Solyanik estimates and local H\"older continuity of halo functions of geometric maximal operators

Let $\mathcal{B}$ be a homothecy invariant basis consisting of convex sets in $\mathbb{R}^n$, and define the associated geometric maximal operator $M_{\mathcal{B}}$ by $$M_{\mathcal{B}} f(x) :=\sup_{x \in R \in \mathcal{B}}\frac{1}{|R|}\int_R |f|$$ and the halo function $\phi_{\mathcal{B}}(\alpha)$ on $(1,\infty)$ by $$\phi_{\mathcal B}(\alpha) :=\sup_{E \subset \mathbb{R}^n :\, 0 < |E| < \infty}\frac{1}{|E|}|\{x\in \mathbb{R}^n : M_{\mathcal{B}} \chi_E (x) >1/\alpha\}|.$$ It is shown that if $\phi_{\mathcal{B}}(\alpha)$ satisfies the Solyanik estimate $\phi_{\mathcal B}(\alpha) - 1 \leq C (1 - \frac{1}{\alpha})^p$ for $\alpha\in(1,\infty)$ sufficiently close to 1 then $\phi_{\mathcal{B}}$ lies in the H\"older class $C^p(1,\infty)$. As a consequence we obtain that the halo functions associated with the Hardy-Littlewood maximal operator and the strong maximal operator on $\mathbb{R}^n$ lie in the H\"older class $C^{1/n}(1,\infty)$.Comment: 19 pages, 1 figure, minor typos corrected, incorporates referee's report, to appear in Adv. Mat

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

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 $\mu$.Comment: 35 pages, 1 figure, minor typos corrected, one reference added, incorporates referee's report; to appear in Trans. Amer. Math. So

Lp(R2)L^p(\mathbb{R}^2) bounds for geometric maximal operators associated to homothecy invariant convex bases

Let $\mathcal{B}$ be a nonempty homothecy invariant collection of convex sets of positive finite measure in $\mathbb{R}^2$. Let $M_\mathcal{B}$ be the geometric maximal operator defined by $$M_\mathcal{B}f(x) = \sup_{x \in R \in \mathcal{B}}\frac{1}{|R|}\int_R |f|\;.$$ We show that either $M_\mathcal{B}$ is bounded on $L^p(\mathbb{R}^2)$ for every $1 < p \leq \infty$ or that $M_\mathcal{B}$ is unbounded on $L^p(\mathbb{R}^2)$ for every $1 \leq p < \infty$. As a corollary, we have that any density basis that is a homothecy invariant collection of convex sets in $\mathbb{R}^2$ must differentiate $L^p(\mathbb{R}^2)$ for every $1 < p \leq \infty$

Weak Type Inequalities for Maximal Operators Associated to Double Ergodic Sums

Given an approach region Γ ∈ Z+2 and a pair U, V of commuting nonperiodic measure preserving transformations on a probability space (Ω, Σ, μ), it is shown that either the associated multiparameter ergodic averages of any function in L1(Ω) converge a.e. or that, given a positive increasing function ϕ on [0,∞) that is o(log x) as x → ∞, there exists a function g ∈ Lϕ(L)(Ω) whose associated multiparameter ergodic averages fail to converge a.e