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

We study a natural question that, apparently, has not been well addressed in the literature. Given functions u with support unit ball B 1 ? R n and gradient Morrey space M p , ? ( ) where < what is largest range of exponents q for which necessarily ? L ? While David R. Adams proved 1975 that this embedding holds ? / ? an article from 2011 claimed larger . Here we disprove last statement by constructing function provides counterexample > The basically negative power distance to set Hausdorff dimension When ? Z fractal. also make detailed radially symmetric case, situation exponent can go up