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 $\mu$.
Categories: math.CA math.FA