On the local boundedness of singular integral operators

Leckband, Mark

doi:10.1090/S0002-9947-1988-0946428-1

On the local boundedness of singular integral operators
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Trans. Amer. Math. Soc. 308 (1988), 39-56 Request permission

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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