Necessary and sufficient conditions for imbeddings in weighted Sobolev spaces. (English) Zbl 0704.46021

Opic, Bohumír

Čas. Pěstování Mat. 114, No. 4, 343-355 (1989).

The author gives necessary and sufficient conditions under which the weighted Sobolev space \(W^ 1_ p(\Omega;v_ 0,v_ 1,...,v_ N)\) with norm \[ \| u\|^ p=\int_{\Omega}| u(x)|^ pv_ 0(x)dx+\sum^{N}_{i=1}\int_{\Omega}| \frac{\partial u}{\partial x_ i}(x)|^ pv_ i(x)dx \] is imbedded into the weighted Lebesgue space \(L_ q(\Omega;w)\) with norm \[ \| u\|^ q=\int_{\Omega}| u'x)|^ qw(x)dx. \] Related work was previously done by P. Gurka, A. Kufner, and the author [see, e.g., Math. Nachr. 133, 63-70 (1987; Zbl 0639.46035), Čas Pěst. Mat. 113, 60-73 (1988)), or Czechosl. Math. J. 39, 78-94 (1989)].

Reviewer: J.Appell

MSC:

46E35

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

Keywords:

weighted Sobolev space; weighted Lebesgue space

Citations:

Zbl 0639.46035