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

We give explicit analytic criteria for two problems associated with the Schr"odinger operator $H=-\mathrm{\Delta }+Q$ on $L^2({\mathbb{R}}^n)$ where $Q\in D^{\mathrm{\prime }}({\mathbb{R}}^n)$ 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 $Q\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^n)$, this property can be expressed in the form of the integral inequality: $$\mathrm{\mid }{\int }_{{\mathbb{R}}^n}\mathrm{\mid }u(x){\mathrm{\mid }}^{2}Q(x)dx\mathrm{\mid }\le ϵ\mathrm{\mid }\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }{\mathrm{\mid }}_{L^2({\mathbb{R}}^n)}^2+C(ϵ)\mathrm{\mid }\mathrm{\mid }u\mathrm{\mid }{\mathrm{\mid }}_{L^2({\mathbb{R}}^n)}^2,\mathrm{\forall }u\in C_0^{\mathrm{\infty }}({\mathbb{R}}^n),$$ for an arbitrarily small $ϵ>0$ and some $C(ϵ)>0$. Secondly, we characterize Trudinger’s subordination property where $C(ϵ)$ 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 $\mathrm{\nabla }(1-\mathrm{\Delta }{)}^{-1}Q$ and $(1-\mathrm{\Delta }{)}^{-1}Q$ 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.