Korean J. Math.  Vol 25, No 4 (2017)  pp.483-494
DOI: http://dx.doi.org/10.11568/kjm.2017.25.4.483

### On singular integral operators involving power nonlinearity

Sevgi Esen Almalı, Gumrah Uysal, Vishnu Narayan Mishra, Ozge Ozalp Guller

#### Abstract

In the current manuscript, we investigate the pointwise convergence of the singular integral operators involving power nonlinearity given in the following form:
\begin{equation*}
T_{\lambda
}(f;x)=\int \limits_{a}^{b}\sum \limits_{m=1}^{n}f^{m}(t)K_{\lambda
,m}(x,t)dt,\text{ }\lambda \in \Lambda ,\text{ }x\in \left( a,b\right) ,
\end{equation*}
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 $f$ $\in L_{1}\left( a,b\right) .$ Here, $f^{m}$ denotes $m-th$ power of the function $f$ and $\left( a,b\right)$ stands for arbitrary bounded interval in $\mathbb{R}$ or $\mathbb{R}$ itself. We also handled the indicated problem under the assumption $f$ $\in L_{1}\left( \mathbb{R}\right) .$

#### Keywords

Pointwise convergence, Nonlinear integral operators, $\mu-$generalized Lebesgue point

#### Subject classification

41A35, 41A25, 47G10, 47A58.

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