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

Emil Vuorinen¹

¹ Department of Mathematics and Statistics P.O.B. 68 (Gustaf Hällströmin Katu 2b) FI-00014 University of Helsinki, Finland

Studia Mathematica, Tome 237 (2017) no. 1, pp. 25-56

Résumé

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.

Détail

DOI : 10.4064/sm8288-9-2016

Keywords: consider two weight inequalities dyadic shifts dyadic square function general exponents infty shown so called quadratic mathscr condition related measures holds family dyadic shifts satisfies two weight estimate mathcal bounded sense only satisfies direct dual quadratic testing condition reduces result hyt nen rez treil volberg dyadic square function satisfies two weight estimate only satisfies quadratic testing condition quadratic mathscr condition holds again recover result nazarov treil volberg example shows general quadratic mathscr condition stronger muckenhoupt type condition