Korean J. Math.  Vol 25, No 4 (2017)  pp.483-494
DOI: http://dx.doi.org/10.11568/kjm.2017.25.4.483

On singular integral operators involving power nonlinearity

Sevgi Esen Almalı, Gumrah Uysal, Vishnu Narayan Mishra, Ozge Ozalp Guller


In the current manuscript, we investigate the pointwise convergence of the singular integral operators involving power nonlinearity given in the following form:
}(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) ,
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 $f$ $\in L_{1}\left( \mathbb{R}\right) .$


Pointwise convergence, Nonlinear integral operators, $\mu-$generalized Lebesgue point

Subject classification

41A35, 41A25, 47G10, 47A58.


Full Text:



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