scholarly journals Some estimates for spectral functions connected with formally hypoelliptic differential operators

1972 ◽  
Vol 10 (1-2) ◽  
pp. 251-275 ◽  
Author(s):  
Nils Nilsson
Keyword(s):  
Differential Operators ◽  
Spectral Functions

On bounds for Titchmarsh–Weyl m-coefficients and for spectral functions for second-order differential operators

1984 ◽  
Vol 97 ◽  
pp. 1-7 ◽  
Author(s):  
F. V. Atkinson
Keyword(s):  
Weight Function ◽  
Differential Operators ◽  
Order Term ◽  
Weyl Function ◽  
Second Order ◽  
Spectral Functions ◽  
Positive Weight ◽  
Nevanlinna Function ◽  
Fixed Sign ◽  
Order Of Magnitude

SynopsisOrder-of-magnitude results are extended to the case of general second-order term, with coefficient not necessarily of fixed sign, with general positive weight-function. The bounds are used to establish the expression for the Titchmarsh–Weyl function m(λ) as a Nevanlinna function in terms of the spectral function.

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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