Weighted estimates for bilinear fractional integral operators: a necessary and sufficient condition for power weights

Abstract

We consider weighted estimates for two bilinear fractional integral operators $I_{2,\alpha }$ and $BI_{\alpha }$. Moen (Collect Math 60:213–238, 2009) obtained a necessary and sufficient condition for $I_{2,\alpha }$. However we know only some sufficient conditions for $BI_{\alpha }$ which is a variant of the bilinear Hilbert transform. Restricted to power weights we obtain a necessary and sufficient condition for $BI_{\alpha }$. We also prove a bilinear Stein–Weiss inequality.

Appendix

We show how to use the interpolation theorem (Lemma 3) to obtain (9):

$$\begin{aligned} \Vert T_{A,B,C}^{\alpha }(f,g) \Vert _{L^r} \le K \Vert f \Vert _{L^p} \Vert g \Vert _{L^q} \end{aligned}$$

from weak type estimates (10).

Let A, B and C be fixed such that $A+B+C \ge 0$. For the sake of simplicity, we consider in the following region:

$$\begin{aligned} {\mathbb {D}} := \left\{ (x,y,z) \in {\mathbb {R}}^3 : x> \alpha /n, y> \alpha /n, x+y < 1, z=x+y - (\alpha -A-B-C)/n \right\} , \end{aligned}$$

where $\alpha < n/2$; see Remarks in Sect. 2.

Let $P:=(1/p,1/q,1/r) \in {\mathbb {D}}$ be fixed, and assume that

$$\begin{aligned} A< n/p', B< n/q' \quad \text {and}\quad C< n/r. \end{aligned}$$

It suffices to find three points $P_i :=(1/p_i,1/q_i,1/r_i) \in {\mathbb {D}}, i=1,2,3$ such that P lies in the interior of the triangle with vertices $P_1, P_2$ and $P_3$, and

$$\begin{aligned} \Vert T_{A,B,C}^{\alpha }(f,g) \Vert _{L^{r_i, \infty }} \le K \Vert f \Vert _{L^{p_i}} \Vert g \Vert _{L^{q_i}}. \end{aligned}$$

Let $p_1=p$, and we take $q_1 >q$ sufficiently near q such that $B< n/q_1'$, and $C< n/r_1$. Similarly let $q_2=q$ and we take $p_2>p$ sufficiently near p such that $A < n/p_2'$ and $C< n/r_2$. Finally we take $p_3 < p$ and $q_3 < q$ sufficiently near p and q respectively such that $A< n/p_3' , B < n/q_3'$, and $C< n/r_3$.