1 Introduction and main results
Let f be a Schwartz function in \(\mathcal{S}(\mathbb{R}^{n})\) and
It is well known that \(S_{t}f(x)\) is the solution of the fractional Schrödinger equation
Here f̂ denotes the Fourier transform of f defined by \(\hat{f}(\xi)=\int_{\mathbb{R}^{n}}e^{-i\xi\cdot x}f(x)\,dx\).
$$S_{t}f(x)=u(x,t)=(2\pi)^{-n} \int_{\mathbb{R}^{n}} e^{ix\cdot\xi+it|\xi|^{a}}\hat{f}(\xi)\,d\xi,\quad (x,t)\in \mathbb{R}^{n}\times\mathbb{R}. $$
$$\begin{aligned} \textstyle\begin{cases} i\partial_{t}u+(-\Delta)^{a/2} u=0,\quad (x,t)\in\mathbb {R}^{n}\times\mathbb{R},\\ u(x,0)=f(x). \end{cases}\displaystyle \end{aligned}$$
(1.1)
We recall the homogeneous Sobolev space \(\dot{H}^{s}(\mathbb{R}^{n})\ (s\in\mathbb{R})\), which is defined by
and the non-homogeneous Sobolev space \(H^{s}(\mathbb{R}^{n})\ (s\in \mathbb{R})\), which is defined by
Maximal operator \(S^{\ast}f\) associated with the family of operators \(\{S_{t}\}_{0< t<1}\) is defined by
$$\dot{H}^{s}\bigl(\mathbb{R}^{n}\bigr)= \biggl\{ f\in \mathcal{S}^{\prime}: \Vert f \Vert _{H^{s}}= \biggl( \int_{\mathbb{R}^{n}} \vert \xi \vert ^{2s} \bigl\vert \hat{f}(\xi) \bigr\vert ^{2}\,d\xi \biggr)^{1/2} < \infty \biggr\} , $$
$$H^{s}\bigl(\mathbb{R}^{n}\bigr)= \biggl\{ f\in \mathcal{S}^{\prime}: \Vert f \Vert _{H^{s}}= \biggl( \int_{\mathbb{R}^{n}} \bigl(1+ \vert \xi \vert ^{2} \bigr)^{s} \bigl\vert \hat{f}(\xi) \bigr\vert ^{2}\,d\xi \biggr)^{1/2}< \infty \biggr\} . $$
$$S^{\ast}f(x)= \sup_{0< t< 1} \bigl\vert S_{t}f(x) \bigr\vert ,\quad x\in\mathbb{R}^{n}. $$
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It is well known that if \(a=2\), u is the solution of the Schrödinger equation
In 1979, Carleson [4] proposed a problem: if \(f\in H^{s}(\mathbb{R}^{n})\) for which s does
$$\begin{aligned} \textstyle\begin{cases} i\partial_{t}u-\Delta u=0,\quad (x,t)\in\mathbb{R}^{n}\times\mathbb {R},\\ u(x,0)=f(x). \end{cases}\displaystyle \end{aligned}$$
(1.2)
$$\begin{aligned} \lim_{t\rightarrow0}u(x,t)=f(x),\quad \mbox{a.e. }x\in \mathbb{R}^{n}. \end{aligned}$$
(1.3)
Carleson first considered this problem for dimension \(n=1\) in [4] and showed that the convergence (1.3) holds for \(f\in H^{s}(\mathbb{R})\) with \(s \geq\frac{1}{4}\), which is sharp was shown by Dahlberg and Kenig [8]. The higher dimensional case of convergence (1.3) has been studied by several authors, see [1, 2, 9, 11‐13, 24, 25, 30, 34, 35] for example. In fact, by a standard argument, for \(f\in H^{s}(\mathbb {R}^{n})\), the pointwise convergence (1.3) follows from the local estimate
for some \(q\geq1\) and \(s\in\mathbb{R}\). Here \(\mathbb{B}^{n}\) is the unit ball centered at the origin in \(\mathbb{R}^{n}\). 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
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.
$$ \bigl\Vert S^{\ast}f \bigr\Vert _{L^{q}(\mathbb {B}^{n})}\leq C \Vert f \Vert _{H^{s}(\mathbb{R}^{n})},\quad f\in H^{s}\bigl( \mathbb{R}^{n}\bigr), $$
(1.4)
$$ \bigl\Vert S^{\ast}f \bigr\Vert _{L^{q}(\mathbb{R}^{n})} \leq C \Vert f \Vert _{H^{s}(\mathbb{R}^{n})}. $$
(1.5)
For \(n\geq2\) and a multiindex \(a=(a_{1},a_{2},\ldots, a_{n})\), with \(a_{j}>1\) and f being a Schwartz function in \(\mathcal {S}(\mathbb{R}^{n})\), we set
where \(t=(t_{1},t_{2},\ldots, t_{n})\in\mathbb{R}^{n}\). For \(n\geq2\), the local maximal operator \(M^{\ast}\) is defined by
and the global maximal operator \(M^{\ast\ast}\) is defined by
The global estimate
and
$$S_{t}f(x)=(2\pi)^{-n} \int_{\mathbb{R}^{n}} e^{ix\cdot\xi}e^{i(t_{1} |\xi_{1}|^{a_{1}}+t_{2}|\xi_{2}|^{a_{2}}+ \cdots+t_{n}|\xi_{n}|^{a_{n}})}\hat{f}(\xi)\,d\xi,\quad x\in\mathbb{R}^{n}, $$
$$M^{\ast}f(x)= \sup_{0< t_{i}< 1} \bigl\vert S_{t}f(x) \bigr\vert ,\quad x\in\mathbb{R}^{n}, $$
$$M^{\ast\ast}f(x) = \sup_{t_{i}\in\mathbb{R}} \bigl\vert S_{t}f(x) \bigr\vert ,\quad x\in\mathbb{R}^{n}. $$
$$ \bigl\Vert M^{\ast\ast}f \bigr\Vert _{L^{q}(\mathbb{R}^{n})} \leq C \Vert f \Vert _{\dot{H}^{s}(\mathbb{R}^{n})} $$
(1.6)
$$ \bigl\Vert M^{\ast}f \bigr\Vert _{L^{q}(\mathbb{R}^{n})} \leq C \Vert f \Vert _{H^{s}(\mathbb{R}^{n})}. $$
(1.7)
In 2014, Sjolin and Soria [32] obtained the following results.
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Theorem A
([32])
Assume
\(n\geq2\). Then, for every
a, inequality (1.6) holds if and only if
\(4\leq q<\infty\)
and
\(s=n(\frac{1}{2}-\frac{1}{q})\).
Theorem B
([32])
Assume
\(n\geq2\). Then, for every
a
and for
\(2< q<4\), inequality (1.7) holds if and only if
\(s\geq\frac {n}{2}-\frac{|a|}{4}+\frac{|a|}{q}-\frac{n}{q}\).
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 integral \(S_{t,\varPhi}\) defined by
Here, \(n\geq2\) and f is a Schwartz function in \(\mathcal{S}(\mathbb {R}^{n})\), \(\varPhi=(\phi_{1},\phi_{2},\ldots,\phi_{n})\), \(\phi_{i}\)
\((i=1,2,3,\ldots, n)\) is a function on \(\mathbb {R}^{+}\rightarrow\mathbb{R}\). For \(n\geq2\), the local maximal operator \(M_{\varPhi}^{\ast}\) is defined by
and the global maximal operator \(M_{\varPhi}^{\ast\ast}\) is defined by
The global estimates of maximal operators \(M_{\varPhi}^{\ast}\) and \(M_{\varPhi}^{\ast\ast}\) are defined by
and
$$S_{t,\varPhi}f(x)=(2\pi)^{-n} \int_{\mathbb{R}^{n}} e^{ix\cdot\xi}e^{i(t_{1} \phi_{1}(|\xi_{1}|)+t_{2}\phi_{2}(|\xi_{2}|)+ \cdots+t_{n}\phi_{n}(|\xi_{n}|))}\hat{f}(\xi)\,d\xi,\quad x\in\mathbb{R}^{n}. $$
$$M_{\varPhi}^{\ast}f(x)= \sup_{0< t_{i}< 1} \bigl\vert S_{t,\varPhi}f(x) \bigr\vert ,\quad x\in\mathbb{R}^{n}, $$
$$M_{\varPhi}^{\ast\ast}f(x)= \sup_{t_{i}\in\mathbb{R}} \bigl\vert S_{t,\varPhi}f(x) \bigr\vert ,\quad x\in\mathbb{R}^{n}. $$
$$ \bigl\Vert M_{\varPhi}^{\ast\ast}f \bigr\Vert _{L^{q}(\mathbb{R}^{n})} \leq C \Vert f \Vert _{\dot{H}^{s}(\mathbb{R}^{n})} $$
(1.8)
$$ \bigl\Vert M_{\varPhi}^{\ast}f \bigr\Vert _{L^{q}(\mathbb{R}^{n})} \leq C \Vert f \Vert _{H^{s}(\mathbb{R}^{n})}. $$
(1.9)
Assume that \(\phi: \mathbb{R}^{+}\rightarrow\mathbb{R}\) satisfies:
(H1)
There exists \(m_{1}>1\) such that \(|\phi^{\prime}(r)|\sim r^{m_{1}-1}\) and \(|\phi^{\prime \prime}(r)|\gtrsim r^{m_{1}-2}\) for all \(0< r<1\);
(H2)
There exists \(m_{2}>1\) such that \(|\phi^{\prime}(r)|\sim r^{m_{2}-1}\) and \(|\phi^{\prime \prime}(r)|\gtrsim r^{m_{2}-2}\) for all \(r\geq1\);
(H3)
Either \(\phi^{\prime\prime}(r)>0\) or \(\phi^{\prime \prime}(r)<0\) for all \(r>0\).
Now we state our main results as follows.
Theorem 1.1
Assume that
\(n\geq2\)
and
\(\phi_{i}\)
\((i=1,2,3,\ldots, n)\)
satisfies (H1)–(H3). If
\(4\leq q<\infty\)
and
\(s=n(\frac{1}{2}-\frac{1}{q})\), then the global estimate (1.8) holds.
Theorem 1.2
Let
\(m=(m_{1,2},m_{2,2},\ldots,m_{n,2})\)
and set
\(|m|=m_{1,2}+m_{2,2}+\cdots+m_{n,2}\). Assume that
\(n\geq2\)
and
\(\phi _{i}\)
\((i=1,2,3,\ldots, n)\)
satisfies (H1)–(H3) with
\(m_{i,1}>1\), \(m_{i,2}>1\). Then, for every
m, inequality (1.9) holds if
\(2< q<4\)
and
\(s\geq\frac{n}{2}-\frac{|m|}{4}+\frac{|m|}{q}-\frac{n}{q}\).
Remark 1.1
There are many elements ϕ satisfying conditions (H1)–(H3), for instance, the fractional Schrödinger equation (\(\phi(r)= r^{a}\)), or \(( \phi(r)=(1+r^{2})^{\frac{a}{2}}), (a\geq1)\), the Beam equation \((\phi(r)=\sqrt{1+r^{4}})\), the fourth-order Schrödinger equation \((\phi(r)=r^{2}+r^{4})\), iBq \((\phi(r)=r\sqrt{1+r^{2}})\), and so on (see [5, 6, 18, 19, 22, 23, 27], and the references therein). Hence, Theorem 1.1 and Theorem 1.2 imply the sufficiency part of Theorem A and Theorem B, respectively. However, due to the complexity of the symbol ϕ, we cannot obtain the necessities of the range of q in Theorem 1.1 and Theorem 1.2.
This paper is organized as follows. The proofs of Theorem 1.1 and Theorem 1.2 are given in Sect. 2 and Sect. 3, respectively. To prove Theorem 1.1 and Theorem 1.2, we next need the following important lemmas, which play a key role in proving Theorem 1.1 and Theorem 1.2, respectively. The proof of Lemma 1.4 is given in Sect. 4.
Lemma 1.3
([26])
Assume that
ϕ
satisfies (H1)–(H3) with
\(m_{1}>1\), \(m_{2}>1\). \(\frac{1}{2}\leq s<1\)
and
\(\mu\in C_{0}^{\infty}(\mathbb {R})\). Then
for
\(x\in\mathbb{R}\setminus\{0\}\), \(t\in\mathbb{R}\), and
\(N=1,2,3,\ldots\) . Here the constant
C
may depend on
s
and
\(m_{1}\), \(m_{2}\), and
μ
but not on
x, t, or
N.
$$\biggl\vert \int_{\mathbb{R}} e^{ix\xi+it\phi( \vert \xi \vert )} \vert \xi \vert ^{-s} \mu \biggl(\frac{\xi}{N} \biggr) \,d\xi \biggr\vert \leq C \frac{1}{ \vert x \vert ^{1-s}} $$
Lemma 1.4
Assume that
ϕ
satisfies (H1)–(H3) with
\(m_{1}>1\), \(m_{2}>1\). \(\frac{1}{2}\leq\alpha\leq\frac{m_{2}}{2}\), \(-1< d<1\), and
\(\mu\in C_{0}^{\infty}(\mathbb{R})\). Then
for
\(x\in\mathbb{R}\setminus\{0\}\)
and
\(N=1,2,3,\ldots\) , where
\(\beta=\frac{\alpha+\frac{m_{2}}{2}-1}{m_{2}-1}\). Here the constant
C
may depend on
α
and
\(m_{1}\), \(m_{2}\), and
μ
but not on
x, d, and
N.
$$ \biggl\vert \int_{\mathbb{R}} \frac{e^{i(d\phi( \vert \xi \vert )-x\xi)}}{(1+\xi ^{2})^{\frac{\alpha}{2}}} \mu\biggl(\frac{\xi}{N} \biggr)\,d\xi \biggr\vert \leq C\frac{1}{ \vert x \vert ^{\beta}} $$
(1.10)
2 The proof of Theorem 1.1
Assume that \(n\geq2\), \(\phi_{i}\)
\((i=1,2,3,\ldots ,n)\) satisfies (H1)–(H3). For \(i=1,2,3,\ldots ,n\), let \(t_{i}(x)\) be a measurable function on \(\mathbb{R}^{n}\) with \(t_{i}(x)\in\mathbb{R}\). Denote \(t(x)=(t_{1}(x),t_{2}(x),\ldots,t_{n}(x))\), we set
For \(4\leq q<\infty\) and \(s=n(\frac{1}{2}-\frac{1}{q})\), that is, \(\frac{n}{4}\leq s<\frac{n}{2}\) and \(q=\frac{2n}{n-2s}\). By linearizing the maximal operator (see [30]) to prove the global estimate (1.8) holds, it suffices to show that
To prove (2.1) it suffices to prove that
Let \(g(\xi)=|\xi_{1}|^{\frac{s}{n}}|\xi_{2}|^{\frac{s}{n}} \cdots|\xi_{n}|^{\frac{s}{n}}\hat{f}(\xi)\), then we have
where
To prove (2.2) it suffices to prove that
for g continuous and rapidly decreasing at infinity. We take a real-valued function \(\rho\in C_{0}^{\infty}(\mathbb{R}^{n})\) such that \(\rho(x)=1\) if \(|x|\leq1\) and \(\rho(x)=0\) if \(|x|\geq2\). And we choose a real-valued function \(\psi\in C_{0}^{\infty}(\mathbb{R})\) such that \(\psi(x)=1\) if \(|x|\leq1\) and \(\psi(x)=0\) if \(|x|\geq2\), and set \(\sigma(\xi )=\psi(\xi_{1})\psi(\xi_{2}) \cdots\psi(\xi_{n})\). For \(\xi\in\mathbb{R}^{n}\) and \(N=1,2,3,\ldots\) , we set \(\rho_{N}(x)=\rho(\frac{x}{N})\) and \(\sigma_{N}(\xi)=\sigma(\frac{\xi}{N})\). For \(x\in\mathbb{R}^{n}\), \(g\in L^{2}(\mathbb{R}^{n})\), and for \(N=1,2,3,\ldots\) , we define
The adjoint of \(R_{N,\varPhi}\) is given by
where \(\xi\in\mathbb{R}^{n}\) and \(h\in L^{2}(\mathbb{R}^{n})\). To prove (2.4) it is sufficient to prove that
By duality, to prove (2.5) it suffices to show that
where \(\frac{1}{q}+\frac{1}{q^{\prime}}=1\). Thus, we have
where
and
where \(i=1,2,\ldots,n\) and \(N=1,2,\ldots\) . Since \(\frac{n}{4}\leq s <\frac{n}{2}\), we have \(\frac{1}{2}\leq \frac{2s}{n}<1\). Therefore, by Lemma 1.3, (2.9), and (2.8), we obtain
We define
\(i=1,2,\ldots,n\). Thus, by (2.7) and (2.10), we obtain
Invoking Hölder’s inequality, we get
Since \(q=\frac{2n}{n-2s}\), it follows that \(q^{\prime}=\frac {2n}{n+2s}\) and the fact \(\frac{1}{q}=\frac{1}{q^{\prime}}-\frac{2s}{n}\). Denote by \(I_{\sigma}\) the Riesz potential of order σ, which is defined by
Applying the fact \(I_{s}\) is bounded from \(L^{q^{\prime}}(\mathbb{R})\) to \(L^{q}(\mathbb{R})\), we have
where \(j=1,2,\ldots,n\). By (2.13) and Minkowski’s inequality, we have
Therefore, (2.6) follows from (2.12) and (2.14). Now we complete the proof of Theorem 1.1.
$$\begin{aligned} S_{t(x),\varPhi}f(x)={}&(2\pi)^{-n} \int_{\mathbb{R}^{n}} e^{ix\cdot\xi}e^{i(t_{1}(x) \phi_{1}(|\xi_{1}|)+t_{2}(x)\phi_{2}(|\xi_{2}|)+ \cdots+t_{n}(x)\phi_{n}(|\xi_{n}|))}\hat{f}(\xi)\,d\xi,\\ & x\in\mathbb{R}^{n}, f\in\mathcal{S}\bigl(\mathbb{R}^{n}\bigr). \end{aligned}$$
$$ \Vert S_{t(x),\varPhi}f \Vert _{L^{q}(\mathbb{R}^{n})}\leq C \Vert f \Vert _{\dot {H}^{s}}=C \biggl( \int_{\mathbb{R}^{n}} \vert \xi \vert ^{2s} \bigl\vert \hat{f}(\xi) \bigr\vert ^{2}\,d\xi \biggr)^{1/2}. $$
(2.1)
$$ \Vert S_{t(x),\varPhi}f \Vert _{L^{q}(\mathbb{R}^{n})}\leq C \biggl( \int_{\mathbb{R}^{n}} \vert \xi_{1} \vert ^{\frac{2s}{n}} \vert \xi_{2} \vert ^{\frac{2s}{n}} |\cdots \vert \xi_{n} \vert ^{\frac{2s}{n}} \bigl\vert \hat{f}(\xi) \bigr\vert ^{2} \,d\xi \biggr)^{1/2}. $$
(2.2)
$$\begin{aligned} S_{t(x),\varPhi}f(x)={}& \int_{\mathbb{R}^{n}} e^{ix\cdot\xi}e^{i(t_{1}(x) \phi_{1}( \vert \xi_{1} \vert )+t_{2}(x)\phi_{2}( \vert \xi_{2} \vert )+ \cdots+t_{n}(x)\phi_{n}( \vert \xi_{n} \vert ))} \vert \xi_{1} \vert ^{-\frac{s}{n}} \vert \xi_{2} \vert ^{-\frac{s}{n}} \cdots \vert \xi_{n} \vert ^{-\frac{s}{n}}g(\xi)\,d \xi \\ ={}&R_{\varPhi}g(x), \end{aligned}$$
(2.3)
$$R_{\varPhi}g(x)= \int_{\mathbb{R}^{n}} e^{ix\cdot\xi}e^{i(t_{1}(x) \phi_{1}( \vert \xi_{1} \vert )+t_{2}(x)\phi_{2}( \vert \xi_{2} \vert )+ \cdots+t_{n}(x)\phi_{n}( \vert \xi_{n} \vert ))} \vert \xi_{1} \vert ^{-\frac{s}{n}} \vert \xi_{2} \vert ^{-\frac{s}{n}} \cdots \vert \xi_{n} \vert ^{-\frac{s}{n}}g(\xi)\,d \xi. $$
$$ \Vert R_{\varPhi}g \Vert _{L^{q}(\mathbb{R}^{n})}\leq C \Vert g \Vert _{L^{2}(\mathbb{R}^{n})} $$
(2.4)
$$\begin{aligned} R_{N,\varPhi}g(x)={}&\rho_{N}(x) \int_{\mathbb{R}^{n}} e^{ix\cdot\xi}e^{i(t_{1}(x) \phi_{1}( \vert \xi_{1} \vert )+t_{2}(x)\phi_{2}( \vert \xi_{2} \vert )+ \cdots+t_{n}(x)\phi_{n}( \vert \xi_{n} \vert ))} \vert \xi_{1} \vert ^{-\frac{s}{n}} \vert \xi_{2} \vert ^{-\frac{s}{n}} \cdots\\ &{}\times \vert \xi_{n} \vert ^{-\frac{s}{n}} \sigma_{N}(\xi)g(\xi)\,d\xi. \end{aligned}$$
$$\begin{aligned} R^{\prime}_{N,\varPhi}h(\xi)={}&\sigma_{N}(\xi) \vert \xi_{1} \vert ^{-\frac{s}{n}} \vert \xi_{2} \vert ^{-\frac{s}{n}} \cdots\\ &{}\times \vert \xi_{n} \vert ^{-\frac{s}{n}} \int_{\mathbb{R}^{2}} e^{-ix\cdot\xi}e^{-i(t_{1}(x) \phi_{1}( \vert \xi_{1} \vert )+t_{2}(x)\phi_{2}( \vert \xi_{2} \vert )+ \cdots+t_{n}(x)\phi_{n}( \vert \xi_{n} \vert ))} \rho_{N}(x)h(x)\,dx, \end{aligned}$$
$$ \Vert R_{N,\varPhi}g \Vert _{L^{q}(\mathbb{R}^{n})}\leq C \Vert g \Vert _{L^{2}(\mathbb{R}^{n})}. $$
(2.5)
$$ \bigl\Vert R_{N,\varPhi}^{\prime}h \bigr\Vert _{L^{2}(\mathbb{R}^{n})}\leq C \Vert h \Vert _{L^{q^{\prime}}(\mathbb{R}^{n})}, $$
(2.6)
$$ \bigl\Vert R_{N,\varPhi}^{\prime}h \bigr\Vert _{L^{2}(\mathbb{R}^{n})}^{2} = \int \bigl\vert R^{\prime}_{N,\varPhi}h(\xi) \bigr\vert ^{2}\,d\xi = \int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}} K_{N}(x,y)\rho_{N}(x) \rho_{N}(y)h(x)\overline{h(y)}\,dx\,dy, $$
(2.7)
$$ K_{N}(x,y)=K^{1}_{N}(x,y) K^{2}_{N}(x,y)\cdots K^{n}_{N}(x,y) $$
(2.8)
$$ K^{i}_{N}(x,y)= \int_{\mathbb{R}} \vert \xi_{i} \vert ^{-\frac{2s}{n}} e^{i(y_{i}-x_{i})\xi_{i}} e^{i(t_{i}(y)-t_{i}(x))\phi( \vert \xi_{i} \vert )} \psi_{N}(\xi_{i})^{2} \,d\xi_{i}, $$
(2.9)
$$ \bigl\vert K_{N}(x,y) \bigr\vert \leq C \frac{1}{ \vert x_{1}-y_{1} \vert ^{1-\frac{2s}{n}}} \frac{1}{ \vert x_{2}-y_{2} \vert ^{1-\frac{2s}{n}}} \cdots\frac{1}{ \vert x_{n}-y_{n} \vert ^{1-\frac{2s}{n}}}. $$
(2.10)
$$P_{i}f(x_{1},x_{2},\ldots, x_{n})= \int_{\mathbb{R}} \frac{1}{ \vert x_{i}-y_{i} \vert ^{1-\frac{2s}{n}}} f(x_{1}, \ldots,x_{i-1},y_{i}, x_{i+1},\ldots,x_{n}) \,dy_{i}, $$
$$\begin{aligned} &\int \bigl\vert R_{N,\varPhi}^{\prime}h(\xi) \bigr\vert ^{2}\,d\xi \\ &\quad \leq C \int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}} \frac{1}{ \vert x_{1}-y_{1} \vert ^{1-\frac{2s}{n}}} \frac{1}{ \vert x_{2}-y_{2} \vert ^{1-\frac{2s}{n}}} \cdots \frac{1}{ \vert x_{n}-y_{n} \vert ^{1-\frac{2s}{n}}} \bigl\vert h(x) \bigr\vert \bigl\vert h(y) \bigr\vert \,dx\,dy \\ &\quad = C \int_{\mathbb{R}^{n}} \biggl( \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}}\frac{1}{ \vert x_{n}-y_{n} \vert ^{1-\frac{2s}{n}}} \frac{1}{ \vert x_{n-1}-y_{n-1} \vert ^{1-\frac{2s}{n}}}\cdots \frac {1}{ \vert x_{3}-y_{3} \vert ^{1-\frac{2s}{n}}} \\ &\qquad{}\times\frac {1}{ \vert x_{2}-y_{2} \vert ^{1-\frac{2s}{n}}} \biggl( \int\frac {1}{ \vert x_{1}-y_{1} \vert ^{1-\frac{2s}{n}}} \bigl\vert h(y_{1},y_{2}, \ldots, y_{n}) \bigr\vert \,dy_{1} \biggr) \,dy_{2}\,dy_{3}\cdots \,dy_{n} \biggr) \bigl\vert h(x) \bigr\vert \,dx \\ &\quad =C \int_{\mathbb{R}^{n}}P_{n}P_{n-1}\cdots P_{2}P_{1} \vert h \vert (x) \bigl\vert h(x) \bigr\vert \,dx. \end{aligned}$$
(2.11)
$$ \int \bigl\vert R_{N,\varPhi}^{\prime}h(\xi) \bigr\vert ^{2}\,d\xi\leq C \bigl\Vert P_{n}P_{n-1}\cdots P_{2}P_{1} \vert h \vert \bigr\Vert _{L^{q}(\mathbb{R}^{n})} \Vert h \Vert _{L^{q^{\prime}}(\mathbb{R}^{n})}. $$
(2.12)
$$I_{\sigma}(f) (u)= \int_{\mathbb{R}}\frac{f(v)}{ \vert u-v \vert ^{1-\sigma}}\,dv. $$
$$ \biggl( \int_{\mathbb{R}} \bigl\vert P_{j}h(x) \bigr\vert ^{q}\,dx_{j} \biggr) ^{1/q}\leq C \biggl( \int_{\mathbb{R}} \bigl\vert h(x) \bigr\vert ^{q^{\prime}} \,dx_{j} \biggr) ^{1/q^{\prime}}, $$
(2.13)
$$ \bigl\Vert P_{n}P_{n-1}\cdots P_{2}P_{1} \vert h \vert \bigr\Vert _{L^{q}(\mathbb{R}^{n})} \leq C \Vert h \Vert _{L^{q^{\prime}}(\mathbb{R}^{n})}. $$
(2.14)
3 The proof of Theorem 1.2
Assume that \(n\geq2\), \(\phi_{i}\)
\((i=1,2,3,\ldots, n)\) satisfies (H1)–(H3) with \(m_{i,1}>1\), \(m_{i,2}>1\). For every \(m=(m_{1,2},m_{2,2},\ldots,m_{n,2})\) and \(2< q<4\), we will prove that inequality (1.9) holds if \(s=\frac{n}{2}-\frac{|m|}{4}+\frac{|m|}{q}-\frac{n}{q}\), where \(|m|=m_{1,2}+m_{2,2}+\cdots+m_{n,2}\). For \(i=1,2,3,\ldots, n\), let \(t_{i}(x)\) be a measurable function on \(\mathbb{R}^{n}\) with \(0< t_{i}(x)<1\). Denote \(t(x)=(t_{1}(x),t_{2}(x),\ldots,t_{n}(x))\), we set
By linearizing the maximal operator, to prove the global estimate (1.9) it suffices to show that
Since \(s=\frac{n}{2}-\frac{|m|}{4}+\frac{|m|}{q}-\frac{n}{q} =n\frac{1}{2}-\frac{m_{1,2}+m_{2,2}+\cdots+m_{n,2}}{4} +\frac{m_{1,2}+m_{2,2}+\cdots+m_{n,2}}{q}-n\frac{1}{q} =:s_{1}+s_{2}+\cdots+s_{n}\), where \(s_{i}=\frac{1}{2}-\frac{m_{i,2}}{4}+ \frac{m_{i,2}}{q}-\frac{1}{q}\), \(i=1,2,\ldots,n\). Therefore, to prove (3.1) it suffices to prove that
$$S_{t(x),\varPhi}f(x)= \int_{\mathbb{R}^{n}} e^{ix\cdot\xi}e^{i(t_{1}(x) \phi_{1}(|\xi_{1}|)+t_{2}(x)\phi_{2}(|\xi_{2}|)+ \cdots+t_{n}(x)\phi_{n}(|\xi_{n}|))}\hat{f}(\xi)\,d\xi,\quad x\in\mathbb{R}^{n}, f\in\mathcal{S}\bigl(\mathbb{R}^{n}\bigr). $$
$$ \Vert S_{t(x),\varPhi}f \Vert _{L^{q}(\mathbb{R}^{n})}\leq C \Vert f \Vert _{H^{s}}=C \biggl( \int_{\mathbb{R}^{n}} \bigl(1+ \vert \xi \vert ^{2} \bigr)^{s} \bigl\vert \hat{f}(\xi) \bigr\vert ^{2}\,d\xi \biggr)^{1/2}. $$
(3.1)
$$ \Vert S_{t(x),\varPhi}f \Vert _{L^{q}(\mathbb{R}^{n})}\leq C \biggl( \int_{\mathbb{R}^{2}} \bigl(1+ \vert \xi_{1} \vert ^{2}\bigr)^{s_{1}}\bigl(1+ \vert \xi_{2} \vert ^{2}\bigr)^{s_{2}} \vert \cdots \vert \bigl(1+ \vert \xi_{n} \vert ^{2}\bigr)^{s_{n}} \bigl\vert \hat{f}(\xi) \bigr\vert ^{2} \,d\xi \biggr)^{1/2}. $$
(3.2)
Let \(g(\xi)=(1+|\xi_{1}|^{2})^{\frac{s_{1}}{2}}(1+|\xi_{2}|^{2}) ^{\frac{s_{2}}{2}} \cdots(1+|\xi_{n}|^{2})^{\frac{s_{n}}{2}}\hat{f}(\xi)\), then we have
where
By (3.3), to prove (3.2) it is sufficient to show that
for g continuous and rapidly decreasing at infinity. We take a real-valued function \(\rho\in C_{0}^{\infty}(\mathbb{R}^{n})\) such that \(\rho(x)=1\) if \(|x|\leq1\) and \(\rho(x)=0\) if \(|x|\geq2\). And we choose a real-valued function \(\psi\in C_{0}^{\infty}(\mathbb{R})\) such that \(\psi(x)=1\) if \(|x|\leq1\) and \(\psi(x)=0\) if \(|x|\geq2\), and set \(\sigma(\xi )=\psi(\xi_{1})\psi(\xi_{2})\cdots\psi(\xi_{n})\) for \(\xi\in \mathbb{R}^{n}\). For \(N=1,2,3,\ldots\) , we set \(\rho_{N}(x)=\rho (\frac{x}{N})\) and \(\sigma_{N}(\xi)=\sigma(\frac{\xi}{N})\). For \(x\in\mathbb{R}^{n}\), \(g\in L^{2}(\mathbb{R}^{n})\), and \(N=1,2,3,\ldots\) , we define
The adjoint of \(R_{N,\varPhi}\) is given by
where \(\xi\in\mathbb{R}^{n}\) and \(h\in L^{2}(\mathbb{R}^{n})\). To prove (3.4) it suffices to prove that
By duality, to prove (3.5) it is sufficient to show that
where \(\frac{1}{q}+\frac{1}{q^{\prime}}=1\). Thus, we have
where
and
where \(i=1,2,\ldots,n\) and \(N=1,2,\ldots\) . Denote \(\alpha_{i}=2s_{i}\), since \(s_{i}=\frac{1}{2}-\frac{m_{i,2}}{4}+ \frac{m_{i,2}}{q}-\frac{1}{q}\), \(i=1,2,\ldots,n\), and \(2< q<4\), it follows that \(\frac{1}{2}<\alpha_{i}<\frac{m_{i,2}}{2}\), \(i=1,2,\ldots,n\). Therefore, by (3.9) and Lemma 1.4, we obtain
where \(\beta_{i}=\frac{\alpha_{i}+\frac{m_{i,2}}{2}-1}{ m_{i,2}-1}\). Denote \(\sigma_{i}=1-\beta_{i}\), we define
\(i=1,2,\ldots,n\). Thus, by (3.7), (3.8), and (3.10), we obtain
Invoking Hölder’s inequality, we get
Since \(\beta_{i}=\frac{\alpha_{i}+\frac{m_{i,2}}{2}-1}{m_{i,2}-1}\) and \(\alpha_{i}=2s_{i}\), \(s_{i}=\frac{1}{2}-\frac{m_{i,2}}{4}+ \frac{m_{i,2}}{q}-\frac{1}{q}\), \(i=1,2,\ldots,n\). It follows that \(\beta_{i}=\frac{2}{q}\) and \(\sigma_{i}=1-\beta_{i}=1-\frac {2}{q}\), \(\frac{1}{q}=\frac{1}{q^{\prime}}-\sigma_{i}\). Thus, estimate (3.6) follows from (3.12) and estimate (2.14) in the proof of Theorem 1.1. Now we complete the proof of Theorem 1.2.
$$ S_{t(x),\varPhi}f(x)=R_{\varPhi}g(x), $$
(3.3)
$$\begin{aligned} R_{\varPhi}g(x)={}& \int_{\mathbb{R}^{n}} e^{ix\cdot\xi}e^{i(t_{1}(x) \phi_{1}( \vert \xi_{1} \vert )+t_{2}(x)\phi_{2}( \vert \xi_{2} \vert )+ \cdots+t_{n}(x)\phi_{n}( \vert \xi_{n} \vert ))} \bigl(1+ \vert \xi_{1} \vert ^{2}\bigr)^{-\frac{s_{1}}{2}}\bigl(1+ \vert \xi_{2} \vert ^{2}\bigr) ^{-\frac{s_{2}}{2}} \cdots\\ &{}\times\bigl(1+ \vert \xi_{n} \vert ^{2}\bigr)^{-\frac{s_{n}}{2}} g(\xi)\,d \xi. \end{aligned}$$
$$ \Vert R_{\varPhi}g \Vert _{L^{q}(\mathbb{R}^{n})}\leq C \Vert g \Vert _{L^{2}(\mathbb{R}^{n})} $$
(3.4)
$$\begin{aligned} R_{N,\varPhi}g(x)={}& \rho_{N}(x) \int_{\mathbb{R}^{n}} e^{ix\cdot\xi}e^{i(t_{1}(x) \phi_{1}( \vert \xi_{1} \vert )+t_{2}(x)\phi_{2}( \vert \xi_{2} \vert )+ \cdots+t_{n}(x)\phi_{n}( \vert \xi_{n} \vert ))} \bigl(1+ \vert \xi_{1} \vert ^{2}\bigr)^{-\frac{s_{1}}{2}}\bigl(1+ \vert \xi_{2} \vert ^{2}\bigr) ^{-\frac{s_{2}}{2}} \\ &{}\times \cdots\times\bigl(1+ \vert \xi_{n} \vert ^{2} \bigr)^{-\frac{s_{n}}{2}} \sigma_{N}(\xi)g(\xi)\,d\xi. \end{aligned}$$
$$\begin{aligned} R^{\prime}_{N,\varPhi}h(\xi)={}& \sigma_{N}(\xi) \bigl(1+ \vert \xi_{1} \vert ^{2}\bigr)^{-\frac{s_{1}}{2}} \bigl(1+ \vert \xi_{2} \vert ^{2}\bigr) ^{-\frac{s_{2}}{2}} \cdots\bigl(1+ \vert \xi_{n} \vert ^{2}\bigr)^{-\frac{s_{n}}{2}} \int_{\mathbb{R}^{2}} e^{-ix\cdot\xi}e^{-it_{1}(x) \phi_{1}( \vert \xi_{1} \vert )} \\ &{}\times e^{-i(t_{2}(x)\phi_{2}( \vert \xi_{2} \vert )+ \cdots+t_{n}(x)\phi_{n}( \vert \xi_{n} \vert ))}\rho_{N}(x)h(x)\,dx, \end{aligned}$$
$$ \Vert R_{N,\varPhi}g \Vert _{L^{q}(\mathbb{R}^{n})}\leq C \Vert g \Vert _{L^{2}(\mathbb{R}^{n})}. $$
(3.5)
$$ \bigl\Vert R_{N,\varOmega}^{\prime}h \bigr\Vert _{L^{2}(\mathbb{R}^{n})}\leq C \Vert h \Vert _{L^{q^{\prime}}(\mathbb{R}^{n})}, $$
(3.6)
$$ \bigl\Vert R_{N,\varPhi}^{\prime}h \bigr\Vert _{L^{2}(\mathbb{R}^{n})}^{2}= \int \bigl\vert R^{\prime}_{N,\varPhi}h(\xi) \bigr\vert ^{2}\,d\xi = \int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}} K_{N}(x,y)\rho_{N}(x) \rho_{N}(y)h(x)\overline{h(y)}\,dx\,dy, $$
(3.7)
$$ K_{N}(x,y)=K^{1}_{N}(x,y) K^{2}_{N}(x,y)\cdots K^{n}_{N}(x,y), $$
(3.8)
$$ K^{i}_{N}(x,y)= \int_{\mathbb{R}}\bigl(1+\xi_{i}^{2} \bigr)^{-s_{i}} e^{i(y_{i}-x_{i})\xi_{i}} e^{i(t_{i}(y)-t_{i}(x))\phi(|\xi_{i}|)} \psi_{N}( \xi_{i})^{2}\,d\xi_{i}, $$
(3.9)
$$ \bigl\vert K^{i}_{N}(x,y) \bigr\vert \leq C\frac{1}{ \vert x_{i}-y_{i} \vert ^{\beta_{i}}}, $$
(3.10)
$$P_{i}f(x_{1},x_{2},\ldots, x_{n})= \int_{\mathbb{R}} \frac{1}{ \vert x_{i}-y_{i} \vert ^{1-\sigma_{i}}} f(x_{1}, \ldots,x_{i-1},y_{i}, x_{i+1},\ldots,x_{n}) \,dy_{i}, $$
$$\begin{aligned} &\int \bigl\vert R_{N,\varPhi}^{\prime}h(\xi) \bigr\vert ^{2}\,d\xi \\ &\quad \leq C \int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}} \frac{1}{ \vert x_{1}-y_{1} \vert ^{1-\sigma_{1}}} \frac{1}{ \vert x_{2}-y_{2} \vert ^{1-\sigma_{2}}} \cdots \frac{1}{ \vert x_{n}-y_{n} \vert ^{1-\sigma_{n}}} \bigl\vert h(x) \bigr\vert \bigl\vert h(y) \bigr\vert \,dx\,dy \\ &\quad = C \int_{\mathbb{R}^{n}} \biggl( \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}}\frac{1}{ \vert x_{n}-y_{n} \vert ^{1-\sigma_{n}}} \frac{1}{ \vert x_{n-1}-y_{n-1} \vert ^{1-\sigma_{n-1}}}\cdots \frac {1}{ \vert x_{3}-y_{3} \vert ^{1-\sigma_{3}}} \\ &\qquad{}\times\frac {1}{ \vert x_{2}-y_{2} \vert ^{1-\sigma_{2}}} \biggl( \int\frac {1}{ \vert x_{1}-y_{1} \vert ^{1-\sigma_{1}}} \bigl\vert h(y_{1},y_{2}, \ldots, y_{n}) \bigr\vert \,dy_{1} \biggr) \,dy_{2}\,dy_{3}\cdots \,dy_{n} \biggr) \bigl\vert h(x) \bigr\vert \,dx \\ &\quad = C \int_{\mathbb{R}^{n}}P_{n}P_{n-1}\cdots P_{2}P_{1} \vert h \vert (x) \bigl\vert h(x) \bigr\vert \,dx. \end{aligned}$$
(3.11)
$$ \int \bigl\vert R_{N,\varPhi}^{\prime}h(\xi) \bigr\vert ^{2}\,d\xi\leq C \bigl\Vert P_{n}P_{n-1}\cdots P_{2}P_{1} \vert h \vert \bigr\Vert _{L^{q}(\mathbb{R}^{n})} \Vert h \Vert _{L^{q^{\prime}}(\mathbb{R}^{n})}. $$
(3.12)
4 The proof of Lemma 1.4
To prove Lemma 1.4, 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
\(F\in C^{\infty}(I)\)
be real-valued and assume that
\(\psi\in C^{\infty}(I)\).
(i)
Assume that
\(|F^{\prime}(x)|\geq\lambda>0\)
for
\(x\in I\)
and that
\(F^{\prime}\)
is monotonic on
I. Then
where C does not depend on
F, ψ, or
I.
$$\biggl\vert \int_{a}^{b}e^{iF(x)}\psi(x)\,dx \biggr\vert \leq C\frac{1}{\lambda }\biggl\{ \bigl\vert \psi(b) \bigr\vert + \int_{a}^{b} \bigl\vert \psi^{\prime}(x) \bigr\vert \,dx\biggr\} , $$
(ii)
Assume that
\(|F^{\prime\prime}(x)|\geq\lambda>0\)
for
\(x\in I\). Then
where C does not depend on
F, ψ, or
I.
$$\biggl\vert \int_{a}^{b}e^{iF(x)}\psi(x)\,dx \biggr\vert \leq C\frac{1}{\lambda ^{1/2}}\biggl\{ \bigl\vert \psi(b) \bigr\vert + \int_{a}^{b} \bigl\vert \psi^{\prime}(x) \bigr\vert \,dx\biggr\} , $$
Proof of Lemma 1.4
By conditions (H1) and (H2), there exist positive constants \(C_{i}\) (\(i=1,2,\ldots,6\)) so that for \(r\geq1\) and \(m_{2}>1\) such that
and for \(0< r<1\) and \(m_{1}>1\) such that
Set
To prove Lemma 1.4, it suffices to show that there exists a constant C such that for \(x\in\mathbb{R}\setminus\{ 0\}\), \(\beta=\frac{\alpha+\frac{m_{2}}{2}-1}{m_{2}-1}\) and \(N\in \Bbb {N}\),
where C depends only on α, \(m_{1}\), \(m_{2}\), \(C_{i}\)
\((i=1,2,\ldots,6)\), and μ.
$$ C_{1}r^{m_{2}-1}\leq \bigl\vert \phi^{\prime}(r) \bigr\vert \leq C_{2}r^{m_{2}-1} \quad\mbox{and}\quad \bigl\vert \phi^{\prime\prime}(r) \bigr\vert \geq C_{3}r^{m_{2}-2}, $$
(4.1)
$$ C_{4}r^{m_{1}-1}\leq \bigl\vert \phi^{\prime}(r) \bigr\vert \leq C_{5}r^{m_{1}-1} \quad\mbox{and}\quad \bigl\vert \phi^{\prime\prime}(r) \bigr\vert \geq C_{6}r^{m_{1}-2}. $$
(4.2)
$$J= \int_{\mathbb{R}} \frac{e^{i(d\phi(|\xi|)-x\xi)}}{(1+\xi ^{2})^{\frac{\alpha}{2}}} \mu\biggl(\frac{\xi}{N} \biggr)\,d\xi. $$
$$ \vert J \vert \leq C\frac{1}{ \vert x \vert ^{\beta}}, $$
(4.3)
Without loss of generality, we may assume \(\xi, d>0\). Denote \(\psi(\xi)={(1+\xi^{2})^{-\frac{\alpha}{2}}} \mu(\frac{\xi}{N})\), then we have
In fact, since \(\mu\in C_{0}^{\infty}(\mathbb{R})\) and \(\frac {1}{2}\leq\alpha\leq\frac{m_{2}}{2}\), we get
Noting that
we have
Since \(\mu\in C_{0}^{\infty}(\mathbb{R})\) and \(\frac{1}{2}\leq \alpha\leq\frac{m_{2}}{2}\), we obtain
and
By (4.7), (4.8), and (4.9), we get
Therefore, (4.4) follows from (4.5) and (4.10).
$$ \max_{\xi\geq0} \bigl\vert \psi(\xi) \bigr\vert + \int_{ 0}^{\infty} \bigl\vert \psi ^{\prime}( \xi) \bigr\vert \,d\xi\leq C. $$
(4.4)
$$ \max_{\xi\geq0} \bigl\vert \psi(\xi) \bigr\vert \leq C. $$
(4.5)
$$ \psi^{\prime}(\xi) =-\alpha\xi{\bigl(1+\xi^{2} \bigr)^{-\frac{\alpha}{2}-1}} \mu \biggl(\frac{\xi}{N} \biggr) +{\bigl(1+ \xi^{2}\bigr)^{-\frac{\alpha}{2}}}\frac{1}{N} \mu^{\prime} \biggl(\frac{\xi}{N} \biggr), $$
(4.6)
$$\begin{aligned} \int_{0}^{\infty} \bigl\vert \psi^{\prime}(\xi) \bigr\vert \,d\xi\leq{}&\alpha \int_{0}^{\infty}\xi{\bigl(1+\xi^{2} \bigr)^{-\frac{\alpha}{2}-1}} \biggl\vert \mu \biggl(\frac{\xi}{N} \biggr) \biggr\vert \,d\xi \\ &{}+ \int_{0}^{\infty}{\bigl(1+\xi^{2} \bigr)^{-\frac{\alpha}{2}}}\frac{1}{N} \biggl\vert \mu^{\prime} \biggl( \frac{\xi}{N} \biggr) \biggr\vert \,d\xi \\ =:{}&G_{1}+G_{2}. \end{aligned}$$
(4.7)
$$ G_{1}\leq C \int_{0}^{\infty}\xi{\bigl(1+\xi^{2} \bigr)^{-\frac{\alpha }{2}-1}}\,d\xi =C \int_{0}^{\infty}{\bigl(1+\xi^{2} \bigr)^{-\frac{\alpha}{2}-1}}\,d\bigl(1+\xi ^{2}\bigr) =C $$
(4.8)
$$ G_{2}\leq C \int_{0}^{\infty}\frac{1}{N} \biggl\vert \mu^{\prime} \biggl(\frac{\xi}{N} \biggr) \biggr\vert \,d\xi\leq C. $$
(4.9)
$$ \int_{ 0}^{\infty} \bigl\vert \psi^{\prime}(\xi) \bigr\vert \,d\xi\leq C. $$
(4.10)
To estimate (4.3), we choose a positive constant M such that \(M=\max\{(\frac{1}{\delta})^{m_{2}-1},2C_{5},2\}\), where δ is a small positive constant such that \(\delta ^{m_{2}-1}C_{2}\leq\frac{1}{2}\). Below, we show (4.3) by dividing two cases \(|x|\geq M\) and \(|x|< M\).
Case (I): \(|x|\geq M\). Let \(F(\xi)=d\phi(\xi)-x\xi\), we have
Denote \(\rho=(\frac{|x|}{d})^{\frac{1}{m_{2}-1}}\), then we have \(\delta\rho\geq1\). In fact, noting that \(|x|\geq(\frac{1}{\delta})^{m_{2}-1}\), \(0< d<1\), \(m_{2}>1\), and \(\frac{|x|}{d}>|x|\), it follows that \(\delta\rho >\delta|x|^{\frac{1}{m_{2}-1}}\geq1\). We choose a large positive constant λ such that \(\lambda\geq\max\{(\frac{2}{C_{1}}) ^{\frac{1}{m_{2}-1}},\delta\}\). Denote
Thus, we obtain
Firstly, we estimate \(J_{1}\). We will show that the following estimate holds:
Now we divide the verification of (4.12) into two cases according to the value of ξ.
$$F^{\prime}(\xi)=d\phi^{\prime}(\xi)-x,\qquad F^{\prime\prime}(\xi )=d \phi^{\prime\prime}(\xi). $$
$$I_{1}=[0,\delta\rho],\qquad I_{2}=[\delta\rho,\lambda\rho],\qquad I_{3}=[\lambda\rho,\infty). $$
$$\begin{aligned} \vert J \vert &= \biggl\vert \int_{0}^{\infty}e^{iF(\xi)}\psi(\xi)\,d\xi \biggr\vert \leq \sum_{j=1}^{3} \biggl\vert \int_{I_{j}}e^{iF(\xi)}\psi(\xi)\,d\xi \biggr\vert =:\sum _{j=1}^{3} J_{j}. \end{aligned}$$
(4.11)
$$ \bigl\vert F^{\prime}(\xi) \bigr\vert \geq \frac{ \vert x \vert }{2},\quad \xi\in[0,\delta \rho]. $$
(4.12)
Case (I-a): \(\xi\in[0,1)\). Since \(m_{1}>1\) and \(0< d<1\), we have
By (4.13), if \(\xi\in[0,1)\), we get
$$ d \bigl\vert \phi^{\prime}(\xi) \bigr\vert \leq C_{5} \,d \xi^{m_{1}-1}\leq C_{5}\leq \frac {M}{2}\leq\frac{ \vert x \vert }{2}. $$
(4.13)
$$ \bigl\vert F^{\prime}(\xi) \bigr\vert \geq \vert x \vert -d \bigl\vert \phi^{\prime}(\xi) \bigr\vert \geq \frac{ \vert x \vert }{2}. $$
(4.14)
Case (I-b): \(\xi\in[1,\delta\rho]\). Since \(m_{2}>1\), we have
By (4.15), we get
Therefore (4.12) follows from (4.14) and (4.16). Since \(\phi^{\prime}\) is monotonic on \(\mathbb{R}^{+}\) by condition (H3) and \(d>0\), it follows that \(F^{\prime}\) is monotonic on \(\xi\in I_{1}\). Thus, by (i) of Lemma 4.1 and estimate (4.12), (4.4), we have
where we use \(|x|\geq2\) and the fact \(\frac{1}{2}\leq\beta\leq1\). Next we prove estimate \(J_{3}\). Since \(\xi\geq\lambda(\frac{|x|}{d})^{\frac{1}{m_{2}-1}}>1\) and \(\lambda\geq(\frac{2}{C_{1}})^{\frac{1}{m_{2}-1}}\),
it follows that
Thus, by (i) of Lemma 4.1 and estimate (4.18), (4.4), we have
where we use \(|x|\geq2\) and the fact \(\frac{1}{2}\leq\beta\leq1\). Now, we give estimate \(J_{2}\). Since \(\xi\in I_{2}\), we have \(|\xi|\geq1\). By (4.1), we obtain
We first prove that the following estimate holds:
In fact, since \(\mu\in C_{0}^{\infty}(\mathbb{R})\) and \(\frac {1}{2}\leq\alpha\leq\frac{m_{2}}{2}\), we get
By (4.6), we have
Since \(\mu\in C_{0}^{\infty}(\mathbb{R})\) and \(\frac{1}{2}\leq \alpha\leq\frac{m_{2}}{2}\), we obtain
and
By (4.23), (4.24), and (4.25), we get
Therefore, (4.21) follows from (4.22) and (4.26). Thus, by (ii) of Lemma 4.1 and estimate (4.20), (4.21), we have
Here in the last inequality we use the fact \(\frac{\alpha-\frac{1}{2}}{m_{2}-1}\geq0\) and \(0< d<1\). Therefore, for \(|x|\geq M\), by estimates (4.11), (4.17), (4.19), and (4.27), it follows (4.3).
$$ d \bigl\vert \phi^{\prime}(\xi) \bigr\vert \leq C_{2} \,d \xi^{m_{2}-1}\leq C_{2}\,d \delta ^{m_{2}-1}\frac{ \vert x \vert }{d}\leq C_{2} \delta^{m_{2}-1} \vert x \vert \leq\frac{ \vert x \vert }{2}. $$
(4.15)
$$ \bigl\vert F^{\prime}(\xi) \bigr\vert \geq \vert x \vert -d \bigl\vert \phi^{\prime}(\xi) \bigr\vert \geq \frac{ \vert x \vert }{2}. $$
(4.16)
$$ \vert J_{1} \vert \leq C\frac{1}{ \vert x \vert } \leq C \frac{1}{ \vert x \vert ^{\beta}}, $$
(4.17)
$$d \bigl\vert \phi^{\prime}(\xi) \bigr\vert \geq C_{1} \,d \xi^{m_{2}-1}\geq C_{1} \,d\lambda ^{m_{2}-1} \frac{ \vert x \vert }{d}\geq2 \vert x \vert , $$
$$ \bigl\vert F^{\prime}(\xi) \bigr\vert \geq2 \vert x \vert - \vert x \vert = \vert x \vert ,\quad \xi\in[\lambda\rho ,\infty). $$
(4.18)
$$ \vert J_{3} \vert \leq C\frac{1}{ \vert x \vert } \leq C \frac{1}{ \vert x \vert ^{\beta}}, $$
(4.19)
$$ \bigl\vert F^{\prime\prime}(\xi) \bigr\vert \geq d \bigl\vert \phi^{\prime\prime}(\xi) \bigr\vert \geq C_{3}\,d \xi^{m_{2}-2} \geq C_{3}\,d \biggl(\frac{ \vert x \vert }{d} \biggr)^{\frac {m_{2}-2}{m_{2}-1}}. $$
(4.20)
$$ \max_{I_{2}} \vert \psi \vert + \int_{I_{2}} \bigl\vert \psi^{\prime} \bigr\vert \,d\xi \leq C \biggl(\frac{ \vert x \vert }{d} \biggr)^{-\frac{\alpha}{m_{2}-1}}. $$
(4.21)
$$ \max_{\xi\in A_{2}} \bigl\vert \psi(\xi) \bigr\vert \leq C(\delta\rho)^{-\alpha }=C\delta^{-\alpha}(\rho)^{-\alpha}= C \delta^{-\alpha} \biggl(\frac{ \vert x \vert }{d} \biggr)^{-\frac{\alpha}{m_{2}-1}}. $$
(4.22)
$$\begin{aligned} \int_{A_{2}} \bigl\vert \psi^{\prime}(\xi) \bigr\vert \,d\xi &\leq\alpha \int_{A_{2}}\xi{\bigl(1+\xi^{2}\bigr)^{-\frac{\alpha}{2}-1}} \biggl\vert \mu \biggl(\frac{\xi}{N} \biggr) \biggr\vert \,d\xi+ \int_{A_{2}}{\bigl(1+\xi^{2}\bigr)^{-\frac{\alpha}{2}}} \frac{1}{N} \biggl\vert \mu^{\prime} \biggl(\frac{\xi}{N} \biggr) \biggr\vert \,d\xi \\ &=:L_{1}+L_{2}. \end{aligned}$$
(4.23)
$$ L_{1}\leq C \int_{A_{2}}\xi{\bigl(1+\xi^{2}\bigr)^{-\frac{\alpha}{2}-1}} \,d\xi \leq C \int_{\delta\rho}^{\lambda\rho}\xi^{-\alpha-1}\,d\xi =C \biggl( \frac{ \vert x \vert }{d} \biggr)^{-\frac{\alpha}{m_{2}-1}} $$
(4.24)
$$ L_{2}\leq C (\delta\rho)^{-\alpha} \int_{A_{2}}\frac{1}{N} \biggl\vert \mu^{\prime} \biggl(\frac{\xi}{N} \biggr) \biggr\vert \,d\xi\leq C \biggl( \frac{ \vert x \vert }{d} \biggr)^{-\frac{\alpha}{m_{2}-1}}. $$
(4.25)
$$ \int_{ 0}^{\infty} \bigl\vert \psi^{\prime}(\xi) \bigr\vert \,d\xi\leq C \biggl(\frac { \vert x \vert }{d} \biggr)^{-\frac{\alpha}{m_{2}-1}}. $$
(4.26)
$$ \vert J_{2} \vert \leq d^{-\frac{1}{2}} \biggl( \frac{ \vert x \vert }{d} \biggr)^{-\frac {m_{2}-2}{2(m_{2}-1)}} \biggl(\frac{ \vert x \vert }{d} \biggr)^{-\frac{\alpha}{m_{2}-1}} =C\frac{d^{\frac{\alpha-\frac{1}{2}}{m_{2}-1}}}{ \vert x \vert ^{\frac{\alpha+\frac{m_{2}}{2}-1}{m_{2}-1}}}\leq C\frac{1}{ \vert x \vert ^{\beta}}. $$
(4.27)
Case (II): \(|x|< M\). Now we divide the verification of (4.3) into three cases according to the value of α for \(|x|< M\).
Case (II-a): \(\alpha>1\). Since \(\mu\in C_{0}^{\infty}(\mathbb{R})\) and \(\alpha>1\), we get
Noting that \(|x|< M\) and \(\frac{1}{2}\leq\beta\leq1\), by (4.28), we have
which follows (4.3).
$$ \vert J \vert = \biggl\vert \int_{0}^{\infty} \frac{e^{i(d\phi( \vert \xi \vert )-x\xi )}}{(1+\xi^{2})^{\frac{\alpha}{2}}} \mu\biggl( \frac{\xi}{N}\biggr)\,d\xi \biggr\vert \leq C \int_{0}^{\infty} \frac {1}{{(1+\xi^{2})^{\frac{\alpha}{2}}}}\,d\xi\leq C. $$
(4.28)
$$\vert J \vert \leq C=C \vert x \vert ^{\beta}\frac{1}{ \vert x \vert ^{\beta}} \leq CM^{\beta}\frac {1}{ \vert x \vert ^{\beta}}=C\frac{1}{ \vert x \vert ^{\beta}}, $$
Case (II-b): \(\frac{1}{2}\leq\alpha<1\). By the mean value theorem, when \(\frac{1}{2}\leq\alpha<1\), we have
By (4.29), we obtain
Noting that \(\frac{1}{2}\leq\alpha<1\), by (4.30), we have
and
By (4.31), we have
By Lemma 1.3, we obtain
Noting that \(\frac{1}{|x|}>\frac{1}{M}\), and the fact \(\beta \geq1-\alpha\), it follows from \(\frac{1}{2}\leq\alpha<1\), \(\frac {1}{2}\leq\beta\leq1 \) that
Therefore, by (4.33) and (4.34), we have
Hence, (4.3) holds from (4.32) and (4.35).
$$ 0< \bigl(1+\xi^{2}\bigr)^{\frac{\alpha}{2}}- \xi^{\alpha} =\bigl(1+\xi^{2}\bigr)^{\frac{\alpha}{2}}-\bigl( \xi^{2}\bigr)^{\frac{\alpha}{2}} \leq\frac{\alpha}{2}\bigl( \xi^{2}\bigr)^{\frac{\alpha}{2}-1}\leq\xi ^{\alpha-2}. $$
(4.29)
$$ \frac{1}{\xi^{\alpha}}-\frac{1}{(1+\xi^{2})^{\frac{\alpha}{2}}} =O \biggl(\frac{1}{\xi^{\alpha+2}} \biggr),\quad \xi\rightarrow\infty. $$
(4.30)
$$ \int_{0}^{\infty} \biggl\vert \frac{1}{\xi^{\alpha}}- \frac{1}{(1+\xi^{2})^{\frac{\alpha}{2}}} \biggr\vert \,d\xi \leq C $$
(4.31)
$$\begin{aligned} \vert J \vert = \biggl\vert \int_{0}^{\infty} \frac{e^{i(d\phi( \vert \xi \vert )-x\xi )}}{(1+\xi^{2})^{\frac{\alpha}{2}}} \mu\biggl( \frac{\xi}{N}\biggr)\,d\xi \biggr\vert \leq{}& \biggl\vert \int_{0}^{\infty}e^{i(d\phi( \vert \xi \vert )-x\xi)} \biggl( \frac{1}{(1+\xi^{2})^{\frac{\alpha}{2}}} -\frac{1}{\xi^{\alpha}} \biggr)\mu\biggl(\frac{\xi}{N}\biggr) \,d\xi \biggr\vert \\ &{} + \biggl\vert \int_{0}^{\infty}e^{i(d\phi( \vert \xi \vert )-x\xi)} \frac{1}{\xi^{\alpha}}\mu \biggl(\frac{\xi}{N}\biggr)\,d\xi \biggr\vert \\ ={}& :K_{1}+K_{2}. \end{aligned}$$
$$ \vert K_{1} \vert \leq C=C \vert x \vert ^{\beta}\frac{1}{ \vert x \vert ^{\beta}}\leq CM^{\beta}\frac {1}{ \vert x \vert ^{\beta}}=C \frac{1}{ \vert x \vert ^{\beta}}. $$
(4.32)
$$ \vert K_{2} \vert \leq C\frac{1}{ \vert x \vert ^{1-\alpha}}. $$
(4.33)
$$ \vert x \vert ^{1-\alpha}= \vert x \vert ^{\beta} \vert x \vert ^{1-\alpha-\beta} = \vert x \vert ^{\beta} \biggl(\frac{1}{ \vert x \vert } \biggr)^{\beta-(1-\alpha)}\geq C \vert x \vert ^{\beta}. $$
(4.34)
$$ \vert K_{2} \vert \leq C\frac{1}{ \vert x \vert ^{\beta}}. $$
(4.35)
Case (II-c): \(\alpha=1\). From the proof of Lemma 1.3, noting that \(M\geq2\), we may get
and
By (4.36) and \(\frac{1}{2}\leq\beta\leq1\), for \(0<|x|\leq\frac{1}{2}\), we have
By (4.37), for \(\frac{1}{2}<|x|<M\), we have
Thus, for \(\alpha=1\), \(|x|< M\), by (4.38) and (4.39), we get
which is just estimate (4.3).
$$ \vert J \vert \leq C \log\biggl(\frac{1}{ \vert x \vert }\biggr) \quad\text{if } 0< \vert x \vert \leq \frac{1}{2} $$
(4.36)
$$ \vert J \vert \leq C \quad\text{if } \frac{1}{2}< \vert x \vert < M. $$
(4.37)
$$ \vert J \vert \leq C \log\biggl(\frac{1}{ \vert x \vert }\biggr)\leq C\frac{1}{ \vert x \vert ^{\beta}}. $$
(4.38)
$$ \vert J \vert \leq C=C \vert x \vert ^{\beta} \frac{1}{ \vert x \vert ^{\beta}}\leq CM^{\beta}\frac {1}{ \vert x \vert ^{\beta}}=C\frac{1}{ \vert x \vert ^{\beta}}. $$
(4.39)
$$\vert J \vert \leq C\frac{1}{ \vert x \vert ^{\beta}}, $$
5 Conclusion
In this paper, by linearizing the maximal operator and duality methods, and applying the results of Lemma 1.3 and Lemma 1.4, we obtain the maximal global \(L^{q}\) inequalities (1.8) and (1.9) for multiparameter oscillatory integral \(S_{t,\varPhi}\). These estimates are apparently good extensions to maximal global \(L^{q}\) inequalities (1.6) and (1.7) for the multiparameter fractional 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.
Competing interests
The authors declare that they have no competing interests.
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