Some weighted approximation properties of nonlinear double integral operators
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Abstract
In this paper, we present some recent results on weighted pointwise convergence and the rate of pointwise convergence for the family of nonlinear double singular integral operators in the following form:
\begin{equation*}
T_{\eta }\left( f;x,y\right) =\underset{\mathbb{R}^{2}}{\iint }K_{\eta
}\left( t-x,s-y,f\left( t,s\right) \right) dsdt,\text{ }\left( x,y\right)
\in \mathbb{R}^{2},\text{ }\eta \in \Lambda ,
\end{equation*}
where the function $f: \mathbb{R}^{2}\rightarrow \mathbb{R}$ is Lebesgue measurable on $\mathbb{R}^{2}$ and $\Lambda $ is a non-empty set of indices. Further, we provide an example to support these theoretical results.
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References
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