In this talk, we focus on the special case of doubling weight functions. We say
a weight function is doubling if for every pair of adjacent intervals of equal
length, the areas under the graph of the weight function on each of the two
intervals are roughly the same.
Certain doubling weight functions arise naturally in the theory of boundedness
of singular integral operators on weighted Lebesgue spaces. Doubling weight
functions are also important in complex analysis as they arise as the boundary
values on the real line of quasiconformal mappings of the upper half-plane to
For the related concept of dyadic doubling weight functions, we require the
same condition to hold but only for a smaller collection of pairs of adjacent
dyadic intervals. Thus doubling implies dyadic doubling. It turns out that
dyadic doubling does not imply doubling. We consider an additional, similarly
defined assumption, known as triadic doubling. In this talk, we answer the
question: Is a measure that is both dyadic doubling and triadic doubling
necessarily also doubling?