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 $$\mu[B(x,r)]\leq Cr^s,$$ 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.