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.
Categories: math.CA math.MG