\(L^2\times L^2 \rightarrow L^1\) boundedness criteria

  • Loukas Grafakos
  • Danqing He
  • Lenka Slavíková


We obtain a sharp \(L^2\times L^2 \rightarrow L^1\) boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the \(L^q\) integrability of this function; precisely we show that boundedness holds if and only if \(q<4\). We discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. Our second result is an optimal \(L^2\times L^2\rightarrow L^1\) boundedness criterion for bilinear operators associated with multipliers with \(L^\infty \) derivatives. This result provides the main tool in the proof of the first theorem and is also manifested in terms of the \(L^q\) integrability of the multiplier. The optimal range is \(q<4\) which, in the absence of Plancherel’s identity on \(L^1\), should be compared to \(q=\infty \) in the classical \(L^2\rightarrow L^2\) boundedness for linear multiplier operators.

Mathematics Subject Classification

42B15 42B20 42B99 



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Department of MathematicsSun Yat-sen (Zhongshan) UniversityGuangzhouChina

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