For a given second-order linear elliptic operator $L$ which admits a positive minimal Green function, and a given positive weight function $W$, we introduce a family of weighted Lebesgue spaces $L^p(ϕ_p)$ with their dual spaces, where $1\leq p\leq \infty$. We study some fundamental properties of the corresponding (weighted) Green operators on these spaces. In particular, we prove that these Green operators are bounded on $L^p(ϕ_p)$ for any $1\leq p\leq \infty$ with a uniform bound. We study the existence of a principal eigenfunction for these operators in these spaces, and the simplicity of the corresponding principal eigenvalue. We also show that such a Green operator is a resolvent of a densely defined closed operator which is equal to $(-W^{-1})L$ on $C_0^\infty$, and that this closed operator generates a strongly continuous contraction semigroup. Finally, we prove that if $W$ is a (semi)small perturbation of $L$, then for any $1\leq p\leq \infty$, the associated Green operator is compact on $L^p(ϕ_p)$, and the corresponding spectrum is $p$-independent.
Form boundedness of the general second order differential operator
给出了一般二阶分布系数为实值或复值的微分算子L在索伯列夫空间W^{1，2}（R^n）到其对偶W^{-1，2}（R^n）上有界的充分充要条件.这使我们能够得到关于L关于L^2（R^n）上的拉普拉斯算子的相对形式有界性、紧性和无穷小形式有界性等基本概念的分析准则。特别地，我们建立了Schroedinger算子（i \nabla + a）^2 + q在磁矢势\in L ^2_{loc}和\in D '（\R^n）下的形式有界性的完整刻画.
On Schrödinger Pseudo-Multipliers and their Commutators
Indian Institute of Science Education and Research Kolkata（印度科学教育研究所加尔各答分校）；National Taiwan Normal University（台湾师范大学）；Louisiana State University（路易斯安那州立大学）
