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arXiv · 1411.2318

An $L^1$-type estimate for Riesz potentials

Armin Schikorra·Daniel Spector·Jean Van Schaftingen
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Abstract

In this paper we establish new $L^1$-type estimates for the classical Riesz potentials of order $α\in (0, N)$: \[ \|I_αu\|_{L^{N/(N-α)}(\mathbb{R}^N)} \leq C \|Ru\|_{L^1(\mathbb{R}^N;\mathbb{R}^N)}. \] This sharpens the result of Stein and Weiss on the mapping properties of Riesz potentials on the real Hardy space $\mathcal{H}^1(\mathbb{R}^N)$ and provides a new family of $L^1$-Sobolev inequalities for the Riesz fractional gradient.

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Armin Schikorra, Daniel Spector, Jean Van Schaftingen. 2016-09-01. An $L^1$-type estimate for Riesz potentials. https://doi.org/10.4171/rmi%2F937

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