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

Abstract

Let $\lambda>-1/2$, $\lambda\neq0$, and let $\Delta_\lambda=-\frac{d^2}{dx^2}-\frac{2\lambda}{x}\frac{d}{dx}$ 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 $$ \|R_\lambda f\|_{L^p(\mathbb R_+,w\,dx)} \le C_{p,\lambda} [w]_{A_{p,\lambda}}^{\max\{1,1/(p-1)\}} \|f\|_{L^p(\mathbb R_+,w\,dx)}. $$ Moreover, for a real-valued function $b$ in the Bessel BMO space ${\rm BMO}_\lambda$, the sharp weighted bound for the Riesz commutator is $$ \|[b,R_\lambda]f\|_{L^p(\mathbb R_+,w\,dx)} \le C_{p,\lambda}\|b\|_{{\rm BMO}_\lambda} [w]_{A_{p,\lambda}}^{2\cdot\max\{1,1/(p-1)\}} \|f\|_{L^p(\mathbb R_+,w\,dx)}. $$ 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 Calder\'on--Zygmund kernel with respect to $\nu_\lambda$.