Muckenhoupt-type weights and the intrinsic structure in Bessel Setting

Fix $\lambda >-1\mathrm{/}2$ and $\lambda ≠0$. Consider the Bessel operator (introduced by Muckenhoupt–Stein) ${\mathrm{△}}_{\lambda }:=-\frac{d^2}{d{x}^2}-\frac{2\lambda }{x}\frac{d}{dx}$ on ${\mathbb{R}}_{+}:=(0,\mathrm{\infty })$ with $d{m}_{\lambda }(x):=x^{2\lambda }dx$ and $dx$ the Lebesgue measure on ${\mathbb{R}}_{+}$. In this paper, we study the Muckenhoupt-type weights which reveal the intrinsic structure in this Bessel setting along the line of Muckenhoupt–Stein and Andersen–Kerman. Besides, exploiting more properties of the weights $A_{p,\lambda }$ introduced by Andersen–Kerman, we introduce a new class ${\tilde{A}}_{p,\lambda }$ such that the Hardy–Littlewood maximal function is bounded on the weighted $L_w^p$ space if and only if $w$ is in ${\tilde{A}}_{p,\lambda }$. Moreover, along the line of Coifman–Rochberg–Weiss, we investigate the commutator $[b,R_{\lambda }]$ with $R_{\lambda }:=\frac{d}{dx}({\mathrm{△}}_{\lambda }{)}^{-\frac{1}{2}}$ to be the Bessel Riesz transform. We show that for $w\in A_{p,\lambda }$, the commutator $[b,R_{\lambda }]$ is bounded on weighted $L_w^p$ if and only if $b$ is in the BMO space associated with ${\mathrm{△}}_{\lambda }$.