贝塞尔里斯变换及其与Andersen–Kerman权的交换子的精确加权估计
Sharp weighted estimates for the Bessel Riesz transform and its commutator with Andersen--Kerman weights
AI总结：
本文针对贝塞尔里斯变换及其与Andersen–Kerman权的交换子，证明了对应的精确定量加权估计，解决了此前未解决的相关精确加权界问题。
AI中文摘要：
设λ>-1/2且λ≠0，Δ_λ=-d²/dx² - 2λ/x · d/dx是Muckenhoupt与Stein（《美国数学会汇刊》1965年）研究的正半轴ℝ₊=(0,∞)上的贝塞尔算子。Andersen与Kerman（《数学研究》1981年）证明，当1<p<∞时，贝塞尔里斯变换R_λ=d/dx Δ_λ^(-1/2)在L^p(ℝ₊,w(x)dx)上有界当且仅当w属于内在类A_{p,λ}，但此前未解决通过[w]_{A_{p,λ}}得到的精确定量加权界问题。本文通过证明精确定量估计‖R_λ f‖_{L^p(ℝ₊,w dx)} ≤ C_{p,λ} [w]_{A_{p,λ}}^{max{1,1/(p-1)}} ‖f‖_{L^p(ℝ₊,w dx)}对该问题给出肯定解答。此外，对贝塞尔BMO空间BMO_λ中的实值函数b，里斯交换子的精确加权界为‖[b,R_λ]f‖_{L^p(ℝ₊,w dx)} ≤ C_{p,λ}‖b‖_{BMO_λ} [w]_{A_{p,λ}}^{2·max{1,1/(p-1)}} ‖f‖_{L^p(ℝ₊,w dx)}。论证采用精确共轭U(x)=x^{p-2λ-1}w(x)、dν_λ=x^{2λ+1}dx，满足[U]_{A_p(dν_λ)}=[w]_{A_{p,λ}}，将Andersen–Kerman估计转化为齐次型空间(ℝ₊,|x-y|,dν_λ)上辅助算子R_λ F(x)=1/x R_λ(yF(y))(x)的A_p加权估计，该算子的核是关于ν_λ的标准Calderón–Zygmund核。
英文摘要：
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 $ν_λ$.
