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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
##### Keywords:
Cauchy type singular operator
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