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Integral Equations and Operator Theory

, Volume 66, Issue 2, pp 253–264 | Cite as

Translation-Invariant Bilinear Operators with Positive Kernels

  • Loukas Grafakos
  • Javier Soria
Article

Abstract

We study L r (or L r, ∞) boundedness for bilinear translation-invariant operators with nonnegative kernels acting on functions on \({\mathbb {R}^n}\). We prove that if such operators are bounded on some products of Lebesgue spaces, then their kernels must necessarily be integrable functions on \({\mathbb R^{2n}}\), while via a counterexample we show that the converse statement is not valid. We provide certain necessary and some sufficient conditions on nonnegative kernels yielding boundedness for the corresponding operators on products of Lebesgue spaces. We also prove that, unlike the linear case where boundedness from L 1 to L 1 and from L 1 to L 1, ∞ are equivalent properties, boundedness from L 1 × L 1 to L 1/2 and from L 1 × L 1 to L 1/2, ∞ may not be equivalent properties for bilinear translation-invariant operators with nonnegative kernels.

Mathematics Subject Classification (2010)

Primary 42A85 Secondary 47A07 

Keywords

Bilinear operators convolution positive kernels 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  1. 1.Dept. of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Dept. Appl. Math. and AnalysisUniversity of BarcelonaBarcelonaSpain

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