The optimal exponent in the embedding into the Lebesgue spaces for functions with gradient in the Morrey space

Xavier Cabre; Fernando Charro

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arXiv · 1907.12982

The optimal exponent in the embedding into the Lebesgue spaces for functions with gradient in the Morrey space

Xavier Cabre·Fernando Charro

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Abstract

We study the following natural question that, apparently, has not been well addressed in the literature: Given functions $u$ with support in the unit ball $B_1\subset\mathbb{R}^n$ and with gradient in the Morrey space $M^{p,\lambda}(B_1)$, where $1 \lambda p/(\lambda-p)$. The function is basically a negative power of the distance to a set of Hausdorff dimension $n-\lambda$. When $\lambda\notin\mathbb{Z}$, this set is a fractal. We also make a detailed study of the radially symmetric case, a situation in which the exponent $q$ can go up to $np/(\lambda-p)$.

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Xavier Cabre, Fernando Charro. 2019-07-30. The optimal exponent in the embedding into the Lebesgue spaces for functions with gradient in the Morrey space. https://arxiv.org/abs/1907.12982

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