Download Topics in Fourier Analysis and Function Spaces by Hans-Jurgen Schmeisser, Hans Triebel PDF

By Hans-Jurgen Schmeisser, Hans Triebel

Covers a number of sessions of Besov-Hardy-Sobolevtype functionality areas at the Euclidean n-space and at the n-forms, specifically periodic, weighted, anisotropic areas, in addition to areas with dominating mixed-smoothness houses. in keeping with the most recent suggestions of Fourier research; the publication is an up to date, revised, and prolonged model of Fourier Analysis and features Spaces by way of Hans Triebel.

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Topics in Fourier Analysis and Function Spaces

Covers numerous periods of Besov-Hardy-Sobolevtype functionality areas at the Euclidean n-space and at the n-forms, specially periodic, weighted, anisotropic areas, in addition to areas with dominating mixed-smoothness houses. in response to the newest strategies of Fourier research; the publication is an up to date, revised, and prolonged model of Fourier research and features areas through Hans Triebel.

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L. Wheeden [1] and R. R. Coifman, C. Fefferman [1] (cf. [F, p. 42] for details). Finally we mention the famous assertion by C. Fefferman [1] that the characteristic function o f a ball is a Fourier multiplier in L p with 1 < p < oo if and only if p = 2. Cf. also B. S. Mitjagin [1] for generalizations. Duality H. On the basis of the results about Fourier multipliers on L p{Rn9 £(*)) and the above heuristical arguments, cf. (2), one can prove the following result. Let 1 < p < oo, — + -i- = 1 and ^(x) = (1 + |*|)d with —— < d < P P P P closed rectangle in R„, then we have (Lp(Q9p>L))f = L~, °( q~19ftf) .

Marcinkiewicz, S. G. Michlin, L. Hormander, E. M. Stein and others. Problems of this type have been studied for their own sake, but they are also of crucial interest in the theory of singular integrals, partial differential equations etc. We shall not go into detail, references may be found in E. M. Stein [1], E. M. Stein, G. 4] and (as far as more recent developments are concerned) R. E. Edwards [l, in particular Chapter 16]. Fourier multipliers have also been studied for other spaces, in particular for (unweighted and unmixed) spaces of Besov-Hardy-Sobolev type.

Peetre [4, pp. 180 and 190]. 44 1. 6. 1. 6. 1/6). We restrict ourselves to n = 2, however the extension to arbitrary natural numbers n > 1 is obvious. Furthermore, we deal only with unweighted mixed spaces. This has the advantage that we do not need the ultra-distributions: The usual distributions are sufficient. , where we fixed our notations and where we recalled the famous Paley-WienerSchwartz theorems. In particular, S = S(R2) and S' = S'(R 2) have the usual meaning, F an d F~1 stand for the Fourier transform and its inverse, respectively.

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