Fix $d\ge 3$ and $1<p<\infty$. Let $V:{\mathbb{R}}^d\to [0,\infty )$ belong to the reverse Hölder class $R{H}_{d/2}$ and consider the Schrödinger operator $L_V:=-\mathrm{\Delta }+V$. In this article, we introduce classes of weights w for which the Riesz transforms $\mathrm{\nabla }{L}_V^{-1/2}$, their adjoints $L_V^{-1/2}\mathrm{\nabla }$ 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 $L_V^{-\alpha /2}$ from $L^p(w)$ to $L^{\nu }(w^{\nu /p})$ for $0<\alpha \le 2$, $1<p<\frac{d}{\alpha }$ and $\frac{1}{\nu }=\frac{1}{p}-\frac{\alpha }{d}$ will also be proved. These weight classes are strictly larger than a class previously introduced by Bongioanni, Harboure and Salinas in [8] 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 $-\mathrm{\Delta }+\mu$, the uniformly elliptic operator with potential $-\mathrm{div}A\mathrm{\nabla }+V$ and the magnetic Schrödinger operator ${(\mathrm{\nabla }-ia)}^{⁎}(\mathrm{\nabla }-ia)+V$ will all be considered.

Finally, this article will investigate necessary conditions that a weight w must satisfy in order for the Riesz transforms or the heat maximal operator to be bounded on $L^p(w)$. To aid in this task, lower bounds for the heat kernel of the standard Schrödinger operator $-\mathrm{\Delta }+V$ will be proved. These estimates provide a lower counterpart to the upper estimates proved in [23].

使固定 $d\ge 3$ 和 $\mathrm{1个}<p<\infty$。让$\mathrm{伏特}：{\mathbb{[R}}^d\to [0，\infty ）$ 属于反向霍尔德类 $\mathrm{[R}{H}_{d/\mathrm{2个}}$ 并考虑Schrödinger运算符 ${\mathrm{大号}}_{\mathrm{伏特}}：=-\mathrm{\Delta }+\mathrm{伏特}$。在本文中，我们介绍了Riesz变换的权重w的类别$\mathrm{\nabla }{\mathrm{大号}}_{\mathrm{伏特}}^{-\mathrm{1个}/\mathrm{2个}}$，他们的陪伴 ${\mathrm{大号}}_{\mathrm{伏特}}^{-\mathrm{1个}/\mathrm{2个}}\mathrm{\nabla }$ 和热量最大算子 ${\mathrm{SUP}}_{Ť>0}{Ë}^{-Ť{\mathrm{大号}}_{\mathrm{伏特}}}|F|$ 限制在加权Lebesgue空间上 ${\mathrm{大号}}^p（w）$。的有界性${\mathrm{大号}}_{\mathrm{伏特}}$-里兹势 ${\mathrm{大号}}_{\mathrm{伏特}}^{-\alpha /\mathrm{2个}}$ 从 ${\mathrm{大号}}^p（w）$ 到 ${\mathrm{大号}}^{\nu }（w^{\nu /p}）$ 为了 $0<\alpha \le \mathrm{2个}$， $\mathrm{1个}<p<\frac{d}{\alpha }$ 和 $\frac{\mathrm{1个}}{\nu }=\frac{\mathrm{1个}}{p}-\frac{\alpha }{d}$也将得到证明。这些权重类别严格大于Bongioanni，Harboure和Salinas先前在[8]中引入的具有这些性质的类别，它们包含指数增长和衰减的权重。

还将考虑与Schrödinger算子的不同广义形式有关的类。特别是具有测量潜力的Schrödinger算子$-\mathrm{\Delta }+\mu$，有可能的均匀椭圆算子 $-\mathrm{股利}\mathrm{一个}\mathrm{\nabla }+\mathrm{伏特}$ 和磁性薛定ding算子 ${（\mathrm{\nabla }-\mathrm{一世}\mathrm{一个}）}^{⁎}（\mathrm{\nabla }-\mathrm{一世}\mathrm{一个}）+\mathrm{伏特}$ 将全部考虑在内。

最后，本文将研究权重w必须满足的必要条件，以使Riesz变换或最大热量算子受到限制${\mathrm{大号}}^p（w）$。为了帮助完成此任务，请为标准Schrödinger算子的热核设定下限$-\mathrm{\Delta }+\mathrm{伏特}$将被证明。这些估计值与[23]中证明的较高估计值相比较低。