Let $$L=-\varDelta +V$$ be a Schrodinger operator acting on $$L^2({\mathbb {R}}^{d})$$, where V belongs to the reverse Holder class $$B_q$$ for some $$q\ge d$$. For $$\alpha , \beta \in [0,1)$$, let $$\varLambda _{\alpha ,\beta }^L({\mathbb {R}}^d)$$ be the Campanato–Sobolev space associated with L. Via the Poisson semigroup $$\{e^{-t\sqrt{L}}\}_{t\ge 0}$$, we extend $$\varLambda _{\alpha ,\beta }^L({\mathbb {R}}^d)$$ to $${\mathcal {T}}^{\alpha ,\beta }_{L}({\mathbb {R}}^{d+1}_{+})$$ which is defined as the set of all distributional solutions u of $$-u_{tt}+Lu=0$$ on the upper half space $${\mathbb {R}}_+^{d+1}$$ satisfying $$\begin{aligned} \sup _{(x_0,r)\in {\mathbb {R}}_+^{d+1}}r^{-(2\alpha +d)}\int _{B(x_0,r)}\int _0^r|\nabla _{x,t}u(x,t)|^2t^{1-2\beta }dtdx<\infty . \end{aligned}$$

中文翻译：

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与薛定谔算子关联的 Campanato-Sobolev 类型空间的扩展

令 $$L=-\varDelta +V$$ 是作用于 $$L^2({\mathbb {R}}^{d})$$ 的薛定谔算子，其中 V 属于反向持有类 $$B_q $$ 一些 $$q\ge d$$。对于 $$\alpha , \beta \in [0,1)$$，让 $$\varLambda _{\alpha ,\beta }^L({\mathbb {R}}^d)$$ 成为 Campanato–与 L 相关的 Sobolev 空间。通过泊松半群 $$\{e^{-t\sqrt{L}}\}_{t\ge 0}$$，我们扩展 $$\varLambda _{\alpha ,\beta }^L({\mathbb {R}}^d)$$ 到 $${\mathcal {T}}^{\alpha ,\beta }_{L}({\mathbb {R}}^{d+ 1}_{+})$$ 定义为 $$-u_{tt}+Lu=0$$ 在上半空间 $${\mathbb {R}}_+ 上的所有分布解 u 的集合^{d+1}$$ 满足 $$\begin{aligned} \sup _{(x_0,r)\in {\mathbb {R}}_+^{d+1}}r^{-(2\ alpha +d)}\int _{B(x_0,r)}\int _0^r|\nabla _{x,t}u(x,t)|^2t^{1-2\beta }dtdx<\无。\end{对齐}$$