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 $u\in L^{n/(n-1)}$.
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1$\begingroup$ I assume by $v\in L^p$ for a distribution $v$ you mean that there is some function $f\in L^p$ such that $\int f\varphi\, dx$ converges for all test functions $\varphi$ (in other words, $f\varphi\in L^1$) and equals $(v,\varphi)$. However, this implies that $f\in L^1$ (if $f$ is also compactly supported). $\endgroup$– Christian RemlingCommented Nov 9, 2016 at 17:12
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$\begingroup$ For $p \ge 1$, $\nabla u \in L^p$ implies $u \in L^p_{\mathrm{loc}}$. At a glance I didn't see how the assumption $p \ge 1$ enters into the proof of that Theorem on p2 of Maz'ya's Sobolev Spaces; maybe it does not. $\endgroup$– anonymousCommented Dec 13, 2016 at 0:48
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1$\begingroup$ @ChristianRemling I'm not sure I follow. What about the function $\frac 1x$ which is in $L^{1/2}[0,1]$ but not $L^1[0,1]$? $\endgroup$– anonymousCommented Dec 13, 2016 at 0:59
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$\begingroup$ @anonymous: As I point out in my comment, $\int (1/x)\varphi$ is not defined for arbitrary test functions, so $1/x$ doesn't define a distribution in this way. However, the OP seems to assume that $\nabla u$ is a distribution (or else how did we define it in the first place?). $\endgroup$– Christian RemlingCommented Dec 14, 2016 at 3:31
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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 $s - \frac np = - \frac nr$.
