Sharp estimates for oscillatory integral operators via polynomial partitioning

Abstract

The sharp range of L p -estimates for the class of Hörmander-type oscillatory integral operators is established in all dimensions under a positivedefinite assumption on the phase. This is achieved by generalising a recent approach of the first author for studying the Fourier extension operator, which utilises polynomial partitioning arguments. The main result implies improved bounds for the Bochner-Riesz conjecture in dimensions n ě 4.3 Strictly speaking, in [9] weaker L 8´Lp bounds are proven, but the methods can be used to establish the L p´Lp strengthening: see, for instance, [14, §9] where the L p´Lp argument appears (although in a slightly disguised form). 4 In particular, Lee [19] proved that for positive-definite phases (1.4) holds for p ě 2¨n`2 n in all dimensions, extending the range in Theorem 1.1 when n is odd.

Sharp Resolvent Estimates Outside of the Uniform Boundedness Range

“…But, the sharp bound for (−∆ − z) −1 p→q with general p, q cannot be deduced from interpolation between previously known estimates. For the purpose we need to make use of L p theory of oscillatory integral operators of Carleson-Sjölin type under the additional elliptic condition ( [11,27,46,36,23], also see Section 2.1 below).…”

“…It is important to obtain the optimal L p -L q bounds for each of the operators which are given by the dyadic decomposition. For the purpose we use the Carleson-Sjölin reduction ( [11,46]), and combine this with Theorem 2.2 in Section 2.1 ( [23]) and bilinear estimate for the extension operator associated to the hypersurfaces of elliptic type ( [48]). For more details, see Section 2 (Corollary 2.12).…”

“…These results are based on multilinear estimates due to Bennett, Carbery and Tao [3] and the method of polynomial partitioning due to Guth [22,21]. We record here the recent sharp result due to Guth, Hickman and Iliopoulou [23].…”

“…The estimate (2.4) with p = q in Theorem 2.2 is uniform under small smooth perturbation of the phase Φ and the amplitude a. In fact, the estimate (2.4) in [23] was obtained by running induction argument over a class of operators while the phase functions are properly normalized. See [23, Lemma 4.1, Definition 11.3].…”

“…. For most recent developments see Bourgain and Guth [6] and Guth, Hickman and Iliopoulou [23]. These results are based on multilinear estimates due to Bennett, Carbery and Tao [3] and the method of polynomial partitioning due to Guth [22,21].…”