Two-weight $L^{p}$-inequalities for dyadic shifts and the dyadic square function
Volume 237 / 2017
Abstract
We consider two-weight $L^{p}\to L^{q}$-inequalities for dyadic shifts and the dyadic square function with general exponents $1 \lt p,q \lt \infty $. It is shown that if a so-called quadratic $\mathscr {A}_{p,q}$-condition related to the measures holds, then a family of dyadic shifts satisfies the two-weight estimate in an $\mathcal {R}$-bounded sense if and only if it satisfies the direct and the dual quadratic testing condition. In the case $p=q=2$ this reduces to the result by T. Hytönen, C. Pérez, S. Treil and A. Volberg (2014).
The dyadic square function satisfies the two-weight estimate if and only if it satisfies the quadratic testing condition, and the quadratic $\mathscr {A}_{p,q}$-condition holds. Again in the case $p=q=2$ we recover the result by F. Nazarov, S. Treil and A. Volberg (1999).
An example shows that in general the quadratic $\mathscr {A}_{p,q}$-condition is stronger than the Muckenhoupt type $A_{p,q}$-condition.
