scholarly journals Extension of holomorphic L2-functions with weighted growth conditions

1992 ◽  
Vol 126 ◽  
pp. 141-157 ◽  
Author(s):  
Klas Diederich ◽  
Gregor Herbort
Keyword(s):  
Fixed Point ◽  
Holomorphic Function ◽  
Lebesgue Measure ◽  
Complex Plane ◽  
Pseudoconvex Domain ◽  
Smooth Boundary ◽  
Small Neighborhood ◽  
Growth Conditions

In this article a new contribution to the following question is given: Let Ω ⊂ ⊂ Cn be a bounded pseudoconvex domain with C∞-smooth boundary, q ∈ ∂Ω a fixed point and H a k-dimensional affine complex plane such that q ∈ H and H intersects ∂Ω at q transversally. Let U be a suitably small neighborhood of q, and denote by r a C∞-defining function of Ω on U. Under which conditions on ∂Ω near q is it possible to find an exponent η>0 > 0 such that every holomorphic function f on Ω′ = H ∩Ω∩ U withwhere dλ′ denotes the Lebesgue-measure on H, can be extended to a holomorphic function ^f on Ω ∩ U such that even

scholarly journals Lp extension of holomorphic functions from submanifolds to strictly pseudoconvex domains with non-smooth boundary

2003 ◽  
Vol 172 ◽  
pp. 103-110
Author(s):  
Kenzō Adachi
Keyword(s):  
Holomorphic Function ◽  
Pseudoconvex Domain ◽  
Smooth Boundary ◽  
Holomorphic Functions ◽  
Pseudoconvex Domains ◽  
Strictly Pseudoconvex Domain

AbstractLet D be a bounded strictly pseudoconvex domain in ℂn (with not necessarily smooth boundary) and let X be a submanifold in a neighborhood of . Then any Lp (1 ≥ p < ∞) holomorphic function in X ∩ D can be extended to an Lp holomorphic function in D.

scholarly journals A remark on boundedness of Bloch functions

1991 ◽  
Vol 44 (3) ◽  
pp. 527-528 ◽  
Author(s):  
Krzysztof Samotij
Keyword(s):  
Holomorphic Function ◽  
Unit Circle ◽  
Lebesgue Measure ◽  
Unit Disk ◽  
Complex Plane ◽  
Hausdorff Measure ◽  
Bloch Space ◽  
Bloch Functions ◽  
Zero Measure

Two consequences of a theorem of Dahlberg are derived. Let f be a holomorphic function in the unit disk D of the complex plane, and let E be an Fσ subset of the unit circle T. Suppose that |f(rw)| ≤ M, ω ∈ T/E, for some constant M.Then f is bounded in either of the two cases:(i) if f is in the Bloch space and E is of zero measure with respect to the Hausdorff measure associated with the function ψ(t) = t log log (2πee/t),(ii) if f is integrable with respect to the planar Lebesgue measure on D and E is of zero measure with respect to the Hausdorff measure associated with the function ψ(t) = t log(2πee/t).

scholarly journals A remark on the theorem of Ohsawa-Takegoshi

2000 ◽  
Vol 158 ◽  
pp. 185-189 ◽  
Author(s):  
Klas Diederich ◽  
Emmanuel Mazzilli
Keyword(s):  
Pseudoconvex Domain ◽  
Complex Analysis ◽  
Holomorphic Functions ◽  
Growth Conditions ◽  
Analytic Subset ◽  
Famous Theorem ◽  
Restriction Operator

If D ⊂ ℂn is a pseudoconvex domain and X ⊂ D a closed analytic subset, the famous theorem B of Cartan-Serre asserts, that the restriction operator r : (D) → (X) mapping each function F to its restriction F|X is surjective. A very important question of modern complex analysis is to ask what happens to this result if certain growth conditions for the holomorphic functions on D and on X are added.

The approximate solution of nonlinear Fredholm implicit integro-differential equation in the complex plane

2021 ◽  
Author(s):  
Samir Lemita ◽  
Sami Touati ◽  
Kheireddine Derbal
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Approximate Solution ◽  
Integral Operator ◽  
Complex Plane ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Numerical Examples ◽  
Integro Differential Equation ◽  
Numerical Process

This paper’s purpose is to study the nonlinear Fredholm implicit integro-differential equation in the complex plane, where the term implicit integro-differential means that the derivative of unknown function is founded inside of the integral operator. Initially, according to Banach fixed point theory, we ensure that the equation has a unique solution under particular conditions. However, we exhibit a numerical process based on the conjunction between Nyström and Picard methods, for the sake of approximating solutions of this equation. In addition to that, the convergence analysis of this numerical process is demonstrated, and some illustrated numerical examples are presented.

Smooth Boundary Values Along Totally Real Submanifolds

1984 ◽  
Vol 36 (2) ◽  
pp. 240-248 ◽  
Author(s):  
Edgar Lee Stout
Keyword(s):  
Pseudoconvex Domain ◽  
Smooth Boundary ◽  
Regularity Result ◽  
Boundary Values ◽  
Real Submanifolds ◽  
Strongly Pseudoconvex Domain ◽  
Real Submanifold ◽  
Totally Real Submanifold ◽  
Class C ◽  
Totally Real

The main result of this paper is the following regularity result:THEOREM. Let D ⊂ CNbe a bounded, strongly pseudoconvex domain with bD of class Ck, k ≧ 3. Let Σ ⊂ bD be an N-dimensional totally real submanifold, and let f ∊ A(D) satisfy |f| = 1 on Σ, |f| < 1 on. If Σ is of class Cr, 3 ≦ r < k, then the restriction fΣ = f|Σ of f to Σ is of class Cr − 0, and if Σ is of class Ck, then fΣ is of class Ck − 1.Here, of course, A(D) denotes the usual space of functions continuous on , holomorphic on D, and we shall denote by Ak(D), k = 1, 2, …, the space of functions holomorphic on D whose derivatives or order k lie in A(D).

The logarithm of the modulus of a holomorphic function as a minorant for a subharmonic function. II. The complex plane

2017 ◽  
Vol 101 (3-4) ◽  
pp. 590-607 ◽  
Author(s):  
T. Yu. Baiguskarov ◽  
B. N. Khabibullin ◽  
A. V. Khasanova
Keyword(s):  
Holomorphic Function ◽  
Complex Plane ◽  
Subharmonic Function

ON FUNCTIONS ATTRACTING POSITIVE ENTROPY

2017 ◽  
Vol 97 (1) ◽  
pp. 69-79 ◽  
Author(s):  
ANNA LORANTY ◽  
RYSZARD J. PAWLAK
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Lebesgue Measure ◽  
Positive Entropy ◽  
Density Point

We examine dynamical systems which are ‘nonchaotic’ on a big (in the sense of Lebesgue measure) set in each neighbourhood of a fixed point $x_{0}$, that is, the entropy of this system is zero on a set for which $x_{0}$ is a density point. Considerations connected with this family of functions are linked with functions attracting positive entropy at $x_{0}$, that is, each mapping sufficiently close to the function has positive entropy on each neighbourhood of $x_{0}$.

scholarly journals Domain Identification for Inverse Problem via Conformal Mapping and Fixed Point Methods in Two Dimensions

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Fagueye Ndiaye
Keyword(s):  
Fixed Point ◽  
Conformal Mapping ◽  
Holomorphic Function ◽  
Unit Disk ◽  
Bounded Domain ◽  
Cauchy Data ◽  
Two Dimensions ◽  
Inverse Eigenvalue Problem ◽  
Unknown Boundary ◽  
Domain Identification

In this paper, we present a survey of the inverse eigenvalue problem for a Laplacian equation based on available Cauchy data on a known part Γ0 and a homogeneous Dirichlet condition on an unknown part Γ0 of the boundary of a bounded domain, Ω⊂ℝN. We consider variations in the eigenvalues and propose a conformal mapping tool to reconstruct a part of the boundary curve of the two-dimensional bounded domain based on the Cauchy data of a holomorphic function that maps the unit disk onto the unknown domain. The boundary values of this holomorphic function are obtained by solving a nonlocal differential Bessel equation. Then, the unknown boundary is obtained as the image of the boundary of the unit disk by solving an ill-posed Cauchy problem for holomorphic functions via a regularized power expansion. The Cauchy data were restricted to a nonvanishing function and to the normal derivatives without zeros. We prove the existence and uniqueness of the holomorphic function being considered and use the fixed-point method to numerically analyze the results of convergence. We’ll calculate the eigenvalues and compare the result with the shape obtained via minimization functional method, as developed in a previous study. Further, we’ll observe via simulations the shape of Γ and if it preserves its properties with varying the eigenvalues.

scholarly journals Recurrence and fixed points of surface homeomorphisms

1988 ◽  
Vol 8 (8) ◽  
pp. 99-107 ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Lebesgue Measure ◽  
Invariant Chain ◽  
Transitive Set ◽  
Surface Homeomorphisms ◽  
A Chain ◽  
Area Preserving ◽  
Twist Condition

AbstractWe prove that if f is a homeomorphism of the annulus which is homotopic to the identity and has a compact invariant chain transitive set L, then either f has a fixed point or every point of L moves uniformly in one direction: clockwise or counterclockwise. If f is area-preserving, then the annulus itself is a chain transitive set, so, in the presence of a boundary twist condition, one obtains a fixed point. The same techniques apply to homeomorphisms of the torus T2. In this setting we show that if f is homotopic to the identity, preserves Lebesgue measure and has mean translation 0, then it has at least one fixed point.

scholarly journals Pointwise Remez inequality

2021 ◽  
Author(s):  
B. Eichinger ◽  
P. Yuditskii
Keyword(s):  
Fixed Point ◽  
Asymptotic Solution ◽  
Lebesgue Measure ◽  
Chebyshev Polynomials ◽  
Extremal Solution ◽  
Fixed Size ◽  
Extremal Polynomials ◽  
Remez Inequality ◽  
Upper Estimate ◽  
Extremal Value

AbstractThe standard well-known Remez inequality gives an upper estimate of the values of polynomials on $$[-1,1]$$ [ - 1 , 1 ] if they are bounded by 1 on a subset of $$[-1,1]$$ [ - 1 , 1 ] of fixed Lebesgue measure. The extremal solution is given by the rescaled Chebyshev polynomials for one interval. Andrievskii asked about the maximal value of polynomials at a fixed point, if they are again bounded by 1 on a set of fixed size. We show that the extremal polynomials are either Chebyshev (one interval) or Akhiezer polynomials (two intervals) and prove Totik–Widom bounds for the extremal value, thereby providing a complete asymptotic solution to the Andrievskii problem.

Sign in / Sign up

Export Citation Format

Share Document