scholarly journals H∞ functional calculus of elliptic operators with C∞ coefficients on Lp spaces of smooth domains

1990 ◽  
Vol 48 (1) ◽  
pp. 113-123 ◽  
Author(s):  
Xuan Thinh Duong
Keyword(s):  
Functional Calculus ◽  
Differential Operators ◽  
Higher Order ◽  
Elliptic Operators ◽  
Partial Differential ◽  
Lp Spaces ◽  
Partial Differential Operators ◽  
Quadratic Estimates

AbstractThe purpose of this paper is to show that higher order elliptic partial differential operators on smooth domains have an H∞ functional calculus and satisfy quadratic estimates in Lp spaces on these domains.

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

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