The Roles Played by Order of Starlikeness and the Bloch Condition in the Roper–Suffridge Extension Operator

2017 ◽  
Vol 12 (1) ◽  
pp. 247-259 ◽  
Author(s):  
Xiaofei Zhang ◽  
Yonghong Xie
Keyword(s):  
Extension Operator ◽  
Order Of Starlikeness

A Continuous Extension Operator for Convex Metrics

2010 ◽  
Vol 53 (4) ◽  
pp. 719-729
Author(s):  
I. Stasyuk ◽  
E. D. Tymchatyn
Keyword(s):  
Continuous Extension ◽  
Extension Operator ◽  
Uniform Topology ◽  
Simultaneous Extension ◽  
Peano Continuum ◽  
Convex Metrics

AbstractWe consider the problem of simultaneous extension of continuous convex metrics defined on subcontinua of a Peano continuum. We prove that there is an extension operator for convex metrics that is continuous with respect to the uniform topology.

Trace/extension operators in rough domains and applications to partial differential equations

2015 ◽  
Author(s):  
◽  
Kevin Brewster
Keyword(s):  
Dirichlet Boundary ◽  
Boundary Value ◽  
Weighted Sobolev Spaces ◽  
Extension Operator ◽  
Extension Theory ◽  
Lipschitz Domains ◽  
Poisson Boundary ◽  
Ellipticity Condition ◽  
University Of Missouri ◽  
The University

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Trace and extension theory lay the foundation for solving a plethora of boundary value problems. In developing this theory, one typically needs well-behaved extension operators from a specified domain to the entire Euclidean space. Historically, three extension operators have developed much of the theory in the setting of Lipschitz domains (and rougher domains); those due to A.P. Calderon, E.M. Stein, and P.W. Jones. In this dissertation, we generalize Stein's extension operator to weighted Sobolev spaces and Jones' extension operator to domains with partially vanishing traces. We then develop a rich trace/extension theory as a tool in solving a Poisson boundary value problem with Dirichlet boundary condition where the differential operator in question is of second order in divergence form with bounded coefficients satisfying the Legendre-Hadamard ellipticity condition.

Loewner Chain Associated with the Modified Roper–Suffridge Extension Operator

2015 ◽  
Vol 16 (2) ◽  
pp. 265-281 ◽  
Author(s):  
Xiaofei Zhang ◽  
Shuxia Feng ◽  
Yongjie Li
Keyword(s):  
Extension Operator ◽  
Loewner Chain

The order of starlikeness of uniformly convex functions

2017 ◽  
Vol 23 (1) ◽  
Author(s):  
Agnieszka Wiśniowska-Wajnryb
Keyword(s):  
Convex Functions ◽  
Uniformly Convex ◽  
Uniformly Convex Functions ◽  
Order Of Starlikeness

AbstractWe investigate classes of

scholarly journals Linear Hahn–Banach extension operators

1989 ◽  
Vol 32 (1) ◽  
pp. 53-57 ◽  
Author(s):  
Brailey Sims ◽  
David Yost
Keyword(s):  
Banach Space ◽  
Model Theory ◽  
General Result ◽  
Extension Operator ◽  
Density Character ◽  
Finite Dimensional ◽  
Skolem Theorem

Given any subspace N of a Banach space X, there is a subspace M containing N and of the same density character as N, for which there exists a linear Hahn–Banach extension operator from M* to X*. This result was first proved by Heinrich and Mankiewicz [4, Proposition 3.4] using some of the deeper results of Model Theory. More precisely, they used the Banach space version of the Löwenheim–Skolem theorem due to Stern [11], which in turn relies on the Löwenheim–Skolem and Keisler–Shelah theorems from Model Theory. Previously Lindenstrauss [7], using a finite dimensional lemma and a compactness argument, obtained a version of this for reflexive spaces. We shall show that the same finite dimensional lemma leads directly to the general result, without any appeal to Model Theory.

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