Measures of polynomial growth and classical convolution inequalities

Abstract

We study $L^p(\mu )\to L^q(\nu )$ mapping properties of the convolution operator $T_{\lambda }f(x)=\lambda \ast (f\mu )(x)$ and of the corresponding maximal operator ${\mathcal{T}}_{\lambda }f(x)={\sup }_{t>0}|{\lambda }_t\ast (f\mu )(x)|$, where $\lambda$ is a tempered distribution, and $\mu$ and $\nu$ are compactly supported measures satisfying the polynomial growth bounds $\mu (B(x,r))\le C{r}^{s_{\mu }}$ and $\nu (B(x,r))\le C{r}^{s_{\nu }}$. 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.

References in corpus (1)

• Sharpness of Falconer's estimate in continuous and arithmetic settings, geometric incidence theorems and distribution of lattice points in convex domains