  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 $np/(\lambda-p)$.
