadic decomposition provided by Proposition 3.2, for each k ∈ N,  x 0 f (x, v) dv = A10 f (x) + B01 g(x), RD



f (x, v) dv = At2k f (x) + tB2t k g(x),

x2k

(6.33)

RD

and we utilize Lemma 6.1 to obtain, in virtue of property (3.30) from Proposition 3.2, that



x

x



1 D α D + C S x g 1 D β D ,  (R ;B (R )) 2

0 f (x, v) dv  C S2 f L (R ;B (R )) L p

RD

x

Lx



x

 k f (x, v) dv

2 RD

p

Lx

p,q

t

α+D(1− p1 )

⎡ ⎢ ⎢ +C⎢ ⎣

t

x

x

 k−1

1

C

v

[2

2kα

1) 1−γ −D(1− p

2kγ

1

log(1+t2k ) q 2kγ 1 2kγ

,2k+1 ]

p,q

v

f L 1 (RD ;B α x

D p,q (Rv ))

⎤ if β + D(1 − p1 ) < 1 ⎥ ⎥ ⎥ if β + D(1 − p1 ) = 1 ⎦ if β + D(1 − p1 ) > 1

× x[2k−1 ,2k+1 ] g L 1 (RD ;B β x

D p,q (Rv ))

,

(6.34)

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

525

where γ = min{β, 1 − D(1 − p1 )}. This concludes the estimate on the low frequencies of the velocity averages  x0 RD f (x, v) dv. Next, in order to control the high frequencies, optimizing in t for each value of k, we fix the interpolation −k (α−γ )+(a−b)

1+α−γ parameter t as tk = 2 (cf. proof of Theorem 4.1 for a heuristic explanation on how to choose this optimal parameter). Note that tk  2−k , for b  a − 1 and 1 + α − γ > 0, and that this choice is independent of 1  p, q  ∞. Furthermore, in the cases β + D(1 − p1 ) > 1 or β + D(1 − p1 ) = 1 and q = 1, we can choose t = ∞, which is, in

fact, more optimal than t2k = 2 This case is discussed later on.

1+b−a k 1+α−γ

, since it eliminates the first term in the right-hand side of the above estimate.

Therefore, denoting s = (1 + b − a)



x

2 2k f (x, v) dv

α+D(1− p1 ) 1+α−γ

+ a − D(1 − p1 ), we find that

ks

p

Lx

RD

ka

x[2k−1 ,2k+1 ] f L α D 1 (RD x ;Bp,q (Rv ))   2kb if β + D(1 − p1 ) = 1

x k−1 k+1 g 1 D β D . 1 +C  (Rx ;Bp,q (Rv )) [2 ,2 ] L 1+b−a k) q if β + D(1 − p1 ) = 1 2kb (1 + 1+α−γ

 C2

(6.35)

Consequently, summing over k, we obtain, in case β + D(1 − p1 ) = 1 or q = 1,

 

ks x

2  k f (x, v) dv

2

∞

p

k=0 q

Lx

RD



 C 2ka x2k f L 1 (RD ;B α

∞

D )) k=−1 q (R p,q v

x



+ C 2kb x2k g L 1 (RD ;B β x

= Cf B a,α

D D 1,p,q (Rx ×Rv )



∞

D p,q (Rv ))

k=−1 q

+ CgB b,β

D D 1,p,q (Rx ×Rv )

(6.36)

,

while, if β + D(1 − p1 ) = 1 and q = 1, we find, for any  > 0,





k(s−) x

2

 k f (x, v) dv

2

∞

p

Lx

RD

k=0 q



 C 2k(a−) x2k f L 1 (RD ;B α



+ C 2kb x2k g L 1 (RD ;B β x



∞

D p,q (Rv ))

x

k=−1 q

∞

D p,q (Rv ))



k=−1 q

= Cf B a−,α (RD ×RD ) + CgB b,β 1,p,q

x

D D 1,p,q (Rx ×Rv )

v

.

(6.37)

We handle now the cases β + D(1 − p1 ) > 1 or β + D(1 − p1 ) = 1 and q = 1, by letting t tend to infinity in (6.34), as mentioned previously. This leads to



x

C

x

 k

1 D β D . f (x, v) dv (6.38)  (Rx ;Bp,q (Rv ))

2

p  2kγ [2k−1 ,2k+1 ] g L Lx RD

Hence, recalling s = 1 + b − D(1 − p1 ) and summing over k yields

 

ks x

2  k f (x, v) dv

2 RD

∞

p

Lx

k=0 q



 C 2kb x2k g L 1 (RD ;B β x

= CgB b,β

D D 1,p,q (Rx ×Rv )

which concludes the proof of the theorem.

2

,

∞

D )) k=−1 q (R p,q v (6.39)

526

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551 p

6.2. Velocity averaging in L1x Lv , homogeneous case Here, we explain how the proofs of the previous section in the inhomogeneous case are adapted to the homogeneous case and we give justifications of Theorems 4.3 and 4.4. Again, for the sake of simplicity as in the previous section and without any loss of generality, we will only consider here the operators Atδ and Bδt as defined for the specific cutoff ρ(s) ˜ = δ{s=1} . Thus, we start by establishing, in Lemma 6.2 below, sharp estimates on the operators Atδ and Bδt based on dispersion. We then proceed to the proofs of Theorems 4.3 and 4.4, which are essentially an interpolation argument based on the estimates from Lemma 6.2. Lemma 6.2. For every 1  p, q  ∞, α, β ∈ R, k ∈ N and t > 0 such that  1 β +D 1− < 1, p

(6.40)

it holds that

t

A k f 2

p

Lx

C

 t

D(1− p1 )

v

 k f t2

p

L1x Lv



C α+D(1− p1 ) kα t 2

f L 1 (RD ;B˙ α

D p,q (Rv ))

x

,

(6.41)

and

t

B k g 2



p Lx

C

j k +1

β+D(1− p1 ) kβ t 2

j =−∞

C

 t

β+D(1− p1 ) kβ 2

2

v2j g L1 Lp

−(jk +1−j )(1−β−D(1− p1 )) jβ

2

gL 1 (RD ;B˙ β x

D p,q (Rv ))

x

v

(6.42)

,

where jk  1 is the largest integer such that 2jk −1  t2k and C > 0 is independent of t and 2k . Furthermore, if β + D(1 − p1 ) = 1 and q = 1, then it still holds that

t

B k g 2

p Lx



jk +1

C C 2jβ v2j g L1 Lp  kβ gL D . 1 (RD ˙β kβ x v x ;Bp,1 (Rv )) t2 t2

(6.43)

j =−∞

Proof. We follow precisely the steps of the proof of Lemma 6.1. Thus, we first have, as a consequence of the dispersive properties of the transport operator, that

t 1  

A k f (x) p  Ct −D(1− p ) v k f (x, v) 1 p Lx Lx Lv t2 2

−α−D(1− p1 ) −kα  Ct 2 f (x, v) L (6.44) 1 (RD ;B˙ α (RD )) , x

which concludes the estimate for As for B2t k , we first obtain

t

B k g(x) 2

p,q

v

At2k . 1 p

Lx

C

 −D(1− p1 )

(st)

 vst2k g (x, v) L1 Lp ds. x

v

(6.45)

0

Then, we further split the above integral into small dyadic intervals  j −1 j   jk −1  2 2 2 Ij = , = ,1 , and I jk t2k t2k t2k

(6.46)

where j, jk ∈ Z, j  jk and jk is the largest integer such that 2jk −1  t2k . Thus, on each dyadic interval Ij , the frequency st2k is between 2j −1 and 2j . Therefore, we deduce that

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551



−D(1− p1 )



vst2k g L1 Lp x v

(st)



−D(1− p1 )

ds =

Ij

527

v[2j −2 ,2j +1 ] vst2k g L1 Lp ds

(st) Ij

 C|Ij |2

v

x

−(j −k)D(1− p1 )

v[2j −2 ,2j +1 ] g L1 Lp v

x

C (j −k)(1−D(1− p1 ))

v j −2 j +1 g 1 p . = 2 Lx Lv [2 ,2 ] t

(6.47)

Thus, on the whole, we have obtained that

t

B k g 2

1 p

Lx

C

−D(1− p1 )

vst2k g L1 Lp ds

(st)

x

v

0 jk 

=C

j =−∞ I



vst2k g L1 Lp ds

−D(1− p1 )

(st)

v

x

j

jk +1

C (j −k)(1−D(1− p1 ))

vj g 1 p 2 Lx Lv 2 t j =−∞

 Ct

j k +1

−β−D(1− p1 ) −kβ

2

2

−(jk +1−j )(1−β−D(1− p1 )) jβ

2

j =−∞

v2j g L1 Lp . x

(6.48)

v

It follows that, if 1 − β − D(1 − p1 ) > 0, a further application of Hölder’s inequality to the preceding estimate yields

t

B k g(x)

p

Lx

2

 Ct

 −j (1−β−D(1− 1 )) ∞ −β−D(1− p1 ) −kβ

g(x, v) 1 D β D , p 2 2  (Rx ;B˙ p,q (Rv )) j =0 q L

which concludes the proof of the lemma.

(6.49)

2

Proof of Theorem 4.3. According to Proposition 3.2, we begin with the following dyadic interpolation formula, for each k ∈ N,  x2k f (x, v) dv = At2k f (x) + tB2t k g(x). (6.50) RD

Then, a direct application of Lemma 6.2 yields the estimate, for every t > 0,



x

 k α v C

 k

 t2 t2k f L1 Lp f (x, v) dv 1

2

p x v α+D(1− p ) kα Lx t 2 D R

+

C

j k +1

β−1+D(1− p1 ) kβ t 2

j =−∞

2

v2j g L1 Lp ,

−(jk +1−j )(1−β−D(1− p1 )) jβ

2

x

v

(6.51)

where jk ∈ Z is the largest integer such that 2jk −1  t2k and C > 0 is independent of t and 2k . Next, optimizing in t for each value of k, we fix the interpolation parameter t as 1

tk = λ 1+α−β 2 where λ =

f L 1 (RD ;B˙ α g1

x

D p,q (Rv ))

β D L (RD x ;B˙ p,q (Rv ))

α−β −k 1+α−β

(cf. proof of Theorem 4.1 for a heuristic explanation on how to choose this op-

timal parameter). Note that 1 + α − β > 0 since α > −D(1 − s=

α+D(1− p1 ) 1+α−β

(6.52)

,

− D(1 − p1 ), noticing that jk = [1 +

log λ (1+α−β) log 2

+

1 p ).

Thus, setting t = tk in (6.51), denoting

k 1+α−β ]

and summing over k, yields that

528

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551





ks x

2  k

f (x, v) dv

2

RD

C

 λ

1) α+D(1− p 1+α−β



∞ p

k=−∞ q

Lx

 k α v



tk 2  k f 1 p ∞

L L k=−∞ q t 2 k

x

v

 j +1

∞ k





−(jk +1−j )(1−β−D(1− p1 )) jβ v

p  + 2 2 g

. 1 j 1) Lx Lv 2 β−1+D(1− p

q j =−∞ k=−∞  1+α−β λ Then, it is readily seen that the first term in the right-hand side above is controlled by f L 1 (RD ;B˙ α C

D p,q (Rv )) 1 −j (1−β−D(1− p )) x

1 1+α−β

tk 2k = (λ2k ) → ±∞ as k → ±∞. Furthermore, writing aj = 2jβ v2j gL1 Lpv and bj = 2 x for all j ∈ Z, we see that a = {aj }j ∈Z ∈ q , b = {bj }j ∈Z ∈ 1 and that

 j +1

∞ k





−(jk +1−j )(1−β−D(1− p1 )) jβ v 2 2 2j g L1 Lp

x v

j =−∞ k=−∞ q

 ∞ = (a ∗ b)jk +1 k=−∞ q  aq b1  CgL 1 (RD ;B˙ β (RD )) . p,q

x

p

RD

x

k=−∞ q

p,q

(6.54)

v

L (Rx ;Bp,q (RD v ))

v

, for

1{j 0} ,

Thus, on the whole, combining the preceding estimates with (6.53), we deduce that 1) 1)



∞  1−β−D(1− p α+D(1− p

ks x

1+α−β 1+α−β

2  k

 Cf  f (x, v) dv 1 (RD ;B˙ α (RD )) × g1 D ˙ β

2

L Lx

(6.53)

,

(6.55)

2

which concludes the proof of the theorem.

Proof of Theorem 4.4. As in the proof of Theorem 4.3, we begin with the dyadic decomposition provided by Proposition 3.2, for each k ∈ N,  x2k f (x, v) dv = At2k f (x) + tB2t k g(x), (6.56) RD

and we utilize Lemma 6.2 to obtain, in virtue of property (3.30) from Proposition 3.2, that



x

x

1

 k

1 D ˙ α D f (x, v) dv

2

p  C α+D(1− 1 ) kα [2k−1 ,2k+1 ] f L (Rx ;Bp,q (Rv )) p Lx t 2 D R

+C

t

1−β−D(1− p1 )

2kβ

x k−1 [2

,2k+1 ]

g L 1 (RD ;B˙ β

D p,q (Rv ))

x

(6.57)

.

Next, optimizing in t for each value of k, we fix the interpolation parameter t as 1

tk = λ 1+α−β 2 where λ =

f B˙ a,α

D D 1,p,q (Rx ×Rv )

g ˙ b,β

D B1,p,q (RD x ×Rv )

−k (α−β)+(a−b) 1+α−β

(6.58)

,

(cf. proof of Theorem 4.1 for a heuristic explanation on how to choose this optimal

parameter). Note that 1 + α − β > 0 since α > −D(1 − p1 ), and that this choice is independent of 1  p, q  ∞. α+D(1− 1 )

Therefore, denoting s = (1 + b − a) 1+α−βp + a − D(1 − p1 ), we find that





x C

 2ks f (x, v) dv 2ka x[2k−1 ,2k+1 ] f L  1 (RD ;B˙ α 1

2k p

RD

Lx

λ

α+D(1− p ) 1+α−β

C

+ λ

1) β−1+D(1− p 1+α−β

x

D p,q (Rv ))

2kb x[2k−1 ,2k+1 ] g L 1 (RD ;B˙ β x

D p,q (Rv ))

.

(6.59)

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

Finally, summing over k, we obtain



∞ 

ks x

2  k f (x, v) dv

2

p

λ

C λ

1) β−1+D(1− p 1+α−β

= Cf 

L

2

1) α+D(1− p 1+α−β

+

which concludes the proof of the theorem.

 ka x

2  k f 1

C



k=−∞ q

Lx

RD



529

D ˙α (RD x ;Bp,q (Rv ))



∞

k=−∞ q

 kb x

2  k g

∞

β D )) k=−∞ q ˙ 1 (RD ; B (R L p,q x v

2

1) 1−β−D(1− p 1+α−β a,α D B˙ 1,p,q (RD x ×Rv )

× g

1) α+D(1− p 1+α−β b,β D B˙ 1,p,q (RD x ×Rv )

(6.60)

,

2

6.3. The classical L2x L2v case revisited We provide now the proofs for the classical Hilbertian case of velocity averaging. Thus, we start by establishing, in Lemma 6.3 below, sharp estimates on the operators Atδ and Bδt , which contain the crucial regularization property by velocity averaging. The proof of Lemma 6.3 is based on the technical Lemmas 6.4 and 6.5. Once Lemmas 6.3, 6.4 and 6.5 are established, we proceed to the proof of Theorem 4.5, which is essentially an interpolation argument based on the estimates from Lemma 6.3. Lemma 6.3. Let φ(v) ∈ C0∞ (RD ). For every 1  q  ∞, α, β ∈ R, β = 12 , k ∈ N and t  2−k , it holds that



1

A (f φ) 2  C x f 2 D α D  Cf 2 D α D , 0 L (Rx ;B (Rv )) 0 L L (R ;B (R )) x

t

A k (f φ) 2

and

L2x



1

(t2k )α+ 2

1

B (gφ) 0

t

B k (gφ) 2

L2x



x

C

x

 k f 2 2

2,q

α D L (RD x ;B2,q (Rv ))

 C x0 g L 2 (RD ;B β

L2x

x

C γ + 12

(t2k )

x

 k g 2

2,q

v

D 2,q (Rv ))



C 1

(t2k )α+ 2

f L α 2 (RD x ;B

 CgL 2 (RD ;B β

β D 2 (RD L x ;B2,q (Rv ))

x



C 1 (t2k )γ + 2

D 2,q (Rv ))

t

B k (gφ) 2

x

L2x

2,q

v

x

(6.61)

,

(6.62)

,

gL 2 (RD ;B β

D 2,q (Rv ))

x

where γ = min{β, 12 } and C > 0 is independent of t and 2k . Furthermore, if β = 12 , then, it holds that

1



B (gφ) 2  C x g 2 D β D  Cg 2 D β  (R ;B (R ))  (R ;B 0 0 L L L x

,

D 2,q (Rv ))

D 2,q (Rv ))

,

1  C log 1 + t2k q x2k g L β D 2 (RD k x ;B2,q (Rv )) t2 1   C  k log 1 + t2k q gL β D . 2 (RD x ;B2,q (Rv )) t2 

(6.63)

The above lemma will be obtained as a consequence of the following Lemmas 6.4 and 6.5. Lemma 6.4. For any χ ∈ S(R) and ρ ∈ C ∞ (R) such that χ ≡ 1 in a neighborhood of the origin and ρˆ is compactly supported and bounded pointwise, and for each λ  1, R > 0 and t  R1 , let h1 (η, v) and h2 (η, v) be defined by     v · η η  v·η  h1 (η, v) = χ v − ρ λ , |η| |η|  |η|    v · η η  h2 (η, v) = 1{ R |η|2R} χ v − ρ(tv · η), (6.64) 2 |η| |η|  for all η, v ∈ RD .

530

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

Then, for each k ∈ N, v0 h1 L2v and v2k h1 L2v are independent of η and, for every 1  q  ∞ and α = 12 , it holds that C h1 B −α (RD )  , v 2,q γ + 12 λ C −α , (6.65) h2 L D  ∞ (RD 1 η ;B2,q (Rv )) (tR)γ + 2 where γ = min{α, 12 } and the constant C > 0 is independent of λ, t and R. Furthermore, if α = 12 , then it holds that 1 C log(1 + λ) q , v 2,q λ 1 C −α log(1 + tR) q . h2 L D ))  ∞ (RD ;B (R η v 2,q tR Finally, if the support of ρˆ doesn’t contain the origin, then, for every α ∈ R, it holds that

h1 B −α (RD ) 

h1 B −α (RD )  v

2,q

C 1

λα+ 2

h2 L ∞ (RD ;B −α (RD ))  η

v

2,q

(6.66)

, C 1

(tR)α+ 2

(6.67)

.

Lemma 6.5. For any χ ∈ S(R) and ρ ∈ C ∞ (R) such that χ ≡ 1 in a neighborhood of the origin and ρˆ is compactly supported and bounded pointwise, let h(η, v) be defined by   h(η, v) = 1{|η|1} χ |v| ρ(v · η), (6.68) for all η, v ∈ RD . Then, for every 1  q  ∞ and α ∈ R, it holds that

 −α  D  ∞ RD h(η, v) ∈ L η ; B2,q Rv .

(6.69)

We defer the proofs of Lemmas 6.4 and 6.5 and proceed now to the proof of Lemma 6.3. Proof of Lemma 6.3. First, notice that, for any ρ(s) ∈ S(R), it holds

 r ρ(s) − ρ(0) F ρ(σ ˆ ) dσ, (r) = 2πρ(0)1{r0} − is

(6.70)

−∞

in the sense of tempered distributions. Indeed, one easily obtains by duality, for any test function ϕ(r) ∈ S(R), that   isσ   1 e −1 ρ(s) − ρ(0) F (r)ϕ(r) dr = ρ(σ ˆ ) dσ ϕ(s) ˆ ds is 2π is R

R



= R

R

1 2π

  σ

=



  σ e

R

0

ist

dt ρ(σ ˆ ) dσ ϕ(s) ˆ ds



ϕ(t) dt ρ(σ ˆ ) dσ R

0

∞∞

0  t ϕ(t)ρ(σ ˆ ) dσ dt −

= 0

t

−∞ −∞

ϕ(t)ρ(σ ˆ ) dσ dt

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

∞ =

 t ρ(σ ˆ ) dσ ϕ(t) dt −

ρ(σ ˆ ) dσ ϕ(t) dt R −∞

0 R

 



t 2πρ(0)1{t0} −

=

531

ρ(σ ˆ ) dσ ϕ(t) dt.

(6.71)

−∞

R

In the particular setting of the present lemma, we impose that ρˆ be compactly supported, supported away from the is smooth near the origin, that origin and that ρ(0) = 1. It then follows that τ (s) = 1−ρ(s) is  τˆ (r) =

e

−isr

1 − ρ(s) ds = is

r ρ(σ ˆ )dσ − 2π 1{r0}

(6.72)

−∞

R

is compactly supported as well and that |τˆ (r)| is bounded pointwise. Notice, however, that the origin is always contained in the support of τˆ . Consider now R > 0 so that supp φ(v) ⊂ {|v|  R} and let χ(r) ∈ C0∞ (R) be a cutoff function satisfying 1{|r|R}  χ(r)  1{|r|2R} . Then, according to the identities (3.28) from Proposition 3.2 and Lemmas 6.4 and 6.5, for any t  2−k , we obtain



1

 

A (f φ) 2 = 1 ψ(η)fˆ(η, v)φ(v)χ |v| ρ(v · η) dv 0 D

2 Lx Lη (2π) 2 D R

x

 

1{|η|1} χ |v| ρ(v · η) ∞ D −α D  0 f φ L α D 2 (RD L (Rη ;B2,q (Rv )) x ;B2,q (Rv ))

x  C  0 f φ L  Cf φL α D , α 2 (RD D 2 (RD x ;B2,q (Rv )) x ;B2,q (Rv )) 

 



t

 

A k (f φ) 2 = 1 ϕ η fˆ(η, v)φ(v)χ v − v · η η  ρ(tv · η) dv D  

2 Lx 2 k 2 |η| |η| Lη (2π) 2 D R



 

x



v · η η 

  2k f φ L 1{2k−1 |η|2k+1 } χ v − ρ(tv · η) α D 2 (RD 

∞ D −α D x ;B2,q (Rv )) |η| |η| L (Rη ;B (Rv ))  

2,q

1

x

 k f φ 2

1

f φL α 2 (RD x ;B

C (t2k )α+ 2 C (t2k )α+ 2

α D L (RD x ;B2,q (Rv ))

2

(6.73)

D 2,q (Rv ))

and, similarly,

1

B (gφ) 0

 C x0 gφ L 2 (RD ;B β

L2x

x

t

B k (gφ) 2

L2x

D 2,q (Rv ))

 CgφL 2 (RD ;B β x

D 2,q (Rv ))



1

C q x gφ log(1 + tR) 1 k 2 2 D 2 (RD t2k L x ;B2,q (Rv ))



1 C log(1 + tR) q gφ , 1 k 2 D 2 (RD t2 L x ;B2,q (Rv ))

,

(6.74)

and, if β = 12 ,

t

B k (gφ) 2

L2x



C γ + 12

(t2k )

x

 k gφ 2

β D 2 (RD L x ;B2,q (Rv ))

where γ = min{β, 12 } and C > 0 is independent of t and 2k .



C 1 (t2k )γ + 2

gφL 2 (RD ;B β x

D 2,q (Rv ))

,

(6.75)

532

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

Then, the conclusion of the lemma easily follows from a straightforward application of the methods of paradifferential calculus to the products f (x, v)φ(v) and g(x, v)φ(v). For the sake of completeness, we have reproduced in Lemma B.1 the precise estimates that are required here in order to conclude the proof. 2 Proof of Lemma 6.4. For each η ∈ RD , considering an orthogonal transformation Rη : RD → RD such that the unit η is mapped onto (0, . . . , 0, 1) and writing v = (v1 , . . . , vD−1 ) so that v = (v , vD ), it holds that vector |η|   1    1 Fv h1 (η, ξ ) = e−iv·(Rη ξ ) χ v  ρ(λvD ) dv = π (Rη ξ )1 , . . . , (Rη ξ )D−1 ρˆ (Rη ξ )D (6.76) λ λ RD

and

 Fv h2 (η, ξ ) = 1{ R |η|2R} 2

  e−iv·(Rη ξ ) χ v  ρ(t|η|vD ) dv

RD

   1 1 ρˆ (Rη ξ )D , = 1{ R |η|2R} π (Rη ξ )1 , . . . , (Rη ξ )D−1 2 t|η| t|η|

 where π(ξ ) = RD−1 e−iξ ·v χ(|v |) dv ∈ S(RD−1 ). Consequently, by Plancherel’s theorem,



v

1

 h1 2 =

ψ(ξ )π(ξ1 , . . . , ξD−1 )ρˆ 1 ξD , 0 D

2 Lv λ Lξ λ(2π) 2

 

v

1

ϕ ξ π(ξ1 , . . . , ξD−1 )ρˆ 1 ξD

 k h1 2 = D

2 Lv 2 k 2 λ Lξ λ(2π) 2 and

(6.78)



ψ(ξ )π(ξ1 , . . . , ξD−1 )ρˆ 1 ξD , = 1 R 2 0 D { 2 |η|2R}

2 Lv t|η| Lξ t|η|(2π) 2

 

v 1

ϕ ξ π(ξ1 , . . . , ξD−1 )ρˆ 1 ξD .

 k h2 2 = 1 R D { 2 |η|2R}

2 Lv 2 k 2 t|η| Lξ t|η|(2π) 2

v

 h2

(6.77)

1

(6.79)

Recall now that supp ψ(ξ ) ⊂ {|ξ |  1} and supp ϕ( 2ξk ) ⊂ {2k−1  |ξ |  2k+1 }. In particular, for every k ∈ N, we have that   ξ ξ ϕ k = ϕ k (1{|ξ |2k−2 } + 1{|ξ |<2k−2 } ) 2 2 (6.80)  1{2k−2 |ξ |2k+1 } 1{|ξD |2k+1 } + 1{|ξ |<2k−2 } 1{2k−2 |ξD |2k+1 } . Hence, we deduce that

v

 h1

v C

 h2 2  C , , (6.81) 0 Lv λ tR and, utilizing that π(ξ ) decays faster than any polynomial and assuming without any loss of generality that ρ(r) ˆ is supported inside {|r|  2}, we further obtain that



v

  1

 k h1 2  1 1 k−2 k+1 π ξ 2 1 ρˆ ξD

2 Lv Lξ {|ξD |2k+1 } 2 λ {2 |ξ |2 } λ Lξ

 D

  1 1 + 1{|ξ |<2k−2 } π ξ L2 ξD 1{2k−2 |ξD |2k+1 } ρˆ

2 λ λ ξ Lξ D

k  k

  22

1 k−2 k+1 π ξ 2 1{|ξ |2} ρˆ 2 ξD  {2 |ξ |2 } D

2 L λ λ ξ L 0

L2v



ξD

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

533

k  k

22 2

+ C 1{ 1 |ξD |2} ρˆ ξD

2 4 λ λ L C

ξD

k 2

k 2

2 2 + 1 2k k(1−α) λ { λ 8} λ(1 + 2 )



(6.82)

and, similarly, that

 k k 22 22 +  C1{ R |η|2R} 1 2k L2v 2 2 t|η|(1 + 2k(1−α) ) t|η| { t|η| 8}  k k 22 22 C + 1 . (6.83) 2k tR(1 + 2k(1−α) ) tR { tR 16}  n+1 Then, recalling the geometric sum formula nk=0 x k = x x−1−1 , valid for any x = 1, it follows that, in the case 1  q < ∞ and α = 12 ,  k( 1 −α) ∞ 



 −kα v ∞

2 

2  k h1 2

qC 2

+ 2k( 12 −α) 1 log(8λ) ∞ q Lv k=0  2 {k log 2 } k=0  λ 1 + 2k(1−α) k=0 q 1  1 C (16λ)( 2 −α)q − 1 q C  , (6.84) 1+  1 1 λ 2( 2 −α)q − 1 λγ + 2

v

 k h2

where γ = min{α, 12 }, and, similarly, that

 −kα v ∞

2  k h2 2

2

Lv k=0 q



C 1

(tR)γ + 2

(6.85)

.

Furthermore, both cases q = ∞ and α = 12 follow from obvious modifications of the preceding estimates. Thus, combining (6.81), (6.84) and (6.85) readily shows that the estimates (6.65) and (6.66) hold. It only remains to handle the case α  12 and 0 ∈ / supp ρ. ˆ Without any loss of generality, we may assume that ρ(r) ˆ is supported inside {1  |r|  2}. In this setting, it is possible to refine estimates (6.82) and (6.83) as to obtain

k  k

v

2  

 k h1 2  2 1 k−2 k+1 π ξ 2 1{|ξ |2} ρˆ 2 ξD {2 |ξ |2 } D

2 Lv L 2 λ λ ξ Lξ D k  k

22 2 +C ξD 1{ 1 |ξD |2} ρˆ

2 4 λ λ L

ξD

 2 2 C 1 1 2k + 1 1 2k λ(1 + 2k ) { 2  λ } λ { 2  λ 8}  2kα 2kα C 1 + 1 k k 1 2 1 2 1 1 λα+ 2 (1 + 2k ) { 2  λ } λα+ 2 { 2  λ 8} k 2

k 2

(6.86)

and, similarly, that

v

 k h2 2

L2v



k

k

22 22 1 1 1 2k k + 1 2 2 t|η|(1 + 2k(1+α) ) { 2  t|η| } t|η| { 2  t|η| 8}  k k 22 22 C + 1 1 k k 1 2 1 2 tR(1 + 2k ) { 4  tR } tR { 4  tR 16}  2kα 2kα C 1 1 2k + 1 1 2k . 1 1 (tR)α+ 2 (1 + 2k ) { 4  tR } (tR)α+ 2 { 4  tR 16}



 C1{ R |η|2R}

It is then readily seen that, for any 1  q  ∞,

(6.87)

534

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

 −kα v ∞

2  k h1 2

Lv k=0 q

2

 

 ∞

1

α+ 1 1 + 2k C

λ

k=0 

2

C 1

λα+ 2



∞

+ {1 log λ } log λ

q { log 2 −1k log 2 +3} k=0 q (6.88)

,

and, similarly, that

 −kα v ∞

2  k h2 2

Lv k=0 q

2



C 1

(tR)α+ 2

(6.89)

.

Finally, notice that, since supp ψ(ξ ) ⊂ {|ξD |  1} and λ  1, tR  1,



v

 h1 2 = 1 ψ(ξ )π(ξ1 , . . . , ξD−1 )ρˆ 1 ξD 1{λ<1} = 0, 0

2 Lv λ λ Lξ



v 1 1

ψ(ξ )π(ξ1 , . . . , ξD−1 )ρˆ

 h2 2 = 1 R ξ D 1{tR<1} = 0, 0 { 2 |η|2R} t|η| Lv t|η| L2

(6.90)

ξ

which readily establishes (6.67) and thus, concludes the proof of the lemma.

2

Proof of Lemma 6.5. For each η ∈ RD , considering an orthogonal transformation Rη : RD → RD such that mapped onto (0, . . . , 0, 1) and writing v = (v1 , . . . , vD−1 ) so that v = (v , vD ), it holds that      Fv h(η, ξ ) = 1{|η|1} e−iv·(Rη ξ ) χ |v| ρ |η|vD dv RD



= 1{|η|1} where π(ξ ) =

R

 RD−1

  1 1 π (Rη ξ )1 , . . . , (Rη ξ )D−1 , (Rη ξ )D − r ρˆ r dr, |η| |η| 

η |η|

is

(6.91)

e−iξ ·v χ(|v|)dv

∈ S(RD ). Consequently,

 

v

 

 h 2 = 1{|η|1} 1 ψ(ξ ) π ξ , ξD − r 1 ρˆ 1 r dr 0 D

2 Lv |η| |η| Lξ (2π) 2 R



 π(ξ ) L2 ρ(ξ ˆ D ) L1 , ξ ξ

D 

v

  1 1

 k h 2 = 1{|η|1} 1 ϕ ξ π ξ , ξ − r ρ ˆ r dr D D

2. Lv 2 k 2 |η| |η| Lξ (2π) 2

(6.92)

R

Recall now that

supp ϕ( 2ξk ) ⊂ {2k−1

 |ξ |  2k+1 }. In particular, for every k ∈ N, we have that   ξ ξ ϕ k = ϕ k (1{|ξ |2k−2 } + 1{|ξ |<2k−2 } ) 2 2  1{2k−2 |ξ |2k+1 } 1{|ξD |2k+1 } + 1{|ξ |<2k−2 } 1{2k−2 |ξD |2k+1 } .

(6.93)

Hence, we deduce, utilizing that π(ξ ) decays faster than any polynomial and assuming without any loss of generality that ρ(r) ˆ is supported inside {|r|  2}, that

v





 k h 2  1 k−2 k+1 π(ξ ) 2 ρ(ξ ˆ D ) 1 + 1 k−2 ˆ D ) 1 π(ξ ) 2 ρ(ξ k+1 2

{2

Lv

 It clearly follows that

|ξ |2

C 2k(1−α)

}

Lξ

Lξ

D

{2

−2|ξD |2

+2}

Lξ

Lξ

D

(6.94)

.

 −kα v

2  k h 2

which concludes the proof of the lemma.

2

∞

q 2 L∞ η Lv k=0 

 2C,

(6.95)

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

535

Now that the technical Lemmas 6.3, 6.4 and 6.5 are established, we may proceed to the proof of the main result Theorem 4.5. Proof of Theorem 4.5. First of all, we easily obtain that



x



0 f (x, v)φ(v) dv

L2x

RD

 C x0 f (x, v) L 2 (RD ;B α x

D 2,q (Rv ))

 C f (x, v) B a,α

D D 2,2,q (Rx ×Rv )



φ(v)

−α D B2,q (Rv )



φ(v)

−α D B2,q (Rv )

,

(6.96)

which concludes the control of the low frequencies. Notice, however, that it will be convenient, in order to carry out interpolation arguments later on, to also estimate the low frequencies with the same decomposition that was used in the proof of Theorem 4.2 and provided by Proposition 3.2, i.e.  x0 f (x, v)φ(v) dv = A10 (f φ)(x) + B01 (gφ)(x). (6.97) RD

This is easily done with an application of Lemma 6.3, thus yielding

1



A (f φ) 2  C x f 2 D α D  Cf  a,α B 0 0 L (R ;B (R )) L x

x

,

(6.98)

.

(6.99)

D D 2,2,q (Rx ×Rv )

v

2,q

and

1

B (gφ) 0

L2x

 C x0 g L 2 (RD ;B β

D 2,q (Rv ))

x

 CgB b,β

D D 2,2,q (Rx ×Rv )

In order to estimate the high frequencies, according to Proposition 3.2, we consider ρ ∈ S(R) a cutoff function 1 1 such that ρ(r) ˜ = 2π ρ(−r) ˆ is compactly supported supp ρ(r) ˜ ⊂ {1  |r|  2} and ρ(0) = 2π ˆ dr = 1, so that, R ρ(r) for any t > 0, we have the dyadic frequency decompositions, for each k ∈ N,  x2k f (x, v)φ(v) dv = At2k (f φ)(x) + tB2t k (gφ)(x). (6.100) RD

Consequently, in virtue of Lemma 6.3, we infer that, if β = 12 or q = 1, for any t  2−k ,



x

t

t

 k



f (x, v)φ(v) dv

2

 A2k (f φ) L2 + t B2k (gφ) L2 x

L2x

RD



C α+ 12

(t2k )

+ 2−k

x

x

 k f 2

α D L (RD x ;B2,q (Rv ))

2

C 1 (t2k )γ − 2

x

 k g 2

β D , 2 (RD L x ;B2,q (Rv ))

(6.101)

where γ = min{β, 12 }, and similarly, if β = 12 and q = 1,



x

t

t

 k



f (x, v)φ(v) dv

2

 A2k (f φ) L2 + t B2k (gφ) L2 RD

x

L2x



where C > 0 is independent of t and 2k .

C α+ 12

x

x

 k f 2 2

L (RD ;B α (RD ))

x v 2,q (t2k )

1  C + k log 1 + t2k q x2k g L β D , 2 (RD x ;B2,q (Rv )) 2

(6.102)

536

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551 k 1+b−a

1+b−a Then, optimizing the interpolation parameter t , we choose t2k = 2 1+α−γ , which is admissible since 1+α−γ 0 −k and thus t  2 . Furthermore, in the cases β > 12 or β = 12 and q = 1, we can choose t = ∞, which is, in fact, more optimal than k 1+b−a

t2k = 2 1+α−γ , since it eliminates the first term in the right-hand side of the above estimates. This case is discussed later on. α+ 12 γ − 12 It follows that, recalling s = (1 + b − a) 1+α−γ + a = (1 + b − a) 1+α−γ + 1 + b, in the case β = 12 or q = 1,





ks x ka x kb x 2k g L 2 2k f (x, v)φ(v) dv (6.103) 2 (RD ;B α (RD )) + C2 2 (RD ;B β (RD )) ,

 C2 2k f L x

L2x

RD

2,q

and similarly, if β = 12 and q = 1, for every  > 0,





k(s−) x ka x 2 f (x, v)φ(v) dv 2 (RD ;B α

 C2 2k f L

2k x

L2x

RD

v

D 2,q (Rv ))

x

2,q

v

+ C2kb x2k g L 2 (RD ;B β x

D 2,q (Rv ))

.

(6.104)

Finally, summing over k ∈ N yields that, in the case β = 12 or q = 1,

∞

 



ks x



2  k

 C 2ka xk f 2 D α D ∞ q f (x, v)φ(v) dv

2

2

q L (Rx ;B2,q (Rv )) k=0  2 Lx k=0 

RD



+ C 2kb x2k g L 2 (RD ;B β

∞

D )) k=0 q , (R v 2,q

x

and similarly, when β = 12 and q = 1,

∞





k(s−) x

2

 k f (x, v)φ(v) dv

2



L2x k=0 q

RD

(6.105)



 C 2ka x2k f L 2 (RD ;B α

∞

D )) k=0 q (R v 2,q

x



+ C 2kb x2k g L 2 (RD ;B β x

∞

D )) k=0 q . (R v 2,q

(6.106)

We handle now the cases β > 12 or β = 12 and q = 1, by letting t tend to infinity in (6.101), as mentioned previously. This leads to





x

 C xk g 2 D β D .

 k f (x, v)φ(v) dv (6.107)  (Rx ;B (Rv ))

2 2k

2 2 L 2,q Lx RD

Hence, recalling s = 1 + b and summing over k yields



∞ 

ks x

2  k

f (x, v)φ(v) dv

2

L2x k=0 q

RD

which concludes the proof of the theorem.



 C 2kb x2k g L 2 (RD ;B β x

∞

D )) k=0 q , (R v 2,q

(6.108)

2

p

6.4. The L1x Lv and L2x L2v cases reconciled We give now the proof of the general Theorem 4.6, which will follow from a simple interpolation procedure. In particular, the following lemma results from the interpolation between Lemma 6.1 and Lemma 6.3. We then proceed to the proof of Theorem 4.6, which is essentially an interpolation argument based on the estimates from Lemma 6.6. Lemma 6.6. Let φ(v) ∈ C0∞ (RD ). For every 1  p, r  ∞, 1  q < ∞, α ∈ R, k ∈ N and t  2−k such that r  p  r ,

(6.109)

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

it holds that

1

A (f φ) 0

p

Lx

537

 Cf L α D r (RD x ;Bp,q (Rv ))

(6.110)

and

t

A k (f φ)

p

Lx

2

where C > 0 is independent of t and 2k . Furthermore, if

C

 t

α+1− 1r +D( 1r − p1 ) k(α+1− 1 ) r

2



1 1 D − r p then, for any β ∈ R, it holds that

1

B (gφ) 0

and, if further β =

1 r

 <

p

Lx

2 −1 r

f L α D , r (RD x ;Bp,q (Rv ))

(6.111)

p = r = 2,

or

 CgL r (RD ;B β x

D p,q (Rv ))

(6.112)

(6.113)

,

− D( 1r − p1 ),

t

B k (gφ) 2

p

Lx

C

 t

γ +1− 1r +D( 1r − p1 ) k(γ +1− 1 ) r

2

gL r (RD ;B β x

D p,q (Rv ))

,

(6.114)

,

(6.115)

where γ = min{β, 1r − D( 1r − p1 )} and C > 0 is independent of t and 2k . Finally, if β =

1 r

− D( 1r − p1 ), then

t

B k (gφ) 2

p Lx



C t2

k(β+1− 1r )

1  log 1 + t2k q gL r (RD ;B β x

D p,q (Rv ))

where C > 0 is independent of t and 2k . Proof. First, if r = 1, then the statement of the present lemma is a straightforward consequence of Lemma 6.1 and Lemma B.1, while the case r = 2 is contained in Lemma 6.3. If 1 < r < 2, then the result will follow from the interpolation of Lemma 6.1 with Lemma 6.3. To this end, we define the interpolation parameter 0 < λ < 1 by λ = 2(1 − 1r ), so that 1 1−λ λ = + , r 1 2

(6.116)

1 1−λ λ + . = p p0 2

(6.117)

and the parameter 1  p0 < ∞ by

It is then easy to check that if D( 1r − p1 ) <

Further notice that

2 r

− 1, then  1 D 1− < 1. p0

(6.118)

   1 1 1 1 λ −D − = (1 − λ) 1 − D 1 − + . r r p p0 2

< 1 1 1 Therefore, if β{ < = } r − D( r − p ), it is always possible to respectively find β0 { = }1 − D(1 − > > that

β = (1 − λ)β0 + λβ1 .

(6.119) 1 p0 )

1 and β1 { < = } 2 such >

(6.120)

538

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

Consequently, defining γ0 = min{β0 , 1 − D(1 −

1 p0 )}

and γ1 = min{β1 , 12 }, it also holds that

γ = (1 − λ)γ0 + λγ1 . Then, in the case β =

1 r

(6.121)

− D( 1r − p1 ), we deduce according to Lemma 6.1 that

1

A (f φ) p0  Cf 1 D α , 0 L (Rx ;Bp ,q (RD L v )) x

t

A k (f φ) 2



p

Lx 0

1

B (gφ) 0

t

B k (gφ) 2



p

Lx 0

0

C t

α+D(1− p1 ) kα 0 2 p

Lx 0

f L α 1 (RD x ;Bp

0 ,q

 Cg1

β

0 D L (RD x ;Bp0 ,q (Rv ))

C γ +D(1− p1 ) kγ 0 2 0 t 0

g1

, (RD v ))

, β

0 D L (RD x ;Bp0 ,q (Rv ))

(6.122)

,

where C > 0 is independent of t and 2k , and employing Lemma 6.3, we infer that

1

A (f φ) 2  Cf 2 D α D , 0 L (Rx ;B (Rv )) L

t

A k (f φ) 2

L2x

0

2

L2x

C



1

B (gφ)

t

B k (gφ)

2,q

x

1

(t2k )α+ 2

L2x

D 2,q (Rv ))

,

 CgL 2 (RD ;B β1 (RD )) , x

C



f L α 2 (RD x ;B

1

(t2k )γ1 + 2

2,q

v

gL 2 (RD ;B β1 (RD )) , x

2,q

(6.123)

v

where C > 0 is independent of t and 2k . In the case β = 1r − D( 1r − p1 ), according to the same Lemmas 6.1 and 6.3, solely the estimates on B2t k should be replaced by

t

B k (gφ) 2

p

Lx 0

t

B k (gφ) 2

1  C log 1 + t2k q g1 D β0 D , kβ L (Rx ;Bp0 ,q (Rv )) 0 t2 1   C  k log 1 + t2k q gL β1 D . 2 (RD x ;B2,q (Rv )) t2



L2x

(6.124)

We are now going to utilize standard results from complex interpolation theory (cf. [6]) in order to obtain new estimates from the interpolation of estimates (6.122) and (6.123). To this end, first recall that the complex interpolation of Lebesgue spaces (cf. [6]) yields that (Lp0 , Lp1 )[θ] = Lp , θ for any 1  p, p0 , p1 < ∞ and 0 < θ < 1 such that p1 = 1−θ p0 + p1 . Furthermore, the complex interpolation of standard Besov spaces (cf. [6]) yields, in particular, that α α , for any α, α , α ∈ R and 1  p, p , p , q < ∞ such that α = (1 − θ )α + θ α and (Bp00,q , Bpα11,q )[θ] = Bp,q 0 1 0 1 0 1 1 1−θ θ = + . It is then possible to smoothly adapt the proof of this standard property to obtain a corresponding p p0 p1 α , for r0 Bpα00,q , L r1 Bpα11,q )[θ] = L r Bp,q result for the Besov spaces introduced in Section 2. Namely, one can show that (L 1 1−θ θ 1 1−θ any α, α0 , α1 ∈ R and 1  p, p0 , p1 , q, r, r0 , r1 < ∞ such that α = (1 − θ )α0 + θ α1 , p = p0 + p1 and r = r0 + rθ1 . Therefore, we deduce from the complex interpolation of (6.122) and (6.123) that, in the case β =

1

A (f φ) p  Cf r D α D , 0 L (Rx ;Bp,q (Rv )) L

t

A k (f φ) 2

− D( 1r − p1 ),

x

p

Lx



C 1 α+D(1− p1 ) kα 1−λ 0 2 ) (t ((t2k )α+ 2 )λ

C

= t and

1 r

α+1− 1r +D( 1r − p1 ) k(α+1− 1 ) r

2

f L α D r (RD x ;Bp,q (Rv ))

f L α D , r (RD x ;Bp,q (Rv ))

(6.125)

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

1

B (gφ) 0

t

B k (gφ) 2

p

Lx

p

Lx

 CgL r (RD ;B β x

D p,q (Rv ))

539

,

C



gL β D r (RD 1 x ;Bp,q (Rv )) 2kγ0 )1−λ ((t2k )γ1 + 2 )λ C = gL β D , r (RD γ +1− 1r +D( 1r − p1 ) k(γ +1− 1 ) x ;Bp,q (Rv )) r t 2 (t

γ0 +D(1− p1 ) 0

(6.126)

where C > 0 is independent of t and 2k . Accordingly, in the case β = 1r − D( 1r − p1 ), only the estimate on B2t k should be replaced by

t

B k (gφ) 2

p

Lx

 =

C (t2kβ0 )1−λ (t2k )λ C t2

k(β+1− 1r )

1  log 1 + t2k q gL r (RD ;B β

D p,q (Rv ))

x

1  log 1 + t2k q gL r (RD ;B β

D p,q (Rv ))

x

(6.127)

,

2

where C > 0 is independent of t and 2k , which concludes the proof of the lemma. Proof of Theorem 4.6. As usual, we begin with the decompositions  x0 f (x, v)φ(v) dv = A10 (f φ)(x) + B01 (gφ)(x)

(6.128)

RD

and  f (x, v)φ(v) dv = At2k (f φ)(x) + tB2t k (gφ)(x)

x2k

(6.129)

RD

provided by Proposition 3.2. We show in Lemma 6.6 that, in the case β = satisfy the bounds

− D( 1r − p1 ) or β =

1 r

1

A (f φ) 0

t

A k (f φ) 2

p

Lx

t

α+1− 1r +D( 1r − p1 )

0

2

p

Lx

p

Lx

1

2k(α+1− r )

t

f L α D , r (RD x ;Bp,q (Rv ))

 CgL r (RD ;B β x

D p,q (Rv ))

C



− D( 1r − p1 ) and q = 1, the operators above

 Cf L α D , r (RD x ;Bp,q (Rv )) C



1

B (gφ)

t

B k (gφ)

p

Lx

1 r

γ +1− 1r +D( 1r − p1 ) k(γ +1− 1 ) r

2

,

gL r (RD ;B β x

D p,q (Rv ))

(6.130)

,

where γ = min{β, 1r − D( 1r − p1 )} and C > 0 is independent of t and 2k . It then follows that, in virtue of the identities (3.30) from Proposition 3.2,







x  

 A1 S x f φ p + B 1 S x gφ p

 f (x, v)φ(v) dv 0 2 0 2

0 L L x

p

RD

Lx

x

 C S2x f L r (RD ;B α x

D p,q (Rv ))

+ C S2x g L r (RD ;B β

a,α  Cf Br,p,q D + Cg b,β (RD B x ×Rv )

D D r,p,q (Rx ×Rv )

which concludes the estimate on the low frequencies.

D p,q (Rv ))

x

,

(6.131)

540

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

Regarding the high frequencies, we obtain



x

t  x

t  x  

 k



f (x, v)φ(v) dv

2

 A2k [2k−1 ,2k+1 ] f φ Lp + t B2k [2k−1 ,2k+1 ] gφ Lp x

p

Lx

RD

x

1

C t

α+1− 1r +D( 1r − p1 ) k(α+1− 1 ) r

x

 k−1

2

1

+C

tr

−γ −D( 1r − p1 )

2

k(γ +1− 1r )

x k−1 [2

,2k+1 ]

[2

,2k+1 ]

f L r (RD ;B α

g L r (RD ;B β

D p,q (Rv ))

x

D p,q (Rv ))

x

(6.132)

.

−k (α−γ )+(a−b)

1+α−γ Next, optimizing in t for each value of k, we fix the interpolation parameter t as tk = 2 . Note that −k tk  2 , for b  a − 1, and that this choice is independent of 1  p, q, r  ∞. Furthermore, in the cases β > 1r − D( 1r − p1 ) or β = 1r − D( 1r − p1 ) and q = 1, we can choose t = ∞, which is, in

fact, more optimal than t2k = 2 This case is discussed later on.

1+b−a k 1+α−γ

, since it eliminates the first term in the right-hand side of the above estimates. α+1− 1 +D( 1 − p1 )

r r Therefore, denoting s = (1 + b − a) 1+α−γ



x

2ks f (x, v)φ(v) dv 

2k

p

Lx

RD

+ a − D( 1r − p1 ), and setting t = tk , we find that

 C2ka x[2k−1 ,2k+1 ] f L r (RD ;B α

D p,q (Rv ))

x

+ C2kb x[2k−1 ,2k+1 ] g L r (RD ;B β

D p,q (Rv ))

x

Hence, summing over k, we obtain

 

ks x

2  k f (x, v)φ(v) dv

2 RD

∞

p

k=0 q

Lx



 C 2ka x2k f L r (RD ;B α

k=−1 q



+ C 2kb x2k g L r (RD ;B β



∞

D p,q (Rv ))

x



∞

D p,q (Rv ))

x

(6.133)

.

k=−1 q

a,α = Cf Br,p,q D + Cg b,β (RD B x ×Rv )

D D r,p,q (Rx ×Rv )

,

(6.134)

1 1 1 r − D( r − p ). = 1r − D( 1r − p1 )

which concludes the proof of the theorem in the case β <

We handle now the cases β > 1r − D( 1r − p1 ) or β and q = 1, by letting t tend to infinity in (6.132), as mentioned previously. This leads to



x

x

C

 k

r D β D . f (x, v)φ(v) dv (6.135)  (Rx ;Bp,q (Rv ))

2

p  k(1−D( 1 − 1 )) [2k−1 ,2k+1 ] g L r p Lx 2 D R

Hence, recalling s = 1 + b − D( 1r − p1 ) and summing over k yields

∞

 



ks x

2  k

 C 2kb xk g r D β f (x, v)φ(v) dv  (R ;B

2

2 L RD

p Lx

k=0 q

which concludes the proof of the theorem in the cases β >

D p,q (Rv ))

x

1 r

− D( 1r − p1 ) or β =

1 r

∞

,

k=−1 q

(6.136)

− D( 1r − p1 ) and q = 1.

As for the case β = 1r − D( 1r − p1 ) and q = 1, employing the corresponding estimate from Lemma 6.6, we find that, as in the proofs of Theorems 4.2 and 4.5, for every  > 0,



∞ 

k(s−) x

2

 k

 Cf  a,α D D + Cg b,β D D , f (x, v)φ(v) dv (6.137) Br,p,q (Rx ×Rv )

2

B (R ×R ) p

Lx

RD

which concludes the proof of the theorem.

2

k=0 q

r,p,q

x

v

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

541

We proceed now to the proof of the most general Theorem 4.7, which will follow from a quite involved interpolation procedure. In particular, the following lemma is a generalization of Lemma 6.6. We will then show the proof of Theorem 4.7, which is essentially an interpolation argument based on the estimates from Lemma 6.7. Lemma 6.7. Let φ(v) ∈ C0∞ (RD ). For every 1  p, q, r  ∞, 0 < m < ∞, α, λ ∈ R and k ∈ N such that

and

r  p  r

(6.138)

 1 1 1 1 =α+1− +D − − λ, m r r p

(6.139)

it holds that

1

{t2−k } t





λ t A2k (f φ) Lp Lm,q x t



C 2

k(α+1− 1r )

f L α D , r (RD x ;Bp,q (Rv ))

(6.140)

where C > 0 is independent of 2k . Furthermore, if

1 1 D − r p

 <

2 −1 r

or

p = r = 2,

(6.141)

then, for any 0 < n < ∞ and β, τ ∈ R such that

and

 1 1 1 β < −D − r r p

(6.142)

 1 1 1 1 =β +1− +D − − τ, n r r p

(6.143)

it holds that

1

{t2−k } t





τ t B2k (gφ) Lp Ln,q x t



C 2

k(β+1− 1r )

gL r (RD ;B β

D p,q (Rv ))

x

,

(6.144)

where C > 0 is independent of 2k . Proof. We proceed by interpolation of the estimates from Lemma 6.6 on the velocity regularity index, which in particular illustrates the importance of systematically dealing with the most general cases of function spaces as possible. To this end, consider α0 < α < α1 , β0 < β < β1 , 0 < m1 < m < m0 < ∞ and 0 < n1 < n < n0 < ∞, such that  α0 + α1 1 β0 + β1 1 1 α= , β= , β1 < − D − , (6.145) 2 2 r r p and, for each i = 0, 1,

 1 1 1 1 = αi + 1 − + D − − λ, mi r r p  1 1 1 1 = βi + 1 − + D − − τ. ni r r p

(6.146)

Then, using Lemma 6.6, it holds that, for each i = 0, 1 and every t  2−k ,

t

A k (f φ) 2

p

Lx

C

 t

αi +1− 1r +D( 1r − p1 ) k(αi +1− 1 ) r

2

αi f L r (RD ;Bp,q (RD )) x

v

(6.147)

542

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

and, if further D( 1r − p1 ) <

− 1 or p = r = 2,

2 r

t

B k (gφ) 2

It follows that

1

{t2−k }

C

 t

βi +1− 1r +D( 1r − p1 ) k(βi +1− 1 ) r

2





λ t A2k (f φ) Lp Lmi ,∞ x t

{t2−k } t

1

p

Lx

t τ B t k (gφ) 2

p

n ,∞

Lx Lt i

C



2

k(αi +1− 1r )

C



1

2k(βi +1− r )

gL r (RD ;B βi x

D p,q (Rv ))

(6.148)

.

αi f L r (RD ;Bp,q (RD )) , x

v

gL r (RD ;B βi x

D p,q (Rv ))

,

(6.149)

where Lp,q denotes the standard Lorentz spaces. Recall now that the real interpolation of Lorentz spaces (cf. [6]) yields, in particular, that (Lm0 ,∞ A, Lm1 ,∞ A) 1 ,q = Lm,q A, where A is any fixed Banach space, for any 0 < m, m0 , m1 , q  ∞ such that 2

= 12 ( m10 + m11 ) and m0 = m1 . Furthermore, the real interpolation of standard Besov spaces (cf. [6]) yields, in particular, that α0 α1 α , for any α, α , α ∈ R and 1  p, q, c  ∞ such that α = α0 +α1 and α = α . It is then pos, Bp,q ) 1 ,c = Bp,c (Bp,q 0 1 0 1 2 2 sible to smoothly adapt the proof of this standard property to obtain a corresponding result for the Besov spaces α0 r α1 α , for any α, α , α ∈ R and r Bp,c r Bp,q , L Bp,q ) 1 ,c = L introduced in Section 2. Namely, one can show that (L 0 1 1 m

2

1 1  p, q, c  ∞ such that α = α0 +α and α0 = α1 . 2 Therefore, we deduce from (6.149) that

λ t

1

{t2−k } t A2k (f φ) Lp Lm,q  x

1

{t2−k } t





C

t

τ t B2k (gφ) Lp Ln,q x t

2

k(α+1− 1r )

C



2

k(β+1− 1r )

f L α D , r (RD x ;Bp,q (Rv )) gL r (RD ;B β x

D p,q (Rv ))

(6.150)

,

which concludes the proof of the lemma. Note that, unfortunately, in the case β  1r − D( 1r − p1 ), we cannot use interpolation methods to improve the results of Lemma 6.6 because γ − of parameters. 2

1 r

+ D( 1r − p1 ) = 0, where γ = min{β, 1r − D( 1r − p1 )}, remains constant over that range

Proof of Theorem 4.7. As usual, we begin with the decompositions  x 0 f (x, v)φ(v) dv = A10 (f φ)(x) + B01 (gφ)(x)

(6.151)

RD

and

 f (x, v)φ(v) dv = At2k (f φ)(x) + tB2t k (gφ)(x)

x2k

(6.152)

RD

provided by Proposition 3.2. We have shown in Lemma 6.7 above that the operators At2k and B2t k satisfy the bounds

1

{t2−k } t

1





λ t A2k (f φ) Lp0 Lq0 x t

{t2−k } t





τ t B2k (gφ) Lp1 Lq1 x t

where C > 0 is independent of 2k and

C

 2

k(α+1− r1 )

C

 2

f L α r0 (RD x ;Bp

, (RD v ))

gL r1 (RD ;B β

,

0 ,q0

0

k(β+1− r1 ) 1

 1 1 1 1 +D − − , r0 r0 p0 q0  1 1 1 1 − − . τ =β +1− +D r1 r1 p1 q1

x

D p1 ,q1 (Rv ))

(6.153)

λ=α+1−

(6.154)

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543

We will now make use of the construction of interpolation spaces known as espaces de moyennes presented in Appendix A. Specifically, in virtue of the property (Lp0 , Lp1 )θ,p = Lp valid for all 1  p, p0 , p1  ∞ and 0 < θ < 1 θ p such that p1 = 1−θ p0 + p1 , we wish to express the Lebesgue space L using the norm (A.7). That is to say, we will employ the norm inf

a=a0 +a1

 −θ

s a0 (s) q0q

p0 D L 0 ((0,∞), ds s ;L (R ,dx))

q 1 + s 1−θ a1 (s) L1q1 ((0,∞), ds ;Lp1 (RD ,dx)) p ,

(6.155)

s

θ 1 1−θ θ which is equivalent to the usual norm on Lp (RD , dx), where p1 = 1−θ p0 + p1 , provided p = q0 + q1 and q0 , q1 < ∞. To this end, for some suitable bijective function t (s) : (0, ∞) → (0, ∞) to be determined later on, we will decompose   x2k f (x, v)φ(v) dv = a = a0 (s) + a1 (s) = x2k f (x, v)φ(v) dv + 0, (6.156) RD

RD

when 0 < t (s) < 2−k , and  x 2k f (x, v)φ(v) dv = a = a0 (s) + a1 (s) = At2k (f φ)(x) + tB2t k (gφ)(x),

(6.157)

RD

when t (s)  2−k . It follows that, using Bernstein’s inequalities,



x

 k

f (x, v)φ(v) dv

2

p Lx

RD





−θ− q1 x

0  C 1{0
2k

q0

p

p0 q0

Lx

RD

Ls

qp0

qp1 −θ− q1 1−θ− q1 0 At (f φ) p0 q + C 1 1 t B t (gφ) p1 q + C 1{t2−k } s s −k {t2 } 0 1 L L 2k 2k x

Ls

x

q0

 1 q0  q p

p s (t) q0 kD( r1 − p1 ) p0 −θ x

0 0  C2 f (x, v)φ(v) dv 2k

r0

1{0
q 0 s(t) Lt

q0

1 q0

p (t) s −θ t

A k (f φ) p0 + C

1{t2−k } s(t) Lx q0 2 s(t)

Ls

Lx

RD

Lt

q1

1 q1

p (t) s 1−θ t + C t B2k (gφ) Lp1 .

1{t2−k } s(t) x q1 s(t)

(6.158)

Lt

Next, we set the dependence of s with respect to t so that we may utilize the estimates (6.153). This amounts to optimizing the value of the interpolation parameter s for each value of t , which also dictates the value of 0 < θ < 1. That is to say, for given coefficients c1 , c2 > 0 independent of k, we wish to find an optimal value for a function of the form

s(t)−θ

c1 2

k(α+1− r1 ) 0

tλ

s (t) s(t)

1

q0



s(t)1−θ

+ c2 2

k(β+1− r1 ) 1

tτ

s (t) s(t)

1

q1

(6.159)

t.

Thus, we define s(t) =

2

k(β+b+1− r1 ) τ + q1 −1

2

1

t

k(α+a+1− r1 ) 0

1

t

λ+ q1 0

=2

k((β+b)−(α+a)+ r1 − r1 ) β−α−1+ r1 − r1 +D( r1 − r1 − p1 + p1 ) 0

1

t

0

1

1

0

1

0

,

(6.160)

544

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

which is admissible since β − α − 1 + terms

s(t)−θ t

λ+ q1 0

results in

s(t)1−θ

and t

τ + q1 −1 1

−

1 r0

+ D( r11 −

1 r1

1 r0

−

1 p1

+

1 p0 ) < 0.

Furthermore, in order that the resulting

be independent of t, we set 0 < θ < 1 so that (1 − θ )(λ +

θ=

α+1−

1 r0

  α + 1 − r10 + D r10 − p10    + D r10 − p10 − β + r11 − D r11 −

1 p1

1 q0 ) + θ (τ

+

1 q1

− 1) = 0, which

.

(6.161)

To be precise, these choices of parameters yield that s(t)−θ = 2−ks 2 s(t)1−θ = 2

k(a+α+1− r1 ) λ+ q1 0

t

0

,

1 1 −ks k(b+β+1− r1 ) τ −1+ q1

2

1 r0 ) + θ (β

t

(6.162)

,

where s = (1 − θ )(α + a + 1 − +b+1− We conclude that, in virtue of the identities (3.30) from Proposition 3.2,

p 

 ks x f (x, v)φ(v) dv  C 2ka x[2k−1 ,2k+1 ] f L 2 2k r0 (RD ;B α

1 r1 ).

p Lx

RD

q0

D p0 ,q0 (Rv ))

x

 + C 2kb x[2k−1 ,2k+1 ] g L r1 (RD ;B β x

Hence, summing over k, we obtain

 

ks x

2  k f (x, v)φ(v) dv

2

∞

p

Lx

RD

k=0 p



 C 2ka x2k f L r0 (RD ;B α



+ C 2kb x2k g L r1 (RD ;B β D D r0 ,p0 ,q0 (Rx ×Rv )

k=−1 

0





q1 ∞ p q D p1 ,q1 (Rv )) k=−1  1 q1 p b,β D Br1 ,p1 ,q1 (RD x ×Rv )

x

q0

(6.163)

.

qp0

q

∞

D p0 ,q0 (Rv ))

x

 Cf Bpa,α

q1

D p1 ,q1 (Rv ))

+ Cg

(6.164)

.

Now, if the above estimate holds true for any f and g, then it must also be valid for all λf and λg, where λ > 0, so that

∞

  q0 q0

ks x

2  k

 Cλ p −1 f  pa,α f (x, v)φ(v) dv D

p

2

p Br0 ,p0 ,q0 (RD x ×Rv ) Lx

RD

k=0 

q1

+ Cλ p

−1

q1

g pb,β

D Br1 ,p1 ,q1 (RD x ×Rv )

(6.165)

.

θ Recalling that p1 = 1−θ q0 + q1 and optimizing in λ concludes the proof of the main estimate for the high frequencies of the velocity average.  Thus, there only remains to control the low frequencies x0 RD f (x, v)φ(v) dv. To this end, if p  r0 , a straightforward application of Bernstein’s inequalities shows that





x





 C x f (x, v)φ(v) dv f (x, v)φ(v) dv 0

0

p

r0 RD

Lx

Lx

RD



∞ 



x v v x v v = C 0 0 f (x, v)S2 φ(v) dv + 0 2k f (x, v)[2k−1 ,2k+1 ] φ(v) dv

k=0

RD



 C x0 v0 f Lr0 Lp0 S2v φ x

 Cf Bra,α,p 0

0 ,q0

v

p Lv 0

+C

∞

x v

  k f 0

k=0

D φB −α (RD ) . (RD x ×Rv ) v p0 ,q0

r

Lx0

RD

2

r

p

Lx0 Lv 0

v

 k−1 [2

,2k+1 ]

φ

p

Lv 0

(6.166)

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

545

Finally, in the case p < r0 , the direct use of Bernstein’s inequalities as above is not allowed and it is therefore necessary to localize the norm in space in order to carry out the preceding argument. Thus, for any χ(x) ∈ C0∞ (RD ), we employ the standard methods of paradifferential calculus, which are used in the proof of Lemma B.1, to decompose, following Bony’s method,   ∞   x x x x x x x x x x 0 f (x, v)χ(x) = 0 0 f S4 χ + 1 f S8 χ + 2 f S16 χ + 2k f [2k−2 ,2k+2 ] χ . (6.167) k=2

Since χ is rapidly decaying, it then follows that, repeating the estimates from (6.166),



x



0 f (x, v)χ(x)φ(v) dv p

Lx

RD

 

 ∞



x x x x x x x x  C 2k f [2k−2 ,2k+2 ] χ φ(v) dv 0 f S4 χ + 1 f S8 χ + 2 f S16 χ +

k=2

RD



x

 C  f (x, v)φ(v) dv 0

r

Lx0

RD

 Cf Bra,α,p 0

0 ,q0

+C

∞



x

2ka  f (x, v)φ(v) dv

2k

k=0

RD

D φB −α (RD ) , (RD x ×Rv ) v

p

Lx

r

Lx0

(6.168)

p0 ,q0

which concludes the proof of the theorem.

2

Appendix A. Real interpolation theory We present here some useful elements from real interpolation theory. We merely discuss properties that are useful to our work and we refer to [6] for more details on the subject. First of all, we briefly recall the K-method of real interpolation. To this end, we consider any couple of normed spaces A0 and A1 compatible in the sense that they are embedded in a common topological vector space. The sum A1 + A0 is the normed space defined by the norm aA1 +A0 =

inf

a0 ∈A0 ,a1 ∈A1 a=a0 +a1

a0 A0 + a1 A1 .

(A.1)

For any 0 < θ < 1, 1  q  ∞, we define the normed space [A0 , A1 ]θ,q ⊂ A0 + A1 by the norm a[A0 ,A1 ]θ,q =

∞ 

t

−θ

q dt K(t, a) t

1 q

,

(A.2)

0

if q < ∞, and a[A0 ,A1 ]θ,∞ = sup t −θ K(t, a),

(A.3)

t>0

if q = ∞, where the K-functional is defined, for any a ∈ A0 + A1 , t > 0, by K(t, a) =

inf

a0 ∈A0 ,a1 ∈A1 a=a0 +a1

a0 A0 + ta1 A1 .

(A.4)

Then, the normed space [A0 , A1 ]θ,q is an exact interpolation space of exponent θ . In other words, this means that, considering another couple of compatible normed spaces B0 and B1 , for any operator T bounded from A0 into B0 and from A1 into B1 , the operator T is bounded from [A0 , A1 ]θ,q into [B0 , B1 ]θ,q as well, with an operator norm satisfying θ T [A0 ,A1 ]θ,q →[B0 ,B1 ]θ,q  T 1−θ A0 →B0 T A1 →B1 .

(A.5)

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D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

It turns out that there are other equivalent methods yielding the same interpolation spaces as the K-method. In particular, we are now briefly presenting the construction of interpolation spaces known as espaces de moyennes (as coined by Lions and Peetre [20]), which is equivalent to the K-method of interpolation but has the advantage of being slightly more general. We refer to [6] for more details on these methods. θ Thus, for any 0 < θ < 1 and 1  q, q0 , q1 < ∞ such that q1 = 1−θ q0 + q1 , and any compatible couple of normed spaces A0 and A1 , we define the following norms

 −θ 

s a0 (s) q (A.6) inf + s 1−θ a1 (s) Lq1 ((0,∞), ds ;A ) L 0 ((0,∞), ds ;A ) a0 (s)∈A0 ,a1 (s)∈A1 a=a0 (s)+a1 (s)

s

0

1

s

and inf

a0 (s)∈A0 ,a1 (s)∈A1 a=a0 (s)+a1 (s)

 −θ

s a0 (s) q0q L

0 ((0,∞), ds ;A0 ) s

q 1 + s 1−θ a1 (s) L1q1 ((0,∞), ds ;A ) q . s

1

(A.7)

It is possible to show (cf. [6, Theorem 3.12.1]) that both norms above are equivalent to a[A0 ,A1 ]θ,q and thus define the same interpolation space. Appendix B. Some paradifferential calculus For the sake of completeness and convenience of the reader, we include below a technical lemma on basic paradifferential calculus. s D r (RD Lemma B.1. Let 1  p, q, r  ∞, s ∈ R and φ(v) ∈ S(RD ). Then, for any f (x, v) ∈ L x ; Bp,q (Rv )), it holds that

f φL s D  Cf L s D , r (RD r (RD x ;Bp,q (Rv )) x ;Bp,q (Rv ))

(B.1)

where C > 0 only depends on φ and on fixed parameters. Proof. We begin formally with the standard Bony’s decomposition (cf. [10])



 ∞ ∞ v v v v f φ = 0 f + 2k f 0 φ + 2k φ = T (f, φ) + T (φ, f ) + R(f, φ), k=0

(B.2)

k=0

where we have denoted the paraproducts T (f, φ) =

∞

v2k f S2vk−3 φ

and

T (φ, f ) =

k=2

∞

v2k φS2vk−3 f,

(B.3)

k=2

and the remainder R(f, φ) = v0 f R0v φ +

∞

v2k f R2vk φ,

k=0

where R0v φ = v0 φ +

1 j =0

R2v φ = v0 φ +

3 j =0

v2j φ

v2j φ,

R1v φ = v0 φ +

and R2vk φ =

2 j =0

2 j =−2

v2k+j φ

v2j φ,

if k  2.

(B.4)

We then estimate T (f, φ), T (φ, f ) and R(f, φ) separately. The control of the paraproduct T (f, φ) proceeds as follows. First, notice that the support of the Fourier transform v φ is contained inside a closed ball of radius 2k−2 centered at the origin and is thus in the velocity variable of Sk−3 separated from the support of the Fourier transform of v2k f by a distance of at least 2k−2 . Therefore, the Fourier

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

547

transform of the general term in the sum of the paraproduct is supported inside an annulus centered at the origin of inner radius 2k−2 and outer radius 9 · 2k−2 . As a consequence, we have that   v0 v2j f S2vj −3 φ ≡ 0 for all j  2   (B.5) and v2k v2j f S2vj −3 φ ≡ 0 if |j − k|  3. We may then proceed with the estimation



∞

T (f, φ) r D s = 2ks v2k T (f, φ) Lr Lp k=0 q )) L (Rx ;Bp,q (RD v v x

 ∞ ∞



 2(k−j )s 2j s v2j f S2vj −3 φ Lr Lp x v

j =2 |j −k|2





k=0 q



∞  C 2ks v2k f S2vk−3 φ Lr Lp k=2 q x v

 ks v 

2  k f r p ∞ q  CφL∞ v L L k=2  2 x

v

 Cf L s D . r (RD x ;Bp,q (Rv ))

(B.6)

Regarding the paraproduct T (φ, f ), we handle it through the following similar calculation, using that φ(v) is rapidly decaying,



∞

T (φ, f ) r D s  C 2ks v2k φS2vk−3 f Lr Lp k=2 q L (Rx ;Bp,q (RD )) v x v

 ks v v

∞



 C 2 2k φ L∞ S2k−3 f Lr Lp k=2 q v x v



∞

 C 2ks v2k φ L∞ k=2 q v0 f Lr Lp v x v

 ∞ k−3

−j (s∧0) j (s∧0) v

ks v

2j f Lr Lp + 2 2k φ L∞ 2 2

v x v

j =0 k=3 q

 ks v ∞ v  C 2 2k φ L∞ k=2 q 0 f Lr Lp v x v



∞  k(s∧0) v



q 2

 k f r p ∞ q + 2k(s+1)−k(s∧0) vk φ ∞ 2

Lv

2

k=3 

Lx Lv k=0 

 Cf L s∧0 D . r (RD x ;Bp,q (Rv ))

(B.7)

Regarding the remainder, we first notice that the Fourier transforms of v0 f R0v φ and v2k f R2vk φ, for all k ∈ N, are supported inside closed balls centered at the origin of respective radii 5 and 10 · 2k . It follows that   v2k v0 f R0v φ ≡ 0 if k  4   (B.8) and v2k v2j f R2vj φ ≡ 0 if k  j + 5. Therefore, using once again that φ ∈ S(RD ),





∞

R(f, φ) r D s  v0 R(f, φ) Lr Lp + 2ks v2k R(f, φ) Lr Lp k=0 q )) L (Rx ;Bp,q (RD v x v x v

 ∞

v



vk f R vk φ r p  C 0 f R0v φ Lr Lp + L L 2 2 x

v

x

k=0

v

 ∞ ∞





ks v 2j f R2vj φ + C 2



r p

j =k−4 Lx Lv k=4 q

∞



v

 k f r p R vk φ  C v f r p R v φ ∞ + 0

Lx Lv

0

Lv

2

k=0

Lx Lv

2

 L∞ v

548

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

+C

∞

v

 j f 2

p

Lrx Lv

v

R j φ

j =0

v

 C 0 f Lr Lp R0v φ L∞ x v v

L∞ v

2

 ks j +4

2

k=4 q



∞ ∞  + C 2ks v2k f Lr Lp k=0 q 2k−k(s∧0) R2vk φ L∞ k=0 q v

x

v

 Cf L s D . r (RD x ;Bp,q (Rv ))

(B.9)

Finally, incorporating (B.6), (B.7) and (B.9) into the decomposition (B.2), we easily deduce that the estimate (B.1) holds true, which concludes the demonstration. 2 Appendix C. The time dependent transport equation We explain now how the strategy exposed in Section 3 can be adapted to the time dependent setting. D To this end, note that, if f (t, x, v), g(t, x, v) ∈ L2 (Rt × RD x × Rv ) satisfy the transport relation (∂t + v · ∇x )f (t, x, v) = g(t, x, v),

(C.1)

then, considering the Fourier transforms fˆ(τ, η, v) and g(τ, ˆ η, v) in the time and space variables only, it holds that i(τ + v · η)fˆ(τ, η, v) = g(τ, ˆ η, v).

(C.2)

It follows that the appropriate microlocal domain, on which the transport operator enjoys some elliptic properties, should now isolate the points where |τ + v · η| is uniformly bounded away from zero. Thus, introducing some cutoff function ρ ∈ S(R) such that ρ(0) = 1 and an interpolation parameter s > 0, we have now the decomposition   1 − ρ(s(τ + v · η)) g(τ, ˆ η, v), fˆ(τ, η, v) = ρ s(τ + v · η) fˆ(τ, η, v) + i(τ + v · η)

(C.3)

which is the analog of (3.3). Following [13,14], each term in the right-hand side may then be estimated locally in L2v , much like in the stationary case, which yields that, for every test function φ(v) ∈ C0∞ , the velocity average 1  2 f (t, x, v)φ(v) dv belongs to Ht,x . On the other hand, following the preceding developments from Section 3 for the stationary case, in order to exploit the dispersive properties of the transport equation (C.1), we have to interpret f (t, x, v) and g(t, x, v) as a trivial solution of  (∂s + ∂t + v · ∇x )f = g, (C.4) f (s = 0) = f. Hence the interpolation formula, s f (t, x, v) = f (t − s, x − sv, v) +

g(t − σ, x − σ v, v) dσ,

(C.5)

0

which is merely Duhamel’s representation formula for the above time dependent transport equation (C.4). Consequently, applying the identities (3.12) to (C.5), we deduce 





f (t, x, v) dv =

x0 RD



f (t, x, v) dv = RD

 (t − s, x − sv, v) dv +

RD

 x2k

Ssv f

RD



 vs2k f (t − s, x − sv, v) dv +

s  0 RD s 

0 RD

Next, for some cutoff function ρ ∈ S(R) such that 1 = ρ(0) = become now

1 2π



 Sσv g (t − σ, x − σ v, v) dv dσ,



 vσ 2k g (t − σ, x − σ v, v) dv dσ.



ˆ ds, R ρ(s)

(C.6)

the identities (3.18) and (3.19)

D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551 −1 Ft,x ρ

  1 s(τ + η · v) Ft,x f (t, x, v) = 2π 1 = 2π



549

−1 ils(τ +η·v) Ft,x e Ft,x f (t, x, v)ρ(l) ˆ dl

R



f (t + ls, x + lsv, v)ρ(l) ˆ dl,

(C.7)

R

and −1 1 − ρ(s(τ Ft,x

+ η · v)) 1 Ft,x g(t, x, v) = − i(τ + η · v) 2π =−

1 2π

 ls

−1 iσ (τ +η·v) Ft,x e Ft,x g(t, x, v) dσ ρ(l) ˆ dl

R 0  ls

g(t + σ, x + σ v, v) dσ ρ(l) ˆ dl.

(C.8)

R 0

Therefore, incorporating now (C.7) and (C.8) into the interpolation formulas (C.3) or (C.5), we obtain, for each s ∈ R, the following refined decomposition f (t, x, v) = TAs f (t, x, v) + sTBs g(t, x, v), where TAs f (t, x, v) = 1 TBs g(t, x, v) = −

1 s 2π



1 2π

 ls R 0



(C.9)

 f (t + ls, x + lsv, v)ρ(l) ˆ dl =

R

f (t − ls, x − lsv, v)ρ(l) ˜ dl, R

1 g(t + σ, x + σ v, v) dσ ρ(l) ˆ dl = − 2π

 l g(t + σ s, x + σ sv, v) dσ ρ(l) ˆ dl R 0

l

=

g(t − σ s, x − σ sv, v) dσ ρ(l) ˜ dl,

(C.10)

R 0

and   Ft,x TAs f (τ, η, v) = ρ s(τ + η · v) Ft,x f (τ, η, v), 1 − ρ(s(τ + η · v)) Ft,x TBs g(τ, η, v) = Ft,x g(τ, η, v), is(τ + η · v)

(C.11)

which shows that formulas (C.3) and (C.5) are in fact equivalent. Finally, combining the property of transfer of space frequencies to velocity frequencies (C.6) with the above representation formulas for the operators TAs and TBs , we arrive at the following proposition, which is a time dependent analog of Proposition 3.2. Thus, the decomposition (C.13) below is the basis for proving time dependent velocity averaging lemmas and it is now clear that each result from Section 4 can be adapted to the time dependent setting. The justification of Proposition C.1 is strictly identical to the proof of Proposition 3.2, and so, we omit it. Proposition C.1. Let f (t, x, v), g(t, x, v) ∈ S(R × RD × RD ) be such that (∂t + v · ∇x )f = g. For all s > 0, δ  0 and for every cutoff function ρ ∈ S(R) such that ρ(0) = consider the decomposition  xδ f (t, x, v) dv = Asδ f (t, x) + sBδs g(t, x), RD

where the operators Asδ and Bδs are defined by

1 2π



ˆ dl R ρ(l)

=



˜ dl R ρ(l)

(C.12) = 1, we

(C.13)

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D. Arsénio, N. Masmoudi / J. Math. Pures Appl. 101 (2014) 495–551

 Asδ f (t, x) = xδ

TAs f (t, x, v) dv, RD



Bδs g(t, x) = xδ

TBs g(t, x, v) dv.

(C.14)

RD

Then it holds that

 Asδ f (t, x) = RD



Bδs g(t, x) =

  −1 Ft,x ρ s(τ + η · v) Ft,x xδ f (t, x, v) dv,   −1 Ft,x τ s(τ + η · v) Ft,x xδ g(t, x, v) dv,

(C.15)

RD

where τ (r) =

1−ρ(r) ir

is smooth, and  xδ f (t − ls, x − lsv, v) dv ρ(l) ˜ dl,

  Asδ f (t, x) = R

  1 

RD

Bδs g(t, x) =



xδ g(t R

− σ ls, x − σ lsv, v) dv dσ l ρ(l) ˜ dl.

(C.16)

0 RD

Furthermore, if the cutoff ρ ∈ S(R) is such that ρ˜ is compactly supported inside [1, 2], then, for every δ > 0,     v f , v g , As0 f = As0 S2x S4s B0s g = B0s S2x S4s     v g . Asδ f = Asδ xδ vsδ f , Bδs g = Bδs xδ S8sδ (C.17) [ 2 ,2δ]

[

2

,4sδ]

[ 2 ,2δ]

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