SCIENCE CHINAMathematics CrossMark

November 2017 Vol. 60 No. 11: 1981–2010

doi: 10.1007/s11425-017-9148-6

c© Science China Press and Springer-Verlag Berlin Heidelberg 2017 math.scichina.com link.springer.com

. ARTICLES .

Traces of weighted function spaces: Dyadic normsand Whitney extensions

Dedicated to the memory of Professor CHENG MinDe on the occasion of the centenary of his birth

KOSKELA Pekka∗ , SOTO Tomás & WANG Zhuang

Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä FI-40014, Finland

Email: pekka.j.koskela@jyu.fi, tomas.a.soto@jyu.fi, zhuang.z.wang@jyu.fi

Received May 4, 2017; accepted July 21, 2017; published online September 25, 2017

Abstract The trace spaces of Sobolev spaces and related fractional smoothness spaces have been an active

area of research since the work of Nikolskii, Aronszajn, Slobodetskii, Babich and Gagliardo among others in

the 1950’s. In this paper, we review the literature concerning such results for a variety of weighted smoothness

spaces. For this purpose, we present a characterization of the trace spaces (of fractional order of smoothness),

based on integral averages on dyadic cubes, which is well-adapted to extending functions using the Whitney

extension operator.

Keywords trace theorems, weighted Sobolev spaces, Besov spaces, Triebel-Lizorkin spaces

MSC(2010) 46E35, 42B35

Citation: Koskela P, Soto T, Wang Z. Traces of weighted function spaces: Dyadic norms and Whitney extensions.

Sci China Math, 2017, 60: 1981–2010, doi: 10.1007/s11425-017-9148-6

1 Introduction

In 1957, Gagliardo [13] gave a characterization of the trace space of the first order Sobolev space W 1,p(Ω),1 < p

Nonlinear Analysis 177 (2018) 586–600

Contents lists available at ScienceDirect

Nonlinear Analysis

www.elsevier.com/locate/na

Controlled di�eomorphic extension of homeomorphisms

Pekka Koskelaa,*, Zhuang Wanga, Haiqing Xub

a Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35, FI-40014 Jyväskylä,Finlandb School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026,PR China

a r t i c l e i n f o

Article history:Received 16 March 2018

Accepted 18 April 2018

Communicated by Enzo Mitidieri

Keywords:Poisson extension

Di�eomorphism

Chord-arc curve

a b s t r a c t

Let ⌦ be an internal chord-arc Jordan domain and Ï : S æ ˆ⌦ be a homeomor-phism. We show that Ï has finite dyadic energy if and only if Ï has a di�eomorphicextension h : D æ ⌦ which has finite energy.

© 2018 Elsevier Ltd. All rights reserved.

1. Introduction

Let ⌦ µ C be a bounded convex domain and suppose that Ï is a homeomorphism from the unit circle Sonto ˆ⌦ . Then, by [7], the complex-valued Poisson extension h of Ï is a homeomorphism from D̄ onto ⌦̄ .This harmonic map h is a di�eomorphism in D but its derivatives are not necessarily uniformly bounded.In 2007, G. C. Verchota [10] proved that the derivatives of h may fail to be square integrable but that theyare necessarily p-integrable over D for any p < 2. In 2009, T. Iwaniec, G. Martin and C. Sbordone improvedon [6] by showing that the derivatives belong to weak-L2 with sharp estimates. In a related work [1] by K.Astala, T. Iwaniec, G. Martin and J. Onninen, it was shown that if additionally ˆ⌦ is a C1-regular Jordancurve, the square integrability of the derivatives of h is equivalent to the requirement that

⁄

ˆ⌦

⁄

ˆ⌦|log|Ï≠1(›) ≠ Ï≠1(÷)|||d›||d÷| < +Œ.

In this note we give a generalization of the aforementioned results. Towards this end, recall that thePoisson extension of a homeomorphism Ï : S æ ˆ⌦ may fail to be injective if ⌦ is not convex. Next, theboundary ˆ⌦ of a bounded convex domain ⌦ is a chord-arc Jordan curve: ˆ⌦ is a rectifiable Jordan curve

* Corresponding author.E-mail addresses: pekka.j.koskela@jyu.fi (P. Koskela), zhuang.z.wang@jyu.fi (Z. Wang), hqxu@mail.ustc.edu.cn (H. Xu).

https://doi.org/10.1016/j.na.2018.04.020

0362-546X/© 2018 Elsevier Ltd. All rights reserved.

https://doi.org/10.1016/j.na.2018.04.020http://www.elsevier.com/locate/nahttp://www.elsevier.com/locate/nahttp://crossmark.crossref.org/dialog/?doi=10.1016/j.na.2018.04.020&domain=pdfmailto:pekka.j.koskela@jyu.fimailto:zhuang.z.wang@jyu.fimailto:hqxu@mail.ustc.edu.cnhttps://doi.org/10.1016/j.na.2018.04.020

ASIAN J. MATH. c� 2019 International PressVol. 23, No. 5, pp. 837–876, October 2019 004

THE Q↵-RESTRICTION PROBLEM⇤

Z. WANG† , J. XIAO‡ , AND Y. ZHOU§

Abstract. Let ↵ 2 [0, 1) and ⌦ be an open connected subset of Rn�2. This paper showsthat the Q↵-restriction problem Q↵|⌦ = Q↵(⌦) is solvable if and only if ⌦ is an Ahlfors n-regulardomain; i.e., vol(B(x, r) \ ⌦) & rn for any Euclidean ball B(x, r) with center x 2 ⌦ and radiusr 2

�0, diam(⌦)

�, thereby not only yielding an exponential Q↵-integrability as a proper adjustment

of the John-Nirenberg type inequality for Q↵ conjectured in [3, Problem 8.1, (8.2)] but also resolvingthe quasiconformal extension problem for Q↵ posed in [3, Problem 8.5].

Key words. Q space, restriction, Ahlfors regular domain, Uniform domain, Minkowski typedimension.

Mathematics Subject Classification. 46E35, 42B35.

1. Introduction. The conformal Poincaré inequality says that

–

Z

B(x0,r0)|u(x)� ūx0,r0 | dx .

Z

B(x0,r0)|ru(x)|n dx

! 1n

holds for any ball B(x0, r0) (with centre x0 and radius r0) of the Euclidean spaceRn�2 and a given u in Ẇ 1,n ⌘ Ẇ 1,n(Rn) which is the conformal Sobolev space of allfunctions with their weak derivatives being n-integrable over Rn; i.e.,

krukLn ⌘✓Z

Rn|ru(x)|n dx

◆ 1n

< 1.

Here and henceforth,

v̄x0,r0 = –

Z

B(x0,r0)v(x) dx = vol

�B(x0, r0)

��1Z

B(x0,r0)v(x) dx

is the integral mean of a function v over B(x0, r0); vol(·) expresses the n-dimensionalLebesgue measure; A . B or B & A means that A cB for a constant c > 0;moreover A ⇡ B is equivalent to both A . B and B . A. Upon writing BMO ⌘BMO(Rn) as the John-Nirenberg class of functions u with bounded mean oscillation:

[u]BMO ⌘ supB(x0,r0)

–

Z

B(x0,r0)|u(x)� ūx0,r0 | dx < 1,

we get that Ẇ 1,n embeds into BMO; i.e. (cf. [17, p.34]),

Ẇ 1,n ,! BMO with [u]BMO . krukLn .⇤Received May 24, 2018; accepted for publication July 5, 2018. Z. Wang and Y. Zhou were

supported by National Natural Science Foundation of China (#11871088); J. Xiao was supported byNSERC of Canada.

†Department of Mathematics and Statistics, P. O. Box 35 (MaD), FI-40014, University ofJyväskylä, Finland (wangzhuangzhao@gmail.com).

‡Department of Mathematics and Statistics, Memorial University, NL A1C 5S7, Canada (jxiao@mun.ca).

§School of Mathematical Science, Beijing Normal University, Beijing 100875, P. R. China (yuan.zhou@bnu.edu.cn).

837

Potential Analysishttps://doi.org/10.1007/s11118-019-09808-5

Dyadic Norm Besov-Type Spaces as Trace Spaceson Regular Trees

Pekka Koskela1 ·ZhuangWang1

Received: 25 March 2019 / Accepted: 9 October 2019 /© The Author(s) 2019

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

Keywords Besov-type space · Regular tree · Trace space · Dyadic norm

Mathematics Subject Classification (2010) 46E35 · 30L99

1 Introduction

Over the past two decades, analysis on general metric measure spaces has attracted a lot ofattention, e.g., [2, 4, 12, 13, 15–17]. Especially, the case of a regular tree and its Cantor-typeboundary has been studied in [3]. Furthermore, Sobolev spaces, Besov spaces and Triebel-Lizorkin spaces on metric measure spaces have been studied in [5, 25, 26] via hyperbolicfillings. A related approach was used in [23], where the trace results of Sobolev spaces andof related fractional smoothness function spaces were recovered by using a dyadic normand the Whitney extension operator.

Dyadic energy has also been used to study the regularity and modulus of continuity ofspace-filling curves. One of the motivations for this paper is the approach in [20]. Given acontinuous g : S1 → Rn, consider the dyadic energy

E(g;p, λ) :=+∞∑

i=1iλ

2i∑

j=1|gIi,j − gÎi,j |

p . (1.1)

! Zhuang Wangzhuang.z.wang@jyu.fi

Pekka Koskelapekka.j.koskela@jyu.fi

1 Department of Mathematics and Statistics, University of Jyväskylä, PO Box 35, FI-40014Jyväskylä, Finland

http://crossmark.crossref.org/dialog/?doi=10.1007/s11118-019-09808-5&domain=pdfhttp://orcid.org/0000-0002-1988-6927mailto:%20zhuang.z.wang@jyu.fimailto:%20pekka.j.koskela@jyu.fi

: Pubblicazioni Scienze publ_sci@sns.it: Paper submitted to Annali SNS: 2020 5 20 17:17: Wang, Zhuang zhuang.z.wang@jyu.fi

Dear Prof. Wang,

We are glad to inform you that your joint paper with Panu Lahti and Xining Li entitled:

Traces of Newtonian-Sobolev, Hajlasz-Sobolev, and BV functions on metric spaces,

submitted to the Annali della Scuola Normale Superiore di Pisa, Classe diScienze, has been accepted for publication.

Please, let us have the TeX/LaTeX source file, which will speed up theproofs setting.We kindly ask you to make sure that the style of your bibliography strictlyadheres to our typographical rules, available in the "Information forauthors" on the journal's home page or at the end of any of our publishedissues (in particular, the "Abbreviations of names of serials" ofMathematical Reviews should be employed when referring to papers).

With best regards,

Umberto Zannier

mailto:Scienzepubl_sci@sns.itmailto:Scienzepubl_sci@sns.itmailto:Zhuangzhuang.z.wang@jyu.fimailto:Zhuangzhuang.z.wang@jyu.fi

(X, d, µ)Tu u

Ω

limr→0+

∫

B(x,r)∩Ω|u− Tu(x)| dµ = 0

x ∈ ∂Ω H

L1

BV(Ω) N1,1(Ω) L1(∂Ω)