scholarly journals Geometric analysis on Cantor sets and trees

2017 ◽  
Vol 2017 (725) ◽  
Author(s):  
Anders Björn ◽  
Jana Björn ◽  
James T. Gill ◽  
Nageswari Shanmugalingam
Keyword(s):  
Sobolev Space ◽  
Besov Space ◽  
Besov Spaces ◽  
Rooted Tree ◽  
Geometric Analysis ◽  
Cantor Sets ◽  
Rooted Trees ◽  
First Order ◽  
Sharp Estimates ◽  
Type Sets

AbstractUsing uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In this setting, we show that the trace of a first-order Sobolev space on the boundary of a regular rooted tree is exactly a Besov space with an explicit smoothness exponent. Further, we study quasisymmetries between the boundaries of two trees, and show that they have rough quasiisometric extensions to the trees. Conversely, we show that every rough quasiisometry between two trees extends as a quasisymmetry between their boundaries. In both directions we give sharp estimates for the involved constants. We use this to obtain quasisymmetric invariance of certain Besov spaces of functions on Cantor type sets.

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.

scholarly journals Trace operators on fractals, entropy and approximation numbers

2011 ◽  
Vol 18 (3) ◽  
pp. 549-575
Author(s):  
Cornelia Schneider
Keyword(s):  
Besov Spaces ◽  
Trace Operator ◽  
Approximation Numbers ◽  
Atomic Decompositions ◽  
Sharp Estimates ◽  
Compact Embeddings ◽  
Trace Space

Abstract First we compute the trace space of Besov spaces – characterized via atomic decompositions – on fractals Γ, for parameters 0 < p < ∞, 0 < q ≤ min(1, p) and s = (n – d)/p. New Besov spaces on fractals are defined via traces for 0 < p, q ≤ ∞, s ≥ (n – d)/p and some embedding assertions are established. We conclude by studying the compactness of the trace operator TrΓ by giving sharp estimates for entropy and approximation numbers of compact embeddings between Besov spaces. Our results on Besov spaces remain valid considering the classical spaces defined via differences. The trace results are used to study traces in Triebel–Lizorkin spaces as well.

scholarly journals The sobolev embeddings are usually sharp

2005 ◽  
Vol 2005 (4) ◽  
pp. 437-448 ◽  
Author(s):  
A. Fraysse ◽  
S. Jaffard
Keyword(s):  
Sobolev Space ◽  
Besov Space ◽  
Hölder Regularity ◽  
Sobolev Embeddings ◽  
Residual Sets ◽  
Holder Regularity

Letx0∈ℝd; we study the Hölder regularity atx0of a generic function of the Sobolev spaceLp,s(ℝd)and of the Besov spaceBps,q(ℝd)fors−d/p>0. The setting for genericity is supplied here by HP-residual sets.

A Note on Homogeneous Dendrites

1968 ◽  
Vol 11 (1) ◽  
pp. 85-93 ◽  
Author(s):  
Z. A. Melzak
Keyword(s):  
Branch Point ◽  
Rooted Tree ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
Graph Theoretic ◽  
Order Branch

In graph - theoretic terms a homogeneous p-dendrite, p ≥ 2, is defined as a finite singly-rooted tree in which the root has valency 1 while every other vertex has valency 1 or p. More descriptively, a homogeneous p-dendrite may be imagined to start from its root as the main, or 0th order, branch which proceeds to the first - order branch point where it gives rise top first - order branches. Each of these either terminates at its other end (which is a second-order branch point) or it splits there again into p branches (which are of third order), and so on. The order of the dendrite is the highest order of a branch present in it. For completeness, a 0-th order dendrite is also allowed, this consists of the 0-th order branch alone.

scholarly journals Sharp Estimates of the Embedding Constants for Besov Spaces

2006 ◽  
Vol 19 (1) ◽  
Author(s):  
W. Desmond Evans ◽  
Georgi E. Karadzhov ◽  
David E. Edmunds
Keyword(s):  
Besov Spaces ◽  
Sharp Estimates

scholarly journals A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function

2002 ◽  
Vol 65 (2) ◽  
pp. 253-258 ◽  
Author(s):  
Hitoshi Tanaka
Keyword(s):  
Sobolev Space ◽  
Maximal Function ◽  
60Th Birthday ◽  
One Dimensional ◽  
Partial Derivatives ◽  
First Order ◽  
Littlewood Maximal Function ◽  
The One ◽  
Derivatives Of

Dedicated to Professor Kôzô Yabuta on the occasion of his 60th birthdayJ. Kinnunen proved that of P > 1, d ≤ 1 and f is a function in the Sobolev space W1,P(Rd), then the first order weak partial derivatives of the Hardy-Littlewood maximal function ℳf belong to LP(Rd). We shall show that, when d = 1, Kinnunen's result can be extended to the case where P = 1.

scholarly journals Extended Cesáro operators between generalized Besov spaces and Bloch type spaces in the unit ball

2009 ◽  
Vol 7 (3) ◽  
pp. 209-223 ◽  
Author(s):  
Ze-Hua Zhou ◽  
Min Zhu
Keyword(s):  
Unit Ball ◽  
Besov Space ◽  
Besov Spaces ◽  
Complex Space ◽  
Bloch Space ◽  
Sufficient Conditions ◽  
Cesàro Operator ◽  
Type Spaces ◽  
Necessary And Sufficient ◽  
Cesaro Operator

Let 𝑔 be a holomorphic of the unit ballBin then-dimensional complex space, and denote byTgthe extended Cesáro operator with symbolg. Let 0 <p< +∞, −n− 1 <q< +∞,q> −1 and α > 0, starting with a brief introduction to well known results about Cesáro operator, we investigate the boundedness and compactness ofTgbetween generalized Besov spaceB(p, q)and 𝛼α- Bloch spaceℬαin the unit ball, and also present some necessary and sufficient conditions.

Toward Automatic Tolerancing of Mechanical Assemblies: First-Order GD&T Schema Development and Tolerance Allocation

2015 ◽  
Vol 15 (4) ◽  
Author(s):  
Payam Haghighi ◽  
Prashant Mohan ◽  
Nathan Kalish ◽  
Prabath Vemulapalli ◽  
Jami J. Shah ◽  
...  
Keyword(s):  
Feature Recognition ◽  
Reference Frames ◽  
Geometric Analysis ◽  
Tolerance Allocation ◽  
First Order ◽  
Geometric Dimensioning And Tolerancing ◽  
Mechanical Assemblies ◽  
Schema Development ◽  
Design Function ◽  
Acceptance Rates

Geometric and dimensional tolerances must be determined not only to ensure proper achievement of design function but also for manufacturability and assemblability of mechanical assemblies. We are investigating the degree to which it is possible to automate tolerance assignment on mechanical assemblies received only as STEP AP 203 (nominal) geometry files. In a previous paper, we reported on the preprocessing steps required: assembly feature recognition, pattern recognition, and extraction of both constraints and directions of control (DoC) for assembly. In this paper, we discuss first-order tolerance schema development, based purely on assemblability conditions. This includes selecting features to be toleranced, tolerance types, datums, and datum reference frames (DRFs), and tolerance value allocation. The approach described here is a combination of geometric analysis and heuristics. The assumption is that this initial geometric dimensioning and tolerancing (GD&T) specification will be sent to a stack analysis module and iterated upon until satisfactory results, such as desired acceptance rates, are reached. The paper also touches upon issues related to second-order schema development, one that takes intended design function into account.

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