I am looking for references on the topic of Sobolev spaces based on $L^p$ with $0<p<1$. For instance, a natural question could be: let $u$ be a (compactly supported) distribution on $\mathbb R^n$ such that $\nabla u\in L^p$ for some $p\in (0,1)$. Does that imply some regularity for $u$? Note that in the case $p=1$, we get $u\in L^{n/(n-1)}$.

## 1 Answer

The paper (and this comment on it)

Peetre, Jaak.

A remark on Sobolev spaces. The case $0<p<1$. Collection of articles dedicated to G. G. Lorentz on the occasion of his sixty-fifth birthday, III. J. Approximation Theory 13 (1975), 218--228. MR0374900.

might be a good start. Among the (unfortunately rather few) publications that cite it is e.g.

Haroske, Dorothee D.; Triebel, Hans.

Embeddings of function spaces: a criterion in terms of differences. Complex Var. Elliptic Equ. 56 (2011), no. 10-11, 931--944. MR2838229

which studies embeddings of $B^s_{p,q}(\mathbb R^n)$ into $L_r(\mathbb R^n)$ with $0 < p$, $p \le \infty$, $1 < r < \infty$, $s > 0$ and $s - \frac np = - \frac nr$.

Sobolev Spaces; maybe it does not. $\endgroup$