带Andersen--Kerman权的Bessel Riesz变换的双权交换子
Two-weight commutators for the Bessel Riesz transform with Andersen--Kerman weights
AI总结：
本文刻画了带Andersen--Kerman权的Bessel Riesz变换交换子的有界性与紧性，建立了其与BMO_{ν,α}、VMO_{ν,α}的等价关系，采用共轭方法转化问题并推导相关估计。
AI中文摘要：
设α>-1/2且α≠0，令Δ_α = -d²/dx² - (2α/x)d/dx为正实数轴ℝ₊=(0,∞)上的Bessel算子，本文刻画了R_α = d/dx Δ_α^{-1/2}的交换子在Andersen--Kerman双权框架下的有界性与紧性。对1<p<∞且μ,λ∈A_{p,α}，取ν=(μ/λ)^{1/p}，dρ_α(x)=x^{2α+1}dx。对实值符号，||[b,R_α]||_{L^p(μdx)→L^p(λdx)}等价于||b||_{BMO_{ν,α}}，其中Bloom振荡基于dρ_α计算；且[b,R_α]:L^p(μdx)→L^p(λdx)紧当且仅当b∈VMO_{ν,α}。证明采用精确共轭U_w(x)=x^{p-2α-1}w(x)，满足[U_w]_{A_p(ρ_α)}=[w]_{A_{p,α}}，将问题转化为正实数轴上商Bessel核的问题，分离出所需的Calderón--Zygmund估计与单侧非退化性。测度dρ_α不同于常规Bessel测度dm_α=x^{2α}dx，它是Andersen--Kerman共轭选取的Bloom基测度，贯穿后续双权理论。
英文摘要：
Let $α>-1/2$, $α\ne0$, and let $$ Δ_α=-\frac{d^2}{dx^2}-\frac{2α}{x}\frac{d}{dx} $$ be the Bessel operator on $\mathbb R_+=(0,\infty)$. We characterize boundedness and compactness of commutators of $R_α=\frac{d}{dx}Δ_α^{-1/2}$ on the Andersen--Kerman two-weight setting. For $1<p<\infty$ and $μ,λ\in A_{p,α}$, put $ ν=\left(\fracμλ\right)^{1/p}, \ dρ_α(x)=x^{2α+1}\,dx. $ For real-valued symbols, $$ \|[b,R_α]\|_{L^p(μ\,dx)\to L^p(λ\,dx)} \simeq \|b\|_{\rm{BMO}_{ν,α}}, $$ where the Bloom oscillation is computed with respect to $dρ_α$. Moreover $$ [b,R_α]:L^p(μ\,dx)\to L^p(λ\,dx) \text{ is compact} \quad\Longleftrightarrow\quad b\in\rm{VMO}_{ν,α}. $$ The proof uses the exact conjugation $ U_w(x)=x^{p-2α-1}w(x), \ [U_w]_{A_p(ρ_α)}=[w]_{A_{p,α}}, $ which transfers the problem to the quotient Bessel kernel on $(\mathbb R_+,|x-y|,ρ_α)$. We isolate the required Calderón--Zygmund estimates and one-sided non-degeneracy in this quotient normalization. The measure $dρ_α$ is distinct from the usual Bessel measure $dm_α=x^{2α}\,dx$. It is the Bloom base measure selected by the Andersen--Kerman conjugation and is used throughout the two-weight theory below.
