The Calderon-Zygmund Operator and Its Relation to Asymptotic Estimates for Ordinary Differential Operators
- Authors: Savchuk AM1
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Affiliations:
- Lomonosov Moscow State University
- Issue: Vol 63, No 4 (2017): Differential and Functional Differential Equations
- Pages: 689-702
- Section: New Results
- URL: https://journals.rudn.ru/CMFD/article/view/22407
- DOI: https://doi.org/10.22363/2413-3639-2017-63-4-689-702
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Abstract
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.
About the authors
A M Savchuk
Lomonosov Moscow State University
Email: artem_savchuk@mail.ru
1 Leninskiye Gory, 119992 Moscow, Russia
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