On the trace space of a Sobolev space with a radial weight

Helmut Abels,Katrin Schumacher,Miroslav Krbec

doi:10.1155/2008/986720

Helmut Abels, Katrin Schumacher + Show 1 more

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https://doi.org/10.1155/2008/986720

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Abstract

Our concern in this paper lies with trace spaces for weighted Sobolev spaces, when the weight is a power of the distance to a point at the boundary. For a large range of powers we give a full description of the trace space.

Highlights

We consider integer order weighted Sobolev spaces with weights equal to a power of the distance to a point of the boundary and more general weights modelled upon such weights
Our concern in this paper lies with a characterization of trace spaces of these weighted Sobolev spaces
260 On the trace space of a Sobolev space with a radial weight of traces has been closely connected with extension of integer order spaces to spaces with non-integer derivatives, and it was one of the motivation for establishing the general theory of Besov spaces

Summary

Introduction

We consider integer order weighted Sobolev spaces with weights equal to a power of the distance to a point of the boundary and more general weights modelled upon such weights. In the following we shall make use of a Fourier analytic approach to Sobolev spaces and their weighted generalizations, we recall the most important definitions and fix the notation. For Lipschitz domains there exists a universal extension operator working on Sobolev, Besov and Bessel potential spaces (and on the LizorkinTriebel spaces, even for all real s, see Rychkov [24]); this means that many relevant properties of spaces on Lipschitz domains follow from the claims on the whole of Rn. That is, one can work either with a formal definition of spaces on domains as factorspaces of spaces on Rn modulo equality on the domain in question or with a space on the domain with a usual intrinsic norm (if it is available).

Preliminary results on weighted function spaces

Proof for positive α

The result for negative α