Compact embedding of anisotropic Sobolev space

Question

I am interested to know if the following result for anisotrpic Sobolev spaces is correctly presented. Also, does anyone have a good reference for this? Or can maybe confirm this result in its current form below. \begin{align} W^{1,\overrightarrow{p},\epsilon}_{0}(\Omega) \Subset L^{q}(\Omega), \end{align} holds for $1 \leq q < p^{\#}$, where $p^{\#}:= \frac{N\bar{p}}{N-\bar{p}}$ and $\frac{1}{\bar{p}} := \frac{1}{N}\sum_{i=1}^{N}\frac{1}{p^{i}}$.

Thanks a lot for any assistance.

Answer 1

You should put more information into your question. Anisotropic Sobolev spaces have different definitions according to different sources, so a definition of the spaces (rather than linking to one in a comment) is a must. Additionally, I'm assuming $\Omega\subset\mathbb{R}^N$ is open and bounded, although you haven't stated as such.

Under certain conditions, the result you stated is true, although it is possible more general versions of this theorem exist. Specifically, assume $p_1\le\ldots\le p_N$ and that $\Omega$ is semirectangular with respect to $\vec{p}$. By this I mean that if $N_1,\ldots,N_L$ are the multiplicities of each of the values in $\vec{p}$, then there exist bounded Lipschitz domains $\Omega_i$ such that $\Omega=\Omega_1\times\ldots\times\Omega_L$. We have the following theorem:

Theorem: Let $\Omega\subset\mathbb{R}^N$ be open, bounded and semirectangular with respect to $\vec{p}$ and assume $N>\bar{p}$. Suppose further that one of the follows conditions holds:

• $p_0\ge p_N$; or
• $p_0\ge p_1$ and for all $k\in\{1,\ldots,N-1\}$ we have $q_k>\max\{p_{k+1},q_{k+1}\}$ where $$q_k:=\begin{cases} \frac{k}{\sum_{i=1}^k\frac1{p_i}-1}&\text{if}\ \sum_{i=1}^k\frac1{p_i}>1,\\ \infty&\text{otherwise.} \end{cases}$$

Then $W_0^{1,\vec{p},p_0}(\Omega)\hookrightarrow L^q(\Omega)$ for all $1\le q<p^\#$.

There are other similar theorems. I would look at "A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems" by Haskovec and Schmeiser as a first reference. You can find it online at Springer.