Global maximal inequality to a class of oscillatory integrals

R E S E A R C H

Open Access

Global maximal inequality to a class of

oscillatory integrals

Ying Xue

1

_{and Yaoming Niu}

1* *_{Correspondence:}

[email protected]

1_{Faculty of Mathematics, Baotou}

Teachers’ College of Inner Mongolia University of Science and Technology, Baotou, P.R. China

Abstract

In the present paper, we give the globalLq_{estimates for maximal operators} generated by multiparameter oscillatory integralSt,Φ, which is defined by

St,Φf(x) = (2

π

)–n

Rne ix·ξ

ei(t1φ1(|ξ1|)+t2φ2(|ξ2|)+···+tnφn(|ξn|))_fˆ₍

_ξ

_)d

_ξ

_{, x}∈Rn_,

wheren≥2 andfis a Schwartz function inS(Rn),t= (t1,t2,. . .,tn),

Φ

= (

φ

1,

φ

2,. . .,

φ

n),

φ

i(i= 1, 2, 3,. . .,n) is a function onR+→R, which has a suitable growth condition. These estimates are apparently good extensions to the results of Sjölin and Soria (J. Math. Anal. Appl 411:129–143,2014) for the multiparameter fractional Schrödinger equation.

MSC: 42B15; 42B25

Keywords: Maximal operator; Local estimate; Global estimate; Multiparameter oscillatory integrals

1 Introduction and main results

Letf be a Schwartz function inS(Rn_{) and}

Stf(x) =u(x,t) = (2π)–n

Rn

eix·ξ+it|ξ|a_fˆ_{(ξ)dξ, (x,t)∈R}n_×R.

It is well known thatStf(x) is the solution of the fractional Schrödinger equation

⎧ ⎨ ⎩

i∂tu+ (–)a/2u= 0, (x,t)∈Rn×R,

u(x, 0) =f(x). (1.1)

Herefˆdenotes the Fourier transform off defined byfˆ(ξ) =_Rne–iξ·xf(x)dx. We recall the homogeneous Sobolev spaceH˙s_(Rn_{) (s∈R), which is defined by}

˙

HsRn= f ∈S:fHs=

Rn|

ξ|2sfˆ(ξ)2dξ 1/2

<∞

,

and the non-homogeneous Sobolev spaceHs_(Rn_{) (s∈R), which is defined by}

HsRn= f ∈S:fHs=

Rn

1 +|ξ|2sfˆ(ξ)2dξ 1/2

<∞

.

Maximal operatorS∗f associated with the family of operators{St}0<t<1is defined by

S∗f(x) = sup

0<t<1

Stf(x), x∈Rn.

It is well known that ifa= 2,uis the solution of the Schrödinger equation ⎧

⎨ ⎩

i∂tu–u= 0, (x,t)∈Rn×R,

u(x, 0) =f(x). (1.2)

In 1979, Carleson [4] proposed a problem: iff∈Hs_(Rn_{) for whichsdoes}

lim

t→0u(x,t) =f(x), a.e.x∈R

n_{. (1.3)}

Carleson first considered this problem for dimensionn= 1 in [4] and showed that the convergence (1.3) holds forf ∈Hs(R) withs≥ 1₄, which is sharp was shown by Dahlberg and Kenig [8]. The higher dimensional case of convergence (1.3) has been studied by sev-eral authors, see [1,2,9, 11–13,24, 25, 30, 34,35] for example. In fact, by a standard argument, forf ∈Hs_(Rn_{), the pointwise convergence (1.3) follows from the local estimate}

S∗f_Lq_(Bn₎≤CfHs_(Rn), f ∈HsRn, (1.4)

for someq≥1 ands∈R. HereBn_{is the unit ball centered at the origin inR}n_{. On the other} hand, the global estimates are of independent interest since they reveal global regularity properties of the corresponding oscillatory integrals. Next, we recall the global estimate

_S∗_f

Lq_(Rn₎≤CfHs_(Rn). (1.5)

Estimate (1.5) and related questions have been well studied in literature, see, e.g., Carbery [3], Cowling [7], Kenig and Ruiz [21], Kenig, Ponce, and Vega [20], Rogers and Villarroya [29], Rogers [28], Sjölin [30–32], and so on.

Forn≥2 and a multiindexa= (a1,a2, . . . ,an), withaj> 1 andfbeing a Schwartz function inS(Rn_{), we set}

Stf(x) = (2π)–n

Rne ix·ξ

ei(t1|ξ1|a1+t2|ξ2|a2+···+tn|ξn|an)ˆ_{f(ξ)dξ, x}∈Rn_,

wheret= (t1,t2, . . . ,tn)∈Rn. Forn≥2, the local maximal operatorM∗is defined by

M∗f(x) = sup

0<ti<1

Stf(x), x∈Rn,

and the global maximal operatorM∗∗is defined by

M∗∗f(x) =sup

ti∈R

The global estimate

M∗∗f_Lq_(Rn₎≤CfH˙s_(Rn₎ (1.6)

and

M∗f_Lq_(Rn₎≤CfHs_(Rn). (1.7)

In 2014, Sjolin and Soria [32] obtained the following results.

Theorem A([32]) Assume n≥2.Then,for every a,inequality(1.6)holds if and only if

4≤q<∞and s=n(1₂–1_q).

Theorem B([32]) Assume n≥2.Then,for every a and for2 <q< 4,inequality(1.7)holds if and only if s≥n₂–|a₄|+|a_q|–n_q.

Multiparameter singular integrals and related operators have been well studied and raised considerable attention in harmonic analysis, which can been seen in the work of Stein and Fefferman in [14–17], and so on. In the present paper, we consider the maximal estimates associated with multiparameter oscillatory integralSt,Φdefined by

St,Φf(x) = (2π)–n

Rne ix·ξ

ei(t1φ1(|ξ1|)+t2φ2(|ξ2|)+···+tnφn(|ξn|))_fˆ_{(ξ)dξ, x}∈Rn_.

Here,n≥2 andf is a Schwartz function inS(Rn),Φ= (φ1,φ2, . . . ,φn),φi(i= 1, 2, 3, . . . ,n) is a function onR+_{→R. Forn≥2, the local maximal operatorM}∗

Φis defined by

M∗Φf(x) = sup

0<ti<1

St,Φf(x), x∈Rn,

and the global maximal operatorM∗∗Φ is defined by

M∗∗Φf(x) =sup

ti∈R

St,Φf(x), x∈Rn.

The global estimates of maximal operatorsM∗ΦandMΦ∗∗are defined by

M∗∗ΦfLq_(Rn₎≤CfH˙s_(Rn₎ (1.8)

and

M∗ΦfLq_(Rn₎≤CfHs_(Rn). (1.9)

Assume thatφ:R+→Rsatisfies:

(H1) There existsm1> 1such that|φ(r)| ∼rm1–1and|φ(r)|rm1–2for all0 <r< 1;

(H2) There existsm2> 1such that|φ(r)| ∼rm2–1and|φ(r)|rm2–2for allr≥1;

(H3) Eitherφ(r) > 0orφ(r) < 0for allr> 0.

Theorem 1.1 Assume that n≥2andφi(i= 1, 2, 3, . . . ,n)satisfies(H1)–(H3).If4≤q<∞

and s=n(1₂–1_q),then the global estimate(1.8)holds.

Theorem 1.2 Let m= (m1,2,m2,2, . . . ,mn,2)and set|m|=m1,2+m2,2+· · ·+mn,2.Assume

that n≥2andφi(i= 1, 2, 3, . . . ,n)satisfies(H1)–(H3)with mi,1> 1,mi,2> 1.Then,for every

m,inequality(1.9)holds if2 <q< 4and s≥n₂ –|m₄|+|m_q|–n_q.

Remark1.1 There are many elementsφ satisfying conditions (H1)–(H3), for instance, the fractional Schrödinger equation (φ(r) =ra_{), or (φ(r) = (1 +r}2₎a_{2), (a≥1), the Beam} equation (φ(r) =√1 +r4_{), the fourth-order Schrödinger equation (φ(r) =r}2_+r4_{), iBq} (φ(r) =r√1 +r2_{), and so on (see [5,6,18,19,22,23,27], and the references therein). Hence,} Theorem1.1and Theorem1.2imply the sufficiency part of TheoremAand TheoremB, respectively. However, due to the complexity of the symbolφ, we cannot obtain the ne-cessities of the range ofqin Theorem1.1and Theorem1.2.

This paper is organized as follows. The proofs of Theorem1.1and Theorem1.2are given in Sect.2and Sect.3, respectively. To prove Theorem1.1and Theorem1.2, we next need the following important lemmas, which play a key role in proving Theorem1.1and Theorem1.2, respectively. The proof of Lemma1.4is given in Sect.4.

Lemma 1.3([26]) Assume thatφ satisfies(H1)–(H3)with m1> 1,m2> 1. 1₂≤s< 1and μ∈C₀∞(R).Then

_Reixξ+itφ(|ξ|)_|ξ|–s_μ

ξ

N

dξ≤C 1

|x|1–s

for x∈R\ {0},t∈R,and N= 1, 2, 3, . . . .Here the constant C may depend on s and m1,m2,

andμbut not on x,t,or N.

Remark1.2 The proof of Lemma1.3is similar to that of Lemma 2.1 in [10].

Lemma 1.4 Assume thatφsatisfies(H1)–(H3)with m1> 1,m2> 1.1₂≤α≤m2₂ , –1 <d< 1,andμ∈C0∞(R).Then

_Re_{(1 +}i(dφ(|ξ_ξ|2)–₎xα2ξ) μ

ξ

N

dξ≤C 1

|x|β (1.10)

for x∈R\ {0}and N= 1, 2, 3, . . . ,whereβ=α+ m2

2 –1

m2–1 .Here the constant C may depend on αand m1,m2,andμbut not on x,d,and N.

Remark1.3 Applying the result of Lemma1.3, the proof of Lemma1.4is similar to that of Lemma 2.2 in [32]. The proof of Lemma1.4will be given in Sect.4.

2 The proof of Theorem1.1

Assume thatn≥2,φi(i= 1, 2, 3, . . . ,n) satisfies (H1)–(H3). Fori= 1, 2, 3, . . . ,n, letti(x) be a measurable function onRn_witht

i(x)∈R. Denotet(x) = (t1(x),t2(x), . . . ,tn(x)), we set

St(x),Φf(x) = (2π)–n

Rne ix·ξ

x∈Rn,f ∈SRn.

For 4≤q<∞ands=n(1₂–1_q), that is,n₄≤s<n₂ andq=_n2_–2n_s. By linearizing the maximal operator (see [30]) to prove the global estimate (1.8) holds, it suffices to show that

St(x),ΦfLq_(Rn₎≤Cf_H˙s=C

Rn|

ξ|2sfˆ(ξ)2dξ 1/2

. (2.1)

To prove (2.1) it suffices to prove that

St(x),ΦfLq_(Rn₎≤C

Rn| ξ1|

2s n|_ξ

2| 2s n| · · · |_ξ

n| 2s

n_fˆ_(ξ)2_dξ 1/2

. (2.2)

Letg(ξ) =|ξ1| s

n|_ξ2|ns· · · |_ξn|nsfˆ(ξ), then we have

St(x),Φf(x) =

Rne ix·ξ

ei(t1(x)φ1(|ξ1|)+t2(x)φ2(|ξ2|)+···+tn(x)φn(|ξn|))|_ξ 1|–

s

n|_ξ2|–ns · · · |_ξn|–ns_g(ξ)dξ

=RΦg(x), (2.3)

where

RΦg(x) =

Rne ix·ξ

ei(t1(x)φ1(|ξ1|)+t2(x)φ2(|ξ2|)+···+tn(x)φn(|ξn|))_|ξ 1|–

s n|_ξ

2|– s n· · · |_ξ

n|– s n_g(ξ)dξ.

To prove (2.2) it suffices to prove that

RΦgLq_(Rn₎≤Cg_L2_(Rn₎ (2.4)

forg continuous and rapidly decreasing at infinity. We take a real-valued functionρ∈

C₀∞(Rn_{) such thatρ(x) = 1 if|x| ≤1 andρ(x) = 0 if|x| ≥2. And we choose a real-valued} functionψ∈C∞0 (R) such thatψ(x) = 1 if|x| ≤1 andψ(x) = 0 if|x| ≥2, and setσ(ξ) = ψ(ξ1)ψ(ξ2)· · ·ψ(ξn). Forξ∈RnandN= 1, 2, 3, . . . , we setρN(x) =ρ(_Nx) andσN(ξ) =σ(_Nξ). Forx∈Rn_,g∈L2_(Rn_{), and forN= 1, 2, 3, . . . , we define}

RN,Φg(x) =ρN(x)

Rne ix·ξ

ei(t1(x)φ1(|ξ1|)+t2(x)φ2(|ξ2|)+···+tn(x)φn(|ξn|))_|ξ 1|–

s

n|_ξ2|–ns· · ·

× |ξn|– s n_σ

N(ξ)g(ξ)dξ.

The adjoint ofRN,Φis given by

R_N,Φh(ξ) =σN(ξ)|ξ1|– s n|_ξ

2|– s n· · ·

× |ξn|– s n

R2

e–ix·ξ_e–i(t1(x)φ1(|ξ1|)+t2(x)φ2(|ξ2|)+···+tn(x)φn(|ξn|))_ρ

N(x)h(x)dx,

whereξ∈Rnandh∈L2(Rn). To prove (2.4) it is sufficient to prove that

By duality, to prove (2.5) it suffices to show that

_R

N,ΦhL2_(Rn₎≤Ch_Lq_(Rn₎, (2.6)

where 1_q+_q1 = 1. Thus, we have

R_N,Φh

2

L2_(Rn₎= RN,Φh(ξ)

2

dξ=

Rn

RnKN(x,y)ρN(x)ρN(y)h(x)h(y)dx dy, (2.7)

where

KN(x,y) =KN1(x,y)KN2(x,y)· · ·KNn(x,y) (2.8)

and

K_Ni(x,y) =

R|ξi|

–2_ns_ei(yi–xi)ξi_ei(ti(y)–ti(x))φ(|ξi|)_ψ

N(ξi)2dξi, (2.9)

wherei= 1, 2, . . . ,nandN= 1, 2, . . . . Since n₄ ≤s<n₂, we have 1₂ ≤2_ns < 1. Therefore, by Lemma1.3, (2.9), and (2.8), we obtain

KN(x,y)≤C 1

|x1–y1|1– 2s n

1

|x2–y2|1– 2s

n · · · 1

|xn–yn|1– 2s n

. (2.10)

We define

Pif(x1,x2, . . . ,xn) =

R 1

|xi–yi|1– 2s n

f(x1, . . . ,xi–1,yi,xi+1, . . . ,xn)dyi,

i= 1, 2, . . . ,n. Thus, by (2.7) and (2.10), we obtain

R_N,Φh(ξ)

2 dξ ≤C Rn Rn 1

|x1–y1|1– 2s n

1

|x2–y2|1– 2s n

· · · 1

|xn–yn|1– 2s

n

h(x)h(y)dx dy

=C Rn R R R R 1

|xn–yn|1– 2s

n

1

|xn–1–yn–1|1– 2s n

· · · 1

|x3–y3|1– 2s

n

× 1

|x2–y2|1– 2s n

1

|x1–y1|1– 2s

n

h(y1,y2, . . . ,yn)dy1

dy2dy3· · ·dyn

h(x)dx

=C

RnPnPn–1· · ·P2P1|h|(x)

h(x)dx. (2.11)

Invoking Hölder’s inequality, we get

_R

N,Φh(ξ)

2

Sinceq=_n2_–2n_s, it follows thatq=_n2₊₂n_s and the fact 1_q = _q1 –2_ns. Denote byIσ the Riesz

potential of orderσ, which is defined by

Iσ(f)(u) =

R

f(v)

|u–v|1–σ dv.

Applying the factIsis bounded fromLq

(R) toLq_{(R), we have}

R

Pjh(x)qdxj 1/q

≤C

R

h(x)qdxj 1/q

, (2.13)

wherej= 1, 2, . . . ,n. By (2.13) and Minkowski’s inequality, we have

PnPn–1· · ·P2P1|h|_Lq_(Rn₎≤Ch_Lq_(Rn₎. (2.14)

Therefore, (2.6) follows from (2.12) and (2.14). Now we complete the proof of Theorem1.1.

3 The proof of Theorem1.2

Assume thatn≥2,φi(i= 1, 2, 3, . . . ,n) satisfies (H1)–(H3) withmi,1> 1,mi,2> 1. For every

m= (m1,2,m2,2, . . . ,mn,2) and 2 <q< 4, we will prove that inequality (1.9) holds ifs=n₂–

|m|

4 +

|m|

q – n

q, where|m|=m1,2+m2,2+· · ·+mn,2. Fori= 1, 2, 3, . . . ,n, letti(x) be a measurable function onRn_{with 0 <t}

i(x) < 1. Denotet(x) = (t1(x),t2(x), . . . ,tn(x)), we set

St(x),Φf(x) =

Rn

eix·ξ_ei(t1(x)φ1(|ξ1|)+t2(x)φ2(|ξ2|)+···+tn(x)φn(|ξn|))_fˆ_{(ξ)dξ, x}∈Rn_,f∈SRn_.

By linearizing the maximal operator, to prove the global estimate (1.9) it suffices to show that

St(x),ΦfLq_(Rn₎≤Cf_Hs=C

Rn

1 +|ξ|2sfˆ(ξ)2dξ 1/2

. (3.1)

Sinces=n 2–

|m| 4 +

|m| q –

n q =n

1 2–

m1,2+m2,2+···+mn,2

4 +

m1,2+m2,2+···+mn,2

q –n

1

q=:s1+s2+· · ·+sn, wheresi=1₂–m₄i,2+m_qi,2–1_q,i= 1, 2, . . . ,n. Therefore, to prove (3.1) it suffices to prove that

St(x),ΦfLq_(Rn₎≤C

R2

1 +|ξ1|2 s1

1 +|ξ2|2 s2

| · · · |1 +|ξn|2 snˆ

f(ξ)2dξ 1/2

. (3.2)

Letg(ξ) = (1 +|ξ1|2) s₁

2_{(1 +}|_ξ2|2₎ s₂

2 · · ·_{(1 +}|_ξn|2₎sn2_fˆ_{(ξ), then we have}

St(x),Φf(x) =RΦg(x), (3.3)

where

RΦg(x) =

Rne ix·ξ

ei(t1(x)φ1(|ξ1|)+t2(x)φ2(|ξ2|)+···+tn(x)φn(|ξn|))_{1 +|ξ} 1|2

–s₁ 2_{1 +}|_ξ

2|2 –s₂

2 · · ·

×1 +|ξn|2 –sn

By (3.3), to prove (3.2) it is sufficient to show that

RΦgLq_(Rn₎≤Cg_L2_(Rn₎ (3.4)

forg continuous and rapidly decreasing at infinity. We take a real-valued functionρ∈

C₀∞(Rn) such thatρ(x) = 1 if|x| ≤1 andρ(x) = 0 if|x| ≥2. And we choose a real-valued functionψ∈C∞₀ (R) such thatψ(x) = 1 if|x| ≤1 andψ(x) = 0 if|x| ≥2, and setσ(ξ) = ψ(ξ1)ψ(ξ2)· · ·ψ(ξn) forξ∈Rn. ForN= 1, 2, 3, . . . , we setρN(x) =ρ(_Nx) andσN(ξ) =σ(_Nξ). Forx∈Rn_,g∈L2_(Rn_{), andN= 1, 2, 3, . . . , we define}

RN,Φg(x) =ρN(x)

Rne

ix·ξ_ei(t1(x)φ1(|ξ1|)+t2(x)φ2(|ξ2|)+···+tn(x)φn(|ξn|))_{1 +}|_ξ1|2– s₁

2_{1 +}|_ξ2|2– s₂

2

× · · · ×1 +|ξn|2 –sn

2_σ

N(ξ)g(ξ)dξ.

The adjoint ofRN,Φis given by

R_N,Φh(ξ) =σN(ξ)

1 +|ξ1|2 –s₁

2_{1 +}|_ξ 2|2

–s₂

2 · · ·_{1 +}|_ξ n|2

–sn 2

R2

e–ix·ξ_e–it1(x)φ1(|ξ1|)

×e–i(t2(x)φ2(|ξ2|)+···+tn(x)φn(|ξn|))_ρ

N(x)h(x)dx,

whereξ∈Rn_andh∈L2_(Rn_{). To prove (3.4) it suffices to prove that}

RN,ΦgLq_(Rn₎≤Cg_L2_(Rn₎. (3.5)

By duality, to prove (3.5) it is sufficient to show that

R_N,Ωh_L2_(Rn₎≤Ch_Lq_(Rn₎, (3.6)

where 1_q+_q1 = 1. Thus, we have

R_N,Φh

2

L2_(Rn₎= RN,Φh(ξ)

2

dξ=

Rn

RnKN(x,y)ρN(x)ρN(y)h(x)h(y)dx dy, (3.7)

where

KN(x,y) =KN1(x,y)KN2(x,y)· · ·KNn(x,y), (3.8)

and

K_Ni(x,y) =

R

1 +ξ_i2–si

ei(yi–xi)ξi_ei(ti(y)–ti(x))φ(|ξi|)_ψ

N(ξi)2dξi, (3.9)

where i= 1, 2, . . . ,nandN = 1, 2, . . . . Denoteαi= 2si, sincesi= ₂1 – m₄i,2 + m_qi,2 – 1_q, i= 1, 2, . . . ,n, and 2 <q< 4, it follows that1₂<αi< m₂i,2,i= 1, 2, . . . ,n. Therefore, by (3.9) and Lemma1.4, we obtain

K_Ni(x,y)≤C 1

|xi–yi|βi

whereβi=

αi+mi2,2–1

mi,2–1 . Denoteσi= 1 –βi, we define

Pif(x1,x2, . . . ,xn) =

R 1

|xi–yi|1–σi

f(x1, . . . ,xi–1,yi,xi+1, . . . ,xn)dyi,

i= 1, 2, . . . ,n. Thus, by (3.7), (3.8), and (3.10), we obtain

R_N,Φh(ξ)

2

dξ

≤C

Rn

Rn 1

|x1–y1|1–σ1 1

|x2–y2|1–σ2 · · · 1

|xn–yn|1–σn

h(x)h(y)dx dy

=C

Rn

R

R

R

R 1

|xn–yn|1–σn

1

|xn–1–yn–1|1–σn–1· · · 1

|x3–y3|1–σ3

×_| 1

x2–y2|1–σ2

1

|x1–y1|1–σ1

h(y1,y2, . . . ,yn)dy1

dy2dy3· · ·dyn

h(x)dx

=C

RnPnPn–1· · ·P2P1|h|(x)

h(x)dx. (3.11)

Invoking Hölder’s inequality, we get

_R

N,Φh(ξ)

2

dξ≤CPnPn–1· · ·P2P1|h|_Lq_(Rn₎h_Lq_(Rn₎. (3.12)

Sinceβi=

αi+mi2,2–1

mi,2–1 andαi= 2si,si= 1 2–

mi,2 4 +

mi,2 q –

1

q,i= 1, 2, . . . ,n. It follows thatβi= 2 q andσi= 1 –βi= 1 –2_q, 1_q=_q1 –σi. Thus, estimate (3.6) follows from (3.12) and estimate (2.14) in the proof of Theorem1.1. Now we complete the proof of Theorem1.2.

4 The proof of Lemma1.4

To prove Lemma1.4, we need to present the following lemma.

Lemma 4.1(see [33], pp. 309–312) Assume that a<b and set I= [a,b].Let F∈C∞(I)be real-valued and assume thatψ∈C∞(I).

(i) Assume that|F(x)| ≥λ> 0forx∈Iand thatFis monotonic onI.Then

_abeiF(x)ψ(x)dx≤C1

λ ψ(b)+ b

a

ψ_(x)dx,

where C does not depend onF,ψ,orI. (ii) Assume that|F(x)| ≥λ> 0forx∈I.Then

b

a

eiF(x)ψ(x)dx≤C 1

λ1/2 ψ(b)+ b

a

ψ_(x)dx,

where C does not depend onF,ψ,orI.

Proof of Lemma1.4 By conditions (H1) and (H2), there exist positive constantsCi(i= 1, 2, . . . , 6) so that forr≥1 andm2> 1 such that

and for 0 <r< 1 andm1> 1 such that

C4rm1–1≤φ(r)≤C5rm1–1 and φ(r)≥C6rm1–2. (4.2)

Set

J=

R

ei(dφ(|ξ|)–xξ) (1 +ξ2₎α2

μ

ξ

N

dξ.

To prove Lemma1.4, it suffices to show that there exists a constantCsuch that forx∈

R\ {0},β=α+ m2

2 –1

m2–1 andN∈N,

|J| ≤C 1

|x|β, (4.3)

whereCdepends only onα,m1,m2,Ci(i= 1, 2, . . . , 6), andμ.

Without loss of generality, we may assumeξ,d> 0. Denoteψ(ξ) = (1 +ξ2)–α_2μ(ξ

N), then we have

max

ξ≥0ψ(ξ)+ ∞

0

ψ_{(ξ)dξ≤C. (4.4)}

In fact, sinceμ∈C₀∞(R) and1₂≤α≤m2₂ , we get

max ξ≥0

ψ(ξ)≤C. (4.5)

Noting that

ψ(ξ) = –αξ1 +ξ2–

α

2–1_μ

ξ

N

+1 +ξ2–

α

2 1

Nμ

ξ

N

, (4.6)

we have ∞

0

ψ_(ξ)dξ≤α ∞

0

ξ1 +ξ2–

α

2–1_μ

ξ

N

dξ

+ ∞

0

1 +ξ2–

α

2 1

N

μ

ξ

N

dξ

=:G1+G2. (4.7)

Sinceμ∈C₀∞(R) and 1₂≤α≤m2₂ , we obtain

G1≤C ∞

0

ξ1 +ξ2–α2–1_dξ=C ∞

0

1 +ξ2–α2–1_{d1 +ξ}2_{=C (4.8)}

and

G2≤C ∞

0 1

N

μ

ξ

N

By (4.7), (4.8), and (4.9), we get

∞

0

ψ(ξ)dξ≤C. (4.10)

Therefore, (4.4) follows from (4.5) and (4.10).

To estimate (4.3), we choose a positive constantMsuch thatM=max{(1_δ)m2–1_{, 2C5, 2},} whereδis a small positive constant such thatδm2–1C2≤1₂. Below, we show (4.3) by dividing two cases|x| ≥Mand|x|<M.

Case(I):|x| ≥M. LetF(ξ) =dφ(ξ) –xξ, we have

F(ξ) =dφ(ξ) –x, F(ξ) =dφ(ξ).

Denoteρ= (|x_d|)m_2–11 _{, then we haveδρ}≥_{1. In fact, noting that}|_x| ≥₍1

δ)

m2–1_{, 0 <d< 1,}

m2> 1, and |x_d|>|x|, it follows thatδρ>δ|x| 1

m2–1≥_{1. We choose a large positive constant} λsuch thatλ≥max{(_C12 )m2–11 _,δ}_{. Denote}

I1= [0,δρ], I2= [δρ,λρ], I3= [λρ,∞).

Thus, we obtain

|J|= ∞

0

eiF(ξ)ψ(ξ)dξ≤ 3

j=1

Ij

eiF(ξ)ψ(ξ)dξ=: 3

j=1

Jj. (4.11)

Firstly, we estimateJ1. We will show that the following estimate holds:

F(ξ)≥|x|

2 , ξ∈[0,δρ]. (4.12)

Now we divide the verification of (4.12) into two cases according to the value ofξ.

Case(I-a):ξ∈[0, 1). Sincem1> 1 and 0 <d< 1, we have

dφ(ξ)≤C5dξm1–1≤C5≤

M

2 ≤

|x|

2 . (4.13)

By (4.13), ifξ∈[0, 1), we get

F(ξ)≥ |x|–dφ(ξ)≥|x|

2 . (4.14)

Case(I-b):ξ∈[1,δρ]. Sincem2> 1, we have

dφ(ξ)≤C2dξm2–1≤C2dδm2–1|

x| d ≤C2δ

m2–1_{|x| ≤}|x|

2 . (4.15)

By (4.15), we get

F(ξ)≥ |x|–dφ(ξ)≥|x|

Therefore (4.12) follows from (4.14) and (4.16). Sinceφis monotonic onR+_{by condition} (H3) andd> 0, it follows thatFis monotonic onξ∈I1. Thus, by (i) of Lemma4.1and estimate (4.12), (4.4), we have

|J1| ≤C 1

|x|≤C

1

|x|β, (4.17)

where we use |x| ≥2 and the fact 1₂ ≤β ≤1. Next we prove estimate J3. Since ξ ≥

λ(|_dx|)m_2–11 _{> 1 andλ}≥₍2 C1)

1 m_2–1,

dφ(ξ)≥C1dξm2–1≥C1dλm2–1|

x| d ≥2|x|,

it follows that

F(ξ)≥2|x|–|x|=|x|, ξ∈[λρ,∞). (4.18)

Thus, by (i) of Lemma4.1and estimate (4.18), (4.4), we have

|J3| ≤C 1

|x|≤C

1

|x|β, (4.19)

where we use|x| ≥2 and the fact 1

2≤β≤1. Now, we give estimateJ2. Sinceξ∈I2, we have|ξ| ≥1. By (4.1), we obtain

F(ξ)≥dφ(ξ)≥C3dξm2–2≥C3d _|

x| d

m_2–2 m2–1

. (4.20)

We first prove that the following estimate holds:

max

I2 |ψ|+

I2

ψ_dξ≤C|x|

d

– α

m_2–1

. (4.21)

In fact, sinceμ∈C₀∞(R) and1₂≤α≤m2₂ , we get

max ξ∈A2

ψ(ξ)≤C(δρ)–α_=Cδ–α_(ρ)–α_=Cδ–α

_|

x| d

– α

m2–1

. (4.22)

By (4.6), we have

A2

ψ_(ξ)dξ≤α

A2

ξ1 +ξ2–α2–1_μ

ξ

N

dξ+

A2

1 +ξ2–α2 1

N

μ

ξ

N

dξ

=:L1+L2. (4.23)

Sinceμ∈C₀∞(R) and 1 2≤α≤

m2

2 , we obtain

L1≤C

A2

ξ1 +ξ2–

α

2–1_dξ≤_C λρ

δρ

ξ–α–1dξ=C

_|

x| d

– α

m2–1

and

L2≤C(δρ)–α A2 1 N μ ξ N dξ≤C

_| x| d – α m2–1 . (4.25)

By (4.23), (4.24), and (4.25), we get

∞

0

ψ(ξ)dξ≤C

_| x| d – α m2–1 . (4.26)

Therefore, (4.21) follows from (4.22) and (4.26). Thus, by (ii) of Lemma4.1and estimate (4.20), (4.21), we have

|J2| ≤d– 1 2

_|

x| d

– m_2–2 2(m_2–1)|_x|

d

– α

m2–1 =C d

α– 1₂ m2–1

|x|

α+m2 2 –1 m_2–1

≤C 1

|x|β. (4.27)

Here in the last inequality we use the fact α–12

m2–1≥0 and 0 <d< 1. Therefore, for|x| ≥M, by estimates (4.11), (4.17), (4.19), and (4.27), it follows (4.3).

Case(II):|x|<M. Now we divide the verification of (4.3) into three cases according to the value ofαfor|x|<M.

Case(II-a):α> 1. Sinceμ∈C₀∞(R) andα> 1, we get

|J|= ∞

0

ei(dφ(|ξ|)–xξ) (1 +ξ2₎α2

μ

ξ

N

dξ≤C

∞

0 1 (1 +ξ2₎α2 d

ξ≤C. (4.28)

Noting that|x|<Mand1₂≤β≤1, by (4.28), we have

|J| ≤C=C|x|β 1

|x|β ≤CM

β 1

|x|β =C

1

|x|β,

which follows (4.3).

Case(II-b):1₂≤α< 1. By the mean value theorem, when1₂≤α< 1, we have

0 <1 +ξ2α2 _–ξα

=1 +ξ2α2 _–ξ2α2 ≤α 2

ξ2α2–1≤_ξα–2_{. (4.29)}

By (4.29), we obtain

1 ξα –

1

(1 +ξ2₎α2 =O

1 ξα+2

, ξ→ ∞. (4.30)

Noting that1₂≤α< 1, by (4.30), we have

∞

0 _ξ1α –

1 (1 +ξ2₎α2

dξ≤C (4.31)

and

|J|= ∞

0

ei(dφ(|ξ|)–xξ) (1 +ξ2₎α2

μ

ξ

N

dξ≤ ∞

0

ei(dφ(|ξ|)–xξ)

1 (1 +ξ2₎α2

+ ∞

0

ei(dφ(|ξ|)–xξ) 1

ξαμ

ξ

N

dξ

= :K1+K2.

By (4.31), we have

|K1| ≤C=C|x|β 1

|x|β ≤CM

β 1

|x|β =C

1

|x|β. (4.32)

By Lemma1.3, we obtain

|K2| ≤C 1

|x|1–α. (4.33)

Noting that_|1_x|>_M1, and the factβ≥1 –α, it follows from1₂≤α< 1,1₂≤β≤1 that

|x|1–α=|x|β|x|1–α–β=|x|β

1

|x|

β–(1–α)

≥C|x|β. (4.34)

Therefore, by (4.33) and (4.34), we have

|K2| ≤C 1

|x|β. (4.35)

Hence, (4.3) holds from (4.32) and (4.35).

Case(II-c):α= 1. From the proof of Lemma1.3, noting thatM≥2, we may get

|J| ≤Clog

1

|x|

if 0 <|x| ≤1

2 (4.36)

and

|J| ≤C if1

2<|x|<M. (4.37)

By (4.36) and1₂≤β≤1, for 0 <|x| ≤1₂, we have

|J| ≤Clog

1

|x|

≤C 1

|x|β. (4.38)

By (4.37), for1₂<|x|<M, we have

|J| ≤C=C|x|β 1

|x|β ≤CM

β 1

|x|β =C

1

|x|β. (4.39)

Thus, forα= 1,|x|<M, by (4.38) and (4.39), we get

|J| ≤C 1

|x|β,

which is just estimate (4.3).

Summing up all the above estimates, we complete the proof of estimate (4.3) and finish

5 Conclusion

In this paper, by linearizing the maximal operator and duality methods, and applying the results of Lemma1.3and Lemma1.4, we obtain the maximal globalLq_{inequalities (1.8)} and (1.9) for multiparameter oscillatory integralSt,Φ. These estimates are apparently good

extensions to maximal globalLq_{inequalities (1.6) and (1.7) for the multiparameter} frac-tional Schrödinger equation in [32].

Acknowledgements

The authors would like to express their deep gratitude to the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions.

Funding

The work is supported by NSFC (No. 11661061, No. 11561062, No. 11761054), Inner Mongolia University scientific research projects (No. NJZZ16234, NJZY17289).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

YN participated in the design of the study and in the discussions of all results. YX participated in the discussions of all results. All authors read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 28 June 2018 Accepted: 13 December 2018

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