$L^p$ estimates for oscillatory integral operators

An endpoint boundedness result is established for a class of oscillatory integral operators. 1. This paper deals with a problem left open in [6]. Let ∆ = {(x, x) : x ∈ R}, K ∈ C∞(R2\∆) and satisfy | ∂ K ∂xj∂yl (x, y)| ≤ Ajl|x− y|−j−l (1) for (x, y) ∈ R2\∆ and j, l ≥ 0. For a, b ≥ 1, define the non-convolutional oscillatory integral operator Ta,b: (Ta,bf)(x) = ∫ R ei|x| a|y|bK(x, y)f(y)dy, (2) initially for f ∈ S(R). Such operators often arise in harmonic analysis (see e.g. [7], [8], [9]). It should be noted that when a = b = 1 and K ≡ 1 the operator in (2) is essentially the Fourier transform. The main problem under investigation concerns the L boundedness of the operators {Ta,b}. A quick examination of the operators Ta,b with K ≡ 1 reveals that the L inequality ‖Ta,bf‖p ≤ Ca,b,p‖f‖p (3) cannot hold (for all f ∈ S(R)) unless p = a+b a . Thus one is led to the problem of determining whether ‖Ta,bf‖ a+b a ≤ Ca,b‖f‖ a+b a (4) holds for f ∈ S(R) and a, b ≥ 1. The following is known: Theorem A. Suppose that K satisfies (1). Let Ta,b be given as in (2). Then: (i) If a > 1 and b > 1, then Ta,b is bounded from L a+b a to itself. (ii) If a = b = 1 (hence a+b a = 2), then Ta,b = T1,1 is bounded from L 2 to itself. Received by the editors January 8, 1998. 1991 Mathematics Subject Classification. Primary 42A38; Secondary 42A50. The second author was supported in part by NSF Grant DMS-9622979. c ©1999 American Mathematical Society 2995 License or copyright restrictions may apply to redistribution; see https://www.ams. org/journal-terms-of-use 2996 LESLIE C. CHENG AND YIBIAO PAN Part (i) of Theorem A was established in [6] (Theorem 3.1 on page 212), while part (ii) can be found in [5] (see also [7]). Apparently the remaining issue is to answer the following question: Question. Is Ta,b bounded from L a+b a to itself when either a = 1 or b = 1 ? It would seem quite reasonable to believe that the answer is “yes”. In fact, this was shown to be true in the special case where K(x, y) = |x − y|iσ with σ ∈ R ([6]). However, for the operators with generic K’s that satisfy (1) the question has remained unanswered. The main purpose of this paper is to completely resolve this issue. Namely we have: Theorem B. Let b > 1, K satisfy (1), and Tb = T1,b be given as in (2). Then there exists a constant Cb > 0 such that ‖Tbf‖b+1 ≤ Cb‖f‖b+1 (5) holds for f ∈ S(R). For a > 1 and b = 1 one obtains (4) from (5) by duality. It follows from Theorems A and B that Ta,b is bounded on L a+b a for all a, b ≥ 1. Part of our argument presented here is similar to the method used in [6], which goes back to [7]. The key new ingredients are reduction to pseudodifferential operators and the establishment of related L estimates. This combination allows us to tackle the above mentioned problem and may have applications elsewhere. For a nonnegative locally integrable function w we shall write ‖f‖p,w for ( ∫ R |f(x)|pw(x)dx)1/p . When w ≡ 1, we shall simply write ‖f‖p for ‖f‖p,w. 2. Let b > 1, η ∈ C∞(R) such that η(x) ≡ 1 when |x| ≤ 1/2 and η(x) ≡ 0 when |x| ≥ 1. For K(x, y) that satisfies (1) and m ∈ N we define Km(·, ·) and ωm(·, ·) by Km(x, y) = [1 − η(x− y)]K(x, y)η(x− y m ), (6)