generalized Radon transforms on fractal measures

Abstract

In the setting of a general Borel measure $μ$ on $R^d$ with the natural ball size condition $$μ[B(x,r)]\leq Cr^s,$$ we establish the $L^p(μ)$-$L^q(μ)$-estimate for the generalized Radon transform $$(Af)(x):=\int_{Φ(x,y)=0}(fμ)(y)ψ(x,y)dσ_x(y),$$ where $Φ$ 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.