Two-weight inequalities and local Tb theorems: quadratic testing and big piece methods

We consider two-weight L-p -> L-q-inequalities for dyadic shifts and the dyadic square function with general exponents 1 <p, q <infinity. It is shown that if a so-called quadratic A(p,q)-condition related to the measures holds, then a family of dyadic shifts satisfies the two-weight estimate in an 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. Hytonen, C. Perez, S. Treil and A. Volberg (2014). The dyadic square function satis fi es the two-weight estimate if and only if it satis fi es the quadratic testing condition, and the quadratic 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 A(p,q)-condition is stronger than the Muckenhoupt type A(p,q)-condition.