On Fatou Type Convergence of Convolution Type Double Singular Integral Operators

In this paper some approximation formulae for a class of convolution type double singular integral operators depending on three parameters of the type $( T_{\lambda }f) ( x,y)=\int_{a}^{b}\int_{a}^{b}f(t,s) K_{\lambda}(t-x,s-y) dsdt, \quad x,y\in (a,b), \quad \lambda \in \Lambda \subset 0,\infty ), $ are given. Here $f$ belongs to the function space $L_{1}( \langle a,b\rangle ^{2}),$ where $\langle a,b\rangle $ is an arbitrary interval in $\mathbb{R}$. In this paper three theorems are proved, one for existence of the operator $( T_{\lambda }f)( x,y) $ and the others for its Fatou-type pointwise convergence to $f(x_{0},y_{0}),$ as $(x,y,\lambda )$ tends to $(x_{0},y_{0},\lambda_{0}).$ In contrast to previous works, the kernel functions $K_{\lambda}(u,v)$ don't have to be $2\pi$-periodic, positive, even and radial. Our results improve and extend some of the previous results of 1,6,8,10,11,13] in three dimensional frame and especially the very recent paper 15].