| Details: |
The classical Sobolev-Morrey embedding results states that a $W^{1,p}$ Sobolev function on $\mathbb{R}^{n}$ is H\"older continuous if $p > n$. On the other hand, if $p \leq n$, such a function need not even be bounded. Thus, the question of finding an intermediate integrability of the gradient that implies merely the continuity of the function is a natural one. In 1981, Stein proved the by now classical `borderline' Sobolev embedding result that if the weak gradient is in the intermediate Lorentz space $L^{n,1},$ then the function is continuous, but not necessarily more regular.
In this talk, we shall discuss how this borderline embedding question can be recast as a regularity question about the Laplacian and then generalised to the nonlinear setting of the $p$-Laplacian and $p$-Laplacian systems. Finally we shall discuss how these results can be generalised for quasilinear elliptic systems of differential forms and how these new results lead to improved conclusions even in the classical Stein theorem setting. |