A sharp exponential inequality for Lorentz-Sobolev spaces on bounded domains

Abstract:

This paper generalizes an inequality of Moser from the case that $\nabla u$ is in the Lebesgue space $L^n$ to certain subspaces, namely the Lorentz spaces $L^{n,q}$, where $1<q\leq n$. The conclusion is that $\exp (\alpha u^p)$ is integrable, where $1/p+1/q=1$. This is a higher degree of integrability than in the Moser inequality when $q<n$. A formula for $\alpha$ is given and it is also shown that no larger value of $\alpha$ works.