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On singular integral operators involving power nonlinearity
Sevgi Esen Almali,Gumrah Uysal,Vishnu Narayan Mishra,Ozge Ozalp Guller 강원경기수학회 2017 한국수학논문집 Vol.25 No.4
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 $f$ $\in L_{1}\left( \mathbb{R}\right) .$