Extension of Campanato–Sobolev type spaces associated with Schrödinger operators

Abstract

Let $L=-\varDelta +V$ be a Schrödinger operator acting on $L^2({\mathbb {R}}^{d})$, where V belongs to the reverse Hölder 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

$$\begin{aligned} \sup _{(x_0,r)\in {\mathbb {R}}_+^{d+1}}r^{-(2\alpha +d)}\int _{B(x_0,r)}\int _0^r|\nabla _{x,t}u(x,t)|^2t^{1-2\beta }dtdx<\infty . \end{aligned}$$