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.
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Niels Jacob, René L. Schilling, Function Spaces as Dirichlet Spaces (About a Paper by Maz'ya and Nagel). Z. Anal. Anwend. 24 (2005), no. 1, pp. 3–28DOI 10.4171/ZAA/1228