On Fatou type convergence of higher derivatives of certain nonlinear singular integral operators

Abstract

The present paper concerns with the Fatou type convergence properties of the \(r-th\) and \((r+1)-th\) derivatives of the nonlinear singular integral operators defined as \[ \left( I_{\lambda}f\right) (x)=\int\limits_{a}^{b}K_{\lambda}(t-x,f(t))\,{\rm d}t,\,\,\,\,\,\,\,x\in\left( a,b\right) , \] acting on functions defined on an arbitrary interval \(\left( a,b\right) ,\) where the kernel \(K_{\lambda}\) satisfies some suitable assumptions. The present study is a continuation and extension of the results established in the paper [7].

Keywords

• nonlinear singular integral operator
• pointwise convergence
• Fatou type convergence