When Is L^r(R) Contained in L^p(R) + L^q(R)?

Abstract

We prove a necessary and sufficient condition on the exponents p, q, r ≥ 1 such that L^r (R) ⊂ L^p(R) + L^q(R). In doing so, we explore the structure of Lp(R) + L^q(R) as a normed vector space.

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Notes on contributors

Jean-Baptiste Hiriart-Urruty

JEAN-BAPTISTE HIRIART-URRUTY has been a professor of mathematics at the Paul Sabatier University since 1981. He recieved his Doctorat ès sciences mathématiques from the Blaise Pascal University (Clermont-Ferrand) in 1977. His current research interests include Variational analysis (convex, nonsmooth, applied) and all the aspects of optimization.

Patrice Lassère

PATRICE LASSÈRE received his Doctorat ès sciences mathématiques from the University Paul Sabatier, Toulouse, France, in 1991. He currently teaches at the University Paul Sabatier. His research interests includes complex analysis in one or several variables and functional analysis.