Measures of polynomial growth and classical convolution inequalities

TL;DR

This paper investigates the mapping properties of convolution operators between different Lebesgue spaces in the context of measures with polynomial growth, extending classical inequalities without relying on the Plancherel formula.

Contribution

It establishes new $L^p$-improving and maximal inequalities for convolution operators involving measures with polynomial growth bounds, generalizing classical results.

Findings

Proves $L^p$-improving inequalities for measures with polynomial growth.

Establishes maximal inequalities in a non-Plancherel setting.

Discusses connections with the David-Semmes conjecture.

Abstract

We study $L^p(\mu) \to L^q(\nu)$ mapping properties of the convolution operator $ T_{\lambda}f(x)=\lambda*(f\mu)(x)$ and of the corresponding maximal operator $ {\mathcal T}_{\lambda}f(x)=\sup_{t>0} |\lambda_t*(f\mu)(x)|$, where $\lambda$ is a tempered distribution, and $\mu$ and $\nu$ are compactly supported measures satisfying the polynomial growth bounds $\mu(B(x,r)) \leq Cr^{s_{\mu}}$ and $\nu(B(x,r)) \leq Cr^{s_{\nu}}$. As a result, we prove variants of the classical $L^p$-improving (Littman; Strichartz) and maximal (Stein) inequalities in a setting where the Plancherel formula is not available. Connections with the David-Semmes conjecture are also discussed.