Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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ON SINGULAR INTEGRAL OPERATORS INVOLVING POWER NONLINEARITY

  • Almali, Sevgi Esen (Department of Mathematics, Kirikkale University) ;
  • Uysal, Gumrah (Department of Computer Technologies, Division of Technology of Information Security, Karabuk University) ;
  • Mishra, Vishnu Narayan (Department of Mathematics, Indira Gandhi National Tribal University) ;
  • Guller, Ozge Ozalp (Department of Mathematics, Ankara University)
  • Received : 2017.06.06
  • Accepted : 2017.11.08
  • Published : 2017.12.30
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Abstract

In the current manuscript, we investigate the pointwise convergence of the singular integral operators involving power nonlinearity given in the following form: $$T_{\lambda}(f;x)={\int_a^b}{\sum^n_{m=1}}f^m(t)K_{{\lambda},m}(x,t)dt,\;{\lambda}{\in}{\Lambda},\;x{\in}(a,b)$$, where ${\Lambda}$ is an index set consisting of the non-negative real numbers, and $n{\geq}1$ is a finite natural number, at ${\mu}$-generalized Lebesgue points of integrable function $f{\in}L_1(a,b)$. Here, $f^m$ denotes m-th power of the function f and (a, b) stands for arbitrary bounded interval in ${\mathbb{R}}$ or ${\mathbb{R}}$ itself. We also handled the indicated problem under the assumption $f{\in}L_1({\mathbb{R}})$.

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References

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