Rev. Mat. Iberoam.
Inserted: 13 nov 2014
Last Updated: 1 sep 2016
Pages: 12
Year: 2014
Doi: arXiv:1411.2318
Abstract:
In this paper we establish new $L^1$-type estimates for the classical Riesz
potentials of order $\alpha \in (0, N)$:
\[
\
I_\alpha u\
_{L^{N/(N-\alpha)}(\mathbb{R}^N)} \leq C
\
Ru\
_{L^1(\mathbb{R}^N;\mathbb{R}^N)}.
\]
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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