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

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 $1<p<\lambda<n$, what is the largest range of exponents $q$ for which necessarily $u\in L^{q}(B_1)$? While David R. Adams proved in 1975 that this embedding holds for $q\leq\lambda p/(\lambda-p)$, an article from 2011 claimed the embedding in the larger range $q<n p/(\lambda-p)$. Here we disprove this last statement by constructing a function that provides a counterexample for $q>\lambda p/(\lambda-p)$. 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 $np/(\lambda-p)$.