Complex Uniform Convexity and Riesz Measures

Blower, G. and Ransford, T. (2004) Complex Uniform Convexity 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≤p≤2 .