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.

scholarly journals On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve

2014 ◽  
Vol 57 (2) ◽  
pp. 381-389
Author(s):  
Adrian Łydka
Keyword(s):  
Functional Equation ◽  
Meromorphic Function ◽  
Elliptic Curve ◽  
Explicit Formula ◽  
Half Plane ◽  
Complex Plane ◽  
Möbius Function ◽  
Mobius Function ◽  
Upper Half Plane ◽  
Explicit Formulae

AbstractWe study analytic properties function m(z, E), which is defined on the upper half-plane as an integral from the shifted L-function of an elliptic curve. We show that m(z, E) analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for m(z, E) in the strip |ℑz| < 2π.

A Short Proof of an Interpolation Theorem

1974 ◽  
Vol 17 (1) ◽  
pp. 127-128 ◽  
Author(s):  
Edward Hughes
Keyword(s):  
Half Plane ◽  
Complex Plane ◽  
Simple Proof ◽  
Real Line ◽  
Short Proof ◽  
Interpolation Theorem ◽  
Operator Interpolation ◽  
Upper Half Plane ◽  
Positive Functions ◽  
Class Of Functions

In this note we give a simple proof of an operator-interpolation theorem (Theorem 2) due originally to Donoghue [6], and Lions-Foias [7].Let be the complex plane, the open upper half-plane, the real line, ℛ+ and ℛ- the non-negative and non-positive axes. Denote by the class of positive functions on which extend analytically to —ℛ-, and map into itself. Denote by ’ the class of functions φ such that φ(x1/2)2 is in .

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.

ELLIPTIC AND AUTOMORPHIC DYNAMICAL SYSTEMS ON SURFACES

2006 ◽  
Vol 16 (04) ◽  
pp. 911-923 ◽  
Author(s):  
S. P. BANKS ◽  
SONG YI
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Half Plane ◽  
Complex Plane ◽  
Fundamental Domain ◽  
Function Theory ◽  
Theta Series ◽  
The Hyperbolic Metric ◽  
Upper Half Plane ◽  
Definition Of

We derive explicit differential equations for dynamical systems defined on generic surfaces applying elliptic and automorphic function theory to make uniform the surfaces in the upper half of the complex plane with the hyperbolic metric. By modifying the definition of the standard theta series we will determine general meromorphic systems on a fundamental domain in the upper half plane the solution trajectories of which "roll up" onto an appropriate surface of any given genus.

scholarly journals Local model for moduli space of affine vortices

2018 ◽  
Vol 29 (03) ◽  
pp. 1850020
Author(s):  
Sushmita Venugopalan ◽  
Guangbo Xu
Keyword(s):  
Half Plane ◽  
Complex Plane ◽  
Moduli Space ◽  
Smooth Manifold ◽  
American Mathematical Society ◽  
Fredholm Theory ◽  
Mathematical Society ◽  
Vortex Equation ◽  
Upper Half Plane ◽  
Funct Anal

We show that the moduli space of regular affine vortices, which are solutions of the symplectic vortex equation over the complex plane, has the structure of a smooth manifold. The construction uses Ziltener’s Fredholm theory results [A Quantum Kirwan Map: Bubbling and Fredholm Theory, Memiors of the American Mathematical Society, Vol. 230 (American Mathematical Society, Providence, RI, 2014), pp. 1–129]. We also extend the result to the case of affine vortices over the upper half plane. These results are necessary ingredients in defining the “open quantum Kirwan map” proposed by Woodward [Gauged Floer theory for toric moment fibers, Geom. Funct. Anal. 21 (2011) 680–749].

Integrable-square solutions of a singular ordinary differential equation

1981 ◽  
Vol 89 (3-4) ◽  
pp. 267-279 ◽  
Author(s):  
Richard C. Gilbert
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Characteristic Polynomial ◽  
Characteristic Equation ◽  
Complex Plane ◽  
Asymptotic Expansions ◽  
Differential Operators ◽  
Singular Ordinary Differential Equation ◽  
Ordinary Differential Operators

SynopsisAbsolutely square integrable solutions are determined for the equation= λywhere the ζn−r(x) are holomorphic in a sector of the complex plane and have asymptotic expansions asxapproaches infinity. It is shown that the number of such solutions depends upon the roots of the characteristic equation and their multiplicity, and upon the sign of the derivative of the characteristic polynomial. Application is made to formally symmetric ordinary differential operators.

scholarly journals Restricted interpolation by meromorphic inner functions

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Alexei Poltoratski ◽  
Rishika Rupam
Keyword(s):  
Half Plane ◽  
Schrödinger Operator ◽  
Differential Operators ◽  
Schrodinger Operator ◽  
Inner Functions ◽  
Spectral Problems ◽  
Upper Half Plane

AbstractMeromorphic Inner Functions (MIFs) on the upper half plane play an important role in applications to spectral problems for differential operators. In this paper, we survey some recent results concerning function theoretic properties of MIFs and show their connections with spectral problems for the Schrödinger operator.

Sign in / Sign up

Export Citation Format

Share Document