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

Let B be a homothecy invariant basis consisting of convex sets in Rn, and define the associated geometric maximal operator MB byMBf(x):=supx∈R∈B⁡1|R|∫R|f| and the halo function ϕB(α) on (1,∞) byϕB(α):=supE⊂Rn:0<|E|<∞⁡1|E||{x∈Rn:MBχE(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,∞).