$L^p$ improving multilinear Radon-like transforms

Abstract

We characterize (up to endpoints) the k-tuples (p 1 , . . . , p k ) for which certain k-linear generalized Radon transforms map the productThis generalizes a result of Tao and Wright.

Restricted convolution inequalities, multilinear operators and applications

“…We point out that there have been other works on restrictions of convolutions (e.g., [2,1]) and estimates for multilinear convolution operators [16,26], but the results there seem to be of a different nature than those obtained here.…”

UniformLp-improving for weighted averages on curves

“…Proof. There are similar results in [3,5,21,24], but without the uniformity, so we give a complete proof. The upper bound, det DΨ J C 0 (B(c0)) ∼ 1 is an immediate consequence of (4.12) for m ≥ 2, by Picard's existence theorem.…”

“…The Hörmander condition is the statement that P x0 = ∅ for each x 0 ∈ U . When the X i are nonvanishing vector fields tangent to the fibers of the π i , this is the finite type hypothesis in [24,21].…”

“…For the unweighted bilinear operator in the 'polynomial-like' case, the endpoint restricted weak type bounds are known and are due to Gressman in [13]; in the multilinear case, the corresponding estimates follow by combining his techniques with arguments in [21]. The deduction of endpoint bounds from the arguments in [13] does not seem to be immediate in the weighted case, and so these questions remain open except for certain special configurations (such as convolution or restricted X-ray transform along polynomial curves).…”

“…where a is a continuous cutoff function and the π i : R d → R d−1 are smooth submersions. In [24,21], near endpoint estimates of the form…”