Fatou type weighted pointwise convergence of nonlinear singular integral operators Depending on two parameters

¹ Karabuk University, Faculty of Science, Department of Mathematics, Karabuk, Turkey
² Baskent University, Faculty of Education, Department of Mathematic Education, Ankara, Turkey

Abstract

In this paper we present some theorems concerning existence and Fatou type weighted pointwise convergence of nonlinear singular integral operators of the form: $$(T_{\lambda }f)(x)={\displaystyle \underset{R}{\int }{K}_{\lambda }}(t-x;f(t))dt,\text{ x}\in R,\lambda \in \Lambda$$ where Λ ≠ ∅ is a set of non-negative indices, at a common generalized Lebesgue point of the functions f ∈ L_1,ϕ (R) and positive weight function φ. Here, L_1,ϕ (R) is the space of all measurable functions for which $$\left|\frac{f}{\varphi }\right|$$ is integrable on R.