Boundary values of functions in a Sobolev space with Muckenhoupt weight on some non-Lipschitz domains

2014 ◽  
Vol 205 (8) ◽  
pp. 1133-1159 ◽  
Author(s):  
A I Tyulenev
Keyword(s):  
Sobolev Space ◽  
Muckenhoupt Weight ◽  
Lipschitz Domains ◽  
Boundary Values

The Sobolev space of half-differentiable functions and quasisymmetric homeomorphisms

2016 ◽  
Vol 23 (4) ◽  
pp. 615-622 ◽  
Author(s):  
Armen Sergeev
Keyword(s):  
Hilbert Space ◽  
Sobolev Space ◽  
Noncommutative Geometry ◽  
Function Theory ◽  
Banach Algebras ◽  
Linear Operators ◽  
Boundary Values ◽  
Differentiable Functions ◽  
Quasisymmetric Homeomorphisms

AbstractIn this paper, we give an interpretation of some classical objects of function theory in terms of Banach algebras of linear operators in a Hilbert space. We are especially interested in quasisymmetric homeomorphisms of the circle. They are boundary values of quasiconformal homeomorphisms of the disk and form a group ${\operatorname{QS}(S^{1})}$ with respect to composition. This group acts on the Sobolev space ${H^{1/2}_{0}(S^{1},\mathbb{R})}$ of half-differentiable functions on the circle by reparameterization. We give an interpretation of the group ${\operatorname{QS}(S^{1})}$ and the space ${H^{1/2}_{0}(S^{1},\mathbb{R})}$ in terms of noncommutative geometry.

scholarly journals On the boundary values of the solutions of linear elliptic equations

1983 ◽  
Vol 27 (1) ◽  
pp. 1-30 ◽  
Author(s):  
J. Chabrowski ◽  
H.B. Thompson
Keyword(s):  
Elliptic Equation ◽  
Sobolev Space ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Sufficient Condition ◽  
Boundary Values ◽  
Linear Elliptic Equation

The purpose of this article is to investigate the traces of weak solutions of a linear elliptic equation. In particular, we obtain a sufficient condition for a solution belonging to the Sobolev space to have an L2-trace on the boundar.

scholarly journals Sobolev boundedness and continuity for commutators of the local Hardy–Littlewood maximal function

2021 ◽  
Vol 47 (1) ◽  
pp. 203-235
Author(s):  
Feng Liu ◽  
Qingying Xue ◽  
Kôzô Yabuta
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Maximal Function ◽  
Pointwise Estimates ◽  
Boundary Values ◽  
First Order ◽  
Littlewood Maximal Function ◽  
Weak Derivatives ◽  
Derivatives Of

Let \(\Omega\) be a subdomain in \(\mathbb{R}^n\) and \(M_\Omega\) be the local Hardy-Littlewood maximal function. In this paper, we show that both the commutator and the maximal commutator of \(M_\Omega\) are bounded and continuous from the first order Sobolev spaces \(W^{1,p_1}(\Omega)\) to \(W^{1,p}(\Omega)\) provided that \(b\in W^{1,p_2}(\Omega)\), \(1<p_1,p_2,p<\infty\) and \(1/p=1/p_1+1/p_2\). These are done by establishing several new pointwise estimates for the weak derivatives of the above commutators. As applications, the bounds of these operators on the Sobolev space with zero boundary values are obtained.

scholarly journals The Poisson equation in homogeneous Sobolev spaces

2004 ◽  
Vol 2004 (36) ◽  
pp. 1909-1921
Author(s):  
Tatiana Samrowski ◽  
Werner Varnhorn
Keyword(s):  
Sobolev Space ◽  
Poisson Equation ◽  
Sobolev Spaces ◽  
Exterior Domain ◽  
Smooth Boundary ◽  
Boundary Values ◽  
Homogeneous Sobolev Space ◽  
The Poisson Equation ◽  
Homogeneous Sobolev Spaces ◽  
External Forces

We consider Poisson's equation in ann-dimensional exterior domainG(n≥2)with a sufficiently smooth boundary. We prove that for external forces and boundary values given in certainLq(G)-spaces there exists a solution in the homogeneous Sobolev spaceS2,q(G), containing functions being local inLq(G)and having second-order derivatives inLq(G)Concerning the uniqueness of this solution we prove that the corresponding nullspace has the dimensionn+1, independent ofq.

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