scholarly journals Traces of Besov, Triebel-Lizorkin and Sobolev Spaces on Metric Spaces

2017 ◽  
Vol 5 (1) ◽  
pp. 98-115 ◽  
Author(s):  
Eero Saksman ◽  
Tomás Soto
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Metric Space ◽  
Besov Spaces ◽  
Metric Spaces ◽  
Function Spaces ◽  
First Order ◽  
Trace Theorems ◽  
Euclidean Setting ◽  
Regular Subset

Abstract We establish trace theorems for function spaces defined on general Ahlfors regular metric spaces Z. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices s < 1, as well as the first order Hajłasz-Sobolev space M1,p(Z). They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset F ⊂ Z are Besov spaces defined intrinsically on F. Our method employs the definitions of the function spaces via hyperbolic fillings of the underlying metric space.

Metric Spaces

2021 ◽  
pp. 103-125
Author(s):  
James Davidson
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Distance Measure ◽  
Function Spaces ◽  
Important Case ◽  
General Context ◽  
Compact Sets ◽  
Definition Of ◽  
Metric Distance

This chapter introduces and illustrates the concept of a metric (distance measure), and the definition of a metric space. Open, closed, and compact sets are discussed in a general context, and the concepts of separability and completeness introduced. It goes on to look at mappings on metric spaces, examines the important case of function spaces, and treats the Arzelà–Ascoli theorem.

scholarly journals Geraghty Type Generalized F-Contractions and Related Applications in Partial b-Metric Spaces

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Deepak Singh ◽  
Varsha Chauhan ◽  
R. Wangkeeree
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Metric Spaces ◽  
Partial Metric Space ◽  
Partial Metric ◽  
First Order ◽  
New Concepts ◽  
Banach Contraction ◽  
Uniqueness Of A Solution ◽  
The Banach Contraction Principle

The purpose of this paper is to introduce new concepts of (α,β)-admissible Geraghty type generalized F-contraction and to prove that some fixed point results for such mappings are in the perspective of partial b-metric space. As an application, we inaugurate new fixed point results for Geraghty type generalized graphic F-contraction defined on partial metric space endowed with a directed graph. On the other hand, one more application to the existence and uniqueness of a solution for the first-order periodic boundary value problem is also provided. Our findings encompass various generalizations of the Banach contraction principle on metric space, partial metric space, and partial b-metric space. Moreover, some examples are presented to illustrate the usability of the new theory.

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.

Real Interpolation of Sobolev Spaces on Subdomains of Rn

1978 ◽  
Vol 30 (01) ◽  
pp. 190-214 ◽  
Author(s):  
R. A. Adams ◽  
J. J. F. Fournier
Keyword(s):  
Sobolev Space ◽  
Fractional Order ◽  
Sobolev Spaces ◽  
Besov Spaces ◽  
Interpolation Method ◽  
A Priori ◽  
Real Interpolation ◽  
Convenient Tool ◽  
Real Interpolation Method ◽  
Nikolskii Spaces

The real interpolation method is a very convenient tool in the study of imbedding relationships among Sobolev spaces and some of their fractional order generalizations, (Besov spaces, Nikolskii spaces etc.) Central to the application of these methods is the a priori determination that a given Sobolev space Wk'p(Ω) belongs to an appropriate class of spaces intermediate between two other “extreme” spaces.

scholarly journals Bounds on Scott ranks of some polish metric spaces

2020 ◽  
Vol 21 (01) ◽  
pp. 2150001
Author(s):  
William Chan
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
First Order ◽  
Order Language ◽  
Scott Rank ◽  
The Church

If [Formula: see text] is a proper Polish metric space and [Formula: see text] is any countable dense submetric space of [Formula: see text], then the Scott rank of [Formula: see text] in the natural first-order language of metric spaces is countable and in fact at most [Formula: see text], where [Formula: see text] is the Church–Kleene ordinal of [Formula: see text] (construed as a subset of [Formula: see text]) which is the least ordinal with no presentation on [Formula: see text] computable from [Formula: see text]. If [Formula: see text] is a rigid Polish metric space and [Formula: see text] is any countable dense submetric space, then the Scott rank of [Formula: see text] is countable and in fact less than [Formula: see text].

scholarly journals Some Function Spaces Relative to Morrey-Campanato Spaces on Metric Spaces

2005 ◽  
Vol 177 ◽  
pp. 1-29 ◽  
Author(s):  
Dachun Yang
Keyword(s):  
Sobolev Spaces ◽  
Maximal Function ◽  
Metric Spaces ◽  
Function Spaces ◽  
Sharp Maximal Function ◽  
Embedding Theorems ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Lizorkin Space

In this paper, the author introduces the Morrey-Campanato spaces Lsp(X) and the spaces Cps(X) on spaces of homogeneous type including metric spaces and some fractals, and establishes some embedding theorems between these spaces under some restrictions and the Besov spaces and the Triebel-Lizorkin spaces. In particular, the author proves that Lsp(X) = Bs∞,∞(X) if 0 < s < ∞ and µ(X) < ∞. The author also introduces some new function spaces Asp(X) and Bsp(X) and proves that these new spaces when 0 < s < 1 and 1 < p < ∞ are just the Triebel-Lizorkin space Fsp,∞(X) if X is a metric space, and the spaces A1p(X) and B1p(X) when 1 < p < ∞ are just the Hajłasz-Sobolev spaces W1p(X). Finally, as an application, the author gives a new characterization of the Hajłasz-Sobolev spaces by making use of the sharp maximal function.

scholarly journals Duality of Variable Exponent Triebel-Lizorkin and Besov Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Takahiro Noi
Keyword(s):  
Sobolev Spaces ◽  
Besov Spaces ◽  
Variable Exponent ◽  
Function Spaces ◽  
Bessel Potential ◽  
Lebesgue Spaces ◽  
Bessel Potential Spaces

We will prove the duality and reflexivity of variable exponent Triebel-Lizorkin and Besov spaces. It was shown by many authors that variable exponent Triebel-Lizorkin spaces coincide with variable exponent Bessel potential spaces, Sobolev spaces, and Lebesgue spaces when appropriate indices are chosen. In consequence of the results, these variable exponent function spaces are shown to be reflexive.

scholarly journals ON METRIZATION OF UNIONS OF FUNCTION SPACES ON DIFFERENT INTERVALS

2012 ◽  
Vol 92 (3) ◽  
pp. 281-297
Author(s):  
TALEB ALKURDI ◽  
SANDER C. HILLE ◽  
ONNO VAN GAANS
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Function Spaces ◽  
General Construction

AbstractThis paper investigates a class of metrics that can be introduced on the set consisting of the union of continuous functions defined on different intervals with values in a fixed metric space, where the union ranges over a family of intervals. Its definition is motivated by the Skorohod metric(s) on càdlàg functions. We show what is essential in transferring the ideas employed in the latter metric to our setting and obtain a general construction for metrics in our case. Next, we define the metric space where elements are sequences of functions from the above mentioned set. We provide conditions that ensure separability and completeness of the constructed metric spaces.

scholarly journals Dyadic Norm Besov-Type Spaces as Trace Spaces on Regular Trees

2019 ◽  
Vol 53 (4) ◽  
pp. 1317-1346 ◽  
Author(s):  
Pekka Koskela ◽  
Zhuang Wang
Keyword(s):  
Sobolev Spaces ◽  
Function Spaces ◽  
First Order ◽  
Type Spaces ◽  
Trace Spaces ◽  
Besov Type Spaces

AbstractIn this paper, we study function spaces defined via dyadic energies on the boundaries of regular trees. We show that correct choices of dyadic energies result in Besov-type spaces that are trace spaces of (weighted) first order Sobolev spaces.

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