Introduction In this article, we study the maximal estimates of solutions for the fractional Schr¨odinger equation with spatial variable coefficient, i∂tv(r, t

Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 139, pp. 1–13.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

MAXIMAL ESTIMATES FOR FRACTIONAL SCHR ¨ODINGER EQUATIONS WITH SPATIAL VARIABLE COEFFICIENT

BO-WEN ZHENG

Communicated by Jerome A. Goldstein

Abstract. Letv(r, t) =Ttv0(r) be the solution to a fractional Schr¨odinger equation where the coefficient of Laplacian depends on the spatial variable.

We prove some weightedL^qestimates for the maximal operator generated by T_twith initial data in a Sobolev-type space.

1. Introduction

In this article, we study the maximal estimates of solutions for the fractional Schr¨odinger equation with spatial variable coefficient,

i∂_tv(r, t) + [−r^p0(∂_rr+p₁ r∂_r−p₂

r²)]^α/2v(r, t) = 0, (r, t)∈R⁺×R, α∈R⁺,

v(r,0) =v0(r), r∈R⁺,

(1.1)

wherevis a complex-valued function,r=|x|, (x∈Rⁿ) is the radius, and the array (p₀, p₁, p₂) satisfies the assumptions

p0<2, p1>1, p2= (2−p0

2 µ)²−(p1−1

2 )², µ≥0. (1.2) The difficulty in this equation comes from the spatial variable coefficient term r^p0 in front of the Laplacian operator. Such ar^p0-factor arises in the problem of the integrability of the inhomogeneous spherically symmetric Heisenberg ferromagnetic spin system (HFSS)

S~t(r, t) =ρ(r)S~×[S~rr+n−1 r

S~r] +ρr(r)[S~×S~r], (1.3) where the spinS~ = (S^x, S^y, S^z) is constrained byS~²= 1, ρ(r) is a scalar function, r=|x|, 0< r <∞.

2010Mathematics Subject Classification. 35B65, 35Q40, 35Q55.

Key words and phrases. Schr¨odinger equation with spatial variable coefficient;

maximal estimate; Hankel-Sobolev space.

c

2018 Texas State University.

Submitted May 28, 2017. Published July 3, 2018.

1

By a known geometrical process [8, 13], the spin evolution equation (1.3) is equivalent to the following generalized nonlinear Schr¨odinger equation

ivt+ρ(vrr+n−1

r vr−n−1

r² v+ 2|v|²v) + 2ρrvr

+ [ρ_rr+n−1 r ρ_r+ 2

Z r

0

ρ_r⁰|v|²dr⁰+ 4(n−1) Z r

0

ρ

r⁰|v|²dr⁰]v= 0,

(1.4)

and the integrability of (1.3) holds for the conditionsρ(r) =₁r^−2(n−1)+₂r⁻⁽ⁿ⁻²⁾, where 1, 2 are arbitrary constants. Obviously, the factor r^p0 corresponds to the termρ(r) in the (1.4).

In the case of the non-fractional (i.e. α = 2) Schr¨odinger equation without the spatial variable coefficient (i.e. p0 = 0), the (1.1) reduces to the classical Schr¨odinger equation with(out) the inverse-square potential under the assumption of the spherical symmetry:

i∂tu(x, t)−∆u(x, t) + a

|x|²u(x, t) = 0, (x, t)∈Rⁿ×R, u(x,0) =f(x), x∈Rⁿ.

(1.5) As we know, whena= 0, there is a large body of literature studying values ofsfor which the estimates

kS^∗fk_Lq(wdx)≤Ckfk_Hs(Rⁿ), (S^∗f)(x) := sup

t∈R

|e^it∆f(x)| (1.6) holds for someqand weightw(x). This has implications for the existence almost ev- erywhere of lim_t→0u(x, t) for its solutionu(x, t) =e^it∆f(x), which can be formally expressed as

e^it∆f(x) = Z

Rⁿ

e^{i(x·ξ−t|ξ|2)}(Ff)(ξ)dξ, (1.7) whereF is the usual spatial Fourier transform defined byFf =R

Rⁿe^−ix·ξf(x)dx.

The maximal estimate (1.6) and related questions were raised by Carleson [4]

who proved convergence fors≥¹₄ whenn= 1. Dahlberg and Kenig [7] showed that this result is sharp. In higher dimension, the question of identifying the optimal exponent s has been studied by several authors and our state of knowledge may be summarized as follows. Forn= 2, the strongest result to date appears in [10]

for s > 3/8. For n ≥ 2, the convergence is shown to hold for s > ²ⁿ⁻¹_4n (see [1, 2]). More generally, it should also be observed that the maximal estimates (1.6) developed for (1.5) witha= 0 can be extended to the case of fractional Schr¨odinger equation without the spatial variable coefficient (i.e. α >0, p0= 0). Some positive partial results were obtained by Sj¨olin [14], Heinig-Wang [9], Cho-Lee-Shim [5, 6]

and Bourgain [1].

In the case whenp06= 0 andα >0, equation (1.1) can be viewed as the general fractional Schr¨odinger equation with spatial variable coefficient proposed by authors in [19], which is a simplified version of (1.4). Inspired by the results of the papers [18, 19] and equation (1.5), we try to explore the maximal estimate for the more general equation (1.1), which seems that there is no previous literature on it. In this paper, we try to derive some maximal estimates of solution to the general equation (1.1).

Letv(r, t) =Ttv0(r) be the solution to (1.1), we define the maximal operatorT^∗ as

(T^∗v0)(r) = sup

t∈R

|Ttv0(r)|. (1.8)

Our aim is to investigate the mapping properties ofT^∗, which are from a Sobolev- type spaceX to a weightedL^q space. The estimates are of the form

kT^∗v0k_L^q_ω,%(R+)≤Ckv0kX, X =W^s,2(R⁺) or ˙H_rad^s (Rⁿ), (1.9) whereW^s,2(R⁺) is the inhomogeneous Hankel-Sobolev space in Definition 1.2 and H˙_rad^s (Rⁿ) is the usual homogeneous Sobolev space

H˙_rad^s (Rⁿ) ={f is radial, kfk²_H˙s

rad(Rⁿ)= Z

Rⁿ

|ξ|^2s|(Ff)(ξ)|²dξ <∞}. (1.10) We also note that the normkFk_L^q_ω,%(R+)is abbreviated by

kFk_L^q_ω,%(R+):=Z

R⁺

|F(r)|^q%(r)dω_r1/q

, (1.11)

where dωr = r^p1−p0dr is the Lebesgue measure. For simplicity, kFk_L^q_ω(R+) :=

kFk_L^q

ω,1(R⁺)andkFk_Lq(R⁺):= (R

R⁺|F(r)|^qdr)^1/q.

For (1.1), the presence of the factorr^p0 makes it difficult to give the expression of the solution by using the usual Fourier transform, which is only a well-suited tool to analyze constant coefficient Schr¨odinger equation such as (1.5). Inspired by [18, 19], we introduce a suitable Hankel transform.

Definition 1.1. Supposef(r) is an integrable function inR⁺, we define the Hankel transform

(Hµf)(λ) = Z

R⁺

(λr)^1−p21Jµ( 2

2−p₀(λr)^2−p20)f(r)dωr, (1.12) whereJµ(z) is the first Bessel function of orderµdefined as

Jµ(z) = (z/2)^µ Γ(µ+¹₂)π^1/2

Z 1

−1

e^izy(1−y²)^µ−12dy.

We define the fractional power of the second-order operator Aµ :=−r^p0(∂rr+

p₁

r∂r−^p_r²2) in (1.1) by

A^α/2_µ g(r) =Hµ[λ^2−p20α(Hµg)(λ)](r). (1.13) It should be noticed that the definition of A^α/2µ can be referred in [12, 18] and makes sense.

For our purpose, we also introduce the Hankel-Sobolev space via the Hankel transform.

Definition 1.2. The homogeneousHankel-Sobolev spaceW˙^s,2(R⁺) consists of tem- pered distributions f for which Hµ[λ^2−p20s(Hµf)(λ)](r) exists and is in L²_ω(R⁺) function. That is,

W˙^s,2(R⁺) =

f ∈ S⁰(R⁺),kfk²_W˙s,2(

R⁺)= Z

R⁺

|λ^2−p20s(Hµf)(λ)|²dω_λ<∞ . We also define theinhomogeneous Hankel-Sobolev space W^s,2(R⁺) as

W^s,2(R⁺) =

f ∈ S⁰(R⁺),kfk²_Ws,2(R⁺)= Z

R⁺

(1 +λ^2−p0)^s|(Hµf)(λ)|²dωλ<∞ .

Note that the space ˙H_rad^s (Rⁿ) is the special case of ˙W^s,2(R⁺) when (p0, p1, p2) = (0, n−1,0). Our first result is to derive an weightedL² estimate for the maximal functionT^∗v0, which is stated as follows.

Theorem 1.3. Suppose p2 = 0. Letb ∈(^2−p₂⁰,2−p0), 2 ≤n < ^2|p_2−p^1−1|

0 + 2 and s∈(1/2,1). Then

kT^∗v0kL²_ω,%(R⁺)≤C(p0, p1, α)kv0kH˙^s_rad(Rⁿ), (1.14) where%(r) = (1 +r)^−b.

As a consequence, we obtain the almost convergence result forv0∈H˙_rad^s (Rⁿ).

Corollary 1.4. Let v0∈H˙_rad^s (Rⁿ)with s∈(¹₂,1) and2≤n < ^2|p_2−p^1−1|

0 + 2. Then

t→0limv(r, t) =v₀(r), a.e. r∈R⁺.

If the initial datav₀ lies in the spaceW^s,2(R⁺), we improve the integrability of the maximal functionT^∗v₀ for (1.1).

Theorem 1.5. For0< α6= 1. If the initial data v0 ∈ W^s,2(R⁺)with s∈[¹₄,¹₂), Then the estimates

kT^∗v0k_L^q_ω(R+)≤C(p0, p1)kv0k_Ws,2(R⁺), (1.15) kT^∗v₀k_L^q_ω,%(R+)≤C(p₀, p₁)kv₀k_Ws,2(R⁺), (1.16) hold for

8(p1−p0+ 1)

4p₁−3p₀+ 2 ≤q < 2(p1−p0+ 1)

p₁−p₀+ 1−(2−p₀)s and q= 2(p1−p0+ 1) p₁−p₀+ 1−(2−p₀)s respectively, where%(r) =r^b(1 +r)^−b, b >0.

The plan of this paper is as follows: Section 2 is devoted to the preliminaries, including the properties of Bessel function and the relation between ˙W^s,2(R⁺) and H˙_rad^s (Rⁿ). In Section 3, through delicate computation, we give the complete ar- gument about the weighted L^q maximal estimates of the (1.1). If not specified, throughout this paper, the notationsM N andM ∼N denoteM ≤C⁻¹N and CM ≤N ≤CM˜ respectively for some large constants C and ˜C. We also denote

≤_β as ≤C(β), whereC(β) denotes various constant that only depends onβ. We abbreviate by writingA+as A+ orA−as A−for 0< 1.

2. Preliminaries

In this section, we collect some basic facts which will be used in the later context.

We recall some asymptotic properties of the first Bessel function Jµ(z) (see [17]).

For fixedµ, ifz1, a simple calculation gives the rough estimate

|J_µ(z)| ≤ Cz^µ

2^µΓ(µ+¹₂)Γ(1/2)(1 + 1

µ+ 1/2), (2.1)

where C is a absolute constant. Another well known asymptotic expansion about the Bessel function is

Jµ(z) =z^−1/2(b₊e^iz+b₋e^−iz) + Φ_µ(z), z1, (2.2)

where |Φµ(z)| ≤ Cz⁻¹, |b±| ≤ C and the constant C is independent of µ. As pointed out in [16], if one seeks a uniform bound for largeµand z, then the best one can do is

|J_µ(z)| ≤Cz^−1/3, z≥1. (2.3)

A simple consequence of the above properties is the following Lemma.

Lemma 2.1. ForR 1, there exists a constant C(p₀) independent ofµ, R such that

Z 2R

R

|Jµ(r^2−p20)|²dr≤C(p₀)R^p0/2.

Next we review some properties of the Hankel transform, which appear in [3, 19].

Lemma 2.2. The Hankel transform Hµ satisfies:

(i) Hµ=H⁻¹_µ ,

(ii) Hµ is anL² isometry, i.e. kHµφkL²_ω(R⁺)=kφkL²_ω(R⁺),

(iii) Hµ(Aµφ)(λ) =λ^2−p0(Hµφ)(λ), where the operator H_µ⁻¹ is the inverse op- erator ofHµ.

For the Hankel-Sobolev space ˙W^σ,2(R⁺), there exists the following embedding theorem with ˙H_rad^σ (Rⁿ), which is proved in the paper [18].

Lemma 2.3. Let n≥2 andµ >ⁿ⁻²₂ . Iff ∈H˙_rad^σ (Rⁿ), 0≤σ < ⁿ₂, then

kfkW˙^σ,2(R⁺)≤C(σ, µ, n)kfkH˙_rad^σ (Rⁿ). (2.4) Proof. We give only an outline of the proof. From the definition of Hankel trans- form, (1.13) and using the integral formula of Bessel function [17, p. 385], we obtain

M[A^σ/2_µ f](z)

= (2−p0)^σ

Γ(^2z−p_2(2−p¹⁺¹

0) +^µ₂) Γ(1−^2z−p_2(2−p¹⁺¹

0) +^µ₂)

Γ(1−^2z−p_2(2−p¹⁺¹

0) +^σ+µ₂ ) Γ(^2z−p_2(2−p¹⁺¹

0) −^σ−µ₂ ) M[f](z−2−p0

2 σ) (2.5) whereM[f(r)](z) =R

R⁺r^z−1f(r)dris the Mellin transform.

Denote B_µ,w^σ := A^σ/2µ A^−σ/2w . Writing ˜z = _2−p^2z

0 and ˜κ = ^p_2−p¹⁻¹

0, by (2.5), we obtain

M[B_µ,w^σ f](z) =Γ((˜z−κ˜+µ)/2)Γ(1−(˜z−σ−˜κ−µ)/2) Γ(1−(˜z−˜κ−µ)/2)Γ((˜z−σ−˜κ+µ)/2)

×Γ((˜z−σ−˜κ+w)/2)Γ(1−(˜z−˜κ−w)/2)

Γ(1−(˜z−σ−˜κ−w)/2)Γ((˜z−˜κ+w)/2)M[f](z) :=F(z)M[f](z).

Forz= ^p1−p₂⁰⁺¹+iy and ˜z= ˜κ+ 1 +_2−p²

0iy, using the following properties of Gamma function Γ(z):

Γ(z) = Γ(¯z), ∀z∈C,

|Γ(x+iy)|= Γ(x)

∞

Y

k=0

(1 + y²

(x+k)²)^−1/2, ∀x >0, ∀y∈R,

we obtain

|F(p₁−p₀+ 1

2 +iy)|=|Γ((µ+σ+ 1−_2−p²

0iy)/2) Γ((µ−σ+ 1 + _2−p²

0iy)/2)

Γ((w−σ+ 1 +_2−p²

0iy)/2) Γ((w+σ+ 1−_2−p²

0iy)/2)|

=|Γ((µ+σ+ 1)/2) Γ((µ−σ+ 1)/2)

Γ((w−σ+ 1)/2) Γ((w+σ+ 1)/2)|

∞

Y

0

[R_k(˜y)]^1/2, where

R_k(˜y) =(1 + ˜y²/(µ−σ+ 1 + 2k)²)(1 + ˜y²/(w+σ+ 1 + 2k)²) (1 + ˜y²/(µ+σ+ 1 + 2k)²)(1 + ˜y²/(w−σ+ 1 + 2k)²)

=(1 + ˜y²/(Mk−σ)²)(1 + ˜y²/(Nk+σ)²) (1 + ˜y²/(Mk+σ)²)(1 + ˜y²/(Nk−σ)²)≤1.

and ˜y= _2−p^2y

0,M_k =µ+ 1 + 2k, N_k =w+ 1 + 2k. Therefore, forn >2σ, we have sup

y

|F(p₁−p₀+ 1

2 +iy)|<∞.

Hence, using [18, Lemma 2.5], we obtain kB_µ,κ^σ fk_L2

ω(R⁺)≤Ckfk_L2 ω(R⁺),

which is the desired result.

At the end of this section, we show the oscillatory integral estimate [6, 15].

Lemma 2.4. Suppose ϕ∈C²(Rⁿ\{0}) is a radial function such that |ϕ^(k)(ξ)| ∼

|ξ|^a−k, k = 0,1,2 for 0 < a 6= 1. Let A, B, σ be the real numbers such that A, B 6= 0, σ ∈[1/2,1), then there exists a constant C(a, σ), independent of A, B, such that

Z

R

ei(Aϕ(ξ)+Bξ)|ξ|^−σdξ

≤C|B|^−(1−σ). (2.6)

3. Proof of main results

Applying the Hankel transform (1.12) to the (1.1), by Definition 1.1, (1.13) and Lemma 2.2 (i), we have

i∂t˜v+λ^2−p20αv˜= 0

˜

v(λ,0) = ˜v0(λ), where

˜v(λ, t) = (H_µv)(λ, t), ˜v₀(λ) = (H_µv₀)(λ).

Solving the ODE and inverting the Hankel transform, we obtain the formal solution Ttv0(r) =

Z

R⁺

(λr)^1−p21Jµ( 2 2−p0

(λr)^2−p20)e^itλ

2−p0 2 α

˜

v0(λ)dωλ. (3.1)

Proof of Theorem 1.3. The key ingredients are the asymptotic behavior of the Bessel function, and the properties of Hankel transform.

By the continuity of the embedding ˙H¹²⁻(R)∩H˙ ¹²⁺(R),→L^∞(R), it suffices to prove

Proposition 3.1. Let p₂ = 0. For b ∈(^2−p₂⁰,2−p₀), and a∈ [¹₂−_(2−p^b

0)α,¹₂+

n

2α−_(2−p^b

0)α), there exists a constant C independent of v0 such that Z

R

Z

R⁺

|∂_t^a(Ttv0(r))|² dωrdt

(1 +r)^b ≤C(p0, p1, a, α)kv0k²_W˙σ,2(

R⁺), whereσ:= ^(2a−1)α₂ +_2−p^b

0.

This proposition, NS Lemma 2.3, yield Theorem 1.3. Indeed, under the assump- tion ofa, babove, we haveσ∈[0,ⁿ₂) and ⁿ⁻²₂ < ^|p_2−p^1−1|

0 , then Z

R

Z

R⁺

|∂_t^a(Ttv0(r))|² dω_rdt

(1 +r)^b ≤C(p0, p1, a, α)kv0k²_H˙σ

rad(Rⁿ). (3.2) Choosinga= ¹₂+ anda= ¹₂−in (3.2), we obtain

Z

R⁺

|T^∗v0(r)|² dωr

(1 +r)^b ≤C(p0, p1, α)kv0k²

H˙

2−pb0 rad (Rⁿ)

. Hence, Theorem 1.3 is proved.

Proof of Proposition 3.1. Using the Plancherel theorem of the usual Fourier trans- form with respect to timet, we have

Z

R

Z

R⁺

|∂_t^a(Ttv0)|² dωrdt (1 +r)^b =

Z

R⁺

Z

R

|τ^a Z

R

e^−itτ(Ttv0)(r, t)dt|² dτ dωr

(1 +r)^b. (3.3) Using (3.1), we obtain

left-hand side of (3.3)

= Z

R⁺

Z

R

τ^a

Z

R

Z

R⁺

(λr)^1−p21Jµ( 2 2−p0

(λr)^2−p20)e^it(λ

2−p0

2 α−τ)˜v0(λ)dωλdt

2

× dτ dωr

(1 +r)^b

≤ Z

R⁺

Z

R

τ^a

Z

R⁺

(λr)^1−p21J_µ( 2

2−p₀(λr)^2−p20)δ(λ^2−p20α−τ)˜v₀(λ)dω_λ

2

dτ dωr

(1 +r)^b

≤_p0,α Z

R⁺

Z

R

τ^ar^1−p21

Z

R⁺

λ

p1−2p0 +3 (2−p0 )α −1

J_µ(2r^2−p20 2−p0

λ^α1)˜v₀(λ^{(2−p20 )α})δ(λ−τ)dλ

2

dτ dω_r (1 +r)^b

≤p0,α

Z

R⁺

Z

R

r

1−p1 2 λ

p1−2p0 +3 (2−p0 )α −1+a

Jµ(2r^2−p20 2−p0

λ^α1)˜v0(λ^{(2−p20 )α})

2 dλdω_r (1 +r)^b, where the above inequality can be rewritten as

C(p₀, α) Z Z

R⁺

r^1−p21λ^{p1−2p20 +2+(2a−1)(2−p40 )α}Jµ( 2 2−p0

(λr)^2−p20)˜v₀(λ)

2 dλdω_r (1 +r)^b.

We make a dyadic decomposition by choosingχ, which is a smoothing function supported in [¹₂,2], and change the variable _M^λ 7→λ, M r7→r,

left-hand side of (3.3)

≤p₀,α

X

M∈2^Z

Z Z

R⁺

λ^{d+˜ 2p1−2p20 +1}(λr)^1−p21Jµ( 2 2−p0

(λr)^2−p20)˜v0(λ)χ( λ M)

2 dωrdλ (1 +r)^b

≤a,α,p₀

X

M∈2^Z

M^{2 ˜d+p1−p0+1} Z Z

R⁺

(λr)^1−p21Jµ( 2

2−p₀(λr)^2−p20)˜v0(M λ)χ(λ)

2

× dωrdωλ

(1 +_M^r)^b

≤a,α,p₀

X

M∈2^Z

X

R∈2^Z

M^{2 ˜d+p1−p0+1}R^p1−p0K^M_R,

(3.4) where ˜d:= (2a−1)^(2−p₄^0)α. By integrating,

K_R^M :=

Z

R⁺

Z 2R

R

|(λr)^1−p21J_µ( 2 2−p0

(λr)^2−p20)˜v₀(M λ)χ(λ)|² drdω_λ (1 +_M^r)^b. Now we need to prove the bound ofK^M_R, which is divided in two cases:

Case 1: R1. Sinceλ∼1, we have (rλ)^2−p20 1. Using the property of Bessel function (2.1), we have

K^M_R ≤p0,µ

Z

R⁺

Z 2R

R

|(λr)^{1−p1 +(22−p0 )µ}v˜₀(M λ)χ(λ)|² drdω_λ (1 +_M^r)^b

≤_p0,µR^{2−p1+(2−p0)µ}min{1,(M R)^b}

Z

R⁺

|˜v₀(M λ)χ(λ)|²dω_λ.

(3.5)

Case 2: R1. Sinceλ∼1, we obtain (rλ)^2−p20 1. ThenK^M_R can be bounded by

C(p1)R^1−p1 Z

R⁺

|˜v0(M λ)χ(λ)|²( Z 2R

R

|Jµ( 2 2−p0

(λr)^2−p20)|² dr

(1 +_M^r)^b)dωλ. Noticing thatλ∼1, so

Z 2R

R

|Jµ( 2 2−p0

(λr)^2−p20)|² dr (1 +_M^r)^b

≤min{1,(M R)^b}

Z 2R

R

|Jµ( 2 2−p0

(λr)^2−p20)|²dr

≤p0min{1,(M

R)^b}R^p0/2,

where the last inequality is obtained by Lemma 2.1. Then K^M_R ≤p₀,p₁R^{p0−2p21 +2}min{1,(M

R)^b} Z

R⁺

|˜v0(M λ)χ(λ)|²dωλ. (3.6) Hence, combining (3.5) and (3.6), forp2= 0, we obtain the estimate

K^M_R ≤p₀,p₁

(R^{2−p1+|p1−1|}M^p0−p1−1min{1,(^M_R)^b}k˜v0(λ)χ(_M^λ)k²_L2

ω(R⁺), R1, R^{p0−2p21 +2}M^p0−p1−1min{1,(^M_R)^b}k˜v₀(λ)χ(_M^λ)k²_L2

ω(R⁺), R1.

Now, returning to (3.4), we have left-hand side of (3.3)

≤a,α,p₀,p₁

X

M∈2^Z

X

R∈2^Z:R1

M^{2 ¯d}R^{2−p0+|p1−1|}min{1,(M

R)^b}k˜v0(λ)χ( λ M)k²_L2

ω(R⁺)

+ X

M∈2^Z

X

R∈2^Z:R1

M^{2 ¯d}R^2−p20 min{1,(M

R)^b}k˜v0(λ)χ( λ M)k²_L2

ω(R⁺)

≤_a,α,p0,p1 X

M∈2^Z

M^{2 ¯d+b}[X

R1

R^{|p1−1|+2−p0−b}+X

R1

R^{2(1−b)−p2 0}]k˜v₀(λ)χ(λ M)k²_L2

ω(R⁺). From the assumption b ∈ (^2−p₂⁰,2−p₀), the above inequality can be further controlled by

X

M∈2^Z

M^{(2a−1)(2−p20 )α+b}k˜v0(λ)χ( λ M)k²_L2

ω(R⁺). Finally, by Lemma 2.2 (ii) and lettingM = 2^j, we have left-hand side of (3.3)≤a,α,p₀,p₁

X

j∈Z

2^j(2−p0)σkHµ[χ(λ

2^j)Hµv0]k²_L2 ω(R⁺)

≤a,α,p₀,p₁k X

j∈Z

|2^{(2−p20 )σj}Hµ[χ(λ

2^j)Hµv0]|^21/2 k²_L2

ω(R⁺), whereσ:= ^(2a−1)α₂ +_2−p^b

0 and Proposition 3.1 follows from Lemma 3.2 below.

Lemma 3.2 ([18]). Let ς ∈R. Then there exists a constant C that depends only onς andβ such that for all f ∈W˙ ^2−p2ς0,2(R⁺), we have

k X

j∈Z

|2^jςHµ[βj(λ)Hµf]|^21/2 k_L2

ω(R⁺)≤Ckfk

W˙ ^2−p2ς0,2, (3.7) where the functionβ ∈C₀^∞(R⁺)is supported in the interval[1/2,2],βj(λ) =β(₂^λj), P

j∈Zβj= 1.

Proof of Theorem 1.5. We consider the solution (3.1):

Ttv0(r) = Z

R⁺

(λr)^1−p21Jµ( 2 2−p0

(λr)^2−p20)e^itλ

2−p0 2 α

˜

v0(λ)dωλ. By changing the variable _2−p²

0λ^2−p20 7→λ, r^2−p20 7→r, the solution becomes 2−p0

2

^p1_2−p^{−p0 +1}

0

Z

R⁺

λ

p1−p0 +1 2−p0 r

1−p1

2−p0Jµ(λr)e^{it(2−p20λ)α}˜v0 (2−p0

2 λ)^2−p20 dλ :=T[˜v0](r),

To prove (1.15) in Theorem 1.5, it suffices to prove that forq = _p ^2(p1−p0+1)

1−p0+1−(2−p0)s⁰

ands⁰∈[¹₄, s), the normkT[˜v0](r)r

2p1−p0

(2−p0 )qk_LqL^∞_t (R⁺×R)is bounded by Z

R⁺

|˜v₀((2−p0

2 λ)^2−p20)[1 + (2−p0

2 λ)²]^s/2λ

2p1−p0

2(2−p0 )|²dλ1/2

, where the normkh(r, t)kL^qL^∞_t (R⁺×R):= sup_t∈Rkh(·, t)kL^q(R⁺).

We write g(λ) = ˜v0((^2−p₂⁰λ)^2−p20)[1 + (^2−p₂⁰λ)²]^s2λ

2p1−p0

2(2−p0 ). Then the above esti- mate is equivalent to

kP(g)|k_LqL^∞_t (R⁺×R)≤q,p₀,p₁kgk_L2(R⁺), where

P(g)(r, t) =C(p0, p1)r

2p1−p0 (2−p0 )q+^1−p_2−p¹

0

Z

R⁺

Jµ(λr)e^{it(2−p20λ)α}g(λ)

×[1 + (2−p0

2 λ)²]^−s/2λ^1/2dλ.

Therefore, utilizing the dual argument, our main task is to prove that

kP^∗(f)k_L2(R⁺)≤q,p₀,p₁ kfk_LpL¹_t(R⁺×R), (3.8) wherep=_p ^2(p1−p0+1)

1−p0+1+(2−p0)s⁰ and

P^∗(f)(λ) =C(p₀, p₁)[1 + (2−p0

2 λ)²]^−s/2λ^1/2

× Z

R

Z

R⁺

Jµ(λr)e^{−it(2−p20λ)α}f(r, t)r

2p1−p0 (2−p0 )q+^1−p_2−p¹

0 dr dt.

Now we decomposeP^∗(f)(λ) =P

j=0,1,2(P_j^∗f)(λ), where (P_j^∗f)(λ) =C(p0, p1)[1 + (2−p₀

2 λ)²]^−s/2λ^1/2

× Z

R

Z

R⁺

J_µ(λr)e^{−it(2−p20λ)α}φ_j(λr

µ)f(r, t)r^12−γdr dt,

and φ_j are smooth cut-off functions such that φ₀ = 1 on {|η| < ¹₄}, φ₀ = 0 on {|η|<1/4}, φ₁= 1 on {|η| ∼1}, φ₁ = 0 otherwise, φ₂= 0 on {|η|<2}, φ₂ = 1 on{|η|>3}, and φ0+φ1+φ2= 1. The symbolγ=^2p_2−p^1−p0

0 (¹₂−¹_q).

Non-endpoint case: s⁰ ∈[¹₄, s). (i)j= 0. Using the property of Bessel function (2.1), ands⁰∈[^2−p₂⁰, s), we obtain

|(P₀^∗f)(λ)| ≤p₀,p₁ [1 + (2−p0

2 λ)²]^−s/2

Z µ/(2λ)

0

(λr)^µ+12r^−γkf(r,·)k_L1 tdr

≤p₀,p₁,µλ^−s0

Z µ/(2λ)

0

r^−γkf(r,·)k_L1 tdr.

From the basic relation thatk(P₀^∗f)(λ)k_L2(R⁺)=k(P₀^∗f)(_λ¹)¹_λk_L2(R⁺), we have k(P₀^∗f)(λ)k_L2(R⁺)≤p₀,p₁,µkµ^1−s0

Z (µλ)/2

0

r^−γ

(^µλ₂ )^1−s0kf(r,·)k_L1

tdrk_L2(R⁺). (3.9) By extending the functionr^−γkf(r,·)k_L1

t to 0 for r≤0, the right hand side of (3.9) can be controlled by the Riesz potential operatorIβ, 0< β <1, where

I_β(g)(λ) =C_β Z

R

|λ−r|^β−1g(r)dr, λ∈R, andCβ is chosen so that (Iβ)^∧(ξ) =|ξ|^−βbg(ξ). So we further obtain

kP₀^∗fk_L2(R⁺)≤_p0,p1,µ,s⁰ kI_s⁰(r^−γkf(r,·)k_L1 t)(µλ

2 )k_L2(R⁺) (3.10)

≤p₀,p₁,µ,s⁰

Z

R

|ξ|^−2s0|(r^−γkf(r,·)k_L1

t)^∧(ξ)|²dξ1/2

(3.11)

≤p₀,p₁,µ,s⁰ kfk_LpL¹_t(R⁺×R) (3.12) where the last inequality is proved by using Pitt’s inequality for the usual Fourier transform:

Lemma 3.3([11]). Ifl≥p,0≤a <1−1/p,0≤d <1/landd=a+ 1/p+ 1/l−1, then

Z

R

|fb(ξ)|^l|ξ|^−dldξ1/l

≤C(

Z

R

|f(x)|^p|x|^apdx)^1/p, wherefb(ξ)is the usual Fourier transform.

(ii)j= 1. The estimate ofP₁^∗f is similar toP₀^∗f. When|λr| ∼µ, the property (2.3) implies|J_µ(λr)φ₁(^λr_µ)| ≤C(λr)^−1/3, we obtain

k(P₁^∗f)(1 λ)1

λk_L2(R⁺)≤p₀,p₁,µ,s⁰ k Z 2µλ

(µλ)/2

r^16−γ

(2µλ)^76−s0kf(r,·)k_L1

tdrk_L2(R⁺)

≤p₀,p₁,µ,s⁰ kIs⁰(r^−γkf(r,·)k_L1

t)(2µλ)k_L2(R⁺)

≤_p0,p1,µ,s⁰ kfk_LpL¹_t(R⁺×R).

(3.13)

(iii)j = 2. Applying the asymptotic expansion of Bessel function (2.2), we write (P₂^∗f)(λ) =C(p0, p1)b_±[1 + (2−p0

2 λ)²]^−s/2

× Z

R

Z

R⁺

e^{i[±λr−t(2−p20λ)α]}φ2(λr

µ)f(r, t)r^−γdr dt +C(p0, p1)[1 + (2−p0

2 λ)²]^−s/2 Z

R

Z

R⁺

e^{−it(2−p20λ)α}(rλ)^1/2Φµ(rλ)

×φ2(λr

µ)f(r, t)r^−γdr dt :=C(p₀, p₁)[(P_±f)(λ) + (Qf)(λ)],

(3.14) where|Φµ(rλ)| ≤C(rλ)⁻¹,|b_±| ≤C and the constantCis independent ofµ.

For the estimate of (P_±f)(λ), it is sufficient to consider (P+f)(λ). We decompose (P₊f)(λ) =S₁(λ) +S₂(λ),

where

S1(λ) =b+[1 + (2−p0

2 λ)²]^−s/2 Z

R

Z

R⁺

e^{i[λr−t(2−p20λ)α]}f(r, t)r^−γdr dt, S2(λ) =b+[1 + (2−p0

2 λ)²]^−s/2 Z

R

Z

R⁺

e^{i[λr−t(2−p20λ)α]}(φ2(λr

µ)−1)f(r, t)r^−γdr dt.

ForS2(λ), arguing asP₀^∗f, we obtain kS2(λ)k_L2(R⁺)≤

λ^−s0 Z ^3µ_λ

0

kf(r,·)k_L1

tr^−γdr _L2(

R⁺)≤ kfk_LpL¹_t(R⁺×R). (3.15) ForS1(λ), we extendS1 toRby setting

S1(ξ) =b+[1 + (2−p0

2 y)²]^−s/2 Z Z

R²

e^{i[ry−t(2−p20|y|)α]}f(r, t)r^−γdrdt, y <0,