This is a contribution to the theory of Lizorkin-Triebel spaces having mixed Lebesgue norms and quasi-homogeneous smoothness. We discuss their characterisation in terms of general quasi-norms based on convolutions. In particular, this covers the case of local means, in Triebel's terminology. The main step is an extension of some crucial inequalities due to Rychkov to the case with mixed norms.
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J. Johnsen, S. Munch Hansen, Winfried Sickel, Characterisation by Local Means of Anisotropic Lizorkin-Triebel Spaces with Mixed Norms. Z. Anal. Anwend. 32 (2013), no. 3, pp. 257–277DOI 10.4171/ZAA/1484