论文标题
在$ l^p $估计中，阳性riesz的转换与Schrödinger运营商有关
On $L^p$ estimates for positivity-preserving Riesz transforms related to Schrödinger operators
论文作者
论文摘要
我们研究$ l^{p}，$ $ 1 \ leqslant p \ leqslant \ infty，$界面的riesz the y $ v^{a}（ - \ frac {1} {2} {2} {2}Δ+v）我们证明，$ v^{a}（ - \ frac {1} {2}δ+v）^{ - a} $在$ l^p（\ mathbb {r}^d）$上限制为$ 1 <p \ p \ leqslant 2 $，只要$ a \ leqslant 1/p。 $ l^{\ infty}（\ Mathbb {r}^d）$有限性如果$ V $满足$ a $ a $ a的积分条件，对小扰动具有抵抗力。在$ l^{1}（\ Mathbb {r}^d）上也获得了对$ V $的更强假设的类似结果。尤其是我们的$ l^{\ infty} $和$ l^1 $结果适用于非阴性潜力$ V $，这些$ V $在全球范围内具有电力增长或指定增长。我们还讨论了一个反例，表明$ l^{\ infty}（\ mathbb {r}^d）$有界可能会失败。
We study the $L^{p},$ $1\leqslant p\leqslant \infty,$ boundedness for Riesz transforms of the form $V^{a}(-\frac{1}{2}Δ+V)^{-a},$ where $a>0$ and $V$ is a non-negative potential. We prove that $V^{a}(-\frac{1}{2}Δ+V)^{-a}$ is bounded on $L^p(\mathbb{R}^d)$ with $1< p\leqslant 2$ whenever $a\leqslant 1/p.$ We demonstrate that the $L^{\infty}(\mathbb{R}^d)$ boundedness holds if $V$ satisfies an $a$-dependent integral condition that is resistant to small perturbations. Similar results with stronger assumptions on $V$ are also obtained on $L^{1}(\mathbb{R}^d).$ In particular our $L^{\infty}$ and $L^1$ results apply to non-negative potentials $V$ which globally have a power growth or an exponential growth. We also discuss a counterexample showing that the $L^{\infty}(\mathbb{R}^d)$ boundedness may fail.
