Extension of Campanato–Sobolev type spaces associated with Schrödinger operators

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