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

Abstract

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:
\begin{equation*}
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 ,
\end{equation*}
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.