Two-weight L ^p- L ^q Bounds for Positive Dyadic Operators: Unified Approach to p ≤ q and p > q

Abstract

We characterize the $L^{p}(\sigma )\to L^{q}(\omega )$ boundedness of positive dyadic operators of the form

$$T(f\sigma)=\sum\limits_{Q\in\mathcal{D}}\lambda_{Q}{\int}_{Q} f\mathrm{d}\sigma\cdot 1_{Q}, $$

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 σ,σ ₁,σ ₂,ω. In the linear case, we unify the existing “Sawyer testing” (for p≤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∈(1,∞). Our characterization covers the previously unknown case $q<\frac {p_{1}p_{2}}{p_{1}+p_{2}}$, where we introduce a new two-measure Wolff potential.