For a given second-order linear elliptic operator $L$ which admits a positiveminimal Green function, and a given positive weight function $W$, we introducea family of weighted Lebesgue spaces $L^p(\phi_p)$ with their dual spaces,where $1\leq p\leq \infty$. We study some fundamental properties of thecorresponding (weighted) Green operators on these spaces. In particular, weprove that these Green operators are bounded on $L^p(\phi_p)$ for any $1\leqp\leq \infty$ with a uniform bound. We study the existence of a principaleigenfunction for these operators in these spaces, and the simplicity of thecorresponding principal eigenvalue. We also show that such a Green operator isa 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 continuouscontraction semigroup. Finally, we prove that if $W$ is a (semi)smallperturbation of $L$, then for any $1\leq p\leq \infty$, the associated Greenoperator is compact on $L^p(\phi_p)$, and the corresponding spectrum is$p$-independent.
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机译：对于一个给定的二阶线性椭圆算子$ L $，它允许一个正的最小格林函数，一个给定的正权函数$ W $，我们引入了一个加权Lebesgue空间族$ L ^ p（\ phi_p）$及其对偶空间，其中$ 1 \ leq p \ leq \ infty $。我们研究了这些空间上相应的（加权）格林算子的一些基本性质。特别是，我们证明了这些Green运算符在$ L ^ p（\ phi_p）$上对于任何$ 1 \ leqp \ leq \ infty $具有统一边界。我们研究了这些算子在这些空间中的本征函数的存在，以及对应的本征特征值的简单性。我们还表明，此类Green运算符是密集定义的封闭运算符的分解物，该封闭运算符等于$ C_0 ^ \ infty $上的$（-W ^ {-1}）L $，并且该封闭运算符生成一个强连续收缩半群。最后，我们证明如果$ W $是$ L $的（半）小扰动，则对于任何$ 1 \ leq p \ leq \ infty $，相关的Greenoperator在$ L ^ p（\ phi_p）$上是紧凑的，并且相应的频谱是独立于$ p $的。
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