Volume 31, Issue 3
On Fatou Type Convergence of Convolution Type Double Singular Integral Operators

Anal. Theory Appl., 31 (2015), pp. 307-320

Published online: 2017-07

Preview Full PDF 553 4452
Export citation

Cited by

• Abstract
In this paper some approximation formulae for a class of convolution typedouble 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 $L_{1}( \langlea,b\rangle ^{2}),$ where $\langle a,b\rangle$ isan arbitrary interval in $\mathbb{R}$. In this paper three theorems areproved, one for existence of the operator $( T_{\lambda }f)( x,y)$ and the others for its Fatou-type pointwise convergenceto $f(x_{0},y_{0}),$ 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. Ourresults improve and extend some of the previous results of [1,6,8,10,11,13] in three dimensional frame andespecially the very recent paper [15].
• Keywords

Fatou-type convergence convolution type double singular integral operators $\mu$-generalized Lebesgue point

41A35 44A35 42A85

@Article{ATA-31-307, author = {H. Karsli}, title = {On Fatou Type Convergence of Convolution Type Double Singular Integral Operators}, journal = {Analysis in Theory and Applications}, year = {2017}, volume = {31}, number = {3}, pages = {307--320}, abstract = {In this paper some approximation formulae for a class of convolution typedouble 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 $L_{1}( \langlea,b\rangle ^{2}),$ where $\langle a,b\rangle$ isan arbitrary interval in $\mathbb{R}$. In this paper three theorems areproved, one for existence of the operator $( T_{\lambda }f)( x,y)$ and the others for its Fatou-type pointwise convergenceto $f(x_{0},y_{0}),$ 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. Ourresults improve and extend some of the previous results of [1,6,8,10,11,13] in three dimensional frame andespecially the very recent paper [15].}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2015.v31.n3.8}, url = {http://global-sci.org/intro/article_detail/ata/4642.html} }
TY - JOUR T1 - On Fatou Type Convergence of Convolution Type Double Singular Integral Operators AU - H. Karsli JO - Analysis in Theory and Applications VL - 3 SP - 307 EP - 320 PY - 2017 DA - 2017/07 SN - 31 DO - http://doi.org/10.4208/ata.2015.v31.n3.8 UR - https://global-sci.org/intro/article_detail/ata/4642.html KW - Fatou-type convergence KW - convolution type double singular integral operators KW - $\mu$-generalized Lebesgue point AB - In this paper some approximation formulae for a class of convolution typedouble 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 $L_{1}( \langlea,b\rangle ^{2}),$ where $\langle a,b\rangle$ isan arbitrary interval in $\mathbb{R}$. In this paper three theorems areproved, one for existence of the operator $( T_{\lambda }f)( x,y)$ and the others for its Fatou-type pointwise convergenceto $f(x_{0},y_{0}),$ 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. Ourresults improve and extend some of the previous results of [1,6,8,10,11,13] in three dimensional frame andespecially the very recent paper [15].