A note on Singular integral

TL;DR

This paper establishes boundedness results for a specific convolution operator involving singular kernels within a certain parameter range, contributing to the understanding of singular integral operators in harmonic analysis.

Contribution

It proves boundedness of a class of singular convolution operators for particular parameter ranges, extending existing knowledge on such operators.

Findings

The operator is bounded from L^p to L^q for specified p and q.

The boundedness depends on the parameter α within a certain range.

Provides conditions under which the convolution operator is bounded.

Abstract

In this paper, we prove that for $\frac{n}{2}+\frac{1}{4}<\alpha \leq\frac{n+1}{2} $, the convolution operator $$S_{\alpha} f(x)=\int_{|y| \geq 1} f(x-y)\left(|y|^{2}-1\right)^{-\alpha} d y$$ is bounded from $L^p$ to $L^q$ for certain values of $p$ and $q$.