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

Abstract

We study a natural question that, apparently, has not been well addressed in the literature. Given functions \upsilon with support in the unit ball B1 \subset \mathbb{R}n and with gradient in the Morrey space Mp,\lambda(B1), where 1 < p < \lambda < n, what is the largest range of exponents q for which necessarily \upsilon \in Lq(B1)? 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 < np/(\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).