generalized Radon transforms on fractal measures

Abstract

In the setting of a general Borel measure $\mu$ on $R^d$ with the natural ball size condition $$\mu [B(x,r)]\le C{r}^s,$$ 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.