The Calderon–Zygmund Operator and its Relation to Asymptotic Estimates of Ordinary Differential Operators

2021 ◽  
Author(s):  
A. M. Savchuk
Keyword(s):  
Differential Operators ◽  
Asymptotic Estimates ◽  
Zygmund Operator ◽  
Ordinary Differential Operators

scholarly journals The Calderon-Zygmund Operator and Its Relation to Asymptotic Estimates for Ordinary Differential Operators

2017 ◽  
Vol 63 (4) ◽  
pp. 689-702 ◽  
Author(s):  
A M Savchuk
Keyword(s):  
Half Plane ◽  
Complex Plane ◽  
Carleson Measure ◽  
Differential Operators ◽  
Fundamental System ◽  
Asymptotic Estimates ◽  
Zygmund Operator ◽  
Ordinary Differential Operators ◽  
Upper Half Plane

We consider the problem of estimating of expressions of the kind Υ(λ)=supx∈[0,1]∣∣∫x0f(t)eiλtdt∣∣. In particular, for the case f∈Lp[0,1], p∈(1,2], we prove the estimate ∥Υ(λ)∥Lq(R)≤C∥f∥Lp for any q>p′, where 1/p+1/p′=1. The same estimate is proved for the space Lq(dμ), where dμ is an arbitrary Carleson measure in the upper half-plane C+. Also, we estimate more complex expressions of the kind Υ(λ) arising in study of asymptotics of the fundamental system of solutions for systems of the kind y′=By+A(x)y+C(x,λ)y with dimension n as |λ|→∞ in suitable sectors of the complex plane.

Eigenvalues of regular differential operators

1978 ◽  
Vol 79 (3-4) ◽  
pp. 299-305 ◽  
Author(s):  
Harold E. Benzinger
Keyword(s):  
Fine Structure ◽  
Large Class ◽  
Fourier Coefficients ◽  
Differential Operators ◽  
Asymptotic Estimates ◽  
Ordinary Differential Operators

SynopsisIt is shown that the fine structure of the asymptotic estimates for the eigenvalues of a large class of ordinary differential operators, can be described in terms of the Fourier coefficients of a function of class L2.

A class of symmetric ordinary differential operators whose deficiency numbers differ by an integer†

1978 ◽  
Vol 82 (1-2) ◽  
pp. 117-134 ◽  
Author(s):  
Richard C. Gilbert
Keyword(s):  
Differential Operator ◽  
Positive Integer ◽  
Asymptotic Expansions ◽  
Differential Operators ◽  
Ordinary Differential Operator ◽  
Ordinary Differential Operators ◽  
Half Axis ◽  
Complex Coefficients

SynopsisFormulas are determined for the deficiency numbers of a formally symmetric ordinary differential operator with complex coefficients which have asymptotic expansions of a prescribed type on a half-axis. An implication of these formulas is that for any given positive integer there exists a formally symmetric ordinary differential operator whose deficiency numbers differ by that positive integer.

18.—Asymptotic Estimates for the Lengths of the Gaps in the Essential Spectrum of Self-adjoint Differential Operators

1976 ◽  
Vol 74 ◽  
pp. 239-252 ◽  
Author(s):  
M. S. P. Eastham ◽  
W. N. Everitt
Keyword(s):  
Essential Spectrum ◽  
Differential Operators ◽  
Second Order ◽  
Asymptotic Estimates ◽  
Image Position ◽  
Differentiability Properties

SynopsisThe paper gives asymptotic estimates of the formas λ→∞ for the length l(μ)of a gap, centre μ in the essential spectrum associated with second-order singular differential operators. The integer r will be shown to depend on the differentiability properties of the coefficients in the operators and, in fact, r increases with the increasing differentiability of the coefficients. The results extend to all r ≧ – 2 the long-standing ones of Hartman and Putnam [10], who dealt with r = 0, 1, 2.

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