论文标题
SchrödingerRiesz的行为转变为平滑度空间
Behaviour of Schrödinger Riesz transforms over smoothness spaces
论文作者
论文摘要
正如沉的所示，与schrödinger操作员$ l =-Δ+ v $相关的riesz变换在$ l^p（\ mathbb {r}^d）$ - 所有$ p，1 <p <\ p <\ infty $的空间上都没有限制。此外，它们仅在$ p $的某个有限间隔（1，p_0）$中以$ p $的限制，因此不能期望它保留规律性的空间。在这项工作中，我们在潜在的潜在上寻找某种最小的其他条件，以便在所有第一和第二阶的riesz转换中获得适当的加权$ bmo $ type规律性空间，即$ \ nabla l^{ - 1/2} $，$ v^{1/2} l^{1/2} l^{1/2} $ vl^{ - 1} $和$ v^{1/2} \ nabla l^{ - 1} $。我们还探讨了此类额外条件在多大程度上也是必要的。
As it was shown by Shen, the Riesz transforms associated to the Schrödinger operator $L=-Δ+ V$ are not bounded on $L^p(\mathbb{R}^d)$-spaces for all $p, 1<p<\infty$, under the only assumption that the potential satisfies a reverse Hölder condition of order $d/2$, $d\geq3$. Furthermore, they are bounded only for $p$ in some finite interval of the type $(1,p_0)$, so it can not be expected to preserve regularity spaces. In this work we search for some kind of minimal additional conditions on the potential in order to obtain boundedness on appropriate weighted $BMO$ type regularity spaces for all first and second order Riesz transforms, namely for the operators $\nabla L^{-1/2}$, $V^{1/2}L^{-1/2}$, $\nabla^2 L^{-1}$, $VL^{-1}$ and $V^{1/2}\nabla L^{-1}$. We also explore to what extent such extra conditions are also necessary.
