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Anisotropic Sobolev spaces and a quasidistance function. (English) Zbl 0736.46020
Let \(\Omega\) be a proper, non-empty, open subset of the Euclidean space \({\mathbb{R}}^ N\), \(\bar k=(k_ 1,\ldots,k_ N)\in{\mathbb{N}}^ N\), \(\omega=\sum^ N_{i=1}1/k_ i\), \(\lambda_ i=\omega k_ i/N\), \(i=1,\ldots,N\); \(\lambda=\max_{1\leq i\leq N}\lambda_ i\), \(\rho(x,y)=\max_{1\leq i\leq N}| x_ i-y_ i|^{\lambda_ i}\); \(d(x)=\inf\{\rho(x,y): x\in\Omega, y\in\partial\Omega\};\) \[ L^ p(\Omega;\varepsilon)=\{f:\| f\|_{p,\varepsilon;\Omega}=\left(\int_ \Omega| f(x)|^ p d(x)^ \varepsilon dx \right)^{1/p}<\infty,\quad \varepsilon\in{\mathbb{R}}\}. \] The anisotropic Sobolev space \(W^{\bar k,p}(\Omega;\varepsilon)\) consists of all functions \(f\in L^ p(\Omega;\varepsilon)\) such that \(D^{k_ i}_ if\in L^ p(\Omega;\epsilon)\), \(i=1,\ldots,N\), with the norm \[ \| f\|_{\bar k,p,\varepsilon;\Omega}=\| f\|_{p,\varepsilon;\Omega}+\sum^ N_{i=1}\| D^{k_ i}_ i f\|_{p,\varepsilon;\Omega}. \] The closure of \(C^ \infty_ 0(\Omega)\) with respect to such norm will be denoted by \(W^{k,p}_ 0(\Omega;\varepsilon)\).
The following theorem is proved:
Theorem. Let \(1\leq p<\infty\). If \(f\in W^{\bar k,p}(\Omega,\varepsilon)\) is such that \(f\cdot d^{-N/\Omega}\in L^ p(\Omega;\varepsilon)\), then \(f\in W^{\bar k,p}_ 0(\Omega;\varepsilon)\).
The alternative proof of this result is given for the case \(1<p<\infty\), using the anisotropic maximal operator.

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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