论文标题
Schrödinger型操作员的指数增长和衰减的权重
Weights of Exponential Growth and Decay for Schrödinger-type operators
论文作者
论文摘要
修复$ d \ geq 3 $和$ 1 <p <\ infty $。令$ v：\ mathbb {r}^{d} \ rightarrow [0，\ infty）$属于反向Hölder类$ rh_ {d/2} $，并考虑schrödingeroperator $ l_ {v}：= - δ+ v $。在本文中，我们介绍了riesz转换$ \ nabla l_ {v}^{ - 1/2} $的权重$ w $的类别 0} e^{ - t l_ {v}} | f | $在加权的lebesgue space $ l^{p}（w）$上。 $ l_ {v} $ - riesz电位$ l_ { \fracα{d} $也将得到证明。这些重量类别严格比B. Bongioanni，E。Harboure和O. Salinas先前引入的类别大，具有这些特性，并且包含指数级增长和衰减的权重。 这些类也将与Schrödinger运营商不同的广义形式有关。特别是，具有测量潜在$-Δ +μ$的Schrödinger操作员，具有潜在$ - \ mathrm {div} a \ nabla + v $的均匀椭圆运算符和磁性schrödingeroperator $（\ nabla-i a）^{2} + v $都将被认为是所有的。可以证明，在适当的条件下，与这些二阶差分运算符相对应的标准运算符在这些类中的权重$ w $的$ l^{p}（w）$上。
Fix $d \geq 3$ and $1 < p < \infty$. Let $V : \mathbb{R}^{d} \rightarrow [0,\infty)$ belong to the reverse Hölder class $RH_{d/2}$ and consider the Schrödinger operator $L_{V} := - Δ+ V$. In this article, we introduce classes of weights $w$ for which the Riesz transforms $\nabla L_{V}^{-1/2}$, their adjoints $L_{V}^{-1/2} \nabla$ and the heat maximal operator $\sup_{t > 0} e^{- t L_{V}} |f|$ are bounded on the weighted Lebesgue space $L^{p}(w)$. The boundedness of the $L_{V}$-Riesz potentials $L_{V}^{-α/2}$ from $L^{p}(w)$ to $L^ν(w^{ν/p})$ for $0 < α\leq 2$ and $\frac{1}ν = \frac{1}{p} - \fracα{d}$ will also be proved. These weight classes are strictly larger than a class previously introduced by B. Bongioanni, E. Harboure and O. Salinas that shares these properties and they contain weights of exponential growth and decay. The classes will also be considered in relation to different generalised forms of Schrödinger operator. In particular, the Schrödinger operator with measure potential $- Δ+ μ$, the uniformly elliptic operator with potential $- \mathrm{div} A \nabla + V$ and the magnetic Schrödinger operator $(\nabla - i a)^{2} + V$ will all be considered. It will be proved that, under suitable conditions, the standard operators corresponding to these second-order differential operators are bounded on $L^{p}(w)$ for weights $w$ in these classes.
