Endpoint uniform Sobolev inequalities for elliptic operators with applications

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ScienceDirect J. Differential Equations 267 (2019) 4609–4625 www.elsevier.com/locate/jde

Endpoint uniform Sobolev inequalities for elliptic operators with applications Shanlin Huang ∗ , Quan Zheng School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, PR China Received 20 October 2018; revised 22 April 2019 Available online 10 May 2019

Abstract In this paper, we obtain sharp restricted weak-type uniform Sobolev inequalities for a class of elliptic operators P (D). In particular, we improve a recent result of Sikora, Yan and Yao [24] to the endpoint case. The sharpness is proved by constructing explicit counterexamples. As applications, we establish Stein-Tomas type inequalities for H = P (D) + V with certain class of potentials V . © 2019 Elsevier Inc. All rights reserved. MSC: 42B37; 47A10; 35J30 Keywords: Uniform Sobolev inequalities; Stein-Tomas inequalities; Spectral theory

1. Introduction In a recent paper of Sikora, Yan and Yao [24], they proved Lp − Lq uniform Sobolev inequalities for a class of elliptic operators of constant coefficients. More specifically, they showed that if the symbol P (ξ ) (ξ ∈ Rn , n ≥ 2) is a real homogeneous elliptic polynomial of order 2m ≥ 2, and P satisfies the following non-degenerate condition det(

∂ 2 P (ξ ) ) = 0, for ξ = 0. ∂ξi ∂ξj

* Corresponding author.

E-mail addresses: [email protected] (S. Huang), [email protected] (Q. Zheng). https://doi.org/10.1016/j.jde.2019.05.005 0022-0396/© 2019 Elsevier Inc. All rights reserved.

(1.1)

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S. Huang, Q. Zheng / J. Differential Equations 267 (2019) 4609–4625

Then there is a constant Cp,q such that for all z ∈ C \ {0}, n

uLq (Rn ) ≤ Cp,q |z| 2m

( p1 − q1 )−1

u ∈ C0∞ (Rn ),

(P (D) − z)uLp (Rn ) ,

(1.2)

where the exponents 1 ≤ p < q ≤ ∞ satisfy ⎧ ⎪ 2m , 2 1 1 ⎨n < − ≤ 1 and 1 < p < q < ∞, n+1 p q ⎪ ⎩ 1,

when 2m < n, when 2m = n, when 2m > n,

(1.3)

and  min

1 1 1 1 − , − p 2 2 q

 >

1 . 2n

(1.4)

In the special case where P (D) = − and n ≥ 3, the estimate (1.2) (under conditions (1.3) and (1.4)) was established in a celebrated work of Kenig, Ruiz and Sogge [19]. Such inequalities were originally motivated by unique continuation problems and turned out to be connected with many other problems as well, including eigenvalue bounds of Schrödinger operators [8,9], limiting absorption principles and applications to scattering theory [11,18]. 2n 1 1 Moreover, in [19], it was also shown that when n ≥ 3, and if n+2 ≤ p ≤ 2(n+1) n+3 , p + p = 1, then there is a constant Cp such that for all z ∈ C \ {0}, n 1 ( p − p1 )−1

uLp (Rn ) ≤ Cp |z| 2

(− − z)uLp (Rn ) ,

u ∈ C0∞ (Rn ),

(1.5)

where the range of the exponent p in (1.5) is optimal. We mention that the endpoint case p = 2(n+1) n+3 is of special interest since it implies (by the Stone’s formula, see (3.1)) the following estimate for the spectral measure associated with the Laplace operator dE− (λ)

1

2(n+1) 2(n+1) L n+3 (Rn )−L n−1 (Rn )

≤ Cλ− n+1 , λ > 0,

(1.6)

which is in turn, via a standard T T ∗ argument, equivalent to the famous Stein-Tomas restriction theorem  1 2(n + 1) ( |fˆ|2 dσ ) 2 ≤ Cf Lp (Rn ) , 1 ≤ p ≤ . (1.7) n+3 S n−1

In view of the estimate (1.5), it’s natural to ask whether this remains true for more general 2(n+1) elliptic operators. Notice that the pair of exponents (p, p ) = ( 2(n+1) n+3 , n−1 ), occurring in (1.5), 2 . The argument in [24, Theorem 4.1], however, does not cover satisfies the condition p1 − q1 = n+1 this endpoint case (see (1.3)). In the first part of this paper, we confirm this in the affirmative by extending the estimate (1.2) to the following situation (the open line segment AA’ in Fig. 1): 1 1 2 − = p q n+1



and

1 1 1 1 min − , − p 2 2 q

 >

1 . 2n

(1.8)

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More precisely, we shall prove the following sharp restricted weak-type inequality. Theorem 1.1. Let n ≥ 3 and P be a homogeneous elliptic operator of order 2m whose symbol satisfying (1.1). Then there is a uniform constant C such that for all z ∈ C \ {0}, n

u L

2n(n+1) , ∞ (n−1)2

≤ C|z| m(n+1) −1 (P (D) − z)u

2n , 1

L n+1

(Rn )

(Rn )

,

u ∈ C0∞ (Rn ),

(1.9)

where Lp, q denotes the Lorentz space. In particular, the estimate (1.2) is valid under the condition (1.8). Remark 1.2. (i) The estimate (1.9) is sharp in the sense that strong type Lp − Lq estimates can not be expected for the pair of exponents (p, q) = (2n/(n + 1), 2n(n + 1)/(n − 1)2 ). Indeed, take P (D) = (−)m , m = 1, 2, . . .. We shall construct ϕλ (see (2.18) in Sect. 2) to show that the following inequality cannot hold uniformly with respect to z ∈ C \ {0} ϕλ 

n

2n(n+1) L (n−1)2 (Rn )

≤ C|z| m(n+1) −1 (−)m − z)ϕλ 

2n

L n+1 (Rn )

.

(1.10)

(ii) As noted above, Kenig, Ruiz and Sogge [19] established the uniform Sobolev inequalities (1.2) (under conditions (1.3) and (1.4)) for P (D) = −. Gutiérrez [13] obtained restricted weak(n−1)2 n2 +4n−1 n−1 type estimates for the Laplacian at points ( n+1 2n , 2n(n+1) ) and ( 2n(n+1) , 2n ). Very recently, the n−3 n+3 n−1 restricted weak-type estimates at points ( n+1 2n , 2n ) and ( 2n , 2n ) were proved by Ren, Xi and Zhang [22]. We also mention that there has been a lot of interest in extending the uniform Sobolev inequalities to manifolds, see e.g. [5,7,16,20] and the references therein.

Now we turn to some applications. Using Theorem 1.1, we can obtain sharp Lp estimates for the spectral measure dEH (λ) of H = P (D) + V (x) under certain small perturbations. Such estimates, in view of the equivalence (1.6) and (1.7), can be regarded as Stein-Tomas inequalities in the setting of nonnegative self-adjoint operators (see Corollary 3.1). Furthermore, in the particular case P (D) = (−)m , we are able to prove uniform Lp resolvent estimates for H with a class of potentials whose Lp norms are finite. One of our motivations comes from the classical result due to Agmon and Hörmander [2] (see also [14, Chapter XIV]), in which they proved uniform resolvent estimates between weighted L2 spaces for a class of higher order differential operators with short range perturbations. Such estimates have found many important applications in scattering and spectral theories related to Schrödinger equations, see [1,14,18,21] just to name a few. Throughout the paper, C and Cj denote absolute positive constants whose dependence will be specified whenever necessary. The value of C may vary from line to line. 2. Endpoint uniform resolvent estimates In this section, we first prove Theorem 1.1 and then construct explicit counterexamples to show that it’s sharp. As a corollary, we also consider the closely related Lp estimates for the Bochner-Riesz means corresponding to P (D) with negative index. Proof of Theorem 1.1. Recall that Lorentz norms satisfy the following dilation identity for all 1 ≤ p < ∞ and 1 ≤ q ≤ ∞,

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f (ax)Lp,q = a

− pn

f Lp,q , a > 0.

Then by homogeneity of P (D) and the fact that the pair p, q in (1.9) satisfying n+1 (n − 1)2 2 − = . 2n 2n(n + 1) n + 1

(2.1)

We deduce that, after a simple scaling argument, it suffices to prove (1.9) for |z| = 1. Furthermore, in the case δ ≤ | arg z| ≤ π , where δ > 0 is some small but fixed number, the standard Sobolev inequalities apply and actually the strong type Lp − Lq estimates hold in this situation. Therefore the key is to show that the estimate is uniform as arg z tends to zero (i.e., as z approaches the point (1, 0) in C). For the sake of brevity, it’s convenient to write z = (1 + iε)2m with 0 < ε 1 (hence |z| ≈ 1 and arg z ≈ 2mε). We shall adapt the approach in [24,19] and the new ingredient is to use an interpolation argument due to Bourgain [4] (see Lemma 2.1). Let ρ ∈ C0∞ (R) be a cut-off function such that ρ(t) = 1,

if t ∈ [−2, 2],

and supp ρ ⊂ [−4, 4].

(2.2)

Denote by Kε the convolution kernel of the resolvent (P (D) − (1 + iε)2m )−1 . Using Fourier transform, we write Kε = F −1 {(P (ξ ) − (1 + iε)2m )−1 } = F −1 {

1/2m (ξ )) ρ(P 1/2m (ξ )) −1 1 − ρ(P } + F { } P (ξ ) − (1 + iε)2m P (ξ ) − (1 + iε)2m

 K1 + K2 . The estimate for K2 is relatively straightforward. Since we have |K2 (x)| ≤ CN |x|2m−n−N , x = 0, N = 1, 2, . . . ,

(2.3)

then it follows from Young’s inequality that K2 ∗ f Lq ≤ Cf Lp ,

(2.4)

1 1 2m n−2m where 0 ≤ p1 − q1 ≤ 2m n and ( p , q ) = ( n , 0) and (1, n ), which is stronger than the desired estimate. To handle K1 we need more explicit information about the kernel. This could be achieved by using the non-degenerate assumption (1.1) and applying the stationary phase principle. Indeed, one has (see e.g. in [24] or [25, p. 51])

K1 (x) =



|x|−

n−1 2

a± (x)eiφ± (x) , |x| ≥ 1,

(2.5)

±

where φ± (x) are smooth homogeneous function of degree one in the region |x| ≥ 1, and a± (x) ∈ C ∞ satisfy

S. Huang, Q. Zheng / J. Differential Equations 267 (2019) 4609–4625

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|∂ β a± (x)| ≤ Cβ |x|−|β| , |x| ≥ 1.

(2.6)

Now choose ϕ ∈ C0∞ (R) such that suppϕ ⊂ [ 12 , 2] and

∞

j =1 ϕ(2

−j +1 t) = 1 for t

≥ 12 . We define

∞

K1, j (x) = K1 (x)ϕ(

j |x| ), j = 1, 2, . . . and K1, 0 = K1 − K1 . j −1 2

(2.7)

j =1

The associated convolution operators are defined by T1, j f = K1, j ∗ f, j = 0, 1, 2, . . . .

(2.8)

Since the kernel K1, 0 is of compact support, then Young’s inequality gives the following better result T1, 0 f Lq ≤ Cf Lp , 1 ≤ p ≤ q ≤ ∞.

(2.9)

In order to obtain sharp Lp estimates for T1, j , j = 1, 2, . . ., the non-degenerate assumption (1.1) plays an essential role. Since it is equivalent to the fact that the hypersurface  = {ξ ∈ Rn , P (ξ ) = 1} has non-zero Gaussian curvature everywhere, which in turn implies that the phase function φ± (x) in (2.5) satisfies the Carleson-Sjölin conditions (see e.g. [25, p. 68]). Hence if the pair of exponents p, q satisfy q=

n+1 p n−1

and 1 ≤ p ≤ 2,

(2.10)

then T1, j f Lq   1/q q = | K1, j (x − y)f (y) dy| dx ≤



2

− n−1 2 j

·2

jn

·2

jn q



 |

− n−1 2

|x − y|

a(2 |x − y|)e j

i2j φ± (|x−y|)

1/q j

q

f (2 y) dy| dx

±

≤ C2− = C2

n−1 2 j

jn

· 2j n · 2 q · 2

n j ( n+1 2 −p)

f Lp ,

− jqn

·2

− jpn

f Lp (2.11)

where in the first inequality, we perform the change of variables (x, y) → (2j x, 2j y), and the second inequality is due to Stein’s oscillatory integral theorem (see [25, Corollary 2.2.3]), which requires the condition (2.10) above.

Take (2.11) into account, we see that the Lq norm of the series ∞ j =1 T1, j f converges when 2n 2n p < n+1 , and fails exactly when p = n+1 . To overcome this technical difficulty, we can exploit an interpolation argument due to Bourgain [4], which was later used successfully in Bak and Seeger’s work [3] on extensions of the Stein-Tomas theorem. See also the recent work of Ren, Xi and Zhang [22].

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Fig. 1. Uniform Lp − Lq resolvent estimates.

Lemma 2.1. (Bourgain [4].) Let ε0 , ε1 > 0, and let {Tj : j ∈ Z} be a sequence of linear operators satisfying Tj f Lq0 ≤ M0 2ε0 j f Lp0 , Tj f Lq1 ≤ M1 2−ε1 j f Lp1 for some 1 ≤ p0 , p1 , q0 , q1 ≤ ∞. Then T 



j ∈Z Tj

is bounded from Lp,1 to Lq,∞ with

Tf Lq,∞ ≤ CM0θ M11−θ f Lp,1 , where θ=

ε1 , ε0 + ε1

and 1 1−θ 1 1−θ θ θ + , + . = = q q0 q1 p p0 p1 To proceed, we choose p = 2, q = 2(n + 1)/(n − 1) in (2.11) to obtain T1, j f 

j

2(n+1) L n−1

≤ 2 2 f L2 .

(2.12)

This corresponds to the estimate at point E in Fig. 1. On the other hand, it follows from the point-wise decay of the kernel K1, j (see (2.5) and (2.7)) that T1, j f L∞ ≤ 2−

n−1 2 ·j

f L1 .

(2.13)

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And this corresponds to the estimate at point O = (1, 0) in Fig. 1. In order to apply Bourgain’s interpolation lemma, we set θ=

n−1 2 1 n−1 + 2 2

=

n−1 , n

(2.14)

and observe that θ 1−θ n+1 n−1 1−θ (n − 1)2 + = , ·θ + = , 2 1 2n 2(n + 1) ∞ 2n(n + 1)

(2.15)

which corresponds to the desired point A in Fig. 1. Then by (2.12), (2.13), (2.15) and Lemma 2.1 we find K1 ∗ f  L

2n(n+1) , ∞ (n−1)2

≤ Cf  (Rn )

2n , 1

L n+1

(Rn )

.

(2.16)

This together with (2.4) and (2.9) completes the proof of (1.9). Finally, by duality and a real interpolation (see [26, Chapt. V]) in Lorentz spaces between re(n−1)2 n2 +4n+1 n−1 stricted weak-type estimates at A = ( n+1 2n , 2n(n+1) ) and its dual A = ( 2n(n+1) , 2n ), we obtain the following inequality n

uLq,s ≤ Cp,q |z| 2m

( p1 − q1 )−1

(P (D) − z)uLp,s ,

(2.17)

where the pair of exponents p, q satisfy (1.8) and p < s < q. In particular, the estimate (1.2) is valid under the condition (1.8), which finishes the proof. 2 Sharpness of (1.9). Now we construct counterexamples to show that, in the case P (D) = (−)m , m = 1, 2, . . ., the strong type Lp − Lq estimate (1.10) may not hold uniformly with respect to the parameter z. More precisely, we consider the dual version of (1.10) and in the following we denote by z = (λ + i)2m , λ ≥ 1. Choose a real-valued even function ρ ∈ C0∞ (R) satisfying (2.2). We set  ϕλ (x) = Rn

ρ(|ξ |/λ) eix·ξ dξ . − (λ + i)2m

|ξ |2m

(2.18)

Note that ϕλ is the inverse Fourier transform of a C0∞ function, hence it belongs to the Schwartz space, and in particular, ϕλ ∈ Lp for all 1 ≤ p ≤ ∞. On one hand, we observe that  ((−)m − (λ + i)2m )ϕλ (x) =

ρ(|ξ |/λ)eix·ξ dξ = λn ρ(λ ˆ · |x|).

R3

Since ρ ∈ C0∞ , then there exists a constant C independent of λ > 0 so that

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((−)m − (λ + i)2m )ϕλ 

2n(n+1) L n2 +4n−1 (Rn )

= λn ρ(λ·) ˆ

2n(n+1) L n2 +4n−1 (Rn )

≤ Cλn−n·

n2 +4n−1 2n(n+1)

(n−1)2

= Cλ 2(n+1) . (2.19)

Hence (recall that z = (λ + i)2m ) (n−1)2

2n

n

|z| m(n+1) −1 ((−)m − (λ + i)2m )ϕλ 

2n L n+3 (Rn )

On the other hand, we have by scaling  n−2m ϕλ (x) = λ ·

≤ Cλ n+1 −2m+ 2(n+1) = Cλ

n+1 2 −2m

(2.20)

.

ρ(|ξ |) eiλx·ξ dξ . |ξ |2m − (1 + i/λ)2m

Rn

Using stationary phase argument (notice that we have φ± (x) = |x| in (2.5) for this particular case, see [25, p. 52]), one has the following  n+1 n−1 2 −2m · |x|− 2 · cos λ|x|, if λ−1 ≤ |x| ≤ 1, ϕλ (x) ≈ Cλ (2.21) n−2m Cλ , if |x| ≤ λ−1 . A direct computation yields 

2n

2n



|ϕλ (x)| n−1 dx ≈ λ(n−2m)· n−1 |x|≤1

dx + λ( |x|≤λ−1

λ

2n (n−2m)· n−1



n+1 2n 2 −2m)· n−1

λ−1 ≤|x|≤1

· λ−n + λ

2n ( n+1 2 −2m)· n−1

2n

| cos λ|x|| n−1 dx |x|

· log λ,

which implies the following lower bound ϕλ 

2n L n−1 (|x|≤1)

λ

n+1 2 −2m

· (log λ)

n−1 2n

.

(2.22)

However, in view of (2.20) and (2.22), it follows that the following inequality ϕλ 

n

2n L n−1 (Rn )

≤ C|z| m(n+1) −1 (−)m − z)ϕλ 

2n(n+1)

L n2 +4n−1 (Rn )

(2.23)

cannot hold uniformly for all λ > 0. Then by duality, the estimate (1.10) does not hold uniformly either. 2 Finally, it’s natural to consider the closely related Bochner-Riesz means with negative index. More precisely, let H0 be a nonnegative self-adjoint operator on L2 and denote Sλ−α (H0 ) =

  1 H0 −α , 1− (α + 1) λ +

λ > 0, α > 0.

(2.24)

Then using slight modification of the arguments for Theorem 1.1, we can also obtain the following

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Corollary 2.2. Let n ≥ 3 and 12 < α < n+1 2 . Suppose that H0 = P (D) satisfies the assumptions of Theorem 1.1. Then there is some constant Cp,q such that n

Sλ−α (H0 )Lp −Lq ≤ Cp,q λ 2m

( p1 − q1 )−α

, λ > 0,

(2.25)

where the pair of exponents p, q satisfy 

1 1 2α − = p q n+1

and

min

1 1 1 1 − , − p 2 2 q

 >

2α − 1 . 2n

(2.26)

Proof. By duality and interpolation in Lorentz space, it suffices to prove restricted weak-type estimates in (one of) the extreme points of the open segment given by (2.26). In the following, n2 −2nα+2α−1 we only prove estimates for ( p1 , q1 ) = ( n+2α−1 ). The reason that one can follow 2n , 2n(n+1) the same lines as the proof of Theorem 1.1 is seen from the identity

S1−α (H0 ) 1 iπα = e (H0 − 1 − i0)−α ρ(H0 ) − e−iπα (H0 − 1 + i0)−α ρ(H0 ) , (α) 2πi where ρ ∈ C0∞ is given by (2.2). Indeed, when α = 1, it reduces to the same arguments in Theorem 1.1. We mention that in the general case, the kernel of (H0 − 1 − i0)−α ρ(H0 ), denoted by K1α (x), has the following expression (see e.g. [24,17]) K1α (x) =



|x|−

n+1 2 +α

a± (x)eiφ± (x) , |x| ≥ 1,

(2.27)

±

α the same way as where a± , φ± satisfy the same properties as before. Similarly, we define K1, j in (2.7). Then using again the Carleson-Sjölin arguments (see (2.11)), we have α K1, j ∗f

L

2(n+1) n−1

≤ 2j ·

2α−1 2

f L2 .

(2.28)

α now yields The point-wise decay of the kernel K1, j α −j · K1, j ∗ f L∞ ≤ 2

n+1−2α ·j 2

f L1 .

(2.29)

Then we set θ=

n+1−2α 2 2α−1 n+1−2α 2 + 2

=

n + 1 − 2α , n

and observe that θ 1−θ n + 2α − 1 n−1 1−θ n2 − 2nα + 2α − 1 + = , ·θ + = . 2 1 2n 2(n + 1) ∞ 2n(n + 1) Finally by (2.28), (2.29), (2.30) and Lemma 2.1 we find

(2.30)

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S. Huang, Q. Zheng / J. Differential Equations 267 (2019) 4609–4625

K1α ∗ f  which finished the proof.

2n(n+1)

L n2 −2nα+2α−1

,∞

(Rn )

≤ Cf 

2n

L n+2α−1

,1

(Rn )

,

(2.31)

2

Remark 2.3. We mention that Sikora, Yan and Yao [24] proved (2.25) under the condition 1 1 2α 1 1 1 1 2α−1 p − q > n+1 and min p − 2 , 2 − q > 2n . For the classical Bochner-Riesz means (i.e., Sλ−α (−)) with negative index, we refer to [6,12] and references therein.

3. Stein-Tomas type estimates for H = P (D) + V In this section, we shall first apply Theorem 1.1 to prove Lp mapping properties of the spectral measure of H = P (D) + V under certain small perturbations. Throughout this section, we assume that 0 ≤ V ∈ L1loc , then it’s well-known that H can be defined as a nonnegative self-adjoint operator on L2 . The idea behind the proof is based on the following Stone’s formula dEH (λ) =

1 (H − (λ + i0))−1 − (H − (λ − i0))−1 , λ > 0 2πi

(3.1)

and the resolvent identity (H − (λ + i))−1 = (P (D) − (λ + i))−1 − (P (D) − (λ + i))−1 V (H − (λ + i))−1 , (3.2) which is hold for any λ > 0,  > 0. We remark that if the potential V is “small” in certain norms, then one can force the inverse (I + (P (D) − (λ + i))−1 V )−1 to be bounded on Lp uniformly with λ > 0, 0 <  < 1, then the situation becomes much easier to handle. Indeed, in [24], Sikora, Yan and Yao showed that if P (D) satisfies the assumptions of Theorem 1.1. Assume in addition that 2m < n(n+3) 2(n+1) and there is a small constant c0 > 0 such that  V 

n L 2m

+ sup

x∈Rn

V (y) dy ≤ c0 , |x − y|n−2m

(3.3)

then the estimate n

dEH (λ)Lp −Lp ≤ Cλ 2m

( p1 − p1 )−1

, λ>0

(3.4)

holds for all 1 ≤ p < 2(n+1) n+3 , which is sharp except for the endpoint in view of the Stein-Tomas Theorem (see (1.6) and (1.7) for the case H = −). However, using Theorem 1.1 and arguments in [24, Theorem 5.8], we can cover the endpoint case p = 2(n+1) n+3 immediately and obtain the following Stein-Tomas type inequalities for H = P (D) + V . Corollary 3.1. Let P (D) satisfy the assumptions in Theorem 1.1 and 0 ≤ V ∈ L1loc (Rn ). Assume that 2m < n(n+3) 2(n+1) and (3.3) is valid for some small constant c0 > 0. Then the spectral measure estimate (3.4) holds for all 1 ≤ p ≤

2(n+1) n+3 .

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The rest of this section is devoted to proving results without the smallness assumption (3.3) on V . As already mentioned in the introduction, Agmon and Hörmander [2] established uniform resolvent estimates between weighted L2 spaces for a class of higher order differential operators with short range potentials. Roughly speaking, the potential V is only assumed to have decay of order O(1/|x|1+ ) at infinity for any  > 0. On the other hand, for the three dimensional Schrödinger operators, Goldberg and Schlag [11] showed the following Lp version of the limiting absorption principle: sup (− + V − (λ + i))−1  0<<1

1

4

L 3 (R3 )−L4 (R3 )

≤ C(λ0 )λ− 4 , λ > λ0 ,

(3.5)

3

where λ0 > 0 is any but fixed number and V ∈ Lp (R3 ) ∩ L 2 (R3 ) for some p > 32 . Their proof relies heavily on the fact that under such conditions on V , the Schrödinger operator H = − + V does not admit positive eigenvalues. In the general case H = P (D) + V , the absence of positive eigenvalues is of interest in its own right. Actually, it can be quite different from the second order case (see Remark 3.3). Nevertheless, in the following we establish a simple criterion which is needed later. Lemma 3.2. Let H = P (D) + V , where P (D) satisfies the assumptions of Theorem 1.1, and V is a real-valued function that is P (D)-bounded with relative bound less than one. Assume that (a) There exists a multiplication operator W on L2 (Rn ) with D(W ) ⊃ D(P (D)) such that for all ψ ∈ D(P (D)), (a − 1)−1 (Va − V )ψ → W ψ in L2 , as a → 1,

(3.6)

where Va (x) = V (ax), a ∈ R. (b) V is repulsive in the sense that V (ax) ≤ V (x) for all a > 1. Then H = P (D) + V has no positive eigenvalues. Proof. The idea is to build a virial identity for eigenfunctions of H (see (3.8) below) by using assumption (a). Then we exploit the fact that P (D) is nonnegative and the repulsive condition (b) to obtain a contradiction. Suppose that ϕ is an eigenfunction of H with positive eigenvalue λ, i.e., (P (D) + V ) ϕ = λϕ, λ > 0.

(3.7)

We claim that the assumption (a) in Lemma 3.2 implies the following so-called virial identity (see e.g. [21, Theorem XIII.59]) for the eigenfunction ϕ. 2mϕ, P (D)ϕ = ϕ, W ϕ,

(3.8)

where ·, · denotes the inner product on L2 (Rn ) and W is the function appeared in (3.6). In fact, if we define the scaling operator Ua on L2 (Rn ) by n

Ua f (x) = a 2 f (ax), a ∈ R+ , x ∈ Rn , then Ua is an isometry in L2 (Rn ) (group of dilations) and a direct computation yields

(3.9)

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Ua P (D)Ua−1 = a −2m P (D).

(3.10)

Thus we can use the relation (3.10) to rewrite the equation (3.7) as

P (D) + a 2m Va ϕa = a 2m · λϕa ,

(3.11)

where Va (x) = V (ax) and ϕa (x) = ϕ(ax). Combining (3.7) and (3.11) gives (a 2m − 1)λϕa , ϕ = (P (D) + a 2m Va )ϕa , ϕ − ϕa , (P (D) + V )ϕ,

(3.12)

which in turn implies (

a 2m − 1 a 2m − 1 Va − V )λϕa , ϕ = Va · ϕa , ϕ + ϕa , · ϕ, a = 1, a−1 a−1 a−1

(3.13)

where we have used the fact that P (D), Va are symmetric operators. Therefore taking the limit as a → 1 in (3.13), we find 2mλϕ, ϕ = 2mV ϕ, ϕ + ϕa , W ϕ,

(3.14)

which is exactly (3.8) in light of (3.7). To finish the proof of Lemma 3.2, we notice that the repulsive condition (b) on the potential implies that, in view of the assumption (3.6), W ≤ 0. Furthermore, since P (D) is nonnegative, then by (3.8) we obtain 0 ≤ 2mϕ, P (D)ϕ = ϕ, W ϕ ≤ 0,

(3.15)

which indicates that ϕ ≡ 0, and the proof is completed. 2 Remark 3.3. Despite the simplicity of the proof, we mention that the situation for the higher order Schrödinger operator H = (−)m + V (m > 1) is very different from the case m = 1. In particular, when m = 2k is even, H may have positive eigenvalues even if V ∈ C0∞ . As a comparison, we mention the classic result of Kato (see e.g. [21, Theorem XIII.58]), which says if V is bounded and satisfies lim|x|→∞ |x| · |V (x)| = 0, then H = − + V has no positive eigenvalues. We consider n = 3 for simplicity. The main observation (we are indebt to Prof. Xiaohua Yao for pointing out the following construction) is that one has −(

e−|x| 2 d e−r d2 e−r ) = (− 2 − )( )=− , r = |x| > 2, |x| dr r dr r r

(3.16)

which implies (−)2k (

e−|x| e−|x| )= , |x| > 2, k = 1, 2, . . . |x| |x|

Choose 0 ≤ χ(r) ∈ C ∞ (R) such that χ(r) = 1, if r ≥ 2, and χ(r) = 0, if r ≤ 1. Now set

(3.17)

S. Huang, Q. Zheng / J. Differential Equations 267 (2019) 4609–4625

ϕ(x) =

4621

e−|x| · χ(|x|) + 1 − χ(|x|), x ∈ R3 . |x|

(3.18)

Thus we have ϕ ∈ C ∞ (R3 ) and ϕ(x) decays exponentially as |x| → ∞. Moreover, one can check −r directly that e r · χ(r) + 1 − χ(r) is strictly positive for all r ∈ R, hence ϕ(x) > 0, x ∈ R3 .

(3.19)

We define the potential V as the following V (x) =

(−)2k ϕ(x) − ϕ(x) . ϕ(x)

(3.20)

Then it follows from (3.17), (3.18) that V (x) = 0, for |x| > 2. By the definition of ϕ in (3.18) we also obtain V ∈ C ∞ and V is real valued. Moreover ϕ ∈ L2 (R3 ) satisfies ((−)2k + V )ϕ = ϕ,

(3.21)

i.e., ϕ is an eigenfunction of (−)2k + V with eigenvalue 1. In the following, we consider perturbations of P (D) = (−)m . Since our strategy, similar to the methods developed in [2,11], is to use Fredholm theory to prove the uniform boundedness of the operator (I + ((−)m − (λ + i))−1 V )−1 on Lp for the endpoint case p = 2(n+1) n−1 . Therefore we need the following n+1

2 . Consider the map A(z) = Lemma 3.4. Let n ≥ 3 and 2 ≤ 2m < n(n−1) 2(n+1) . Assume V ∈ L m −1 ((−) − z) V , and let C+ be the closed upper half plane C. Then

(i) A(z) is a compact operator on L

2(n+1) n−1

(Rn ) for any z ∈ C+ \ {0};

(ii) A(z) is continuous from C+ \ {0} to the space of bounded operators on L Proof. Since V ∈ L

2(n+1) n−1

(Rn ).

n+1 2 2(n+1) n−1

, using Hölder’s inequality and applying Theorem 1.1, we obtain that

A(λ) is bounded on L (Rn ) for any λ ∈ C+ \ {0}. To prove that it’s a compact operator, we follow the methods in [11], where m = 1, n = 3 is treated. It suffices to restrict the potential V to the following case V ∈ L∞ , and supp V ⊂ {x; |x| ≤ R}. Indeed, take Vk = χ(|x| ≤ k)χ(|V | ≤ k)V , then Vk − V  that Ak (λ) − A(λ)

L

2(n+1) 2(n+1) n−1 −L n−1

L

n+1 2

→ 0 as k → ∞, which indicates

→ 0 as k → ∞, where Ak (λ) = (λ − (−)m )−1 Vk . Hence

if each Ak (λ) is shown to be compact, then A(λ) is also compact. 2(n+1)

n−1 Now suppose {fj }∞ converges weakly to zero, then it’s easy to see that j =1 ⊂ L supj A(λ)fj  2(n+1) < ∞. Notice that

L

n−1

((−)m + 1)A(λ) = V + (λ + 1)A(λ),

(3.22)

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S. Huang, Q. Zheng / J. Differential Equations 267 (2019) 4609–4625 2(n+1)

then A(λ) is bounded from L n−1 to the Sobolev space W 2m, 2(n+1)/(n−1) . Hence by Rellich compactness theorem, for any fixed R0 > 0, one can choose a subsequence fj k such that χ(|x| ≤ R0 )A(λ)fj k 

L

2(n+1) n−1

→ 0.

(3.23)

On the other hand, in view of the expression of the Green function of (− − z)−1 (see e.g. [19, p. 338]) and the identity ((−)m − z)−1 f =

m−1 1 zk (− − zk )−1 f, mz

f ∈ C0∞ (Rn ),

(3.24)

k=0

1

where zk = z m ei estimates

2kπ m

(k = 0, 1, . . . m − 1) are the k-th root of z. We have the following point-wise  m −1

|(λ − (−) )

(x, y)| ≤

C|x − y|2m−n , |x − y| ≤ 1, n−1 C|x − y|− 2 , |x − y| ≥ 1,

(3.25)

which implies that n+3

|A(λ)fj (x)| ≤ C · R n− 2(n+1) V L∞ |x|−

Then for any  > 0, choose R0 > R

n(n+3) n−1

2(n+1)

V Ln−1 − ∞

n−1 2

fj 

2(n+1) n−1

χ(|x| ≥ R0 )A(λ)fj k 

L

L

2(n+1) n−1

, |x| > 2R.

(3.26)

, we obtain

2(n+1) n−1

≤ .

(3.27)

Now combining (3.23) and (3.27) and passing to the diagonal subsequence shows that A(λ) is 2(n+1)

compact on L n−1 . To prove (ii), in view of the identity (3.24), it’s enough to prove the case m = 1. And this is contained in [15, Theorem 1.2] (see also [11, Lemma 10]), thus we omit the details. 2 Having established the two lemmas, we are in a position to prove the main result of this Section. Theorem 3.5. Let n ≥ 3 and H = (−)m + V . Assume 2 ≤ 2m < n+1 2

n(n−1) 2(n+1)

and 0 ≤ V ∈ L

P (D) = (−)m ).

L . Furthermore, if V satisfies the assumption of Lemma 3.2 (with any fixed λ0 > 0, there exists a constant C(λ0 , V ) depending on λ0 , V such that dEH (λ)

n

2(n+1) 2(n+1) L n+3 −L n−1

≤ C(λ0 , V )λ m(n+1) −1 , λ ≥ λ0 .

n+1 3

∩

Then for

(3.28)

Proof. In view of the Stone’s formula (3.1), it suffices to prove the following Lp version of the limiting absorption principle for H = (−)m + V :

S. Huang, Q. Zheng / J. Differential Equations 267 (2019) 4609–4625

sup (H − (λ + i))−1  0<<1

4623

n

2(n+1) 2(n+1) L n+3 −L n−1

≤ C(λ0 , V )λ m(n+1) −1 , λ ≥ λ0 ,

(3.29)

where λ0 > 0 is any but fixed number. The proof is a perturbative approach, which is based on Theorem 1.1 and the resolvent identity (3.2). More specifically, the estimate (3.29) follows if one can prove that for all λ > 0, 0 <  < 1, (I + R0 (λ + i)V )−1 exists as a bounded operator on L

2(n+1) n−1

and for any λ0 > 0, there is some constant C(λ0 , V ) depending on λ0 , V such that sup (I + R0 (λ + i)V )−1  0<<1

L

2(n+1) 2(n+1) n−1 −L n−1

≤ C(λ0 , V ), λ ≥ λ0 ,

(3.30)

where R0 (λ + i) = ((−)m − λ − i)−1 . Now we divide it into the following two steps. Step 1: Existence of (I + R0 (λ + i)V )−1 on L

2(n+1) n−1

. We have seen from (i) of Lemma 3.4 that

2(n+1) n−1

A(z) = ((−)m − z)−1 V

is compact on L for any z ∈ C+ \ {0}. Hence the boundedness of (I + R0 (λ + i)V )−1 follows if one can prove the following (I + R0 (λ + i)V )f = 0, f ∈ L

2(n+1) n−1

=⇒ f ≡ 0.

(3.31)

To this end, we set g = Vf , then we have f = −R0 (λ + i)g. Since V is real valued and V ∈ L (f, g) makes sense. Moreover,

n+1 2

(3.32)

, Holder’s inequality gives g ∈ L 

0 = Im(f, Vf ) = − Im(R0 (λ + i)g, g) = Rn

(|ξ |2m

2(n+1) n+3

, thus the inner product

 · |g(ξ ˆ )|2 dξ. − λ)2 +  2

(3.33)

If  > 0, then from (3.33), we have gˆ = 0 hence f = g = 0. In the limiting case  = 0, notice that  Im(R0 (λ + i0)g, g) = |g(ξ ˆ )|2 dσSλ = 0, (3.34) Sλ

where Sλ = {ξ ∈ Rn ; |ξ | = λ1/2m }. Meanwhile, our assumption V ∈ L 2(n+1) n+5

n+1 3

implies that g ∈

L . Then by (3.32), (3.34) and a recent result of Goldberg [10, Theorem 2], we obtain that f ∈ L2 (Rn ). Therefore, in view of the equation in (3.31), f is an eigenfunction of H with eigenvalue λ > 0. However, by Lemma 3.2, we have f ≡ 0, which proves (3.31). Step 2: Uniform boundedness of the norm (I + R0 (λ + i)V )−1  for λ ≥ λ0 . Next we shall prove the uniform boundedness (3.30). Since by Lemma 3.4, A(λ) = ((−)m − λ)−1 V is con2(n+1)

tinuous from C+ \ {0} to B(L n−1 , L that for any z, z1 ∈ C+ \ {0}, z = z1 ,

2(n+1) n−1

), the space of bounded operators on L

2(n+1) n−1

. Notice

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(I + R0 (z)V )−1 − (I + R0 (z1 )V )−1 = (I + R0 (z)V )−1 (R0 (z1 )V − R0 (z)V (I + R0 (z1 )V )−1 , then we obtain that (I + R0 (z)V )−1 is a continuous function of z for z ∈ C+ \ {0}. In particular, for any but fixed λ0 , R such that 0 < λ0 < R, we deduce sup (I + R0 (λ + i)V )−1  0<<1

L

2(n+1) 2(n+1) n−1 −L n−1

≤ C(λ0 , V ), λ0 < λ ≤ R.

On the other hand, in view of Hölder’s inequality, our assumption V ∈ Lp (with p = that

(3.35) n+1 2 )

yields

V L2(n+1)/(n−1) −L2(n+1)/(n+3) ≤ C. In addition, Theorem 1.1 gives n

R0 (z)L2(n+1)/(n+3) −L2(n+1)/(n−1) ≤ C|z| m(n+1) −1 . Note that

n m(n+1)

− 1 < 0, we can choose the constant R in (3.35) large enough such that

R0 (z)V L2(n+1)/(n−1) −L2(n+1)/(n−1) < 12 provided z ∈ C+ with |z| ≥ R. Then an application of the Neumann series expansion implies sup (I + R0 (z)V )−1 

|z|≥R

L

2(n+1) 2(n+1) n−1 −L n−1

≤ 2.

(3.36)

Therefore the desired estimate (3.30) follows from (3.35) and (3.36), and the proof of the Theorem is complete. 2 Remark 3.6. In the case m = 2, Sikora, Yan and Yao [23, Proposition 6.7] considered similar problems for bi-harmonic operators with rough potentials. More precisely, suppose that H = 2 +V on R3 with a real-valued 0 ≤ V ∈ L1 (R3 ) ∩L2 (R3 ), they proved Lp −Lp (1 ≤ p ≤ 4/3) estimates for spectral measure dEH (λ) for all λ ≥ λ0 for some λ0 > 0. Furthermore, they are able to establish spectral multipliers results for such H [23, Theorem 6.8] by applying the above spectral measure estimates. Acknowledgments We are grateful to the anonymous referee for his/her careful reading and valuable comments that helped us to improve the manuscript. This work was supported by the National Natural Science Foundation of China (Grant No. 11801188) and the Fundamental Research Funds for the Central Universities (Grant No. 2018KFYYXJJ041). References [1] S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Sc. Norm. Super. Pisa 2 (1975) 151–218. [2] S. Agmon, L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math. 30 (1976) 1–38. [3] J.G. Bak, A. Seeger, Extensions of the Stein-Tomas theorem, Math. Res. Lett. 18 (04) (2011) 767–781.

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