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Register a new userA note on local Holder continuity of weighted Tauberian functions
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Let $mathsf M$ and $mathsf M _{mathsf S}$ respectively denote the Hardy-Littlewood maximal operator with respect to cubes and the strong maximal operator on $mathbb{R}^n$, and let $w$ be a nonnegative locally integrable function on $mathbb{R}^n$. We define the associated Tauberian functions $mathsf{C}_{mathsf{HL},w}(alpha)$ and $mathsf{C}_{mathsf{S},w}(alpha)$ on $(0,1)$ by [ mathsf{C}_{mathsf{HL},w}(alpha) :=sup_{substack{E subset mathbb{R}^n 0 < w(E) < infty}} frac{1}{w(E)}w({x in mathbb{R}^n : mathsf M chi_E(x) > alpha}) ] and [ mathsf{C}_{mathsf{S},w}(alpha) := sup_{substack{E subset mathbb{R}^n 0 < w(E) < infty}} frac{1}{w(E)}w({x in mathbb{R}^n : mathsf M _{mathsf S}chi_E(x) > alpha}). ] Utilizing weighted Solyanik estimates for $mathsf M$ and $mathsf M_{mathsf S}$, we show that the function $mathsf{C}_{mathsf{HL},w} $ lies in the local Holder class $C^{(c_n[w]_{A_{infty}})^{-1}}(0,1)$ and $mathsf{C}_{mathsf{S},w} $ lies in the local Holder class $C^{(c_n[w]_{A_{infty}^ast})^{-1}}(0,1)$, where the constant $c_n>1$ depends only on the dimension $n$.
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This paper concerns the smoothness of Tauberian constants of maximal operators in the discrete and ergodic settings. In particular, we define the discrete strong maximal operator $tilde{M}_S$ on $mathbb{Z}^n$ by [ tilde{M}_S f(m) := sup_{0 in R subset mathbb{R}^n}frac{1}{#(R cap mathbb{Z}^n)}sum_{ jin R cap mathbb{Z}^n} |f(m+j)|,qquad min mathbb{Z}^n, ] where the supremum is taken over all open rectangles in $mathbb{R}^n$ containing the origin whose sides are parallel to the coordinate axes. We show that the associated Tauberian constant $tilde{C}_S(alpha)$, defined by [ tilde{C}_S(alpha) := sup_{substack{E subset mathbb{Z}^n 0 < #E < infty} } frac{1}{#E}#{m in mathbb{Z}^n:, tilde{M}_Schi_E(m) > alpha}, ] is Holder continuous of order $1/n$. Moreover, letting $U_1, ldots, U_n$ denote a non-periodic collection of commuting invertible transformations on the non-atomic probability space $(Omega, Sigma, mu)$ we define the associated maximal operator $M_S^ast$ by [ M^ast_{S}f(omega) := sup_{0 in R subset mathbb{R}^n}frac{1}{#(R cap mathbb{Z}^n)}sum_{(j_1, ldots, j_n)in R}|f(U_1^{j_1}cdots U_n^{j_n}omega)|,qquad omegainOmega. ] Then the corresponding Tauberian constant $C^ast_S(alpha)$, defined by [ C^ast_S(alpha) := sup_{substack{E subset Omega mu(E) > 0}} frac{1}{mu(E)}mu({omega in Omega :, M^ast_Schi_E(omega) > alpha}), ] also satisfies $C^ast_S in C^{1/n}(0,1).$ We will also see that, in the case $n=1$, that is in the case of a single invertible, measure preserving transformation, the smoothness of the corresponding Tauberian constant is characterized by the operator enabling arbitrarily long orbits of sets of positive measure.
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