generalized Radon transforms on fractal measures

TL;DR

This paper establishes sharp $L^p$-$L^q$ estimates for generalized Radon transforms acting on fractal measures with specific size conditions, expanding the understanding of such transforms in fractal and measure-theoretic contexts.

Contribution

It introduces new $L^p$-$L^q$ bounds for generalized Radon transforms on fractal measures, under broad conditions including Sobolev bounds, generalizing previous results.

Findings

Established sharp $L^p$-$L^q$ estimates for Radon transforms on fractal measures.

Provided conditions under which the bounds hold, including Sobolev bounds.

Extended the theory of Radon transforms to measures with fractal support.

Abstract

In the setting of a general Borel measure $\mu$ on $R^d$ with the natural ball size condition $$\mu[B(x,r)]\leq Cr^s,$$ we establish the $L^p(\mu)$-$L^q(\mu)$-estimate for the generalized Radon transform $$(Af)(x):=\int_{\Phi(x,y)=0}(f\mu)(y)\psi(x,y)d\sigma_x(y),$$ where $\Phi$ is a smooth function away from the diagonal. Among other reasonable assumptions, an $L^2$-Sobolev bound on $A$ on $R^d$ is imposed. This bound is satisfied in many natural situations. The main result is, in general, sharp up to endpoints.