Equivalent definitions of dyadic Muckenhoupt and reverse Hölder classes in terms of Carleson sequences, weak classes, and comparability of dyadic $L$ log $L$ and $A_{\infty }$ constants

Abstract

In the dyadic case the union of the reverse Hölder classes, ${\cup }_{p>1}R{H}_p^d$, is strictly larger than the union of the Muckenhoupt classes, ${\cup }_{p>1}{A}_p^d=A_{\infty }^d$. We introduce the $R{H}_1^d$ condition as a limiting case of the $R{H}_p^d$ inequalities as $p$ tends to $1$ and show the sharp bound on the $R{H}_1^d$ constant of the weight $w$ in terms of its $A_{\infty }^d$ constant.

We also examine the summation conditions of the Buckley type for the dyadic reverse Hölder and Muckenhoupt weights and deduce them from an intrinsic lemma which gives a summation representation of the bumped average of a weight. We also obtain summation conditions for continuous reverse Hölder and Muckenhoupt classes of weights and both continuous and dyadic weak reverse Hölder classes. In particular, we prove that a weight belongs to the class $R{H}_1$ if and only if it satisfies Buckley's inequality. We also show that the constant in each summation inequality of Buckley type is comparable to the corresponding Muckenhoupt or reverse Hölder constant. To prove our main results we use the Bellman function technique.