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On the boundedness of Cauchy singular operator from the space \(L_ p\) to \(L_ q, p>q\geq1\). (English) Zbl 0813.47034
It is proved that for a Cauchy type singular operator, given by the equality \[ S_ \Gamma(f)(t)= {1\over \pi i}\int_ \Gamma {f(\tau)d\tau\over \tau- t},\quad f\in L_ p(\Gamma),\quad t\in \Gamma, \] to be bounded from the Lebesgue space \(L_ p(\Gamma)\) to \(L_ q(\Gamma)\), as \[ \Gamma= \bigcup^ \infty_{n=1} \Gamma_ n,\quad \Gamma_ n= \{z: | z|= r_ n\}, \] it is necessary and sufficient that either condition \[ \sum^ \infty_{n= 1} \left({\sum^ \infty_{k= n} r_ k\over r_ n}\right)^ \sigma r_ n < \infty;\quad \text{or}\quad\sum^ \infty_{n= 1} n^ \sigma r_ n < \infty \] be fulfilled (where \(\sigma= pq/(p- q)\)).
MSC:
47B38 Linear operators on function spaces (general)
45E05 Integral equations with kernels of Cauchy type
47G10 Integral operators
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References:
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