On singular integral operators involving power nonlinearity
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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 $f$ $\in L_{1}\left( \mathbb{R}\right) .$
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 $f$ $\in L_{1}\left( \mathbb{R}\right) .$
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