Vol. 9, No. 5, 2016

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Bohnenblust–Hille inequalities for Lorentz spaces via interpolation

Andreas Defant and Mieczysław Mastyło

Vol. 9 (2016), No. 5, 1235–1258
Abstract

We prove that the Lorentz sequence space 2m(m+1),1 is, in a precise sense, optimal among all symmetric Banach sequence spaces satisfying a Bohnenblust–Hille-type inequality for m-linear forms or m-homogeneous polynomials on n . Motivated by this result we develop methods for dealing with subtle Bohnenblust–Hille-type inequalities in the setting of Lorentz spaces. Based on an interpolation approach and the Blei–Fournier inequalities involving mixed-type spaces, we prove multilinear and polynomial Bohnenblust–Hille-type inequalities in Lorentz spaces with subpolynomial and subexponential constants. An application to the theory of Dirichlet series improves a deep result of Balasubramanian, Calado and Queffélec.

Keywords
Bohnenblust–Hille inequality, Dirichlet polynomials, Dirichlet series, homogeneous polynomials, interpolation spaces, Lorentz spaces
Mathematical Subject Classification 2010
Primary: 46B70, 47A53
Milestones
Received: 14 January 2016
Revised: 12 February 2016
Accepted: 30 March 2016
Published: 29 July 2016
Authors
Andreas Defant
Institut für Mathematik
Carl von Ossietzky Universität
Postfach 2503
D-26111 Oldenburg
Germany
Mieczysław Mastyło
Faculty of Mathematics and Computer Science
Adam Mickiewicz University in Poznań
Umultowska 87
61-614 Poznań
Poland