Schrodinger算子的无穷小形式有界性和Trudinger从属性
Infinitesimal form boundedness and Trudinger's subordination for the Schrödinger operator
摘要：
我们给出了与薛定谔算子$H = -Δ+ Q$有关的两个问题的显式解析准则，其中$Q\in D'(\R^n)$是任意实值或复值势。首先，我们得到了关于$Q$的充要条件，使得二次型$<Q \cdot, \cdot>$关于拉普拉斯算子有零相对界。对于$Q\in L^1_{\rm loc}(\R^n)$，这个性质可以用积分不等式的形式表示：$$ | \int_{\R^n} |u(x)|^2 Q(x) dx | \leq ε||\nabla u||^2_{L^2(\R^n)} + C(ε) ||u||^2_{L^2(\R^n)}, \quad \forall u \in C^\infty_0(\R^n), $$表示任意小的$ε>0$和一些$C(ε)> 0$。其次，我们刻画了Trudinger的从属性质，其中上述不等式中的$C(ε)$服从条件$C(ε) \le c {ε^{-β}}$（$β>0$）为$ε\to +0$。这样的二次型不等式可以完全在Morrey-Campanato空间的框架中理解，使用球或立方体上的$\nabla (1-Δ)^{-1} Q$和$(1-Δ)^{-1} Q$的平均振荡。因此，我们刻画了一类$Q$满足乘法二次纳什型不等式。
英文摘要：
We give explicit analytic criteria for two problems associated with the Schrödinger operator $H = -Δ+ Q$ 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: $$ | \int_{\R^n} |u(x)|^2 Q(x) dx | \leq ε||\nabla u||^2_{L^2(\R^n)} + C(ε) ||u||^2_{L^2(\R^n)}, \quad \forall u \in C^\infty_0(\R^n), $$ 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 $C(ε) \le c {ε^{-β}}$ ($β>0$) as $ε\to +0$. Such quadratic form inequalities can be understood entirely in the framework of Morrey--Campanato spaces, using mean oscillations of $\nabla (1-Δ)^{-1} Q$ and $(1-Δ)^{-1} Q$ 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.
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