Muckenhoupt-Type Weights in Bessel Setting

Fix \(\lambda >-1/2\) $\lambda >-1/2$ and \(\lambda \not =0\) $\lambda \ne 0$ . Consider the Bessel operator (introduced by Muckenhoupt–Stein) \(\triangle _\lambda :=-\frac{d^2}{dx^2}-\frac{2\lambda }{x} \frac{d}{dx}\) ${▵}_{\lambda }:=-\frac{d^2}{d{x}^2}-\frac{2\lambda }{x}\frac{d}{\mathit{dx}}$ on \(\mathbb {R_+}:=(0,\infty )\) ${\mathbb{R}}_{+}:=(0,\infty )$ with \(dm_\lambda (x):=x^{2\lambda }dx\) $d{m}_{\lambda }(x):=x^{2\lambda }dx$ and dx the Lebesgue measure on \(\mathbb {R}_+\) ${\mathbb{R}}_{+}$ . In this paper, we study the Muckenhoupt-type weights in this Bessel setting along the line of Muckenhoupt–Stein and Andersen–Kerman. Besides, exploiting more properties of the weights \(A_{p,\lambda }\) $A_{p,\lambda }$ introduced by Andersen–Kerman, we introduce a new class \(\widetilde{A}_{p,\lambda }\) ${\tilde{A}}_{p,\lambda }$ such that the Hardy–Littlewood maximal function is bounded on the weighted \(L^p_w\) $L_w^p$ space if and only if w is in \(\widetilde{A}_{p,\lambda }\) ${\tilde{A}}_{p,\lambda }$ . Moreover, along the line of Coifman–Rochberg–Weiss, we investigate the commutator \([b,R_\lambda ]\) $[b,R_{\lambda }]$ with \(R_\lambda :=\frac{d}{dx}(\triangle _\lambda )^{-\frac{1}{2}}\) $R_{\lambda }:=\frac{d}{\mathit{dx}}{({▵}_{\lambda })}^{-\frac{1}{2}}$ to be the Bessel Riesz transform. We show that for \(w\in A_{p,\lambda }\) $w\in A_{p,\lambda }$ , the commutator \([b, R_\lambda ]\) $[b,R_{\lambda }]$ is bounded on weighted \(L^p_w\) $L_w^p$ if and only if b is in the BMO space associated with \(\triangle _\lambda \) ${▵}_{\lambda }$ .