Two weight $L^{p}$-inequalities for dyadic shifts and the dyadic square function

We consider two weight $L^{p}\to L^{q}$-inequalities for dyadic shifts and the dyadic square function with general exponents $1<p,q<\infty$. 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. Hyt\"onen, C. P\'erez, S. Treil and A. Volberg. 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. An example shows that in general the quadratic $\mathscr{A}_{p,q}$-condition is stronger than the Muckenhoupt type $A_{p,q}$-condition.