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We characterize the $L^p(\sigma)\to L^q(\omega)$ boundedness of positivedyadic operators of the form $  T(f\sigma)=\sum_{Q\in\mathscr{D}}\lambda_Q\int_Q f\,\mathrm{d}\sigma\cdot1_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 allcases. We extend these ideas to the bilinear case, obtaining both sequentialtesting and potential type characterizations for the bilinear operator and all$p_1,p_2,q\in(1,\infty)$. Our characterization covers the previously unknowncase $q<\frac{p_1p_2}{p_1+p_2}$, where we introduce a new two-measure Wolffpotential.

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