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On the local boundedness of singular integral operators


Author: Mark Leckband
Journal: Trans. Amer. Math. Soc. 308 (1988), 39-56
MSC: Primary 42B25; Secondary 47G05
DOI: https://doi.org/10.1090/S0002-9947-1988-0946428-1
MathSciNet review: 946428
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Abstract: The class of singular integral operators whose kernels satisfy the usual smoothness conditions is studied. Let such an operator be denoted by $ K$. We establish necessary conditions that imply $ K$ has local (weighted) $ {L^p}$ norm inequalities.

The underlying principle is as follows. If $ {\chi _Q}$ is the characteristic function of a fixed cube $ Q$ of $ {R^n}$, or all of $ {R^n}$, then $ K{\chi _Q}$ and (the adjoint of $ K$) $ {K^{\ast}}{\chi _Q}$ determine the boundedness properties of $ K$ for functions supported in a proper fraction of $ Q$.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0946428-1
Article copyright: © Copyright 1988 American Mathematical Society

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