Two-weight $L^p$-$L^q$ bounds for positive dyadic operators: unified approach to $p\leq q$ and $p>q$

Abstract: We characterize the $L^p(\sigma)\to L^q(\omega)$ boundedness of positive dyadic operators of the form $ T(f\sigma)=\sum_{Q\in\mathscr{D}}\lambda_Q\int_Q f\,\mathrm{d}\sigma\cdot 1_Q, $ and the $L^{p_1}(\sigma_1)\times L^{p_2}(\sigma_2)\to L^q(\omega)$ boundedness of their bilinear analogues, for arbitrary locally finite measures $\sigma,\sigma_1,\sigma_2,\omega$. In the linear case, we unify the existing "Sawyer testing" (for $p\leq q$) and "Wolff potential" (for $p>q$) characterizations into a new "sequential testing" characterization valid in all cases. We extend these ideas to the bilinear case, obtaining both sequential testing and potential type characterizations for the bilinear operator and all $p_1,p_2,q\in(1,\infty)$. Our characterization covers the previously unknown case $q<\frac{p_1p_2}{p_1+p_2}$, where we introduce a new two-measure Wolff potential.