Sharp weighted estimates for the Bessel Riesz transform and its commutator with Andersen--Kerman weights

Abstract

Let $\lambda >-1/2$, $\lambda \ne 0$, and let ${\Delta }_{\lambda }=-\frac{d^2}{d{x}^2}-\frac{2\lambda }{x}\frac{d}{dx}$ be the Bessel operator on ${\mathbb{R}}_{+}=(0,\infty )$ studied by Muckenhoupt and Stein (TAMS 1965). Andersen and Kerman (Studia Math. 1981) proved that for $1<p<\infty$, the Bessel Riesz transform $R_{\lambda }=\frac{d}{dx}{\Delta }_{\lambda }^{-1/2}$ is bounded on $L^p({\mathbb{R}}_{+},w(x)dx)$ if and only if $w$ is in the intrinsic class $A_{p,\lambda }$. However, the sharp quantitative weighted bound via $[w{]}_{A_{p,\lambda }}$ was not addressed before. In this paper, we give a positive answer to this question by proving the sharp quantitative estimate $$\Vert R_{\lambda }f{\Vert }_{L^p({\mathbb{R}}_{+},wdx)}\le C_{p,\lambda }[w{]}_{A_{p,\lambda }}^{\max \{1,1/(p-1)\}}\Vert f{\Vert }_{L^p({\mathbb{R}}_{+},wdx)}.$$ Moreover, for a real-valued function $b$ in the Bessel BMO space ${\mathrm{BMO}}_{\lambda }$, the sharp weighted bound for the Riesz commutator is $$\Vert [b,R_{\lambda }]f{\Vert }_{L^p({\mathbb{R}}_{+},wdx)}\le C_{p,\lambda }\Vert b{\Vert }_{{\mathrm{BMO}}_{\lambda }}[w{]}_{A_{p,\lambda }}^{2\cdot \max \{1,1/(p-1)\}}\Vert f{\Vert }_{L^p({\mathbb{R}}_{+},wdx)}.$$ The argument uses the exact conjugation $U(x)=x^{p-2\lambda -1}w(x),d{\nu }_{\lambda }=x^{2\lambda +1}dx,[U{]}_{A_p(d{\nu }_{\lambda })}=[w{]}_{A_{p,\lambda }},$ which reduces the Andersen--Kerman estimate to an $A_p$ weighted estimate for the auxiliary operator $${\mathcal{R}}_{\lambda }F(x)=\frac{1}{x}{R}_{\lambda }(yF(y))(x)$$ on the space of homogeneous type $({\mathbb{R}}_{+},|x-y|,d{\nu }_{\lambda })$, whose kernel is a standard Calderón--Zygmund kernel with respect to ${\nu }_{\lambda }$.