Measures of polynomial growth and classical convolution inequalities

We study $L^p(μ) \to L^q(ν)$ mapping properties of the convolution operator $T_λf(x)=λ*(fμ)(x)$ and of the corresponding maximal operator ${\mathcal T}_λf(x)=\sup_{t>0} |λ_t*(fμ)(x)|$, where $λ$ is a tempered distribution, and $μ$ and $ν$ are compactly supported measures satisfying the polynomial growth bounds $μ(B(x,r)) \leq Cr^{s_μ}$ and $ν(B(x,r)) \leq Cr^{s_ν}$. 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.

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