Korean J. Math. Vol. 24 No. 3 (2016) pp.545-565
DOI: https://doi.org/10.11568/kjm.2016.24.3.545

A generic research on nonlinear non-convolution type singular integral operators

Main Article Content

Gumrah Uysal
Vishnu Narayan Mishra
Ozge Ozalp Guller
Ertan Ibikli


In this paper, we present some general results on the pointwise convergence of the non-convolution type nonlinear singular integral operators in the following form:
T_{\lambda }(f;x)=\underset{\Omega }{\int }K_{\lambda }\left( t,x,f\left(t\right) \right) dt,\text{ }x\in \Psi ,\text{ }\lambda \in \Lambda ,
where $\Psi =\left \langle a,b\right \rangle $ and $\Omega =\left \langle A,B\right \rangle $ stand for arbitrary closed, semi-closed or open bounded intervals in $\mathbb{R}$ or these set notations denote $\mathbb{R} $, and $\Lambda $ is a set of non-negative numbers, to the function $f\in L_{p,w}\left( \Omega \right) $, where $L_{p,w}\left( \Omega \right) $ denotes the space of all measurable functions $f$ for which $\left \vert \frac{f}{w}\right \vert ^{p}$ $(1\leq p<\infty )$ is integrable on $\Omega ,$ and $w: \mathbb{R} \rightarrow \mathbb{R}^{+}$ is a weight function satisfying some conditions.

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