I am reading Hildebrandt's and Widman's 1975 paper on "Some regularity results of quasilinear elliptic systems of second order".

Theorem 3.1 is the first time in their paper that the function space $H^1_{m, \sigma}(\Omega',\mathbb{R}^N)$ for $\sigma\in (0, 1)$ appears.

Any idea what this space is? I know that $H^1_m$ is the Sobolev space $W^{1, m}$, but what does it mean when there is a $\sigma\in (0, 1)$?


I have not seen this notation before, but after looking at Widman's cited result http://www.ams.org/mathscinet-getitem?mr=296484, I am fairly sure that what is meant is something like $$\int_{B_r} |\nabla u|^m \,dx \le Cr^\sigma $$ for $r$ so small that $B_{2r}\subset \Omega$. This is a typical kind of "Dirichlet growth" result that can then be used to deduce Hölder regularity.

The Corollary to Theorem 3.1 uses that for $m=n$, $H^{1}_{n,\sigma}$ embeds into $H^1_{2,n-2+2\sigma/n}$. I think this follows from my proposed definition above and Hölder's inequality.


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