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

Abstract

We characterize the $L^p(\sigma )\to L^q(\omega )$ boundedness of positive dyadic operators of the form $T(f\sigma )={\sum }_{Q\in \mathcal{D}}{\lambda }_Q{\int }_{Q}f\mathrm{d}\sigma \cdot 1_Q,$ 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 $\sigma ,{\sigma }_1,{\sigma }_2,\omega$. In the linear case, we unify the existing "Sawyer testing" (for $p\le q$) 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 $q<\frac{p_1{p}_2}{p_1+p_2}$, where we introduce a new two-measure Wolff potential.

27 pages

References in corpus (1)

• Operator-valued dyadic shifts and the T(1) theorem

Cited by in corpus (3)

• $A_p$-$A_{\infty }$ estimates for multilinear maximal and sparse operators
• A Two-weight inequality between $L^p({\ell }^2)$ and $L^p$
• Weighted $L^p\to L^q$-boundedness of commutators and paraproducts in the Bloom setting