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 $np/(\lambda-p)$.