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On Clarkson’s inequality, type and cotype for the Edmunds-Triebel logarithmic spaces. (English) Zbl 1049.26016

A Banach space \(X\) is said to satisfy the \((p,p')\) Clarkson inequality (where \(1<p\leq2\) and \(p'\) is the conjugate index) if, for all \(x,y\in X\), \[ \left(\| x+y\| _X^{p'}+\| x-y\| _X^{p'}\right)^{1/p'} \leq 2^{1/p'}\left(\| x\| _X^ p+\| y\| _X^ p\right)^{1/p}. \] The Clarkson inequality is closely related to uniform convexity. It is known that if it holds for some \(p\in(1,2]\), then it also holds for every \(r\in(1,p]\). It has been studied for a number of function spaces. In the paper under review, the authors show that it holds for the logarithmic spaces \(A_{\theta}(\log A)_{b,q}\) studied by D. E. Edmunds and H. Triebel [“Function spaces, entropy numbers, differential operators” (1996; Zbl 0865.46020)] for suitable parameters, and that the Zygmund class \(L_p(\log L)_b\) has an equivalent norm that satisfies the inequality, too. Related results including various chains of inclusions, Rademacher type and cotype of spaces, etc., are pointed out.

MSC:

26D15 Inequalities for sums, series and integrals
46B20 Geometry and structure of normed linear spaces
46B70 Interpolation between normed linear spaces

Citations:

Zbl 0865.46020
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