Two weight norm inequalities for the bilinear fractional integrals

Abstract

In this paper, we give a characterization of the two weight strong and weak type norm inequalities for the bilinear fractional integrals. Namely, we give the characterization of the following inequalities, \[ \|\mathcal I_\alpha (f_1\sigma_1, f_2\sigma_2)\|_{L^q(w)} \le \mathscr N \prod_{i=1}^2\|f_i\|_{L^{p_i}(\sigma_i)} \] and \[ \|\mathcal I_\alpha (f_1\sigma_1, f_2\sigma_2)\|_{L^{q,\infty}(w)} \le \mathscr N_{\textup{weak}} \prod_{i=1}^2\|f_i\|_{L^{p_i}(\sigma_i)}, \] when $q\ge p_1, p_2>1$ and $p_1+p_2\ge p_1p_2$.