Global maximal inequality to a class of oscillatory integrals

In the present paper, we give the global Lq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{q}$\end{document} estimates for maximal operators generated by multiparameter oscillatory integral St,Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S_{t,\varPhi}$\end{document}, which is defined by St,Φf(x)=(2π)−n∫Rneix⋅ξei(t1ϕ1(|ξ1|)+t2ϕ2(|ξ2|)+⋯+tnϕn(|ξn|))fˆ(ξ)dξ,x∈Rn,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{t,\varPhi}f(x)=(2\pi)^{-n} \int_{\mathbb{R}^{n}} e^{ix\cdot\xi}e^{i(t_{1} \phi_{1}(|\xi_{1}|)+t_{2}\phi_{2}(|\xi_{2}|)+ \cdots+t_{n}\phi_{n}(|\xi_{n}|))}\hat{f}(\xi)\,d\xi,\quad x\in\mathbb{R}^{n}, $$\end{document} where n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\geq2$\end{document} and f is a Schwartz function in S(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{S}(\mathbb {R}^{n})$\end{document}, t=(t1,t2,…,tn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t=(t_{1},t_{2},\ldots,t_{n})$\end{document}, Φ=(ϕ1,ϕ2,…,ϕn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varPhi=(\phi_{1},\phi_{2},\ldots,\phi_{n})$\end{document}, ϕi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi_{i}$\end{document}(i=1,2,3,…,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(i=1,2,3,\ldots, n)$\end{document} is a function on R+→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{+}\rightarrow\mathbb{R}$\end{document}, 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.


Introduction and main results
Let f be a Schwartz function in S(R n ) and S t f (x) = u(x, t) = (2π) -n R n e ix·ξ +it|ξ | af (ξ ) dξ , (x, t) ∈ R n × R.
It is well known that S t f (x) is the solution of the fractional Schrödinger equation (1.1) Heref denotes the Fourier transform of f defined byf (ξ ) = R n e -iξ ·x f (x) dx.
We recall the homogeneous Sobolev spaceḢ s (R n ) (s ∈ R), which is defined bẏ |ξ | 2s f (ξ ) 2 dξ 1/2 < ∞ , and the non-homogeneous Sobolev space H s (R n ) (s ∈ R), which is defined by Maximal operator S * f associated with the family of operators {S t } 0<t<1 is defined by It is well known that if a = 2, u is the solution of the Schrödinger equation (1.2) In 1979, Carleson [4] proposed a problem: if f ∈ H s (R n ) for which s does Carleson first considered this problem for dimension n = 1 in [4] and showed that the convergence (1.3) holds for f ∈ H s (R) with s ≥ 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 ∈ H s (R n ), the pointwise convergence (1.3) follows from the local estimate S * f L q (B n ) ≤ C f H s (R n ) , f ∈ H s R n , (1.4) for some q ≥ 1 and s ∈ R. Here B n is the unit ball centered at the origin in 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][31][32], and so on. For n ≥ 2 and a multiindex a = (a 1 , a 2 , . . . , a n ), with a j > 1 and f being a Schwartz function in S(R n ), we set where t = (t 1 , t 2 , . . . , t n ) ∈ R n . For n ≥ 2, the local maximal operator M * is defined by and the global maximal operator M * * is defined by The global estimate and In 2014, Sjolin and Soria [32] obtained the following results. 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][15][16][17], and so on. In the present paper, we consider the maximal estimates associated with multiparameter oscillatory integral S t,Φ defined by Here, n ≥ 2 and f is a Schwartz function in S(R n ), Φ = (φ 1 , φ 2 , . . . , φ n ), φ i (i = 1, 2, 3, . . . , n) is a function on R + → R. For n ≥ 2, the local maximal operator M * Φ is defined by and the global maximal operator M * * Φ is defined by The global estimates of maximal operators M * Φ and M * * Φ are defined by and Assume that φ : R + → R satisfies: (H1) There exists m 1 > 1 such that |φ (r)| ∼ r m 1 -1 and |φ (r)| r m 1 -2 for all 0 < r < 1; (H2) There exists m 2 > 1 such that |φ (r)| ∼ r m 2 -1 and |φ (r)| r m 2 -2 for all r ≥ 1; (H3) Either φ (r) > 0 or φ (r) < 0 for all r > 0. Now we state our main results as follows. ) and set |m| = m 1,2 + m 2,2 + · · · + m n,2 . Assume that n ≥ 2 and φ i (i = 1, 2, 3, . . . , 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 ≥ n 2 -|m| 4 + |m| q -n q .
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.
Here the constant C may depend on α and m 1 , m 2 , and μ but not on x, d, and N .

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 ∈ C ∞ (I) be real-valued and assume that ψ ∈ C ∞ (I).
(i) Assume that |F (x)| ≥ λ > 0 for x ∈ I and that F is monotonic on where C does not depend on F, ψ, or I.
where C does not depend on F, ψ, or I.
Case (II): |x| < M. Now we divide the verification of (4.3) into three cases according to the value of α for |x| < M.

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,Φ . 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].