Weighted Solyanik Estimates for the Hardy–Littlewood Maximal Operator and Embedding of A_∞ into A_p

Let w denote a weight in ℝⁿ which belongs to the Muckenhoupt class A_∞ and let M_w denote the uncentered Hardy–Littlewood maximal operator defined with respect to the measure w(x)dx. The sharp Tauberian constant of M_w with respect to α, denoted by C_w(α), is defined by (Formula presented.). In this paper, we show that the Solyanik estimate (Formula presented.) holds. Following the classical theme of weighted norm inequalities we also consider the sharp Tauberian constants defined with respect to the usual uncentered Hardy–Littlewood maximal operator M and a weight w: (Formula presented.). We show that we have (Formula presented.) if and only if w ∈ A_∞. As a corollary of our methods we obtain a quantitative embedding of A_∞ into A_p.