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Singular integrals in the Cesàro sense. (English) Zbl 0959.42008

In the study of singular integrals \(\int K(x,y)f(y)dy\), the operators are often considered in the principal value sense, \[ Tf(x)=\lim_{\varepsilon\rightarrow 0}\int_{|x-y|>\varepsilon}K(x,y)f(y) dy. \] In this paper the authors instead consider singular integrals in the Cesàro sense, which means that they study the existence of \[ Tf(x)=\lim_{\varepsilon\rightarrow 0}\int_{|x-y|>\varepsilon} \biggl(1-{\varepsilon\over |x-y|} \biggr)^\alpha K(x,y)f(y) dy. \] This is done for \(-1<\alpha <0\) and for weighted spaces.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
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References:

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