Tauberian conditions for geometric maximal operators

Abstract

Let B \mathcal {B} be a collection of measurable sets in R n \mathbb {R}^{n} . The associated geometric maximal operator M B M_{\mathcal {B}} is defined on L 1 ( R n ) L^{1}(\mathbb {R}^n) by M B f ( x ) = sup x ∈ R ∈ B 1 | R | ∫ R | f | M_{\mathcal {B}}f(x) = \sup _{x \in R \in \mathcal {B}}\frac {1}{|R|}\int _{R}|f| . If α > 0 \alpha > 0 , M B M_\mathcal {B} is said to satisfy a Tauberian condition with respect to α \alpha if there exists a finite constant C C such that for all measurable sets E ⊂ R n E \subset \mathbb {R}^n the inequality | { x : M B χ E ( x ) > α } | ≤ C | E | |\{x : M_{\mathcal {B}} \chi _{E}(x) > \alpha \}| \leq C|E| holds. It is shown that if B \mathcal {B} is a homothecy invariant collection of convex sets in R n \mathbb {R}^{n} and the associated maximal operator M B M_{\mathcal {B}} satisfies a Tauberian condition with respect to some 0 > α > 1 0 > \alpha > 1 , then M B M_\mathcal {B} must satisfy a Tauberian condition with respect to γ \gamma for all γ > 0 \gamma > 0 and moreover M B M_{\mathcal {B}} is bounded on L p ( R n ) L^{p}(\mathbb {R}^{n}) for sufficiently large p p . As a corollary of these results it is shown that any density basis that is a homothecy invariant collection of convex sets in R n \mathbb {R}^{n} must differentiate L p ( R n ) L^{p}(\mathbb {R}^{n}) for sufficiently large 𝑝.