  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.
