The Bellman functions and two-weight inequalities for Haar multipliers

Abstract

We give necessary and sufficient conditions for two-weight norm inequalities for Haar multiplier operators and for square functions. The conditions are of the type used by Eric Sawyer in characterizing the boundedness of the wide class of operators with positive kernel. The difference is that our operator is essentially singular. We also show how to separate two Sawyer’s conditions (even for positive kernel operators) by finding which condition is responsible for which estimate.