Abstract
The following subexponential estimate for commutators is proved
where \(c\) and \(\alpha \) are absolute constants, \(T\) is a Calderón–Zygmund operator, \(M\) is the Hardy Littlewood maximal function and \(f\) is any function supported on the cube \(Q\subset \mathbb{R }^n\). We also obtain that
where \(m_f(Q)\) is the median value of \(f\) on the cube \(Q\) and \(M_{\lambda _n;Q}^\#\) is Strömberg’s local sharp maximal function with \(\lambda _n=2^{-n-2}\). As a consequence we derive Karagulyan’s estimate:
from [21] improving Buckley’s theorem [3]. A completely different approach is used based on a combination of “Lerner’s formula” with some special weighted estimates of Coifman–Fefferman type obtained via Rubio de Francia’s algorithm. The method is flexible enough to derive similar estimates for other operators such as multilinear Calderón–Zygmund operators, dyadic and continuous square functions and vector valued extensions of both maximal functions and Calderón–Zygmund operators. In each case, \(M\) will be replaced by a suitable maximal operator.
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Acknowledgments
C. Pérez is supported by the Spanish Ministry of Science and Innovation grant MTM2009-08934, C. Pérez and E. Rela are also supported by the Junta de Andalucía, grant FQM-4745.
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Ortiz-Caraballo, C., Pérez, C. & Rela, E. Exponential decay estimates for singular integral operators. Math. Ann. 357, 1217–1243 (2013). https://doi.org/10.1007/s00208-013-0940-3
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DOI: https://doi.org/10.1007/s00208-013-0940-3