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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.

##### MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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