Modulation spaces with scaling symmetry

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Applied and Computational Harmonic Analysis www.elsevier.com/locate/acha

Letter to the Editor

Modulation spaces with scaling symmetry Árpád Bényi a,∗ , Tadahiro Oh b a

Department of Mathematics, Western Washington University, 516 High Street, Bellingham, WA 98225, USA b School of Mathematics, The University of Edinburgh and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK

a r t i c l e

i n f o

Article history: Received 21 June 2018 Received in revised form 26 March 2019 Accepted 16 April 2019 Available online xxxx Communicated by Karlheinz Gröchenig

a b s t r a c t We indicate how to construct a family of modulation spaces that have a scaling symmetry. We also illustrate the behavior of the Schrödinger multiplier on such function spaces. © 2019 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

MSC: 42B35 42B15 35Q41 Keywords: Modulation spaces Besov spaces Modulation symmetry Dilation symmetry Schrödinger multiplier

1. Reconciling the modulation and dilation scalings The Besov spaces and the modulation spaces are both examples of so-called decomposition spaces that stem from either a dyadic covering or uniform covering of the underlying frequency space. Roughly speaking, both of these classes of function spaces are defined by imposing appropriate decay conditions on a sequence of   the form F −1 (ψk f)Lpx k , where {ψk }k∈Z is a dyadic partition of unity in the Besov case, while {ψk }k∈Zd is a uniform partition of unity in the case of the modulation spaces. Moreover, one can connect the two classes of spaces via the so-called α-modulation spaces of Gröbner [17] for α ∈ [0, 1]; the modulation spaces (and the Besov spaces, respectively) are obtained at the end-point: α = 0 (and α = 1, respectively). The * Corresponding author. E-mail addresses: [email protected] (Á. Bényi), [email protected] (T. Oh). https://doi.org/10.1016/j.acha.2019.04.005 1063-5203/© 2019 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

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α-modulation spaces are still decomposition spaces obtained via a “geometric interpolation” method that now asks for a covering {Qα k }k∈I of the frequency space in which the sets satisfy a condition of the form α α |Qα | ∼ ξ for all ξ ∈ Q ; see [9]. k k In this paper, however, we are interested in a different connection between the two classes of function spaces than that alluded to above. It is clear from their definitions that the Besov spaces enjoy the dilation symmetry: f (x) → f (2k x), k ∈ Z, while the modulation spaces enjoy the modulation symmetry: f (x) → e2πik·x f (x), k ∈ Zd . A natural question then is if there exists a class of function spaces that is sufficiently rich that would enjoy both such symmetries. As we shall see, reconciling the modulation and dilation symmetries is possible by considering scaled versions of the modulation spaces and then appropriately amalgamating them with a “good” vector weight. In many respects, such a class of function spaces would be more enticing to consider from the perspective of someone working in PDEs. Our note stems from these natural considerations and should be seen as a contribution to the general program of constructing new function spaces from given ones, as well as deriving the essential properties of the new spaces (i) from those of the modulation spaces on which they are rooted and (ii) from the characteristics of the weight, as proposed in [5]. p,q,r 2. Definition of the Mw -spaces and their scaling property p,q,r 2.1. The Mw -spaces

Let p, q, r ∈ [1, ∞) be fixed. We denote by w a vector weight {wj }j∈Z with wj ≥ 0. Generally speaking, we aim to define a space which captures the modulations of all scales, while the choice of the weight w is made according to the task for which the space is needed. Let us first recall next the definition of modulation spaces M p,q ; see [7,8]. Let ψ ∈ S(Rd ) such that supp ψ ⊂ Q0 := [−1, 1]d

and



ψ(ξ − k) ≡ 1.

(2.1)

k∈Zd

Then, the modulation space M p,q is defined as the collection of all tempered distributions f ∈ S  (Rd ) such that f M p,q < ∞, where the M p,q -norm is defined by   f M p,q (Rd ) = ψ(D − k)f Lpx (Rd ) q (Zd ) .

(2.2)

k

 Here, ψ(D − k)f (x) = Rd ψ(ξ − k)f(ξ)e2πix·ξ dξ. Clearly, the support of ψ renders the domain of the integration of the previous integral to be ξ ∈ Q0 + k, and in particular we have 

f=

ψ(D − k)f.

(2.3)

k∈Zd

In what follows, for ease of notation and unless specifically stated otherwise, we assume that the underp,q by the lying space is Rd . Now, for a (fixed) dyadic scale j ∈ Z, we define the scaled modulation space M[j] norm:    p,q := ψ f M[j] , d j,k (D)f Lp x (R ) q (Zd ) k

where ψj,k (D) is defined by  ψj,k (D)f (x) = Rd

ψ(2−j ξ − k)f(ξ)e2πix·ξ dξ.

(2.4)

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Namely, while the usual modulation spaces M p,q are adapted to the modulation symmetry of the unit scale, p,q p,q the M[j] -spaces are adapted to the modulation symmetry of scale 2j . When j = 0, we have M[0] = M p,q . Note also that, as in (2.3), we have 

f=

ψj,k (D)f

(2.5)

k∈Zd

for all j ∈ Z. Now, in view of the decompositions (2.3) and (2.5), we have f=

 

wj ψj,k (D)f

j∈Z k∈Zd

w 1 = 1. This motivates the following definition of the modulation for any vector weight w = {wj }j∈Z with w spaces with scaling symmetry. Definition 1. Let 1 ≤ p, q, r < ∞ and w = {wj }j∈Z be a vector weight with wj ≥ 0. We define the space p,q,r Mw (Rd ) as the collection of all tempered distributions f ∈ S  (Rd ) such that f Mwp,q,r < ∞, where the p,q,r Mw -norm is defined by f 

p,q,r Mw

  p,q  := f M[j] = w) r (w



j

wjr f rM p,q [j]

r1 .

(2.6)

j∈Z p,q,r Note that the definition of the space Mw heavily depends on the choice of the weight w and we need to p,q,r make sure that smooth functions belong to Mw for suitable weights w . We first claim that the Schwartz p,q,r d class S(R ) is contained in Mw , provided that w = {wj }j∈Z is a “good” vector weight, that is, for some ε > 0 we have

wj 

 2−εj

if j ≥ 0,

−( pd − d q −ε)j

2

(2.7)

if j < 0.

Note that we allow some growth for j < 0, when q > p . For j ≥ 0, it suffices to impose summability of {wj }j≥0 . Let us first prove this claim. Let ψj,k denote the multiplier of ψj,k (D) defined in (2.4), corresponding to the smoothed version of 12j (Q0 +k) . Now, for j < 0, it follows from Young’s inequality that ψj,k (D)f Lpx ≤ F

−1

     (ψj,k )Lpx  ψj,k+i (D)f  

L1x

|i|≤1

    j pd  ∼2  ψj,k+i (D)f  

,

L1x

|i|≤1

where the implicit constant is independent of k ∈ Zd . By computing the qk -norm, we have    d   p,q = ψ f M[j]  2j p ψj,k (D)f L1x q (Zd ) j,k (D)f Lp x q (Zd ) k

≤2

j pd

 j −s  2 k ψj,k (D)∇s f L1  x

qk (Zd )

k

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j pd

f Wxs,1

 k∈Zd

1 2j ksq

q1

 2jd( p − q ) f Wxs,1 1

1

(2.8)

1

for any s > dq . Here, x = (1 + |x|2 ) 2 and we used the Riemann sum approximation in the last step. Then, p,q,r from (2.7) and (2.8), the contribution to the Mw -norm for j < 0 is estimated by 

wjr f rM p,q [j]

r1





j<0

2εjr

r1

f Wxs,1 < ∞.

j<0

When j ≥ 0, proceeding as above, we have p,q  f  f M[j] Wxs,p

(2.9)

for any s > dq . Hence, from (2.7) and (2.9), we obtain 

wjr f rM p,q [j]

j≥0

r1





2−εjr

r1

f Wxs,p < ∞.

j≥0

We point out that the condition (2.7) is sharp. Let j < 0. Suppose that there exists J ∈ N such that d d wj  2−( p − q )j for any j ≤ −J. Let f ∈ S(Rd ). Then, there exist N ∈ N and k0 ∈ Zd such that |f(ξ)| ≥

1  f L∞ 2

for all ξ ∈ 2−N (Q0 + k0 ) and f essentially behaves like a constant on the cube 2−N (Q0 + k0 ). This in particular implies that ψj,k (D)f Lpx  fL∞ 2j p , d

for any j ≤ −N and k ∈ Zd such that 2j (Q0 + k) ⊂ 2−N (Q0 + k0 ).

(2.10)

Then, it is easy to see that f Mwp,q,r = ∞ by simply computing the contribution from j < 0 and k ∈ Zd satisfying (2.10). The condition (2.7) for j ≥ 0 can also be seen to be essentially sharp by noting that p,q ≥ ψ f M[j]  f Lpx j,0 (D)f Lp x

for all sufficiently large j ≥ 0. p,q,r Lastly, we wish to summarize the essential properties of the function spaces Mw which bear a strong resemblance with those of the classical modulation spaces. The dilation property will be treated separately in the next subsection. See also Section 3 for a connection to other spaces stemming from PDEs. Theorem 2.1. Let 1 ≤ p, q, r < ∞ and assume that the vector weight w satisfies (2.7). Then, we have the following: p,q,r (i) S(Rd ) ⊂ Mw (Rd ),

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p,q,r p,q (ii) Mw (Rd ) ⊂ M p,q (Rd ) = M[0] (Rd ),  p,q,r d  (iii) Mw (R ),  · Mwp,q,r is a Banach space, p1 ,q1 ,r1 p2 ,q2 ,r2 (Rd ) ⊂ Mw (Rd ) for 1 ≤ p1 ≤ p2 , 1 ≤ q1 ≤ q2 , and 1 ≤ r1 ≤ r2 . (iv) Mw

Remark 2.2. A natural and interesting question arises, concerning the duality of our function spaces. Let 1 ≤ p, q, r < ∞ and fix a vector weight w = w (p, q) = {wj }j∈Z satisfying the condition (2.7). Then, by setting         f, g := w j wj ψj,k (D)f (x) ψj,k (D)g (x)dx k∈ZdRd

j∈Z p,q,r for f ∈ Mw , Hölder’s inequality yields

|f, g| ≤ f Mwp,q,r gMp ,q ,r , w

where w  = w  (p , q  ) = {wj }j∈Z is a vector weight satisfying the condition (2.7) with (p , q  ) in place of  p,q,r   p ,q  ,r  (p, q). We call such a vector weight w  a dual weight. This shows that w  Mw ⊂ Mw , where the  union is taken over all the dual weights w  (for a given pair (p, q)). It is, however, not clear to us how to  p,q,r  determine the actual dual space Mw in this formulation. Let us discuss an alternative approach. Let m be a counting measure on Z and write (2.6) as f Mwp,q,r := ψj,k (D)f Lr (Z,wjr dm)qk (Zd )Lpx (Rd ) . Then, a duality pairing may be given by

       (f, g) := ψj,k (D)f (x) ψj,k (D)g (x)dx wjr dm k∈ZdRd

In this case, Hölder’s inequality (viewing wjr dm as a measure on Z) yields |(f, g)| ≤ f Mwp,q,r gMp ,q ,r , w

 p,q,r  p ,q  ,r  where w  = {wjr−1 }j∈Z . This shows Mw ⊂ Mw for this particular weight w  . Note that this weight   w may not satisfy the condition (2.7) unless we impose a further assumption on w . 2.2. Scaling property p,q,r Let us now investigate the scaling property of the new modulation spaces Mw . Define a translation w )j = wj+1 . For k ∈ N, write τ k (and τ −k , operator τ on a vector weight w = {wj }j∈Z by setting (τw respectively) for τ composed with itself k times (and τ −1 composed with itself k times, respectively). Let us fix a dyadic scale λ = 2j0 for some j0 ∈ Z and set fλ (x) = f (λx). First, note that



 ψj,k (D)fλ (x) =



ψ(2−j ξ − k)λ−d f(λ−1 ξ)e2πix·ξ dξ

ξ∈2j (Q0 +k)



= ξ∈2j−j0 (Q0 +k)

ψ(2−j+j0 ξ − k)f(ξ)e2πiλx·ξ dξ

  = ψj−j0 ,k (D)f (λx).

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Hence, we have ψj,k (D)fλ Lpx = λ− p ψj−j0 ,k (D)f Lpx = 2−j0 p ψj−j0 ,k (D)f Lpx . d

d

Therefore, we obtain   p,q  fλ Mwp,q,r = fλ M[j] w) rj (w      = ψj,k (D)fλ Lpx (Rd ) q (Zd )  k

w) rj (w

   d   = 2−j0 p ψj−j0 ,k (D)f Lpx (Rd ) q (Zd )  k

   d   = 2−j0 p ψj,k (D)f Lpx (Rd ) q (Zd )  k

−d p

=λ

w) rj (w

rj (τ j0 w )

f Mp,q,r . j τ 0w

Note that the vector weight on the right-hand side is now given by τ j0 w . Namely, in the general case, the scaling has an effect of translating the weight by log2 λ. Let us now assume further that the vector weight w is multiplicative; that is, wi+j = wi wj for all i, j ∈ Z, or equivalently that wj = w1j , j ∈ Z. Under this extra assumption, we claim that the new modulation spaces p,q,r Mw in Definition 1 enjoy a scaling symmetry. Indeed, by repeating the computation above, we obtain    d   fλ Mwp,q,r = 2−j0 p ψj−j0 ,k (D)f Lpx (Rd ) q (Zd )  r k

w) j (w

k

w) rj (w

   d   = 2−j0 p wj0 ψj,k (D)f Lpx (Rd ) q (Zd )  log2 (w1 )− d p

=λ

f Mwp,q,r .

(2.11)

p,q,r This shows the scaling property of the Mw -spaces, when the vector weight w is multiplicative.

Remark 2.3. (i) The basic property of the Fourier transform states that modulations on the physical side correspond to translations on the Fourier side. Unit-scale modulations, that is, unit-scale translations on the Fourier side, yield the (usual) modulation spaces M p,q and the decomposition (2.3). On the other hand, p,q,r the new modulation spaces Mw with scaling are generated by modulations of all dyadic scales, that is, by translations of all dyadic scales on the Fourier side. Recalling that a wavelet basis is generated by all dyadic dilations and translations of a given (nice) function on the physical side, it is natural that the family   ψj,k (·) = ψ(2−j · −k) j∈Z,k∈Zd , which is essentially a wavelet basis but on the Fourier side, appears in Definition 1. (ii) As it is well known, the modulation spaces M p,q have an equivalent characterization via the short-time (or windowed) Fourier transform (STFT). Given a non-zero window function φ ∈ S(Rd ), we define the STFT Vφ f of a tempered distribution f ∈ S  (Rd ) with respect to φ by1  Vφ f (x, ξ) = Rd

Then, we have the equivalence of norms: 1

This is essentially the Wigner transform of f and φ.

f (y)φ(y − x)e−2πiy·ξ dy.

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  f M p,q ∼φ |||f |||M p,q := Vφ f Lqξ Lpx = Vφ f Lpx Lq , ξ

where the implicit constants depend on the window function φ. Given j ∈ Z, let δj f (x) = 2−jd f (2−j x) and φj (x) = φ(2j x). Then, a direct computation (see (4.4) below) shows that d

d

d

j j p p,q = 2 p δ f  f M[j] |||δj f |||M p,q = 2j p Vφ (δj f )Lqξ Lpx . j M p,q ∼φ 2

(2.12)

Moreover, a straightforward calculation shows that 

 Vφ (δj f ) (x, ξ) = (Vφj f )(2−j x, 2j ξ).

(2.13)

Then, from (2.12) and (2.13), we obtain jd( p + p − q ) p,q ∼ f M[j] Vφj f Lqξ Lpx = 2j q Vφj f Lqξ Lpx . φ 2 1

1

1

d

Based on these considerations, if one prefers to use the ||| · |||M p,q -norm via the STFT to define the p,q,r new modulation spaces Mw with scaling, the only difference will be in exchanging the vector weight d w = {wj }j∈Z with the vector weight σ = {2j q wj }j∈Z . Namely, if we define p,q := V j f  q p , |||f |||M[j] φ Lξ Lx

then we obtain the following equivalence of norms:   p,q  f Mwp,q,r ∼ |||f |||Mσp,q,r := |||f |||M[j] . σ) r (σ j

(iii) The boundedness property of the dilation operator: f (x) → fλ (x) = f (λx) on the modulation spaces was studied in [20]. In particular, it was shown in [20, Theorem 1.1] that there exist c1 , c2 > 0, depending only on d, p, and q, such that λc1 f M p,q  fλ M p,q  λc2 f M p,q

(2.14)

for all λ ≥ 1. When 0 < λ < 1, the estimate (2.14) holds after switching the exponents c1 and c2 . The p,q,r point of our construction is that the modulation spaces Mw are not biased toward any particular scale. p,q,r Moreover, when the vector weight is multiplicative, the Mw -spaces enjoy the exact dyadic scaling (2.11), which is relevant, for example, in the analysis of the nonlinear Schrödinger equation; see also Sections 3 and 4. (iv) There are variants of the estimate (2.14) in the settings of weighted modulation spaces [4] or α-modulation spaces [15] which are interesting in their own right. Naturally, one could consider a similar construction of scale invariant modulation spaces in these contexts. We, however, do not pursue this issue here and leave it for interested readers. (v) Let 1 ≤ p0 < ∞. Then, for p > p0 > q  > 1, the vector weight w = w (p, p0 ) = {wj }j∈Z with wj = 2jd( p − p0 ) 1

1

is a multiplicative weight satisfying (2.7) with 0 < ε < d min{ q1 − weight, (2.11) implies that, for a fixed dyadic scale λ = 2j0 ,

1 1 p0 , p0

− p1 }. Moreover, for this choice of

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fλ Mwp,q,r = λ− p0 f Mwp,q,r . d

p,q,r p0 d Namely, Mw (p,p0 ) scales like L (R ). See also Section 3.

3. A “good” weight and L2 -embedding p,q,q p,q In the discussion below, we restrict our attention to the case q = r and we simply write Mw as Mw . p,q We will show that, for a particular good vector weight w and p, q > 2, the space Mw is sufficiently large. Given p > 2, fix a vector weight w (p) = {wj }j∈Z with

wj = 2jd

p −2 2p

.

(3.1)

We see that w (p) is a multiplicative vector weight and is a “good” weight in the sense of the conditions (2.7), provided that q > 2. Before proceeding further, let us recall the space Xp,q defined in [2, p. 5260] via the norm: f Xp,q =



2

jd p−2 2p q



ψj,k f qLp

q1 .

(3.2)

k∈Zd

j∈Z

The Xp,q -spaces appear in the improvement of the Strichartz estimates for the linear Schrödinger equation (see (4.8) below) and, in particular, play an important role in the study of the mass-critical nonlinear Schrödinger equations. See also [3, Proposition 2.1] and [19, Theorem 4.2] for the one-dimensional and  p−2  two-dimensional versions of the Xp,q -spaces. Note that the vector weight 2jd 2p j∈Z appearing in (3.2) is precisely w (p ) defined in (3.1). p,q Let us now come back to the Mw (p) -space with w (p) = {wj }j∈Z defined in (3.1). For p > 2, it follows from Hausdorff-Young’s inequality that f Mwp,q(p) =



wjq

≤

ψj,k (D)f qLpx

q1

k∈Zd

j∈Z







2

jd p2p−2  q



ψj,k fqLp

q1

= fXp ,q

k∈Zd

j∈Z

=: f FXp ,q .

(3.3)

This shows that p,q FXp ,q ⊂ Mw (p) .

In particular, it follows from [2, Theorem 1.3] and Plancherel’s theorem that for all p, q > 2, we have f Mwp,q(p)  f L2 for all f ∈ L2 (Rd ). Namely, we have the following embedding: p,q d L2 (Rd ) ⊂ Mw (p) (R ).

Moreover, this embedding is strict; indeed, if we set f = F −1 g, where

(3.4)

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− 1 d  g(ξ) = |ξ|− 2  ln |ξ| 2 · 1(0, 12 )d (ξ), p,q d 2 d we see that f ∈ Mw (p) (R ) \ L (R ) when p, q > 2; see again [2, p. 5267].

Remark 3.1. In the study of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces, the Fourier-amalgam spaces m  p,q (Rd ), defined by the norm:    (Rd )  q d ,  f m p,q (Rd ) = ψ(ξ − k)f Lp  (Z ) ξ k

were considered in [13]. We note that the Fourier-amalgam space m  p,q (Rd ) is simply the Fourier image of the usual Wiener amalgam space W (Lp , q )(Rd ), as described in [6,14], defined by the norm:   f W (Lp ,q )(Rd ) = ψ(x − k)f Lpx (Rd ) q (Zd ) . k



Comparing this with the definition (2.2) of the modulation spaces M p,q , we see that m  p ,q “scales like” p p,q d p d M (in the sense that Lξ (R ) scales in the same manner as Lx (R )). In Definition 1, we defined the new p,q,r modulation spaces Mw with scaling for a good vector weight w satisfying (2.7). In an analogous manner, p,q,r w we can also define the new Fourier-amalgam spaces m adapted to scaling by the norm:   p,q,r := f  p,q  f m = w m [j] r (w w)



wjr f rm p,q

r1

[j]

j

,

j∈Z

where the scaled Fourier-amalgam spaces m  p,q [j] are defined by   f m := ψj,k fLpξ (Rd ) q (Zd ) . p,q [j] k

p,q,r As in the case of the Mw -spaces, we need to impose some conditions on the vector weight w so that p,q,r w the resulting space m contains smooth functions. Recalling the scaling property of m  p,q , we impose the following condition on the vector weight w = {wj }j∈Z :

wj 

 2−εj 2

if j ≥ 0,

d −( d p − q −ε)j

(3.5)

if j < 0.

Namely, we replace p in (2.7) by p. Arguing as in Subsection 2.2, we see that these new Fourier-amalgam p,q,r p,q,r w spaces m enjoy a scaling property analogous to that for the Mw -spaces. In particular, if the vector weight w is multiplicative, then we have the following scaling property; if we let λ = 2j0 for some j0 ∈ Z and set fλ (x) = f (λx) as before, then d

log2 (w1 )− p p,q,r = λ p,q,r . fλ m f m w w p,q,q w We conclude this remark by pointing out that the FXp,q -space discussed above is simply m (p ) with the  jd p−2   2p vector weight w (p ) = 2 . It is easy to see that this vector weight satisfies the condition (3.5) for j∈Z q > 2 and it is also multiplicative, thus satisfying the following scaling property for λ = 2j0 for some j0 ∈ Z: d p,q,q = λ fλ FXp,q = fλ m w 

p−2 d 2p − p

w (p )

−d 2

=λ

p,q,q f m w 

w (p )

−d 2

p,q,q = λ f m w

f FXp,q .

Namely, the FXp,q -spaces (and hence the Xp,q -spaces) scale like L2 (Rd ) as it is expected.

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Remark 3.2. In studying the mass-subcritical generalized KdV equation, Masaki-Segata [18, Definition 2.1]  p p q,r considered the so-called generalized Morrey spaces Mq,r (with q ≤ p) and M = FMqp ,r (with p ≤ q and r > p ). The usual Morrey spaces Mqp correspond to r = ∞ in the scale of generalized Morrey spaces, that is p Mqp = Mq,∞ . These spaces generalize Xp,q and FXp ,q defined in (3.2) and (3.3) and in particular we have q  ,r,r 2 2 . We point out that, when r < ∞, the space M p is nothing but m w Xp,q = Mp,q and FXp ,q = M p,q q,r (p,q)  jd( 1 − 1 )  q,r,r 2 p q p  w  (p,q) with a specific vector weight given by w (p, q) = 2 , while Mq,r is the Fourier image of m j∈Z  jd( 1 − 1 )   p q with the vector weight w (p, q) = 2 . For more on Morrey-type spaces, see also [10–12]. j∈Z 4. An application to the Schrödinger multiplier Let S(t) = e−itΔ be the Schrödinger operator defined as a Fourier multiplier operator with a multiplier e . Our main interest here is to discuss a boundedness property of the Schrödinger operator S(t) = p,q,r e−itΔ on appropriate Mw -spaces. Let ψ be as in (2.1). A direct computation shows that 4π 2 it|ξ|2

 ψj,k (D)(S(t)f )(x) =

ψ(2−j ξ − k)e4π

Rd

 ψ(ξ − k)e4π

= 2jd

f(ξ)e2πix·ξ dξ

2

it|ξ|2

2

i(22j t)|ξ|2

j f(2j ξ)e2πi2 x·ξ dξ

Rd

=2

jd



 ψ(D − k)(S(22j t)δj f ) (2j x),

(4.1)

where, as before, we wrote δj for the dilation operator given by δj f (x) = 2−jd f (2−j x). Thus, we have d

ψj,k (D)(S(t)f )Lpx = 2j p ψ(D − k)(S(22j t)δj f )Lpx . Hence, we obtain d

j 2j p,q = 2 p S(2 S(t)f M[j] t)δj f M p,q .

(4.2)

Interpolating [1, Theorem 14] for p = 1 or p = ∞ with the trivial bound for p = 2, we obtain S(22j t)δj f M p,q  22j td | 2 − p | δj f M p,q . 1

1

(4.3)

Lastly, by a computation analogous to (4.1), we have ψ(D − k)(δj f )(x) = 2−jd ψj,k (D)f (2−j x) and hence  d d p,q . δj f M p,q = 2−jd 2j p ψj,k (D)f Lpx q = 2−j p f M[j]

(4.4)

k

Putting (4.2), (4.3), and (4.4) together, we obtain 2j d | 2 − p | p,q  2 p,q . S(t)f M[j] t f M[j] 1

2

Under r > p , this vector weight satisfies the condition (3.5) for q < 2.

1

(4.5)

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Note that, when j = 0, this estimate recovers the boundedness of S(t) on the modulation spaces M p,q . We also point out that, when p = 2, the divergent behavior of the constant as j → ∞ is consistent with the unboundedness of S(t) on Lp (Rd ). Now, let w = {wj }j∈Z be a “good” vector weight, satisfying (2.7), and define a vector weight σ = {σj }j∈Z by setting

σj =

 2 2−jd |1− p | wj

if j ≥ 0, if j < 0.

wj

p,q,r Note that such σ also satisfies (2.7). Then, from (4.5), we obtain the boundedness of S(t) : Mw → Mσp,q,r with a quantitative bound:

S(t)f Mσp,q,r  td | 2 − p | f Mwp,q,r 1

1

for any t ∈ R. Let us further comment on an alternate estimate when p ≥ 2. In this case, the Schrödinger operator is known to also satisfy the following inequality (see [16, Proposition 4.1]): S(t)f M p,q  t−d ( 2 − p ) f M p ,q . 1

1



Note that the decay at infinity in this estimate is now consistent with the one in the Lp → Lp boundedness of S(t). Thus, instead of (4.3), we can now write S(22j t)δj f M p,q  22j td ( 2 − p ) δj f M p ,q . 1

1

(4.6)

Since, by (4.4), we have δj f M p ,q = 2−j p f M p ,q , (4.2) and (4.6) yield d

[j]

2j −d ( 2 − p ) jd ( p − p ) p,q  2 S(t)f M[j] t 2 f M p ,q 1

1

1

1

[j]

= 22j t−d

1 ( 12 − p )

22jd

1 ( 12 − p )

f M p ,q .

(4.7)

[j]

Thus, by considering again w = {wj }j∈Z to be a “good” vector weight, satisfying (2.7), from (4.7) we obtain p ,q,r p,q,r now the boundedness S(t) : Mw → Mw with a quantitative bound: S(t)f Mwp,q,r  t−d ( 2 − p ) f Mp ,q,r 1

1

w

for any p ≥ 2 and t ∈ R. We conclude our note with a comment on the following improvement of the Strichartz estimate proved 4 in [2, Theorem 1.2]; if 2 < p < 2 + d(d+3) and q = 2(d+2) , then d S(t)f Lqt,x (R×Rd )  f FXp ,q (Rd ) .

(4.8)

The proof of this inequality is highly non-trivial; it uses the bilinear restriction estimate in [21, Theorem 1.1] and an appropriate orthogonality lemma for functions with disjoint Fourier supports [22, Lemma 6.1]. Considering the embedding (3.4) stated in the previous section, it would be interesting to know if the p,q FXp ,q -norm in (4.8) can be replaced by the Mw (p) -norm.

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Acknowledgments The first author was partially supported by a grant from the Simons Foundation (No. 246024). The second author is supported by the European Research Council (grant no. 637995 “ProbDynDispEq”). The authors would like to thank the anonymous referees for several useful comments that have improved the presentation of this paper. References [1] Á. Bényi, K. Gröchenig, K.A. Okoudjou, L.G. Rogers, Unimodular Fourier multipliers for modulation spaces, J. Funct. Anal. 246 (2) (2007) 366–384. [2] P. Bégout, A. Vargas, Mass concentration phenomena for the L2 -critical nonlinear Schrödinger equation, Trans. Amer. Math. Soc. 359 (11) (2007) 5257–5282. [3] R. Carles, S. Keraani, On the role of quadratic oscillations in nonlinear Schrödinger equations II. The critical case, Trans. Amer. Math. Soc. 359 (1) (2007) 33–62. [4] E. Cordero, K.A. Okoudjou, Dilation properties for weighted modulation spaces, J. Funct. Spaces Appl. (2012) 145491. [5] H.G. Feichtinger, Choosing function spaces in harmonic analysis, in: Excursions in Harmonic Analysis, in: Appl. Numer. Harmon. Anal., vol. 4, Birkhäuser/Springer, Cham, 2015, pp. 65–101. [6] H.G. Feichtinger, Banach convolution algebras of Wiener type, in: Functions, Series, Operators, vol. I, II, Budapest, 1980, in: Colloq. Math. Soc. János Bolyai, vol. 35, North-Holland, Amsterdam, 1983, pp. 509–524. [7] H.G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. I, J. Funct. Anal. 86 (1989) 307–340. [8] H.G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. II, Monatsh. Math. 108 (1989) 129–148. [9] H.G. Feichtinger, F. Voigtlaender, From Frazier-Jawerth characterizations of Besov spaces to wavelets and decomposition spaces, in: Functional Analysis, Harmonic Analysis, and Image Processing: A Collection of Papers in Honor of Björn Jawerth, in: Contemp. Math., vol. 693, Amer. Math. Soc., Providence, RI, 2017, pp. 185–216. [10] I. Fofana, Continuité de l’intégrale fractionnaire et espace (Lq , lp )α , C.R. Acad. Sci. Paris, Sér. I 308 (1989) 525–527. [11] I. Fofana, Étude d’une classe d’espaces de fonctions contenant les espaces de Lorentz, Afr. Mat. 2 (1988) 29–50. [12] J. Feuto, Intrinsic square functions on functions spaces including weighted Morrey spaces, Bull. Korean Math. Soc. 50 (2013) 1923–1936. [13] J. Forlano, T. Oh, Normal form approach to the one-dimensional cubic nonlinear Schrödinger equation in Fourier-amalgam spaces, preprint. [14] J.J.F. Fournier, J. Stewart, Amalgams of Lp and lq , Bull. Amer. Math. Soc. (N.S.) 13 (1) (1985) 1–21. [15] J. Han, B. Wang, α-Modulation spaces (I) scaling, embedding and algebraic properties, J. Math. Soc. Japan 66 (4) (2014) 1315–1373. [16] H. Hudzik, B. Wang, The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations 232 (1) (2007) 36–73. [17] P. Gröbner, Banachräume glatter funktionen und zerlegungsmethoden, PhD thesis, University of Vienna, 1992, 99 pp. [18] S. Masaki, J. Segata, Existence of a minimal non-scattering solution to the mass-subcritical generalized Kortweg-de Vries equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2) (2018) 283–326. [19] A. Moyua, A. Vargas, L. Vega, Restriction theorems and maximal operators related to oscillatory integrals in R3 , Duke Math. J. 96 (3) (1999) 547–574. [20] M. Sugimoto, N. Tomita, The dilation property of modulation spaces and their inclusion relation with Besov spaces, J. Funct. Anal. 248 (1) (2007) 79–106. [21] T. Tao, A sharp bilinear restriction estimate for paraboloids, Geom. Funct. Anal. 13 (2003) 1359–1384. [22] T. Tao, A. Vargas, L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998) 967–1000.