Complex uniformly convex Banach spaces and Riesz measures.

Blower, Gordon and Ransford, T. J. (2004) Complex uniformly convex Banach spaces and Riesz measures. Canadian Journal of Mathematics, 56 (2). pp. 225-245. ISSN 0008-414X

Abstract

The norm on a Banach space gives rise to a subharmonic function on the complex plane for which the distributional Laplacian gives a Riesz measure. This measure is calculated explicitly here for Lebesgue Lp spaces and the von~Neumann-Schatten trace ideals. Banach spaces that are q-uniformly PL-convex in the sense of Davis, Garling and Tomczak-Jaegermann are characterized in terms of the mass distribution of this measure. This gives a new proof that the trace ideals cp are 2-uniformly PL-convex for 1 leq p leq 2.