Muckenhoupt and reverse Hölder classes of weights play an important role in
harmonic analysis, PDEs and quasiconformal mappings. In 1974, Coifman
and Fefferman showed that a weight belongs to a Muckenhoupt class
for
some
if and only if it belongs to a reverse Hölder class
for
some
.
In 2009, Vasyunin found the exact dependence between
,
and the corresponding characteristic of the weight using the Bellman
function method. The result of Coifman and Fefferman works for the
dyadic classes of weights under an additional assumption that the weights
are dyadically doubling. We extend Vasyunin’s result to the dyadic reverse
Hölder and Muckenhoupt classes and obtain the dependence between
,
,
the doubling constant and the corresponding characteristic of
the weight. More precisely, given a dyadically doubling weight in
on a given
dyadic interval
,
we find an upper estimate on the average of the function
(with
) over the
interval
.
From the bound on this average, we can conclude, for example, that
belongs to the
corresponding
-class
or that
is in
for some
values of
.
We obtain our results using the method of Bellman functions. The main
novelty of this paper is how we use dyadic doubling in the Bellman function
proof.
