Uniform $L_x^p - L^q_{x,r}$ Improving for Dilated Averages over Polynomial Curves

TL;DR

This paper extends the analysis of convolution operators over polynomial curves by considering both translations and dilations, establishing sharp Lebesgue space bounds in various dimensions using refined techniques.

Contribution

It introduces a new variant of the averaging operator over polynomial curves that includes dilations, and determines the optimal Lebesgue space estimates for this operator.

Findings

Sharp Lebesgue space bounds obtained for the operator in all dimensions except the endpoint.

Extension of the refinement method of Christ to handle dilations in averaging.

Demonstrates additional smoothing effects from averaging over dilates.

Abstract

Numerous authors have considered the problem of determining the Lebesgue space mapping properties of the operator $\mathcal{A}$ given by convolution with affine arc-length measure on some polynomial curve in Euclidean space. Essentially, $\mathcal{A}$ takes weighted averages over translates of the curve. In this paper a variant of this problem is discussed where averages over both translates and dilates of a fixed curve are considered. The sharp range of estimates for the resulting operator is obtained in all dimensions, except for an endpoint. The techniques used are redolent of those previously applied in the study of $\mathcal{A}$. In particular, the arguments are based upon the refinement method of Christ, although a significant adaptation of this method is required to fully understand the additional smoothing afforded by averaging over dilates.