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Zeitschrift für Analysis und ihre Anwendungen

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Volume 24, Issue 1, 2005, pp. 3–28
DOI: 10.4171/ZAA/1228

Published online: 2005-03-31

Function Spaces as Dirichlet Spaces (About a Paper by Maz'ya and Nagel)

Niels Jacob[1] and René L. Schilling[2]

(1) University of Wales, Swansea, UK
(2) Technische Universität Dresden, Germany

V. G. Maz'ya and J. Nagel [Beitr. Anal. 12 (1978) 7--17] found for certain classes of weighted Sobolev norms (defined using the Fourier transform) equivalent Slobodeckij-type difference representations. We extend these considerations to a wider class of anisotropic norms which arise in the theory of Markov processes. In particular we show that these Sobolev norms are equivalent to Dirichlet norms.

Keywords: Dirichlet space, anisotropic Sobolev space, Slobodeckij norm, negative definite function, Bernstein function

Jacob Niels, Schilling René: Function Spaces as Dirichlet Spaces (About a Paper by Maz'ya and Nagel). Z. Anal. Anwend. 24 (2005), 3-28. doi: 10.4171/ZAA/1228