Translation-Invariant Bilinear Operators with Positive Kernels |
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Authors: | Loukas Grafakos Javier Soria |
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Institution: | 1. Dept. of Mathematics, University of Missouri, Columbia, MO, 65211, USA 2. Dept. Appl. Math. and Analysis, University of Barcelona, E-08007, Barcelona, Spain
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Abstract: | We study L
r
(or L
r, ∞) boundedness for bilinear translation-invariant operators with nonnegative kernels acting on functions on
\mathbb Rn{\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 R2n{\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. |
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Keywords: | |
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