Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted 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 $$w(x\in {\mathbb{R}}^n:M_{\mathfrak{B}}(1_E)(x)>1/2)<cw(E).$$ 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 $$\mu (x\in {\mathbb{R}}^n:M_{\mathfrak{R},\mu }(1_E)(x)>1/2)<c\mu (E)$$ 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 $\mu$.

35 pages, 1 figure, minor typos corrected, one reference added, incorporates referee's report; to appear in Trans. Amer. Math. Soc

References in corpus (3)

• Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, II
• The multilinear strong maximal function
• The endpoint Fefferman-Stein inequality for the strong maximal function

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• Weighted Solyanik estimates for the strong maximal function
• Solyanik estimates and local Hölder continuity of halo functions of geometric maximal operators
• Hölder continuity of Tauberian constants associated with discrete and ergodic strong maximal operators
• A note on local Hölder continuity of weighted Tauberian functions
• On the Boundedness of Multilinear Fractional Strong Maximal Operator with multiple weights
• Geometric maximal operators and BMO on product bases
• Sharp inequalities for one-sided Muckenhoupt weights
• Beyond local maximal operators