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

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• Abstract

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].

• Keywords

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

41A35, 44A35, 42A85

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@Article{ATA-31-307, author = {}, 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 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].

}, 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 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, convolution type double singular integral operators, $\mu$-generalized Lebesgue point. AB -

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].

H. Karsli. (1970). On Fatou Type Convergence of Convolution Type Double Singular Integral Operators. Analysis in Theory and Applications. 31 (3). 307-320. doi:10.4208/ata.2015.v31.n3.8
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