Sobolev-Lorentz spaces in the Euclidean setting and counterexamples

This paper studies the inclusions between different Sobolev-Lorentz spaces $W^{1,(p,q)}(\Omega)$ defined on open sets $\Omega \subset {\mathbf{R}^n},$ where $n \ge 1$ is an integer, $1<p<\infty$ and $1 \le q \le \infty.$ We prove that if $1 \le q<r \le \infty,$ then $W^{1,(p,q)}(\Omega)$ is strictly included in $W^{1,(p,r)}(\Omega).$ We show that although $H^{1,(p,\infty)}(\Omega) \subsetneq W^{1,(p,\infty)}(\Omega)$ where $\Omega \subset {\mathbf{R}}^n$ is open and $n \ge 1,$ there exists a partial converse. Namely, we show that if a function $u$ in $W^{1,(p,\infty)}(\Omega), n \ge 1$ is such that $u$ and its distributional gradient $\nabla u$ have absolutely continuous $(p,\infty)$-norm, then $u$ belongs to $H^{1,(p,\infty)}(\Omega)$ as well. We also extend the Morrey embedding theorem to the Sobolev-Lorentz spaces $H_{0}^{1,(p,q)}(\Omega)$ with $1 \le n<p<\infty$ and $1 \le q \le \infty.$ Namely, we prove that the Sobolev-Lorentz spaces $H_{0}^{1,(p,q)}(\Omega)$ embed into the space of H\"{o}lder continuous functions on $\overline{\Omega}$ with exponent $1-\frac{n}{p}$ whenever $\Omega \subset {\mathbf{R}}^n$ is open, $1 \le n<p<\infty,$ and $1 \le q \le \infty.$