Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases

TL;DR

This paper characterizes Muckenhoupt weights for convex set bases and explores their implications for differentiation properties and maximal functions in weighted measure spaces.

Contribution

It provides a new characterization of Muckenhoupt weights via Tauberian conditions for convex bases and extends results to doubling measures, impacting differentiation theory.

Findings

Characterization of $A_{ infty, eta}$ weights via Tauberian conditions.

Boundedness of strong maximal functions on $L^p( u)$ spaces.

Tauberian conditions imply differentiation of $L^ infty( u)$ by the bases.

Abstract

We give an alternative characterization of the class of Muckenhoupt weights $A_{\infty, \mathfrak B}$ for homothecy invariant Muckenhoupt bases $\mathfrak B$ consisting of convex sets. In particular we show that $w\in A_{\infty, \mathfrak B}$ if and only if there exists a constant $c>0$ such that for all measurable sets $E\subset \mathbb R^n$ we have $$ w({x\in \mathbb R^n: M_{\mathfrak B} (\mathbf {1}_E)(x)>1/2}) < c w(E).$$ This applies for example to the collection $\mathfrak R$ of rectangles with sides parallel to the coordinate axes, giving a new characterization of strong (multiparameter) Muckenhoupt weights. We also show versions of these results under the presence of a doubling measure. Thus the strong maximal function $M_{\mathfrak R,\mu}$, defined with respect to a product-doubling measure $\mu$, is bounded on $L^p(\mu)$ for some $p>1$ if and only if $$\mu({x\in \mathbb R^n: M_{\mathfrak R,\mu} (\mathbf{1}_E)(x)>1/2}) < c \mu(E)$$ for all measurable sets $E\subset \mathbb R^n$. Finally we discuss applications in differentiation theory, proving among other things that Tauberian conditions as above imply that the corresponding bases differentiate $L^\infty(\mu)$, with respect to the measure $\mu$.