Global maximal inequality to a class of oscillatory integrals.

In the present paper, we give the global L q estimates for maximal operators generated by multiparameter oscillatory integral S t , Φ , which is defined by S t , Φ f ( x ) = ( 2 π ) - n ∫ R n e i x ⋅ ξ e i ( t 1 ϕ 1 ( | ξ 1 | ) + t 2 ϕ 2 ( | ξ 2 | ) + ⋯ + t n ϕ n ( | ξ n | ) ) f ˆ ( ξ ) d ξ , x ∈ R n , where n ≥ 2 and f is a Schwartz function in S ( R n ) , t = ( t 1 , t 2 , … , t n ) , Φ = ( ϕ 1 , ϕ 2 , … , ϕ n ) , ϕ i ( i = 1 , 2 , 3 , … , n ) is a function on R + → 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.

Introduction and main results

Let f be a Schwartz function in and It is well known that is the solution of the fractional Schrödinger equation Here f̂ denotes the Fourier transform of f defined by .

We recall the homogeneous Sobolev space , which is defined by and the non-homogeneous Sobolev space , which is defined by Maximal operator associated with the family of operators is defined by

It is well known that if , u is the solution of the Schrödinger equation In 1979, Carleson [4] proposed a problem: if for which s does

Carleson first considered this problem for dimension in [4] and showed that the convergence (1.3) holds for with , 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 , the pointwise convergence (1.3) follows from the local estimate for some and . Here is the unit ball centered at the origin in . 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.

For and a multiindex , with and f being a Schwartz function in , we set where . For , the local maximal operator is defined by and the global maximal operator is defined by The global estimate and

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

Theorem A

([32])

Assume . Then, for every a, inequality (1.6) holds if and only if and .

Theorem B

([32])

Assume . Then, for every a and for , inequality (1.7) holds if and only if .

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 defined by Here, and f is a Schwartz function in , , is a function on . For , the local maximal operator is defined by and the global maximal operator is defined by The global estimates of maximal operators and are defined by and

Assume that satisfies:

There exists such that and for all ;

There exists such that and for all ;

Either or for all .

Now we state our main results as follows.

Theorem 1.1

Assume that and satisfies (H1)–(H3). If and , then the global estimate (1.8) holds.

Theorem 1.2

Let and set . Assume that and satisfies (H1)–(H3) with , . Then, for every m, inequality (1.9) holds if and .

Remark 1.1

There are many elements ϕ satisfying conditions (H1)–(H3), for instance, the fractional Schrödinger equation (), or , the Beam equation , the fourth-order Schrödinger equation , iBq , 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 , . and . Then for , , and  . Here the constant C may depend on s and , , and μ but not on x, t, or N.

Remark 1.2

The proof of Lemma 1.3 is similar to that of Lemma 2.1 in [10].

Lemma 1.4

Assume that ϕ satisfies (H1)–(H3) with , . , , and . Then for and  , where . Here the constant C may depend on α and , , and μ but not on x, d, and N.

Remark 1.3

Applying the result of Lemma 1.3, the proof of Lemma 1.4 is similar to that of Lemma 2.2 in [32]. The proof of Lemma 1.4 will be given in Sect. 4.

The proof of Theorem 1.1

Assume that , satisfies (H1)–(H3). For , let be a measurable function on with . Denote , we set For and , that is, and . 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 , 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 such that if and if . And we choose a real-valued function such that if and if , and set . For and  , we set and . For , , and for  , we define The adjoint of is given by where and . To prove (2.4) it is sufficient to prove that By duality, to prove (2.5) it suffices to show that where . Thus, we have where and where and  . Since , we have . Therefore, by Lemma 1.3, (2.9), and (2.8), we obtain We define . Thus, by (2.7) and (2.10), we obtain Invoking Hölder’s inequality, we get Since , it follows that and the fact . Denote by the Riesz potential of order σ, which is defined by Applying the fact is bounded from to , we have where . 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.

The proof of Theorem 1.2

Assume that , satisfies (H1)–(H3) with , . For every and , we will prove that inequality (1.9) holds if , where . For , let be a measurable function on with . Denote , we set By linearizing the maximal operator, to prove the global estimate (1.9) it suffices to show that Since , where , . Therefore, to prove (3.1) it suffices to prove that

Let , 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 such that if and if . And we choose a real-valued function such that if and if , and set for . For  , we set and . For , , and  , we define The adjoint of is given by where and . To prove (3.4) it suffices to prove that By duality, to prove (3.5) it is sufficient to show that where . Thus, we have where and where and  . Denote , since , , and , it follows that , . Therefore, by (3.9) and Lemma 1.4, we obtain where . Denote , we define . Thus, by (3.7), (3.8), and (3.10), we obtain Invoking Hölder’s inequality, we get Since and , , . It follows that and , . 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.

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 and set . Let be real-valued and assume that .

Assume that for and that is monotonic on I. Then where C does not depend on F, ψ, or I.

Assume that for . Then where C does not depend on F, ψ, or I.

Proof of Lemma 1.4

By conditions (H1) and (H2), there exist positive constants () so that for and such that and for and such that Set To prove Lemma 1.4, it suffices to show that there exists a constant C such that for , and , where C depends only on α, , , , and μ.

Without loss of generality, we may assume . Denote , then we have In fact, since and , we get Noting that we have Since and , we obtain and By (4.7), (4.8), and (4.9), we get Therefore, (4.4) follows from (4.5) and (4.10).

To estimate (4.3), we choose a positive constant M such that , where δ is a small positive constant such that . Below, we show (4.3) by dividing two cases and .

Case (I): . Let , we have Denote , then we have . In fact, noting that , , , and , it follows that . We choose a large positive constant λ such that . Denote Thus, we obtain Firstly, we estimate . 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.

Case (I-a): . Since and , we have By (4.13), if , we get

Case (I-b): . Since , we have By (4.15), we get Therefore (4.12) follows from (4.14) and (4.16). Since is monotonic on by condition (H3) and , it follows that is monotonic on . Thus, by (i) of Lemma 4.1 and estimate (4.12), (4.4), we have where we use and the fact . Next we prove estimate . Since and , it follows that Thus, by (i) of Lemma 4.1 and estimate (4.18), (4.4), we have where we use and the fact . Now, we give estimate . Since , we have . By (4.1), we obtain We first prove that the following estimate holds: In fact, since and , we get By (4.6), we have Since and , 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 and . Therefore, for , by estimates (4.11), (4.17), (4.19), and (4.27), it follows (4.3).

Case (II): . Now we divide the verification of (4.3) into three cases according to the value of α for .

Case (II-a): . Since and , we get Noting that and , by (4.28), we have which follows (4.3).

Case (II-b): . By the mean value theorem, when , we have By (4.29), we obtain Noting that , by (4.30), we have and By (4.31), we have By Lemma 1.3, we obtain Noting that , and the fact , it follows from , that Therefore, by (4.33) and (4.34), we have Hence, (4.3) holds from (4.32) and (4.35).

Case (II-c): . From the proof of Lemma 1.3, noting that , we may get and By (4.36) and , for , we have By (4.37), for , we have Thus, for , , by (4.38) and (4.39), we get which is just estimate (4.3).

Summing up all the above estimates, we complete the proof of estimate (4.3) and finish the proof of Lemma 1.4. □

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 inequalities (1.8) and (1.9) for multiparameter oscillatory integral . These estimates are apparently good extensions to maximal global inequalities (1.6) and (1.7) for the multiparameter fractional Schrödinger equation in [32].