Some estimates for the bilinear fractional integrals on the Morrey space

TL;DR

This paper investigates the boundedness of a bilinear fractional integral operator on Morrey spaces, providing new weighted estimates and an Olsen type inequality, advancing the understanding of such operators in harmonic analysis.

Contribution

The paper establishes weighted boundedness of the bilinear fractional integral operator on Morrey spaces and introduces an Olsen type inequality for it, which are novel results.

Findings

Proved weighted boundedness of $B\mathcal{I}_\alpha$ on Morrey spaces.

Established an Olsen type inequality for the operator.

Extended the theory of bilinear fractional integrals in Morrey space context.

Abstract

In this paper, we are interested in the following bilinear fractional integral operator $B\mathcal{I}_\alpha$ defined by \[ B\mathcal{I}_{\alpha}({f,g})(x)=\int_{% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-\alpha}}dy, \] with $0< \alpha<n$. We prove the weighted boundedness of $B\mathcal{I}_\alpha$ on the Morrey type spaces. Moreover, an Olsen type inequality for $B\mathcal{I}_\alpha$ is also given.