Abstract
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.
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Communicated by Y. Giga.
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Loukas Grafakos was supported by the Simons Foundation and by the University of Missouri Research Board and Council. Danqing He was supported by NNSF of China (no. 11701583), Guangdong Natural Science Foundation (no. 2017A030310054) and the Fundamental Research Funds for the Central Universities (no. 17lgpy11).
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Grafakos, L., He, D. & Slavíková, L. \(L^2\times L^2 \rightarrow L^1\) boundedness criteria. Math. Ann. 376, 431–455 (2020). https://doi.org/10.1007/s00208-018-1794-5
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DOI: https://doi.org/10.1007/s00208-018-1794-5