Solyanik estimates and local Hölder continuity of halo functions of geometric maximal operators

الأصل · EN

Let B be a homothecy invariant basis consisting of convex sets in Rⁿ, and define the associated geometric maximal operator MB by MB f(x):=ₓ ∈ ᵣ ∈ B1/|R|∫ᵣ |f| and the halo function ϕB(α) on (1,∞) by ϕB(α):=ₑ ⊂ ᵣₙ: ₀ < |ₑ| < ∞1/|E||{x∈ Rⁿ: MB χₑ (x) >1/α}|. It is shown that if ϕB(α) satisfies the Solyanik estimate ϕB(α) - 1 ≤ C (1 - 1α)ᵖ for α∈(1,∞) sufficiently close to 1 then ϕB lies in the Hölder class Cᵖ(1,∞). As a consequence we obtain that the halo functions associated with the Hardy-Littlewood maximal operator and the strong maximal operator on Rⁿ lie in the Hölder class C¹/ⁿ(1,∞).