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تسجيل مستخدم جديدA 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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