论文标题
Riesz在$ AX+B $组上转换
Riesz transforms on $ax+b$ groups
论文作者
论文摘要
我们证明了一阶Riesz的所有$ p \ in（1，\ infty）$的$ l^p $ bugnedness $ x_j \ nathcal {l}^{ - 1/2} $与laplacian $ \ natcal $ \ nathcal {l} \ mathbb {r}^n \ rtimes \ mathbb {r} $;这里$ x_0 $和$ x_1，\ dots，x_n $是$ g $在因子的方向上$ g $ $ \ mathbb {r} $和$ \ mathbb {r}^n $的左右不变的矢量字段。这解决了Hebisch和Steger先前的工作（证明了$ P \ leq 2 $）和Gaudry和Sjögren（仅考虑$ n = 1 = J $）的结果。这里的主要新颖性是我们可以对待（2，\ infty）$中的情况$ p \，并将Riesz转换置于$ \ Mathbb {r} $的方向上； $ \ mathbb {r}^n $上的运算符量型乘数定理结果是实现此目的的关键。我们还建立了一个弱类型的$（1,1）$端点，Riesz沿$ \ Mathbb {r}^n $的方向变换。通过转移，我们的结果暗示着一阶Riesz的$ p $ bundedness（1，1，\ infty）$的$ p \ in theschrödingeroperator $ - \ partial_s^2 + e^{2s} $。
We prove the $L^p$-boundedness for all $p \in (1,\infty)$ of the first-order Riesz transforms $X_j \mathcal{L}^{-1/2}$ associated with the Laplacian $\mathcal{L} = -\sum_{j=0}^n X_j^2$ on the $ax+b$-group $G = \mathbb{R}^n \rtimes \mathbb{R}$; here $X_0$ and $X_1,\dots,X_n$ are left-invariant vector fields on $G$ in the directions of the factors $\mathbb{R}$ and $\mathbb{R}^n$ respectively. This settles a question left open in previous work of Hebisch and Steger (who proved the result for $p \leq 2$) and of Gaudry and Sjögren (who only considered $n=1=j$). The main novelty here is that we can treat the case $p \in (2,\infty)$ and include the Riesz transform in the direction of $\mathbb{R}$; an operator-valued Fourier multiplier theorem on $\mathbb{R}^n$ turns out to be key to this purpose. We also establish a weak type $(1,1)$ endpoint for the adjoint Riesz transforms in the direction of $\mathbb{R}^n$. By transference, our results imply the $L^p$-boundedness for $p \in (1,\infty)$ of the first-order Riesz transforms associated with the Schrödinger operator $-\partial_s^2 + e^{2s}$ on the real line.
