David Cruz-Uribe
The University of Alabama
Title
The fine properties of Muckenhoupt weights in the variable Lebesgue spaces
Abstract
The theory of Muckenhoupt $A_p$ weights, $1\le p<\infty$, has been an important area of harmonic analysis for more than 50 years. As part of this, a rich structure theory for these weights has been developed, including the reverse H\"older inequality, left-openness, Jones factorization, and Rubio de Francia extrapolation.
In the past 15 years this theory has been extended to the variable Lebesgue spaces, with the introduction of the $\mathcal A_{p(\cdot)}$ weights by Cruz-Uribe, Fiorenza, and Neugebauer. Given an exponent function $p(\cdot)$, we say that a weight $w$ is in $\mathcal A_{p(\cdot)}$ if $$[w]_{\mathcal A_{p(\cdot)}}=\sup_Q |Q|^{-1}\|w\|_{L^{p(\cdot)}(Q)}\|w^{-1}\|_{L^{p'(\cdot)}(Q)}<\infty.$$
These weights give sufficient (and in some cases necessary) conditions for maximal operators, singular integrals, and other operators of classical harmonic analysis to be bounded on the weighted variable Lebesgue space $L^{p(\cdot)}(w)$.
However, unlike in the constant exponent case, very little is known about the structural properties of $\mathcal A_{p(\cdot)}$ weights, beyond a theory of extrapolation developed by Cruz-Uribe and Wang. In this talk I will discuss recent progress in this area, proving a reverse H\"older inequality for weights in $\mathcal A_{p(\cdot)}$. As an application, I will use it to prove left and right-openness of these weight classes, a result which we can also prove for matrix weights. If time permits I will also discuss other partial results on the structure of $\mathcal A_{p(\cdot)}$ weights.
This research is joint with my PhD student, Michael Penrod.
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