Singular integrals in the Cesàro sense
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.
scientific article; zbMATH DE number 1445618