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

Abstract

Let $λ>-1/2$, $λ\neq0$, and let $Δ_λ=-\frac{d^2}{dx^2}-\frac{2λ}{x}\frac{d}{dx}$ 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 $R_λ=\frac{d}{dx}Δ_λ^{-1/2}$ is bounded on $L^p(\mathbb R_+,w(x)\,dx)$ if and only if $w$ is in the intrinsic class $A_{p,λ}$. However, the sharp quantitative weighted bound via $[w]_{A_{p,λ}}$ was not addressed before. In this paper, we give a positive answer to this question by proving the sharp quantitative estimate $$ \|R_λf\|_{L^p(\mathbb R_+,w\,dx)} \le C_{p,λ} [w]_{A_{p,λ}}^{\max\{1,1/(p-1)\}} \|f\|_{L^p(\mathbb R_+,w\,dx)}. $$ Moreover, for a real-valued function $b$ in the Bessel BMO space ${\rm BMO}_λ$, the sharp weighted bound for the Riesz commutator is $$ \|[b,R_λ]f\|_{L^p(\mathbb R_+,w\,dx)} \le C_{p,λ}\|b\|_{{\rm BMO}_λ} [w]_{A_{p,λ}}^{2\cdot\max\{1,1/(p-1)\}} \|f\|_{L^p(\mathbb R_+,w\,dx)}. $$ The argument uses the exact conjugation $U(x)=x^{p-2λ-1}w(x), \ dν_λ=x^{2λ+1}\,dx, \ [U]_{A_p(dν_λ)}=[w]_{A_{p,λ}},$ which reduces the Andersen--Kerman estimate to an $A_p$ weighted estimate for the auxiliary operator $$ \mathcal R_λF(x)=\frac{1}{x}R_λ(yF(y))(x) $$ on the space of homogeneous type $(\mathbb R_+,|x-y|,dν_λ)$, whose kernel is a standard Calderón--Zygmund kernel with respect to $ν_λ$.