Infinitesimal form boundedness and Trudinger's subordination for the Schr\"odinger operator

Abstract

We give explicit analytic criteria for two problems associated with the Schr\"odinger operator $H = -\Delta + Q$ on $L^2(\R^n)$ where $Q\in D'(\R^n)$ is an arbitrary real- or complex-valued potential. First, we obtain necessary and sufficient conditions on $Q$ so that the quadratic form $<Q \cdot, \cdot>$ has zero relative bound with respect to the Laplacian. For $Q\in L^1_{\rm loc}(\R^n)$, this property can be expressed in the form of the integral inequality: $$ | \int_{\R^n} |u(x)|^2 Q(x) dx | \leq \epsilon ||\nabla u||^2_{L^2(\R^n)} + C(\epsilon) ||u||^2_{L^2(\R^n)}, \quad \forall u \in C^\infty_0(\R^n), $$ for an arbitrarily small $\epsilon >0$ and some $C(\epsilon)> 0$. Secondly, we characterize Trudinger's subordination property where $C(\epsilon)$ in the above inequality is subject to the condition $C(\epsilon) \le c {\epsilon^{-\beta}}$ ($\beta>0$) as $\epsilon\to +0$. Such quadratic form inequalities can be understood entirely in the framework of Morrey--Campanato spaces, using mean oscillations of $\nabla (1-\Delta)^{-1} Q$ and $(1-\Delta)^{-1} Q$ on balls or cubes. As a consequence, we characterize the class of those $Q$ which satisfy a multiplicative quadratic from inequality of Nash's type.