Boundedness and compactness of a class of integral operators with power and logarithmic singularity when pleq q

Akbota Abylayeva,Bolat Seilbekov,Ryskul Oinarov

doi:10.1186/s13660-022-02758-4

Abstract

In this paper, necessary and sufficient conditions for the boundedness and compactness of one class of integral operators with power and logarithmic singularities in weighted Lebesgue spaces are obtained.

Highlights

Let I = (0, ∞) and let v, u be almost everywhere positive and locally integrable functions on the interval I.Let 1 < p, q < ∞, and p = p p–1 Let us denote byLp,v ≡ Lp(v, I) the set of measurable functions f on I for which f p,v = ∞f (x) pv(x) dx 1 p < ∞.Let W be a positive, strictly increasing, and locally absolutely continuous function on the interval I
The boundedness and compactness of operator (1.2) from Lp,w to Lq,v is obtained in the paper
The main goal of the paper is to establish the criteria for the boundedness and compactness of operator (1.1) from Lp,w to Lq,v for the following relations of the space parameters 1 < p ≤ q < ∞

Summary

Introduction

The boundedness and compactness of operator (1.2) from Lp,w to Lq,v is obtained in the paper The boundedness and compactness of operator (1.2) were obtained in the paper [8] when the upper limit of the integral is a function. The main goal of the paper is to establish the criteria for the boundedness and compactness of operator (1.1) from Lp,w to Lq,v for the following relations of the space parameters 1 < p ≤ q < ∞. 3 Boundedness of the operator Tα,β The main result of this section is the following.

Objectives

Results

Conclusion