By Hans Triebel

This ebook bargains with homogeneous functionality areas of Besov Sobolev style in the framework of tempered distributions in Euclidean n-space in response to Gauss Weierstrass semi-groups. comparable Fourier-analytical descriptions and characterizations by way of derivatives and ameliorations are included after as so-called family norms. This procedure avoids the standard ambiguities modulo polynomials while homogeneous functionality areas are thought of within the context of homogeneous tempered distributions. those notes are addressed to graduate scholars and mathematicians having a operating wisdom of easy components of the idea of functionality areas, specifically of Besov Sobolev variety. specifically, the e-book can be of curiosity for researchers facing (nonlinear) warmth and Navier Stokes equations in homogeneous functionality areas. A book of the ecu Mathematical Society (EMS). allotted in the Americas by way of the yank Mathematical Society.

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80/81], [Tem03, p. 387] and [T10, pp. 22/23], where one finds historical comments and related literature. 2] and the recent surveys [Schm07, ScS04, Vyb06]. All this has been done in the framework of the dual pairing S(Rn ), S (Rn ) in the context of inhomogeneous spaces. 78) below. 71). 3, pp. 107–111]). We indicate as a proposal how a corresponding theory of homogeneous and, in particular, tempered homogeneous spaces with dominating mixed smoothness may look. But it remains to justify, modify, prove or disprove what follows.

84). 151) (maybe by real interpolation with some caution). In any case, the above comments show that it makes sense to ask for tempered ho∗ mogeneous hybrid spaces LrAsp,q (Rn ), in generalization of Chapter 3 dealing with ∗ tempered homogeneous global spaces Asp,q (Rn ). 3 Dominating mixed smoothness. The theory of classical (inhomogeneous) Besov and Sobolev spaces with dominating mixed smoothness was developed in the 1960s by the Russian school. We refer the reader to the relevant parts in [Nik77] (first edition, 1969) and [BIN75].

167). 171) one has rotation invariance. We collect arguments in favour of such an undertaking and indicate what could come out and what one has to study in detail. 172) q>1 2 Spaces on S˙ (Rn ) 38 0 < p < ∞, complemented by L∞ (Rn , w) = L∞ (Rn ), expecting that the related spaces are independent of admitted resolutions of unity {ϕ j }∞j=−∞ and {ϕ j }∞j=0 . This is the case and may be found in [Bui82, Bui84]. 30). Similarly for the inhomogeneous spaces Asp,q (Rn , w). 11, can be transferred to the above weighted case.