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

Abstract

Let $\lambda >-1/2,\ne 0$. 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 ${\tilde{A}}_{\infty ,\lambda }$ 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 $x^{2\lambda }dx$. Among these characterizations, we obtain the sharp reverse Hölder inequality (RHI) for ${\tilde{A}}_{\infty ,\lambda }$ weights. As applications of RHI, we derive a precise open property of ${\tilde{A}}_{p,\lambda }$ weights, and obtain a sharp weighted bounds for the Hardy-Littlewood maximal operator ${\mathcal{M}}_{\lambda }$ associated to ${\Delta }_{\lambda }$, which was missing in the previous papers. Next, we also establish several extrapolation theorems related to ${\tilde{A}}_{p,\lambda }$ weights. Furthermore, we characterize the restricted weighted weak type estimates for ${\mathcal{M}}_{\lambda }$ via a new class of weights ${\tilde{A}}_{p,\lambda }^{\mathcal{R}}$, which is strictly larger than ${\tilde{A}}_{p,\lambda }$. 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 ${\Delta }_{\lambda }$.

arXiv comment52 pages