Bounded $H_\infty$-calculus for elliptic operators
Bounded $H_\infty$-calculus for elliptic operators
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椭圆运算符的有界 $H_infty$ 演算
DOI：
10.57262/die/1370267697
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发表时间：
1994
影响因子：
1.4
通讯作者：
G. Simonett
中科院分区：
文献类型：
--
作者：
H. Amann;Matthias Hieber;G. Simonett
它示出，特别是，L p-实现的一般椭圆系统的Rn或紧流形上没有边界具有有界的虚功率，提供相当温和的正则性条件得到满足。此外，对具有有界H_（00）-演算的算子给出了一些新的扰动定理。0.导论.本文的主要目的是在弱正则性假设下证明作用于JRn上向量值函数或无边界紧致流形上向量丛截面上的椭圆型微分算子的Lp-实现具有有界虚幂.事实上，我们将证明一个更一般的结果，保证给定任何椭圆算子A具有足够大的零阶项，使得其主符号的谱包含在形式为8&0：= {z E C; I atgz！：; eo} U {0}，且给定任意有界全纯函数f：S ~（-7）C，则可定义Lp上的有界线性算子j（A），并得到形式为π f（A）（π. α Lp）：c π f（A）的估计。这意味着椭圆算子具有Mcintosh [16]意义下的有界R 00-演算.特别地，选择f（z）：=zit fortE JR，可以得出A具有有界虚幂（参见。第2节更精确的陈述）。我们对这个问题感兴趣主要有两个原因。首先，它是已知的（cf。[22]，[24]）证明了当0 < e < 1时，复插值空间[E，D（A）]&与分数幂A &的域重合，只要A是Banach空间E上具有有界虚幂的稠定线性算子.其次，根据Dore和Venni [1 OJ]的结果，A具有有界虚幂的事实与形式为u +Au = f（t）的抽象发展方程的“极大正则性结果”密切相关。这两个结果在1993年8月出版的《收稿》中有很大的用处。AMS学科分类：35 J 45、47 F05
It is shown, in particular, that L p-realizations of general elliptic systems on Rn or on compact manifolds without boundaries possess bounded imaginary powers, provided rather mild regularity conditions are satisfied. In addition, there are given some new perturbation theorems for operators possessing a bounded H00-calculus. 0. Introduction. It is the main purpose of this paper to prove under mild regularity assumptionsthat Lp-realizations of elliptic differential operators acting on vector valued functions over JRn or on sections of vector bundles over compact manifolds without boundaries possess bounded imaginary powers. In fact, we shall prove a more general result guaranteeing that, given any elliptic operator A with a sufficiently large zero order term such that the spectrum of its principal symbol is contained in a sector of the form 8&0 := {z E C; I atgz!::::; eo} U {0} for some 0 e0 E [0, n), and given any bounded holomorphic function f: S& ---7 C for some e E (e0 , n), we can define a bounded linear operator j(A) on Lp, and an estimate of the form llf(A)II.ccLp) ::::; c llflloo is valid. This means that elliptic operators possess a bounded R 00-calculus in the sense of Mcintosh [16]. Choosing, in particular, f(z) :=zit fortE JR, it follows that A possesses bounded imaginary powers ( cf. Section 2 below for more precise statements). There are two main reasons for our interest in this problem. First, it is known (cf. [22], [24]) that the complex interpolation spaces [E, D(A)]& coincide with the domains of the fractional powers A & for 0 < e < 1, provided A is a densely defined linear operator on the Banach space E possessing bounded imaginary powers. Second, by a result of Dore and Venni [1 OJ, the fact that A possesses bounded imaginary powers is intimately connected with 'maximal regularity results' for abstract evolution equations of the form u +Au = f (t). Both these results are of great use in the Received for publication August 1993. AMS Subject Classifications: 35J45, 47F05