Convergence Almost Everywhere of Non-convolutional Integral Operators in Lebesgue Spaces

Yakhshiboev M U

doi:10.13189/ms.2020.080611

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 <img src=image/13420650_01.gif> 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 <img src=image/13420650_01.gif> of nonconvolutional integral operators take into account different behaviors at <img src=image/13420650_02.gif> and <img src=image/13420650_03.gif> depending on <img src=image/13420650_04.gif> (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 <img src=image/13420650_05.gif> as <img src=image/13420650_06.gif>. 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.

Highlights

A modification of Liouville fractional integro-differentiation on a straight line R is considered in [6], ”attached” to a certain fixed point c ∈ R, |c| < ∞, and convenient in a way that it can be applied to functions set on the entire straight line that can have any increment at infinity: x (Icαφ)(x) := φ(t)Γ(α) (x−t)1−α c c φ(t)Γ(α) (t−x)1−α dt, x > c, dt, x < c, (1)x where x ∈ R, α > 0
The kernels Kε(x, y) of nonconvolutional integral operators take into account different behaviors at |y| → 0 and |y| → ∞ depending on ε → 0 and cover the situation in the particular case of convolutional and non-convolutional integral operators
We prove the convergence almost everywhere of the non-convolutional integral operator (3) in space Lp (Rn)

Summary

Introduction

A modification of Liouville fractional integro-differentiation on a straight line R is considered in [6], ”attached” to a certain fixed point c ∈ R, |c| < ∞, and convenient in a way that it can be applied to functions set on the entire straight line that can have any increment at infinity: x (Icαφ)(x) :=. In [3], [6], three different ways of reduction of the ChenMarchaud constructions for fractional differentiation of Dcα, are considered, and these different options for reduction are used to describe and invert the fractional integrals (1) of functions from Llpoc (R). Non-convolutional integral operators in space Lp (R) were poorly studied. In this regard, one class of non-convolutional averaging is considered in [8]. Multi-dimensional non-convolutional integral operators are studied averaging was considered; our condition 3) in Theorem 2 being proved is much more general (including the convolutional operators) and covers a wide class of kernels. We prove the convergence almost everywhere of the non-convolutional integral operator (3) in space Lp (Rn)

Auxiliary statements

The case of one-dimensional nonconvolutional integral operators

The case of multidimensional nonconvolutional integral operators

Conclusions