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We study the following natural question that, apparently, has not been welladdressed 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 ofexponents $q$ for which necessarily $u\in L^{q}(B_1)$? While David R. Adamsproved in 1975 that this embedding holds for $q\leq\lambda p/(\lambda-p)$, anarticle from 2011 claimed the embedding in the larger range $q<np/(\lambda-p)$. Here we disprove this last statement by constructing a functionthat provides a counterexample for $q>\lambda p/(\lambda-p)$. The function isbasically 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 alsomake a detailed study of the radially symmetric case, a situation in which theexponent $q$ can go up to $np/(\lambda-p)$.

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