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

Abstract:

Let $\mathfrak {B}$ be a homothecy invariant collection of convex sets in $\mathbb {R}^{n}$. Given a measure $\mu$, the associated weighted geometric maximal operator $M_{\mathfrak {B}, \mu }$ is defined by \begin{align*} M_{\mathfrak {B}, \mu }f(x) := \sup _{x \in B \in \mathfrak {B}}\frac {1}{\mu (B)}\int _{B}|f|d\mu . \end{align*} It is shown that, provided $\mu$ satisfies an appropriate doubling condition with respect to $\mathfrak {B}$ and $\nu$ is an arbitrary locally finite measure, the maximal operator $M_{\mathfrak {B}, \mu }$ is bounded on $L^{p}(\nu )$ for sufficiently large $p$ if and only if it satisfies a Tauberian condition of the form \begin{align*} \nu \big (\big \{x \in \mathbb {R}^{n} : M_{\mathfrak {B}, \mu }(\textbf {1}_E)(x) > \frac {1}{2} \big \}\big ) \leq c_{\mu , \nu }\nu (E). \end{align*} As a consequence of this result we provide an alternative characterization of the class of Muckenhoupt weights $A_{\infty , \mathfrak {B}}$ for homothecy invariant Muckenhoupt bases $\mathfrak {B}$ consisting of convex sets. Moreover, it is immediately seen that the strong maximal function $M_{\mathfrak {R}, \mu }$, defined with respect to a product-doubling measure $\mu$, is bounded on $L^{p}(\nu )$ for some $p > 1$ if and only if \begin{align*} \nu \big (\big \{x \in \mathbb {R}^{n} : M_{\mathfrak {R}, \mu }(\textbf {1}_E)(x) > \frac {1}{2}\big \}\big ) \leq c_{\mu , \nu }\nu (E)\; \end{align*} holds for all $\nu$-measurable sets $E$ in $\mathbb {R}^{n}$. In addition, we discuss applications in differentiation theory, in particular proving that a $\mu$-weighted homothecy invariant basis of convex sets satisfying appropriate doubling and Tauberian conditions must differentiate $L^{\infty }(\nu )$.