Extension of Campanato–Sobolev type spaces associated with Schrödinger operators

Abstract

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