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arXiv · 1410.1436

Measures of polynomial growth and classical convolution inequalities

Alex Iosevich·Ben Krause·Eric Sawyer·Krystal Taylor·Ignacio Uriarte-Tuero
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Abstract

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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Alex Iosevich, Ben Krause, Eric Sawyer, Krystal Taylor, Ignacio Uriarte-Tuero. 2015-11-14. Measures of polynomial growth and classical convolution inequalities. https://arxiv.org/abs/1410.1436

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