@MISC{Fennicae94weightedsobolev, author = {Scientiarum Fennicae and Tero Kilpeläinen}, title = {Weighted Sobolev Spaces And Capacity}, year = {1994} }

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Abstract

. We discuss the role of capacity in the pointwise definition of functions in Sobolev spaces involving weights of Muckenhoupt's A p -class. In particular, it is shown that Sobolev functions possess Lebesgue points quasieverywhere with respect to an appropriate capacity. Introduction Let\Omega be an open set in R n and 1 ! p ! 1 . In this paper we consider the theory of weighted Sobolev spaces H 1;p with weight function in Muckenhoupt's A p -class. Our main purpose is to provide a coherent exposition of the behavior of functions in weighted Sobolev spaces and this leads us to use a concept of capacity. The motivation arises from the theory of partial differential equations, see e.g. [F], [HKM]. Most of the results we present are probably not new but according to our knowledge they have not yet appeared in printed form. We define the weighted Sobolev space H 1;p w) to be the completion of C 1 (R n ) with respect to the norm k'k 1;p;w = `Z \Omega j'j w(x) dx ' 1=p + `Z...