Details of talk
|Title||Doubling, dyadic doubling and triadic doubling weight functions|
|Presenter||Stephanie Mills (University of South Australia)|
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 itself. 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?