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

Gumrah Uysal, Vishnu Narayan Mishra, Ozge Ozalp Guller, Ertan Ibikli

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.


Keywords


$\mu $-generalized Lebesgue point, Pointwise convergence, Rate of convergence.

Subject classification

41A35, 41A25, 45P05

Sponsor(s)



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