Mathematics and Statistics Vol. 8(6), pp. 705 - 710
DOI: 10.13189/ms.2020.080611
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Convergence Almost Everywhere of Non-convolutional Integral Operators in Lebesgue Spaces

Yakhshiboev M. U. ^*
Faculty of Mathematics, National University of Uzbekistan, Tashkent, 100174, Tashkent, Uzbekistan

ABSTRACT

The case of one-dimensional and multidimensional non-convolutional integral operators in Lebesgue spaces is considered in this paper. The convergence in the norm and almost everywhere of non-convolution integral operators in Lebesgue spaces was insufficiently studied. The kernels of non-convolutional integral operators do not need to have a monotone majorant, so the well-known results on the convergence almost everywhere of convolutional averages are not applicable here. The kernels of nonconvolutional integral operators take into account different behaviors at and depending on (which is important in applications) and cover the situation in the particular case of convolutional and non-convolutional integral operators. We are interested in the behavior of function as . Theorems on convergence almost everywhere in the case of one-dimensional and multidimensional nonconvolution integral operators in Lebesgue spaces are proved. The theorems proved are more general ones (including for convolutional integral operators) and cover a wide class of kernels.

KEYWORDS
Non-convolutional Integral Operator, Convergence Almost Everywhere, Chen-Marchaud Fractional Derivatives, Non-convolutional Averaging

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Yakhshiboev M. U. , "Convergence Almost Everywhere of Non-convolutional Integral Operators in Lebesgue Spaces," Mathematics and Statistics, Vol. 8, No. 6, pp. 705 - 710, 2020. DOI: 10.13189/ms.2020.080611.

(b). APA Format:
Yakhshiboev M. U. (2020). Convergence Almost Everywhere of Non-convolutional Integral Operators in Lebesgue Spaces. Mathematics and Statistics, 8(6), 705 - 710. DOI: 10.13189/ms.2020.080611.