Weighted norm inequalities for operators with new Muckenhoupt type weights in the Bessel setting and applications

Abstract

Let $λ>-1/2,\neq0$. Consider the Bessel operators \begin{align*} Δ_λ:=-\frac{d^2}{dx^2}-\frac{2λ}{x}\frac{d}{dx} \end{align*} on $\mathbb{R}_+$. In this paper, we first establish several characterizations for $\widetilde{A}_{\infty,λ}$ weights introduced by Li et al. (Math. Z. 309(1) (2025), Paper No. 13, 29pp), which are different from the previous $A_p$ class with respect to the measure $x^{2λ}dx$. Among these characterizations, we obtain the sharp reverse Hölder inequality (RHI) for $\widetilde{A}_{\infty,λ}$ weights. As applications of RHI, we derive a precise open property of $\widetilde{A}_{p,λ}$ weights, and obtain a sharp weighted bounds for the Hardy-Littlewood maximal operator $\mathcal{M}_λ$ associated to $Δ_λ$, which was missing in the previous papers. Next, we also establish several extrapolation theorems related to $\widetilde{A}_{p,λ}$ weights. Furthermore, we characterize the restricted weighted weak type estimates for $\mathcal{M}_λ$ via a new class of weights $\widetilde{A}_{p,λ}^{\mathcal{R}}$, which is strictly larger than $\widetilde{A}_{p,λ}$. Meanwhile, we obtain quantitative restricted weighted weak type $(p,p)$ bounds for sparse operators. As applications, we derive various kinds of weighted inequalities for Riesz transforms, maximal operators, square functions, variation operators and their commutators associated to $Δ_λ$.