NEW WEIGHTED ESTIMATES FOR BILINEAR FRACTIONAL INTEGRAL OPERATORS

We prove a plethora of weighted estimates for bilinear fractional integral operators of the form BI alpha(f, g)(x) = integral(Rn) f(x - t)g(x + t)/vertical bar t vertical bar(n - alpha) dt, 0 < alpha < n. When the target space has an exponent greater than one, many weighted estimates follow trivially from Holder'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.