Restricted convolution inequalities, multilinear operators and applications

Abstract

Abstract. For functions F, G on R n , any k-dimensional affine subspace H ⊂ R n , 1 ≤ k < n, and exponents p, q, r ≥ 2 with 1 p + 1 q + 1 r = 1, we prove the estimateHere, the mixed norms on the right are defined in terms of the Fourier transform byLebesgue measure on the affine subspace H ⊥ ξ := ξ + H ⊥ . Dually, one obtains restriction theorems for the Fourier transform on affine subspaces. We use this, and a maximal variant, to prove results for a variety of multilinear convolution operators, including L p -improving bounds for measures; bilinear variants of Stein's spherical maximal theorem; estimates for m-linear oscillatory integral operators; Sobolev trace inequalities; and bilinear estimates for solutions to the wave equation.

Lp estimates for multilinear convolution operators defined with spherical measure

“…Observe that the operator A for d = 1 is same as T for n = 2. The operator A and the corresponding bilinear maximal function were introduced and studied in [7]. Later, in [3,9] authors established partial results obtaining…”

On families between the Hardy–Littlewood and spherical maximal functions

“…When m=1, S m reduces to S in (1). The bilinear analogue of Stein's spherical maximal function (when m = 2) was first introduced in [10] by Geba, Greenleaf, Iosevich, Palsson, and Sawyer who obtained the first bounds for it but later improved bounds were provided by [3], [12], [15] and [17]. A multilinear (non-maximal) version of this operator when all input functions lie in the same space L p (R) was previously studied by Oberlin [21].…”

Discrete multilinear maximal functions and number theory

“…A prototypical curved object is the sphere, and maximal spherical averaging operators arise naturally in many contexts. From the multilinear view, optimal Lebesgue space bounds for (multilinear) spherical maximal functions had been pursued in many papers, such as [6,10,11,22], building upon work of Stein [23] and Bourgain [7]. From a discrete perspective, Magyar-Stein-Wainger showed optimal bounds that were both different from the continuous ones and heavily employed number theoretic techniques [19].…”