New weighted estimates for bilinear fractional integral operators

Abstract

We prove a plethora of weighted estimates for bilinear fractional integral operators of the form \[ B I α ( f , g ) ( x ) = ∫ R n f ( x − t ) g ( x + t ) | t | n − α d t , 0 > α > n . BI_\alpha (f,g)(x)=\int _{\mathbb {R}^n}\frac {f(x-t)g(x+t)}{|t|^{n-\alpha }}\,dt, \qquad 0>\alpha >n. \] When the target space has an exponent greater than one, many weighted estimates follow trivially from Hölder’s inequality and the known linear theory. We address the case where the target Lebesgue space is at most one and prove several interesting one and two weight estimates. As an application we formulate a bilinear version of the Stein-Weiss inequality for fractional integrals.