Weighted estimates for diffeomorphic extensions of homeomorphisms

TL;DR

This paper investigates how weighted integrability conditions on derivatives of diffeomorphic extensions relate to double integrals involving boundary homeomorphisms and their inverses in chord-arc domains.

Contribution

It establishes new connections between weighted derivative integrability of extensions and boundary double integrals for homeomorphisms in chord-arc domains.

Findings

Weighted integrability of derivatives is characterized by boundary double integrals.

Results apply to diffeomorphic extensions in internal chord-arc domains.

Provides new tools for analyzing boundary homeomorphisms and their extensions.

Abstract

Let $\Omega \subset \mbr^2$ be an internal chord-arc domain and $\varphi : \mbs^1 \rightarrow \partial \Omega$ be a homeomorphism. Then there is a diffeomorphic extension $h : \mbd \rightarrow \Omega$ of $\varphi .$ We study the relationship between weighted integrability of the derivatives of $h$ and double integrals of $\varphi$ and of $\varphi^{-1} .$

Equations

$${\int }_{\mathbb{D}}|Dh(z){|}^{2}dz\approx {\int }_{{\mathbb{S}}^1}{\int }_{{\mathbb{S}}^1}\frac{|\phi (\xi )-\phi (\eta ){|}^2}{|\xi -\eta {|}^2}|d\xi ||d\eta |,$$

$${\int }_{\mathbb{D}}|Dh(z){|}^{2}dz<\infty \iff {\int }_{\partial \Omega }{\int }_{\partial \Omega }|\log |{\phi }^{-1}(\xi )-{\phi }^{-1}(\eta )|||d\xi ||d\eta |<\infty .$$

$$I_1(p,\alpha ,\lambda ,h)={\int }_{\mathbb{D}}|Dh(z){|}^p{\delta }^{\alpha }(z){\log }^{\lambda }(2{\delta }^{-1}(z))dz,$$

$$I_2(p,\alpha ,\lambda ,h)={\int }_{\mathbb{D}}|Dh(z){|}^p{\log }^{\lambda }(e+|Dh(z)|){\delta }^{\alpha }(z)dz,$$

$$\mathcal{U}(p,\alpha,\lambda,\varphi)=\int_{\mathbb{S}^{1}}\int_{\mathbb{S}^{1}}\frac{\lambda^{p}_{\Omega}(\varphi(\xi),\varphi(\eta))}{|\xi-\eta|^{p-\alpha}}\log^{\lambda}\big{(}e+\frac{\lambda_{\Omega}(\varphi(\xi),\varphi(\eta))}{|\xi-\eta|}\big{)}\,|d\eta|\,|d\xi|,$$

$${\mathcal{A}}_{p,\alpha ,\lambda }(t)={\int }_1^t-x^{1+\alpha -p}{\log }_2^{\lambda }(x^{-1})dx\forall t\ge 0,$$

$$\mathcal{V}(p,\alpha,\lambda,\varphi)=\int_{\partial\Omega}\big{(}\int_{\partial\Omega}\mathcal{A}_{p,\alpha,\lambda}(|\varphi^{-1}(\xi)-\varphi^{-1}(\eta)|)\,|d\eta|\big{)}^{p-1}\,|d\xi|.$$

$$\text{mbox}both{I}_1(p,\alpha ,\lambda ,h)\text{mbox}and{I}_2(p,\alpha ,\lambda ,h)\text{mbox}arefinite.$$

$$\text{mbox}both{I}_1(p,\alpha ,\lambda ,h)\text{mbox}and{I}_2(p,\alpha ,\lambda ,h)\text{mbox}arecomparableto\mathcal{U}(p,\alpha ,\lambda ,\phi ).$$

$$\text{mbox}both{I}_1(p,\alpha ,\lambda ,h)\text{mbox}and{I}_2(p,\alpha ,\lambda ,h)\text{mbox}dominate\mathcal{V}(p,\alpha ,\lambda ,\phi ),$$

$$\mathcal{V}(p,\alpha ,\lambda ,\phi )\text{mbox}controlsboth{I}_1(p,\alpha ,\lambda ,h)\text{mbox}and{I}_2(p,\alpha ,\lambda ,h)$$

$$\ell (w_1,w_2)\le C{\lambda }_{\Omega }(w_1,w_2),$$

$${\lambda }_{\Omega }(w_1,w_2)=\underset{\alpha }{\inf }\ell (\alpha ),$$

$$\Delta =\{(x,y)\in {\mathbb{R}}^2:(x^2+y^2{)}^2-4x(x^2+y^2)-4y^2<0\}.$$

$$I_{j,k}=[2\pi (k-1)2^{-j},2\pi k{2}^{-j}],{\Gamma }_{j,k}=\{e^{i\theta }:\theta \in I_{j,k}\}.$$

$$\ell ({\Gamma }_{j,k})\approx 2^{-j}\forall j\in \mathbb{N}\text{mbox}andk=1,...,2^j.$$

$$Q_{j,k}=\left\{r{e}^{i\theta }:1-2^{1-j}\le r\le 1-2^{-j}\text{mbox}and\theta \in I_{j,k}\right\}.$$

$${\mathcal{L}}^2(Q_{j,k})\approx 2^{-2j}\approx \ell ({\Gamma }_{j,k}{)}^2\forall j\in \mathbb{N}\text{mbox}andk=1,...,2^j.$$

$$B_{j,k}\subset Q_{j,k}\subset C{B}_{j,k}.$$

$$\frac{1}{{\mathcal{L}}^{2}(B)}\int_{B}w(x)\,dx\leq C\big{(}\frac{1}{{\mathcal{L}}^{2}(B)}\int_{B}w(x)^{\frac{1}{1-p}}\,dx\big{)}^{1-p}.$$

$$\frac{1}{{\mathcal{L}}^2(B)}{\int }_{B}w(z)dz\le Cw(x)$$

$$w_{\alpha ,\lambda }(x)=\{\begin{array}{ll}\delta (x{)}^{\alpha }{\log }^{\lambda }\left(2{\delta }^{-1}(x)\right)& 0\le |x|\le 2,\\ {\log }^{\lambda }(2)& |x|\ge 2.\end{array}$$

$$w_1(x)=\{\begin{array}{ll}\delta (x{)}^{{\alpha }_1}{\log }^{{\lambda }_1}\left(2{\delta }^{-1}(x)\right)& 0\le |x|\le 2,\\ {\log }^{{\lambda }_1}(2)& |x|\ge 2,\end{array}$$

$$w_2(x)=\{\begin{array}{ll}\delta (x{)}^{{\alpha }_2}{\log }^{{\lambda }_2}\left(2{\delta }^{-1}(x)\right)& 0\le |x|\le 2,\\ {\log }^{{\lambda }_2}(2)& |x|\ge 2.\end{array}$$

$$\displaystyle\mathop{}\mkern-3.0mu\mathchoice{\hbox to0.0pt{$\displaystyle\vbox{\hbox to5.83331pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox to0.0pt{$\textstyle\vbox{\hbox to5.83331pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox to0.0pt{$\scriptstyle\vbox{\hbox to4.5833pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox to0.0pt{$\scriptscriptstyle\vbox{\hbox to4.0434pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B}w_{1}(x)dx\lesssim\underset{x\in B}{\mbox{inf}}w_{1}(x)$$

$$d_B\le \delta (x)\le 3d_B\forall x\in B.$$

$$w_1(x)={\log }^{{\lambda }_1}(2)\text{mbox}wheneverx\in B_2.$$

$${\left[t^{{\alpha }_1}{\log }^{{\lambda }_1}\left(2t^{-1}\right)\right]}^{\prime }=t^{{\alpha }_1-1}{\log }^{{\lambda }_1}(2t^{-1})\left({\alpha }_1-\frac{{\lambda }_1}{\log (2t^{-1})}\right)<0,$$

$$w_1(x)\le d_B^{{\alpha }_1}{\log }^{{\lambda }_1}(2d_B^{-1})\le \frac{{\log }^{{\lambda }_1}(6)}{3^{{\alpha }_1}}\forall x\in B_1.$$

$$\displaystyle\displaystyle\mathop{}\mkern-3.0mu\mathchoice{\hbox to0.0pt{$\displaystyle\vbox{\hbox to5.83331pt{\hss$\textstyle{-}$\hss}}$\hss}}{\hbox to0.0pt{$\textstyle\vbox{\hbox to5.83331pt{\hss$\scriptstyle{-}$\hss}}$\hss}}{\hbox to0.0pt{$\scriptstyle\vbox{\hbox to4.5833pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}{\hbox to0.0pt{$\scriptscriptstyle\vbox{\hbox to4.0434pt{\hss$\scriptscriptstyle{-}$\hss}}$\hss}}\mkern-3.0mu\int_{B}w_{1}(x)dx$$

Full text

Keywords: Poisson extension, diffeomorphism, internal chord-arc domain.

1. Introduction

Let $\Omega\subset{\mathbb{R}}^{2}$ be a bounded convex domain. Suppose that $\varphi$ is a homeomorphism from the unit circle $\mathbb{S}^{1}$ onto $\partial\Omega.$ Then, by Radó [12], Kneser [7], Choquet [3] and Lewy [9], the complex-valued Poisson extension $h$ of $\varphi$ is a diffeomorphism from $\mathbb{D}$ onto $\Omega$. We are interested in the integrability degrees of the derivatives of $h.$ In $2007$, G. C. Verchota [13] proved that the derivatives of $h$ may fail to be square integrable but that they are necessarily $p$-integrable over $\mathbb{D}$ for all $p<2$. In $2009$, T. Iwaniec, G. J. Martin and C. Sbordone improved on [5] by showing that the derivatives belong to weak-$L^{2}$ with sharp estimates. Actually

[TABLE]

since harmonic functions minimize the $L^{2}$-energy and the right-hand side of (1.1) is the trace norm of $\dot{W}^{1,2}({\mathbb{D}}).$ In [1], it was further shown that if additionally $\partial\Omega$ is a $C^{1}$-regular Jordan curve then

[TABLE]

All the above results require the target domain to be convex.

If $\Omega$ is a bounded non-convex Jordan domain, then there exists a homeomorphism $\varphi:{\mathbb{S}}^{1}\rightarrow\partial\Omega$ for which the harmonic extension fails to map ${\mathbb{D}}$ homeomorphically onto $\Omega,$ see [7, 3]. Hence we cannot use the harmonic extension to produce a diffeomorphic extension. Nevertheless, (weighted) analogs of the results as (1.2) for diffeomorphic extensions in the case of an internal chord-arc Jordan domain exist, see [8]. For the definition of (internal) chord-arc domains, we refer to Definition 2.1. Notice that each bounded convex Jordan domain is a chord-arc domain. In this paper, we generalize the results in [8] to the weighted $L^{p}$-setting.

Let $\Omega$ be an internal chord-arc Jordan domain with the internal distance $\lambda_{\Omega}.$ Assume that $h:{\mathbb{D}}\rightarrow\Omega$ is a diffeomorphism and $\varphi:{\mathbb{S}}^{1}\rightarrow\partial\Omega$ is a homeomorphism. Set $\delta(z)=1-|z|.$ Given $p>1,\alpha\in\mathbb{R},\lambda\in\mathbb{R},$ we define

[TABLE]

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[TABLE]

[TABLE]

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Our main result is the following theorem.

Theorem 1.1**.**

Let $\Omega\subset{\mathbb{R}}^{2}$ be an internal chord-arc Jordan domain and $\varphi:\mathbb{S}^{1}\rightarrow\partial\Omega$ be a homeomorphism. There is a diffeomorphic extension $h:\mathbb{D}\rightarrow\Omega$ of $\varphi$ for which, for any $p>1,$ we have that

1. (1)

if either $\alpha\in(p-2,+\infty)$ and $\lambda\in\mathbb{R}$ or $\alpha=p-2$ and $\lambda\in(-\infty,-1),$

[TABLE] 2. (2)

if either $\alpha\in(-1,p-2)$ and $\lambda\in\mathbb{R}$ or $\alpha=p-2$ and $\lambda\in[-1,+\infty),$

[TABLE]

Moreover whenever $p\in(1,2]$

[TABLE]

while

[TABLE]

for all $p\in[2,+\infty).$ Furthermore both $I_{1}(p,\alpha,\lambda,h)$ and $I_{2}(p,\alpha,\lambda,h)$ are in general comparable to $\mathcal{V}(p,\alpha,\lambda,\varphi)$ only for $p=2.$

*For any $p>1,$ there is no a homeomorphic extension $h:{\mathbb{D}}\rightarrow\Omega$ of $\varphi$ for which $I_{1}(p,\alpha,\lambda,h)<+\infty$ for either $\alpha\in(-\infty,-1)$ and $\lambda\in\mathbb{R}$ or $\alpha=-1$ and $\lambda\in[-1,+\infty);$ and for which $I_{2}(p,\alpha,\lambda,h)<+\infty$ for all $\alpha\in(-\infty,-1]$ and each $\lambda\in\mathbb{R}.$ *

Motivated by (1.2), one could hope to use $\mathcal{V}(p,\alpha,\lambda,\varphi)$ to control both $I_{1}(p,\alpha,\lambda,h)$ and $I_{2}(p,\alpha,\lambda,h).$ Example 4.2 together with Example 4.3 shows that $\mathcal{V}(p,\alpha,\lambda,\varphi)$ is comparable to $I_{1}(p,\alpha,\lambda,h)$ or $I_{2}(p,\alpha,\lambda,h)$ only when $p=2.$ Theorem 1.1 does not cover the case where $p>1$, $\alpha=-1$ and $\lambda\in(-\infty,-1).$ We will return to this case in a future paper.

The structure of this paper is the following. In the next section, we give some preliminaries. Section $3$ is the proof of Theorem 1.1. The final section contains several examples related to Theorem 1.1 ($2$).

2. Preliminaries

By $s\gg 1$ and $t\ll 1$ we mean that $s$ is sufficiently large and $t$ is sufficiently small, respectively. By $f\lesssim g$ we mean that there exists a constant $C>0$ such that $f(x)\leq Cg(x)$ for every $x$. If $f\lesssim g$ and $g\lesssim f$ we may denote $f\approx g$. By $\mathbb{N}$ and $\mathbb{R}$ we denote the set of all positive integers and the set of all real numbers. Let $\mathcal{L}^{2}$ (respectively $\mathcal{L}^{1}$) be the $2$-dimensional ($1$-dimensional) Lebesgue measure. For sets $E\in\mathbb{R}^{2}$ and $F\in\mathbb{R}^{2},$ let ${\rm diam}\,(E)$ be the diameter of $E,$ and ${\rm dist}\,(E,F)$ be the Euclidean distance between $E$ and $F.$ Let $B(p,r)$ be the disk with center $P$ and radius $r.$

Definition 2.1**.**

A Jordan domain $\Omega\subset{\mathbb{R}}^{2}$ is an internal chord-arc Jordan domain if $\partial\Omega$ is rectifiable and there is a constant $C>0$ such that for all $w_{1},w_{2}\in\partial\Omega$,

[TABLE]

where $\ell(w_{1},w_{2})$ is the arc length of the shorter arc of $\partial\Omega$ joining $w_{1}$ to $w_{2}$, and $\lambda_{\Omega}(w_{1},w_{2})$ is the internal distance between $w_{1},w_{2}$, which is defined as

[TABLE]

where the infimum is taken over all rectifiable arcs $\alpha\subset\Omega$ joining $w_{1}$ and $w_{2}$; if there is no rectifiable curve joining $w_{1}$ and $w_{2}$, we set ${\lambda_{\Omega}(w_{1},w_{2})}=\infty$; cf. [11, Section 3.1] or [2, Section 2].

If (2.1) holds for the Euclidean distance instead of the internal distance, we call $\Omega$ be a chord-arc domain. Naturally, every chord-arc Jordan domain is an internal chord-arc domain, but there are internal chord-arc domains that fail to be chord-arc; e.g. the standard cardioid domain

[TABLE]

2.1. Dyadic decomposition

Given $j\in\mathbb{N}$ and $k=1,...,2^{j},$ let

[TABLE]

Then $\{I_{j,k}\}$ is a dyadic decomposition of $[0,2\pi]$ and $\{\Gamma_{j,k}\}$ is a dyadic decomposition of ${\mathbb{S}}^{1}.$ We call $\Gamma_{j,k}$ a $j$-level dyadic arc. Moreover we have that

[TABLE]

Based on (2.2), there is a decomposition of the unit disk $\mathbb{D}$ given by $\{Q_{j,k}:j\in\mathbb{N}\mbox{ and }k=1,...,2^{j}\},$ where

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By (2.3) it follows that

[TABLE]

Moreover there is a uniform constant $C>0$ such that for any $Q_{j,k}$ there is a disk $B_{j,k}$ satisfying

[TABLE]

2.2. $A_{p}$ weights

Definition 2.2**.**

For a given $p\in(1,+\infty),$ a locally integrable function $w:\mathbb{R}^{2}\rightarrow[0,+\infty)$ is an $A_{p}$ weight if there is a constant $C>0$ such that for any disk $B\subset\mathbb{R}^{2}$ we have that

[TABLE]

Next, $w$ is an $A_{1}$ weight if there is a constant $C>0$ such that

[TABLE]

for each disk $B\subset\mathbb{R}^{2}$ and all $x\in B.$

For more information on $A_{p}$ weights, we recommend [4, 10, 6]. Let $\delta(x)=\mbox{dist}(\mathbb{S}^{1},x).$ Given $\alpha\in(-1,p-1)$ and $\lambda\in\mathbb{R},$ we define

[TABLE]

It is well known that $w_{\alpha,0}$ belongs to $A_{p}.$ We now generalize this to all $\lambda\in\mathbb{R}.$

Proposition 2.3**.**

Let $p\geq 1$ and $w_{\alpha,\lambda}$ be as in (2.7). Then $w_{\alpha,\lambda}$ is an $A_{p}$ weight for all $\alpha\in(-1,p-1)$ and $\lambda\in\mathbb{R}.$

Proof.

The idea of proof is to use the Jones factorization of $A_{p}$ weights (see [6]), i.e. we should prove $w_{\alpha,\lambda}=w_{1}w_{2}^{1-p}$ for two $A_{1}$ weights $w_{1}$ and $w_{2}.$

We first consider the case $\lambda\geq 0.$ For a given $\alpha\in(-1,p-1),$ there unique exist $a_{1}\in(0,1)$ and $a_{2}\in(0,1)$ such that $\alpha=a_{1}(-1)+a_{2}(p-1).$ Set $\alpha_{1}=-a_{1}$, $\alpha_{2}=-a_{2}$, $\lambda_{1}=p\lambda$ and $\lambda_{2}=\lambda.$ We define

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and

[TABLE]

We next prove that $w_{1}$ is an $A_{1}$ weight, i.e.

[TABLE]

for all disk $B\subset\mathbb{R}^{2}.$ Let $d_{B}=\mbox{dist}(B,\mathbb{S}^{1}).$

Case 1: $d_{B}\geq\mbox{diam}(B)/2.$ We have that

[TABLE]

If $1\leq d_{B},$ then $\delta(x)\geq 1$ for all $x\in B.$ Therefore $w_{1}(x)=\log^{\lambda_{1}}(2)$ whenever $x\in B.$ Of course (2.10) holds now. If $3d_{B}\leq 1,$ then $w_{1}(x)=\delta(x)^{\alpha_{1}}\log^{\lambda_{1}}\left(2\delta^{-1}(x)\right)$ for all $x\in B.$ By (2.11) it hence follows that $w_{1}(x)\approx d^{\alpha_{1}}_{B}\log^{\lambda_{1}}\left(2d^{-1}_{B}\right)$ whenever $x\in B.$ Therefore (2.10) holds. If $d_{B}<1<3d_{B},$ let $B_{1}=\{x\in B:d_{B}<\delta(x)<1\}$ and $B_{2}=\{x\in B:1\leq\delta(x)<3d_{B}\}.$ Then $B=B_{1}\cup B_{2}$ and

[TABLE]

Since

[TABLE]

for all $t\in(0,1],$ we have that

[TABLE]

Combining (2.12) and (2.14) implies that

[TABLE]

Case 2: $d_{B}<\mbox{diam}(B)/2$ and $\mbox{diam}(B)\leq 2/3.$ Pick $x^{\prime}\in\partial B$ and $x_{0}\in\mathbb{S}^{1}$ such that $\mbox{dist}(B,\mathbb{S}^{1})=|x^{\prime}-x_{0}|.$ Let $r_{B}=3\mbox{diam}(B)/2.$ Since

[TABLE]

for all $x\in B,$ we have $B\subset B(x_{0},r_{B}).$ Let $E=\{x\in\mathbb{R}^{2}:\mbox{dist}(x,\mathbb{S}^{1})<r_{B}\}.$ Then $B(x_{0},r_{B})\subset E.$ Since ${\mathcal{L}}^{2}(B(x_{0},r_{B}))=\pi r^{2}_{B}$ and ${\mathcal{L}}^{2}(E)=4\pi r_{B},$ the maximal number of pairwise disjoint open disks $B(x,r_{B})$ with $x\in\mathbb{S}^{1}$ is less than $4r^{-1}_{B}.$ We have that

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Notice that

[TABLE]

Since $\underset{t\rightarrow 0^{+}}{\mbox{lim}}\alpha_{1}+1-\frac{\lambda_{1}}{\log\left(2t^{-1}\right)}=\alpha_{1}+1$ and $\alpha_{1}+1-\frac{\lambda_{1}}{\log\left(2t^{-1}\right)}$ is decreasing with respect to $t>0,$ there exists $\epsilon\in(0,1)$ determined by $\alpha_{1}$ and $\lambda_{1}$ such that $\alpha_{1}+1-\frac{\lambda_{1}}{\log(2\epsilon^{-1})}\geq(\alpha_{1}+1)/2.$ We then obtain from (2.16) that

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for all $t\in[0,\epsilon r_{B}].$ Therefore

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Moreover by (2.13) we have that

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Combining (2.2), (2.17) with (2.18) implies that

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Together with

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we hence obtain (2.10).

Case 3: $d_{B}<\mbox{diam}(B)/2$ and $\mbox{diam}(B)>2/3.$ Let $x^{\prime}$ and $x_{0}$ be as in Case $2.$ Then $|x^{\prime}|=1+\mbox{dist}(x^{\prime},\mathbb{S}^{1})\leq 1+\mbox{diam}(B)2^{-1}.$ Together with the fact that $|x-x^{\prime}|\leq\mbox{diam}(B)$ for all $x\in B,$ we have $B\subset B(0,1+r_{B}).$ Moreover by (2.17) and (2.18), we obtain that

[TABLE]

Therefore

[TABLE]

Moreover by the monotonicity of $t^{\alpha_{1}}\log^{\lambda_{1}}(2t^{-1})$ on $(0,+\infty),$ we have that

[TABLE]

By combining (2.2) with (2.20), we obtain (2.10).

By the analogous arguments as for (2.10), we obtain $w_{2}\in A_{1}.$ Therefore the Jones factorization theorem implies that $w_{\alpha,\lambda}\in A_{p}$ for all $\alpha\in(-1,p-1)$ and $\lambda\geq 0.$

When $\lambda<0,$ define $w_{1}$ and $w_{2}$ as in (2.8) and (2.9) with $\lambda_{1}=-\lambda,\ \lambda_{2}=2\lambda(1-p)^{-1}$ and both $\alpha_{1}$ and $\alpha_{2}$ invariant. By the same arguments as for the case $\lambda\geq 0,$ we obtain that $w_{\alpha,\lambda}\in A_{p}$ whenever $\alpha\in(-1,p-1)$ and $\lambda<0.$ ∎

2.3. A class of functions

Definition 2.4**.**

We say that a function $f:[0,+\infty)\rightarrow[0,+\infty)$ satisfies the $\Delta_{2}$-condition if there is a constant $C>0$ such that

[TABLE]

for all $t\in[0,+\infty).$

Given $p>1$ and $\lambda\in\mathbb{R},$ set

[TABLE]

Proposition 2.5**.**

Let $\Phi_{p,\lambda}$ be as in (2.21) with $p>1$ and $\lambda\geq 0.$ Then

1. (P-1)

$\Phi_{p,\lambda}(t)$* is strictly increasing, continuous and convex on $[0,+\infty),$* 2. (P-2)

$\Phi_{p,\lambda}(t)$* satisfies the $\Delta_{2}$-condition on $[0,+\infty),$* 3. (P-3)

there is a constant $C>0$ such that $\Phi_{p,\lambda}^{\prime}(t)\leq C\Phi_{p,\lambda}(t)/t$ for all $t\in(0,+\infty),$ 4. (P-4)

there is a constant $C>0$ such that $\Phi_{p,\lambda}(st)\leq Cs^{r}\Phi_{p,\lambda}(t)$ holds for all $s\in[0,1],\ t\in[0,+\infty)$ and each $r\in(0,p).$

Proof.

Simple calculations show that

[TABLE]

and

[TABLE]

where $R_{p,\lambda}(t)=\lambda(\lambda-1)(t(e+t)^{-1})^{2}+\lambda(2p-1)t(e+t)^{-1}\log(e+t).$ Since $p\log(e+t)+\lambda t(e+t)^{-1}>0$ for all $t\in(0,+\infty),$ it follows from (2.22) that $\Phi_{p,\lambda}^{\prime}(t)>0$ for all $t>0.$ Therefore $\Phi_{p,\lambda}$ is strictly increasing on $[0,\infty).$ If $\lambda\geq 1,$ we have that

[TABLE]

Whenever $0\leq\lambda<1,$ since $t/(e+t)<1$ and $\log(e+t)\geq 1$ for all $t\geq 0$ we have that

[TABLE]

for all $t\geq 0.$ By (2.3), (2.24) and (2.3), we have that $\Phi^{\prime\prime}_{p,\lambda}(t)\geq 0$ for all $t\geq 0.$ Therefore $\Phi_{p,\lambda}$ is convex on $[0,+\infty).$

Since both $t^{p}$ and $\log^{\lambda}(e+t)$ satisfy the $\Delta_{2}$-condition on $[0,+\infty),$ (P-2) then holds.

Since $p\log(e+t)+\lambda t(e+t)^{-1}\leq(p+\lambda)\log(e+t)$ for all $t\in[0,+\infty),$ from (2.22) we obtain that

[TABLE]

for all $t>0.$ Hence (P-3) holds.

In order to prove (P-4), it suffices to prove that $s^{p}\log^{\lambda}(e+st)\leq s^{r}\log^{\lambda}(e+t)$ for all $s\in[0,1],\ t\in[0,+\infty)$ and each $r\in(0,p).$ In fact, for any $0\leq s\leq 1$ and $t\geq 0,$ we have that $\log^{\lambda}(e+st)\leq\log^{\lambda}(e+t).$ Hence for any $r\in(0,p),$ it follows that $s^{p}\log^{\lambda}(e+st)\leq s^{r}\log^{\lambda}(e+t)$ for all $s\in[0,1]$ and $t\in[0,+\infty).$ ∎

We define the Hardy-Littlewood maximal function for a Lebesgue measurable function $f$ in $\mathbb{R}^{2}$ as

[TABLE]

where the supremum is taken over all disks $B\subset\mathbb{R}^{2}$ containing $x.$ The following lemma shows the value of Proposition 2.5.

Lemma 2.6**.**

For $p>1,$ let $w$ be an $A_{p}$ weight and $\Phi_{p,\lambda}(t)$ be as in (2.21) with $\lambda\geq 0.$ Given a Lebesgue measurable function $f,$ we have that

[TABLE]

Proof.

By the open-end property of $A_{p}$ weights, we have that $w$ is an $A_{r_{0}}$ weight for some $r_{o}<p.$ For $t\in[0,+\infty),$ set $g_{t}(x)=|f(x)|\chi_{\{x\in{\mathbb{R}}^{2}:2|f(x)|>t\}}.$ Muckenhoupt’s theorem implies that

[TABLE]

for all $t\geq 0.$ Moreover, by Chebyshev’s inequality we obtain that

[TABLE]

Since $\{x\in{\mathbb{R}}^{2}:M_{f}(x)>t\}\subset\{x\in{\mathbb{R}}^{2}:2M_{g_{t}}(x)>t\}$ for all $t>0,$ we obtain that

[TABLE]

Combining (2.26), (2.3) with (2.28) gives that

[TABLE]

for all $t>0.$ Notice that $\Phi_{p,\lambda}(0)=0.$ By Fubini’s theorem and (P-3) in Proposition 2.5, we derive from (2.29) that

[TABLE]

Moreover by Fubini’s theorem, a change of variables, and (P-2) and (P-4) in Proposition 2.5, there is $r\in(r_{0},p)$ such that

[TABLE]

Combining (2.3) with (2.3) implies that

[TABLE]

∎

Let $\Phi_{p,\lambda}$ be as in (2.21) with $p>1$ and $\lambda<0.$ By (2.22) and (2.3), we have that both monotonicity and convexity of $\Phi_{p,\lambda}$ may fail whenever $t\ll 1,$ but still hold for all $t\gg 1.$ We modify $\Phi_{p,\lambda}$ in a neighborhood of the origin so as to ensure (P-1)-(P-4) of Proposition 2.5.

Since $2^{-1}(p+1)\log(e+t)\leq p\log(e+t)+\lambda t(e+t)^{-1}$ whenever $t\gg 1,$ by (2.22) there is a constant $t_{2}\gg 1$ such that

[TABLE]

for all $t\geq t_{2}.$ Without loss of generality, we assume that $\Phi_{p,\lambda}$ is strictly increasing and convex on $[t_{2},\infty).$ Since $pt_{2}t^{p-1}+t^{p}\leq(p+1)t_{2}t^{p-1}\leq t^{p}_{2}\log^{\lambda}(e+t_{2})$ for any $t\ll 1,$ we have that

[TABLE]

for all $t\ll 1.$ Moreover when $t\leq t_{2}(p-1)/(p+1),$ we have that

[TABLE]

Therefore by (2.32), (2.33) and (2.34), there exists a constant $t_{1}\ll 1$ such that

[TABLE]

Let $k=(\Phi_{p,\lambda}(t_{2})-t^{p}_{1})/(t_{2}-t_{1}).$ Given $p>1$ and $\lambda<0,$ we define

[TABLE]

Proposition 2.7**.**

The function $\Psi_{p,\lambda}$ enjoys the four properties in Proposition 2.5.

Proof.

It is easy to see that $\Psi_{p,\lambda}$ is strictly increasing, continuous and convex on $[0,+\infty).$ In order to prove (P-2) for $\Psi_{p,\lambda}$, it suffices to prove that

[TABLE]

for all $t\in[0,+\infty).$ (2.36) is trivial if either $t\geq t_{2}$ or $2t<t_{1}.$ Whenever $t\in[t_{1}/2,t_{2}],$ by the monotonicity of $\Psi_{p,\lambda}$ we have that

[TABLE]

We next prove (P-3) for $\Psi_{p,\lambda}.$ For any $t\in(0,t_{1}),$ we have that

[TABLE]

Since $p\log(e+t)+\lambda t(e+t)^{-1}\leq p\log(e+t)$ for all $t\in[t_{2},+\infty),$ from (2.22) we have that

[TABLE]

for all $t\in[t_{2},+\infty).$ Moreover, since $t^{p}_{1}-t_{1}k<0$ it follows from (2.35) that

[TABLE]

for all $t\in[t_{1},t_{2}).$ By (2.37), (2.38) and (2.39), we finish (P-3) for $\Psi_{p,\lambda}.$

We next prove (P-4). Let $s\in(0,t_{1}/t_{2}).$ If $t\in(0,t_{1}),$ from (2.35) we have that

[TABLE]

If $t\in[t_{1},t_{2}),$ from (2.35) we have that

[TABLE]

Whenever $t\in[t_{2},t_{1}/s),$ it follows from (2.35) that

[TABLE]

If $t\in[t_{1}/s,t_{2}/s),$ we obtain from (2.35) that

[TABLE]

For any $t\in[t_{2}/s,+\infty),$ from (2.35) we have that

[TABLE]

Moreover by the monotonicity of function $(\log(s)+\log(e+\cdot))\log^{-1}(e+\cdot),$ it follows that

[TABLE]

for all $t\geq t_{2}/s.$ Hence we derive from (2.44) that

[TABLE]

for all $t\in[t_{2}/s,+\infty).$ Combining (2.40), (2.41), (2.42), (2.43) with (2.45) implies that there is a constant $C>0$ such that

[TABLE]

for all $t\in(0,+\infty)$ and $0<s<t_{1}/t_{2}.$ For a given $r\in(0,p),$ since $C\frac{s^{p}}{\log^{\lambda}(e+t_{2}s^{-1})}\leq s^{r}$ for all $s\ll 1,$ it follows from (2.46) that there is $s_{0}>0$ such that

[TABLE]

for all $t\in[0,+\infty)$ and $s\in[0,s_{0}).$

Whenever $s\in[s_{0},1],$ by the monotonicity of $\Psi_{p,\lambda}$ we have that

[TABLE]

for all $t\in[0,+\infty)$ and $r>0.$ Combining (2.47) with (2.48) concludes that there is a constant $C>0$ such that

[TABLE]

for all $t\in[0,+\infty)$ $s\in[0,1]$ and $r\in(0,p).$ ∎

Remark 2.8*.*

For $p>1$ and $\lambda<0,$ let $\Psi_{p,\lambda}$ be as in (2.35) and $\Phi_{p,\lambda}$ be as in (2.21). Under the same assumptions as in Lemma 2.6, we have that

[TABLE]

Since $\lim_{t\rightarrow 0+}\Psi_{p,\lambda}(t)/\Phi_{p,\lambda}(t)=1,$ it follows that

[TABLE]

whenever $t\in[0,+\infty).$ Hence we derive from (2.49) that

[TABLE]

3. Proof of Theorem 1.1

We begin by proving the following special case of Theorem 1.1.

Theorem 3.1**.**

Let $\varphi:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ be a homeomorphism, and $h=P[\varphi]:\mathbb{D}\rightarrow\mathbb{D}$ be the harmonic extension of $\varphi.$ For any $p>1,$ we have that

1. (1)

if either $\alpha\in(p-2,+\infty)$ and $\lambda\in\mathbb{R},$ or $\alpha=p-2$ and $\lambda\in(-\infty,-1),$

[TABLE] 2. (2)

if either $\alpha\in(-1,p-2)$ and $\lambda\in\mathbb{R},$ or $\alpha=p-2$ and $\lambda\in[-1,+\infty),$ then

[TABLE]

Moreover whenever $p\in(1,2]$

[TABLE]

while

[TABLE]

for all $p\in[2,+\infty).$ Furthermore both $I_{1}(p,\alpha,\lambda,h)$ and $I_{2}(p,\alpha,\lambda,h)$ are in general comparable to $\mathcal{V}(p,\alpha,\lambda,\varphi)$ only for $p=2.$ 3. (3)

if either $\alpha\in(-\infty,-1)$ and $\lambda\in\mathbb{R},$ or $\alpha=-1$ and $\lambda\in[-1,+\infty),$ we have that $I_{1}(p,\alpha,\lambda,h)=\infty$. While $I_{2}(p,\alpha,\lambda,h)=\infty$ for all $\alpha\in(-\infty,-1]$ and $\lambda\in\mathbb{R}.$

Let $\varphi:{\mathbb{S}}^{1}\rightarrow{\mathbb{S}}^{1}$ be a homeomorphism. Given $p>1,\alpha\in\mathbb{R},\lambda\in\mathbb{R},$ we define

[TABLE]

and

[TABLE]

where $\Phi_{p,\lambda}(t)$ is from (2.21).

Lemma 3.2**.**

Let $\varphi:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ be a homeomorphism. For any $p>1,\alpha\in(-1,+\infty)$ and every $\lambda\in\mathbb{R},$ the dyadic energy $\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ and $\mathcal{E}_{2}(p,\alpha,\lambda,\varphi)$ are equivalent.

Proof.

We first consider the case $\lambda\geq 0$. Let $\Phi_{p,\lambda}$ be as in (2.21). Since $\ell(\varphi(\Gamma_{j,k}))\leq 2\pi$ and $\ell(\Gamma_{j,k})\approx 2^{-j}$ for all $j\in\mathbb{N}$ and $k\in\{1,...,2^{j}\},$ by the monotonicity and $\Delta_{2}$-property of the standard logarithm we have that

[TABLE]

Hence

[TABLE]

Given $p>1$ and $\alpha\in(-1,+\infty),$ there is $\beta\in(0,1)$ such that $\alpha>(1-\beta)p-1>-1.$ Define

[TABLE]

We decompose $\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ as

[TABLE]

Whenever $\ell(\varphi(\Gamma_{j,k}))\geq 2^{-j\beta},$ by (2.3) we have $j^{\lambda}\lesssim\log^{\lambda}\left(e+\ell(\varphi(\Gamma_{j,k}))\ell(\Gamma_{j,k})^{-1}\right).$ Therefore

[TABLE]

Moreover, by (2.3) we have that

[TABLE]

We conclude from (3), (3) and (3.4) that there is a constant $C>0$ such that

[TABLE]

From (3.1) and (3.5) it follows that

[TABLE]

Analogously to (3.6), we have that $\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ and $\mathcal{E}_{2}(p,\alpha,\lambda,\varphi)$ are comparable whenever $\lambda<0.$ ∎

Lemma 3.3**.**

Let $\varphi:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ be a homeomorphism, and $h=P[\varphi]:\mathbb{D}\rightarrow\mathbb{D}$ be the Poisson homeomorphic extension of $\varphi.$ For any $p>1$, we have that $I_{1}(p,\alpha,\lambda,h)\lesssim\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ whenever $\alpha\in(-1,+\infty)$ and $\lambda\in\mathbb{R},$ while $\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ is controlled by $I_{1}(p,\alpha,\lambda,h)$ for all $\alpha\in(-1,p-1)$ and $\lambda\in\mathbb{R}.$

Proof.

We first prove that $I_{1}(p,\alpha,\lambda,h)\lesssim\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ for all $\alpha>-1$ and all $\lambda\in\mathbb{R}.$ Let $w_{\alpha,\lambda}$ be as in (2.7). For any $j\in\mathbb{N}$ and $1\leq k\leq 2^{j},$ by (2.4) and (2.3) we have that

[TABLE]

for all $z\in Q_{j,k}.$ Hence

[TABLE]

Let $\mathcal{P}(\Gamma_{j,k})$ be the technical decomposition of ${\mathbb{S}}^{1}$ based on $\Gamma_{j,k}$ in [8, Section 2.1]. As shown in [8, Proof (iv) $\Rightarrow$ (i)], for any $j\in\mathbb{N}$ and $k=1,...,2^{j}$ we have that

[TABLE]

for all $z\in Q_{j,k}.$ Here $\Gamma_{n,m}\in\mathcal{P}(\Gamma_{j,k})$ and $\sharp i_{n}\leq 3$ for all $n\leq j,$ see [8, Section 2.1]. Let $\alpha>-1.$ There is $q_{0}>1$ such that $p/q_{0}-1-\alpha<0.$ Denote by $p_{0}$ the exponent conjugate to $q_{0}.$ Via Hölder’s inequality we derive from (3.9) that

[TABLE]

for all $z\in Q_{j,k}.$ By (2.5), (3.8) and (3) we have that

[TABLE]

Moreover given a dyadic arc $\Gamma_{n,m},$ for any $j\geq n$ it is shown in [8, Section 2.1] that

[TABLE]

From Fubini’s theorem and (3.12) we obtain that

[TABLE]

Moreover when $p/q_{0}-1-\alpha<0$ we have that

[TABLE]

By (3.11), (3), (3.14) and (2.3), we conclude that

[TABLE]

We next prove that $\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ is controlled by $I_{1}(p,\alpha,\lambda,h)$ for all $\alpha\in(-1,p-1)$ and $\lambda\in\mathbb{R}.$ By [8, ($3.17$)], there is $j_{0}>1$ such that

[TABLE]

for all $j\geq j_{0}$ and $k\in\{1,...,2^{j}\}.$ Set $H(z)=|Dh(z)|\chi_{\mathbb{D}}(z).$ By (2.6) we have that

[TABLE]

where the last inequality comes from the fact that \displaystyle\mathop{}\mkern-3.0mu\mathchoice{\hbox to0.0pt{\displaystyle\vbox{\hbox to22.74162pt{\hss$\textstyle{-}$\hss}}\hss}}{\hbox to0.0pt{\textstyle\vbox{\hbox to22.74162pt{\hss$\scriptstyle{-}$\hss}}\hss}}{\hbox to0.0pt{\scriptstyle\vbox{\hbox to17.97928pt{\hss$\scriptscriptstyle{-}$\hss}}\hss}}{\hbox to0.0pt{\scriptscriptstyle\vbox{\hbox to17.97928pt{\hss$\scriptscriptstyle{-}$\hss}}\hss}}\mkern-3.0mu\int_{CC^{\prime}B_{j,k}}H(z)dz\leq M_{H}(w) for all $w\in Q_{j,k}.$ Combining (3.15) with (3.16) implies that

[TABLE]

for all $j\geq j_{0}$ and $k=1,...,2^{j}.$ By Jensen’s inequality and (2.5), we deduce from (3.17) that

[TABLE]

By (3.7) and (3.18), there is then a constant $C>0$ such that

[TABLE]

Moreover, for any $\alpha\in(-1,p-1)$ and $\lambda\in\mathbb{R},$ from Proposition 2.3 and Lemma 2.6 it follows that

[TABLE]

By (3.19) and (3.20) we conclude that $\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ is controlled by $I_{1}(p,\alpha,\lambda,h)$ for all $\alpha\in(-1,p-1)$ and $\lambda\in\mathbb{R}.$ ∎

Lemma 3.4**.**

Let $\varphi:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ be a homeomorphism, and $h=P[\varphi]:\mathbb{D}\rightarrow\mathbb{D}$ be the Poisson homeomorphic extension of $\varphi.$ For any $p>1$, we have that $I_{2}(p,\alpha,\lambda,h)\lesssim\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ whenever $\alpha\in(-1,+\infty)$ and $\lambda\in\mathbb{R},$ while $\mathcal{E}_{2}(p,\alpha,\lambda,\varphi)$ is controlled by $I_{2}(p,\alpha,\lambda,h)$ for all $\alpha\in(-1,p-1)$ and $\lambda\in\mathbb{R}.$

Proof.

We first consider that case $\lambda\geq 0.$ Let $\Phi_{p,\lambda}$ be as in (2.21). As (P-1) and (P-2) in Proposition 2.5 show that $\Phi_{p,\lambda}(t)$ is increasing and satisfies $\Delta_{2}$-property on $[0,+\infty).$ From (3.7) and (3.9) we have that

[TABLE]

Moreover since $\ell({\varphi(\Gamma_{n,m})})\leq 2\pi$ for all $n\in\mathbb{N}$ and $m=1,...,2^{n}$, it follows that

[TABLE]

for any $j\geq 1.$ Therefore

[TABLE]

for all $j\geq 1.$ By (3) and (3.22) we obtain that

[TABLE]

The analogous arguments as for $I_{1}(p,\alpha,\lambda,h)\lesssim\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ in Lemma 3.3 imply that

[TABLE]

We conclude from (3.23) and (3.24) that $I_{2}(p,\alpha,\lambda,h)\lesssim\mathcal{E}_{1}(p,\alpha,\lambda,\varphi).$

Applying $\Phi_{p,\lambda}$ to the both sides of (3.17), via (P-1) and (P-2) in Proposition 2.5 and Jensen’s inequality we have that

[TABLE]

for all $j\geq j_{0}$ and $k\in\{1,...,2^{j}\}.$ By (2.5), (3.7) and (3.25), we then obtain that

[TABLE]

Moreover, for any $\alpha\in(-1,p-1)$ it follows from Lemma 2.6 that

[TABLE]

By (3) and (3.27) we conclude that $\mathcal{E}_{2}(p,\alpha,\lambda,\varphi)\lesssim I_{2}(p,\alpha,\lambda,h).$

We next consider the case $\lambda<0.$ Let $\Psi_{p,\lambda}$ be as in (2.35). By the analogous arguments as for (3), we have that

[TABLE]

Set $S_{j,k}=\sum_{n\leq j}\sum_{m\in i_{n}}\frac{\ell(\varphi(\Gamma_{n,m}))}{2^{-n}}.$ It follows from (2.50) and (3.28) that

[TABLE]

Since $\alpha>-1,$ there is $\beta>0$ such that $\beta p\leq 1+\alpha.$ Define

[TABLE]

Then

[TABLE]

Since $\log^{\lambda}(e+S_{j,k})\leq\log^{\lambda}(e)=1,$ we have that

[TABLE]

Whenever $S_{j,k}\geq 2^{j\beta},$ it follows that $\log^{\lambda}(e+S_{j,k})\lesssim j^{\lambda}.$ Via the analogous arguments as for $I_{1}(p,\alpha,\lambda,h)\lesssim\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ in Lemma 3.3, it then follows that

[TABLE]

From (3.29), (3), (3.31) and (3.32), we conclude that there is a constant $C>0$ such that $I_{2}(p,\alpha,\lambda,h)\lesssim C+\mathcal{E}_{1}(p,\alpha,\lambda,\varphi).$

By the analogous arguments as for $\mathcal{E}_{2}(p,\alpha,\lambda,\varphi)\lesssim I_{2}(p,\alpha,\lambda,h)$ whenever $\lambda\geq 0,$ we have that

[TABLE]

It follows from (2.50) that $\mathcal{E}_{2}(p,\alpha,\lambda,\varphi)\lesssim I_{2}(p,\alpha,\lambda,h).$ ∎

Proof of Theorem 3.1 (1).

From Lemma 3.3 and Lemma 3.4, we have that both $I_{1}(p,\alpha,\lambda,h)$ and $I_{2}(p,\alpha,\lambda,h)$ are dominated by $\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ for all $p>1,\alpha\in(-1,+\infty)$ and each $\lambda\in\mathbb{R}.$ Moreover since $\ell(\varphi(\Gamma_{j,k}))\leq 2\pi$ for all $j\geq 1$ and $1\leq k\leq 2^{j},$ we have that

[TABLE]

Therefore both $I_{1}(p,\alpha,\lambda,h)$ and $I_{2}(p,\alpha,\lambda,h)$ are controlled by $\sum_{j=1}^{\infty}2^{j(p-2-\alpha)}j^{\lambda}$ whenever $\alpha\in(-1,+\infty)$ and $\lambda\in\mathbb{R}.$ Notice that $\sum_{j=1}^{\infty}2^{j(p-2-\alpha)}j^{\lambda}<\infty$ whenever either $p-2<\alpha$ and $\lambda\in\mathbb{R},$ or $p-2=\alpha$ and $\lambda<-1.$ We hence complete Theorem 3.1 ($1$). ∎

By Example 4.4, there are homeomorphisms $\varphi:{\mathbb{S}}^{1}\rightarrow{\mathbb{S}}^{1}$ such that, for their harmonic extensions $P[\varphi]$, both $I_{1}(p,\alpha,\lambda,P[\varphi])$ and $I_{2}(p,\alpha,\lambda,P[\varphi])$ may be finite or infinite for either some $\alpha\in(-1,p-2)$ and $\lambda\in\mathbb{R}$ or some $\alpha=p-2$ and $\lambda\in[-1,+\infty).$ How can we characterize both $I_{1}(p,\alpha,\lambda,P[\varphi])<\infty$ and $I_{2}(p,\alpha,\lambda,P[\varphi])<\infty$? As shown in [8], double integrals of the inverse mapping over the boundary are potential choices.

Lemma 3.5**.**

Let $\varphi:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ be a homeomorphism. For any $\alpha\in\mathbb{R}$ and $\lambda\in\mathbb{R},$ $\mathcal{V}(p,\alpha,\lambda,\varphi)$ is dominated by $\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ whenever $p\in(1,2]$; while $\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ is controlled by $\mathcal{V}(p,\alpha,\lambda,\varphi)$ if $p\in[2,+\infty)$.

Proof.

We first consider the case $p\in(1,2].$ Given $\xi\in\mathbb{S}^{1}$ and $t\geq 0,$ set

[TABLE]

By Fubini’s theorem we have that

[TABLE]

Moreover, from Jensen’s inequality and Minkowski’s inequality it follows that

[TABLE]

Combining (3) with (3) implies that

[TABLE]

Since $E_{2^{1-j}}(\xi)\subset\cup_{i=k-1}^{k+1}\varphi(\Gamma_{j,i})$ for all $j\in\mathbb{N},\ k=1,...,2^{j}$ and all $\xi\in\varphi(\Gamma_{j,k}),$ we have that

[TABLE]

Moreover by Young’s inequality we have that

[TABLE]

and

[TABLE]

Combining (3), (3) with (3) implies that

[TABLE]

for all $j\in\mathbb{N}.$ Let

[TABLE]

For any $j\in\mathbb{N},$ we have that

[TABLE]

It follows (3.41) and (2.3) that

[TABLE]

By combining (3), (3.40) with (3.42), we conclude that

[TABLE]

We next consider the case $p\in[2,+\infty).$ By the analogous arguments as for (3) we have that

[TABLE]

Since $\varphi(\Gamma_{j,k})\subset E_{\pi 2^{1-j}}(\xi)$ for all $j\geq 5$, $k\in\{1,...,2^{j}\}$ and all $\xi\in\varphi(\Gamma_{j,k}),$ we have that

[TABLE]

By (3.42), (3.43) and (3), there is a constant $C>0$ such that

[TABLE]

∎

We next prove Theorem 3.1 ($2$).

Lemma 3.6**.**

Let $\varphi:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ be a homeomorphism. For any $p\in(1,+\infty),\ \alpha\in(-1,p-1)$ and $\lambda\in\mathbb{R},$ we have that $\mathcal{U}(p,\alpha,\lambda,\varphi)$ and $\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ are comparable.

Proof.

We first prove that $\mathcal{U}(p,\alpha,\lambda,\varphi)$ is controlled by $\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$. Given $j\geq 1$ and $\xi\in\mathbb{S}^{1},$ set

[TABLE]

and $A_{j}(\xi)=\{\eta\in\mathbb{S}^{1}:(\xi,\eta)\in A_{j}\}.$ Notice that $\lambda_{{\mathbb{D}}}$ is the Euclidean distance. We have that

[TABLE]

Notice that

[TABLE]

for all $j\in\mathbb{N}$, $k\in\{1,...,2^{j}\},$ $\xi\in\Gamma_{j,k}$ and $\eta\in A_{j}(\xi).$ It then follows that

[TABLE]

for all $\lambda\geq 0$, $\xi\in\Gamma_{j,k}$ and $\eta\in A_{j}(\xi).$ Since

[TABLE]

for all $j\in\mathbb{N}$, $k=1,...,2^{j}$ and $\xi\in\Gamma_{j,k},$ we derive from (3) and (3) that

[TABLE]

whenever $\lambda\geq 0.$

Since $\varphi$ is homeomorphic, for any $j\in\mathbb{N}$ and $k\in\{1,...,2^{j}\}$ there are $\xi^{\prime}_{j,k}\in\Gamma_{j,k}$ and $\eta^{\prime}_{j,k}\in A_{j}(\xi^{\prime}_{j,k})$ such that

[TABLE]

Since $0<\alpha+1<p,$ there is $\beta\in(-1,0)$ such that $0<(1+\beta)p<\alpha+1.$ Define

[TABLE]

From (3), (3), (3.48), (2.3) and (3.46), we obtain that

[TABLE]

Since $\log^{\lambda}(e+|\varphi(\xi^{\prime}_{j,k})-\varphi(\eta^{\prime}_{j,k})||\xi^{\prime}_{j,k}-\eta^{\prime}_{j,k}|^{-1})\leq 1$ for all $\lambda<0,$ $j\in\mathbb{N}$ and $1\leq k\leq 2^{j},$ by (3.46) and (2.3) we have that

[TABLE]

Moreover we derive from (3.46) that

[TABLE]

for all $\lambda<0.$ Combining (3), (3.51) with (3) implies that there is a constant $C>0$ such that $\mathcal{U}(p,\alpha,\lambda,\varphi)\lesssim C+\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ for all $\lambda<0.$

We next prove that $\mathcal{U}(p,\alpha,\lambda,\varphi)$ dominates $\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$. Given $\xi\in{\mathbb{S}}^{1}$ and $\eta\in{\mathbb{S}}^{1},$ let $\ell(\xi\eta)$ be the arc length of the shorter arc in ${\mathbb{S}}^{1}$ connecting $\xi$ and $\eta.$ Given $j\geq 3$ and $\xi\in{\mathbb{S}}^{1},$ set

[TABLE]

and $B_{j}(\xi)=\{\eta\in\mathbb{S}^{1}:(\xi,\eta)\in B_{j}\}.$ We have that

[TABLE]

Since $\varphi$ is homeomorphic, for any $j\geq 3$ and $1\leq k\leq 2^{j}$ there are $\xi^{\prime\prime}_{j,k}\in\Gamma_{j,k-1}$ and $\eta^{\prime\prime}_{j,k}\in B_{j}(\xi^{\prime\prime}_{j,k})$ such that

[TABLE]

Notice that

[TABLE]

whenever $j\geq 3$ and $k\in\{1,...,2^{j}\}.$ Since ${\mathcal{L}}^{1}(B_{j}(\xi))\approx\ell(\Gamma_{j,k})$ for all $j\geq 3,\ k=1,...,2^{j}$ and $\xi\in\mathbb{S}^{1},$ it follows from (3.53), (3) and (3.55) that

[TABLE]

Moreover, for any $\lambda\leq 0$ we obtain from (3.55) that

[TABLE]

for all $j\in\mathbb{N}$ and all $k=1,...,2^{j}.$ From (3.55), (3.56) and (3.57), there is a constant $C>0$ such that

[TABLE]

for all $\lambda\leq 0.$ For any $\lambda>0,$ by (3.55) and (3.56) there is a constant $C>0$ such that

[TABLE]

Let $\beta$ be same as in (3). Set

[TABLE]

We have that

[TABLE]

Moreover

[TABLE]

and

[TABLE]

From (3), (3.60), (3.61) and (3.58), we have that $\mathcal{U}(p,\alpha,\lambda,\varphi)$ controls $\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ whenever $\lambda>0.$ ∎

Proof of Theorem 3.1 ($2$).

By Lemma 3.2, Lemma 3.3 and Lemma 3.4, for any $p>1$ we have that both $I_{1}(p,\alpha,\lambda,h)$ and $I_{2}(p,\alpha,\lambda,h)$ are comparable to $\mathcal{E}_{1}(p,\alpha,\lambda,\varphi)$ whenever $\alpha\in(-1,p-1)$ and $\lambda\in\mathbb{R}.$ By Lemma 3.6, we hence conclude comparability of both $I_{1}(p,\alpha,\lambda,h)$ and $I_{2}(p,\alpha,\lambda,h)$ with $\mathcal{U}(p,\alpha,\lambda,\varphi)$ for all $p>1,\alpha\in(-1,p-1)$ and every $\lambda\in\mathbb{R}.$ By Lemma 3.5, we can dominate $\mathcal{V}(p,\alpha,\lambda,\varphi)$ by either $I_{1}(p,\alpha,\lambda,h)$ or $I_{2}(p,\alpha,\lambda,h)$ whenever $p\in(1,2],$ while both $I_{1}(p,\alpha,\lambda,h)$ and $I_{2}(p,\alpha,\lambda,h)$ are controlled by $\mathcal{V}(p,\alpha,\lambda,\varphi)$ for all $p\in[2,+\infty).$ Moreover from Example 4.2 and Example 4.3, we have that $\mathcal{V}(p,\alpha,\lambda,\varphi)$ is comparable to either $I_{1}(p,\alpha,\lambda,h)$ or $I_{2}(p,\alpha,\lambda,h)$ only when $p=2.$ ∎

On the proof of Theorem 3.1 ($3$), we have the following general result.

Lemma 3.7**.**

Let $\Omega\subset{\mathbb{R}}^{2}$ be a Jordan domain and $\varphi:{\mathbb{S}}^{1}\rightarrow\partial\Omega$ be a homeomorphism. For any $p>1,$ there is no a homeomorphic extension $h:{\mathbb{D}}\rightarrow\Omega$ of $\varphi$ for which $I_{1}(p,\alpha,\lambda,h)<+\infty$ for either $\alpha\in(-\infty,-1)$ and $\lambda\in\mathbb{R}$ or $\alpha=-1$ and $\lambda\in[-1,+\infty);$ and for which $I_{2}(p,\alpha,\lambda,h)<+\infty$ for all $\alpha\in(-\infty,-1]$ and $\lambda\in\mathbb{R}.$

Proof.

Assume that there is a homeomorphic extension $h:{\mathbb{D}}\rightarrow\Omega$ of $\varphi$ for which $I_{1}(p,\alpha,\lambda,h)<+\infty$ for either $\alpha\in(-\infty,-1)$ and $\lambda\in\mathbb{R}$ or $\alpha=-1$ and $\lambda\in[-1,+\infty).$ Then $h\in W^{1,p}({\mathbb{D}},\Omega).$ Let

[TABLE]

By the ACL-property of Sobolev mappings, we have that

[TABLE]

for ${\mathcal{L}}^{1}$-a.e. $r\in[0,1).$ By Jensen’s inequality we derive from (3.62) that

[TABLE]

Let ${\mathbb{D}}_{r}=\{z\in{\mathbb{R}}^{2}:|z|<r\}.$ Since $h$ is a homeomorphism, we have $\mbox{osc}_{{\mathbb{D}}_{r}}h=\mbox{osc}_{{\mathbb{S}}_{r}}h.$ Hence

[TABLE]

Moreover $w_{\alpha,\lambda}(1-r)\approx 2^{-\alpha j}j^{\lambda}$ for all $j\geq 0\mbox{ and }r\in(1-2^{-j},1-2^{-j-1}].$ By (3), (3.64) and Fubini’s theorem, we obtain that

[TABLE]

By the assumption at the beginning, we derive from (3) that

[TABLE]

for either $\alpha<-1$ and $\lambda\in\mathbb{R}$ or $\alpha=-1$ and $\lambda\geq-1.$ Hence by (3.64) we have that $\mbox{osc}_{\mathbb{S}_{1-2^{-j}}}h=0$ for all $j\geq 1.$ Therefore there is a constant $C$ such that $h(z)=C$ for all $z\in\mathbb{D}.$ This contradicts the homeomorphicity of $h$. We conclude that the assumption at the beginning cannot hold.

We next assume that there is a homeomorphic extension $h:{\mathbb{D}}\rightarrow\Omega$ of $\varphi$ for which $I_{2}(p,\alpha,\lambda,h)<+\infty$ for all $\alpha\in(-\infty,-1]$ and $\lambda\in\mathbb{R}.$ It is not difficult to see that $h\in W^{1,1}({\mathbb{D}},\Omega).$ We first let $\lambda\geq 0.$ The property (P-1) of Proposition 2.5 shows that $\Phi_{p,\lambda}$ is convex. Analogously to (3), we have

[TABLE]

Analogous arguments as below (3.66) imply that there is a contradiction under the above assumption. We next let $\lambda<0.$ Proposition 2.7 shows that $\Psi_{p,\lambda}$ is convex. Analogously to (3.67), we obtain from (2.50) that

[TABLE]

Analogous arguments as below (3.66) imply that there is a contradiction under the above assumption.

∎

Proof of Theorem 1.1.

Let $\lambda_{\Omega}$ be the internal distance and $|\cdot|$ be the Euclidean distance. As the proof of [8, Theorem 1] shows that there exist a bi-Lipschitz mapping $g:({\mathbb{S}}^{1},|\cdot|)\rightarrow(\partial\Omega,\lambda_{\Omega})$ and a diffeomorphic bi-Lipschitz extension $\tilde{g}:({\mathbb{D}},|\cdot|)\rightarrow(\Omega,\lambda_{\Omega})$ of $g.$ Let $h=\tilde{g}\circ P[g^{-1}\circ\varphi].$ Then $h:{\mathbb{D}}\rightarrow\Omega$ is a diffeomorphic extension of $\varphi.$ Moreover

[TABLE]

[TABLE]

Hence Theorem 1.1 ($1$) and ($2$) follow from Theorem 3.1. By Lemma 3.7, we complete the proof of Theorem 1.1. ∎

4. Examples

In this section, we give examples related to Theorem 3.1 ($2$). We first decompose $[0,1].$ For a given $s\in(0,+\infty),$ let

[TABLE]

be the largest integer less than $2^{n/s}.$ There is $n^{s}_{0}\geq 1$ such that

[TABLE]

$\mathit{Step\ 1}$. Let

[TABLE]

Renumber the elements in $T_{1}=\{0,1\}\cup\partial I_{1}$ as $\{b_{1,i_{1}}:i_{1}=1,...,4\}$ such that $b_{1,i^{\prime}_{1}}<b_{1,i^{\prime\prime}_{1}}$ if $i^{\prime}_{1}<i^{\prime\prime}_{1}.$

$\mathit{Step\ 2}$. Let

[TABLE]

Set $I_{2}=\cup_{i=1}^{2}I_{2,i}$, and renumber the elements in $T_{2}=T_{1}\cup\partial I_{2}$ as $\{b_{2,i_{2}}:i_{2}=1,...,8\}$ such that $b_{2,i^{\prime}_{2}}<b_{2,i^{\prime\prime}_{2}}$ if $i^{\prime}_{2}<i^{\prime\prime}_{2}.$

After Step (n-$1$), we have $\{I_{n-1,k_{n-1}}:k_{n-1}=1,...,2^{n-2}\},$ $I_{n-1}=\cup_{k_{n-1}=1}^{2^{n-2}}I_{n-1,k_{n-1}}$ and $T_{n-1}:=T_{n-2}\cup\partial I_{n-1}=\{b_{n-1,i_{n-1}}:i_{n-1}=1,...,2^{n}\}$ where $b_{n-1,i^{\prime}_{n-1}}<b_{n-1,i^{\prime\prime}_{n-1}}$ if $i^{\prime}_{n-1}<i^{\prime\prime}_{n-1}.$ In the following Step n, set

[TABLE]

and $I_{n}=\cup_{k_{n}=1}^{2^{n-1}}I_{n,k_{n}}.$ After renumbering the elements in $T_{n}=T_{n-1}\cup\partial I_{n}$ as above, we can proceed to Step (n+$1$). Moreover we must replace $I_{n,k_{n}}$ in (4.3) by

[TABLE]

whenever $n\geq n^{s}_{0}.$ Let $I=\cup_{n=1}^{\infty}I_{n}$ and $R=[0,1]\setminus I.$ Then $R\neq\emptyset.$ We finally decompose $[0,1]$ as

[TABLE]

We next give an estimate on the length of $I_{n,k_{n}}.$ Since ${\mathcal{L}}^{1}(I_{n,k_{n}})=2^{-j_{n-1}}-2^{1-j_{n}}$ for all $n\geq n_{0}+1$ and $k_{n}\in\{1,...,2^{n-1}\},$ by the first inequality in (4.2) we have that

[TABLE]

for all $n\geq n_{0}+1$ and $k_{n}\in\{1,...,2^{n-1}\}.$ When $n=n_{0},$ from (4.4) and the second estimate in (4.2) we have that

[TABLE]

for all $k_{n}=1,...,2^{n-1}.$ Whenever $1\leq n\leq n_{0}-1$ and $k_{n}\in\{1,...,2^{n-1}\},$ we have ${\mathcal{L}}^{1}(I_{n,k_{n}})=4^{-n}.$ Let $C_{1}(s)=\min\{2^{j_{n-1}-2n}:1\leq n\leq n_{0}\}.$ Then

[TABLE]

for all $1\leq n\leq n_{0}$ and $k_{n}\in\{1,...,2^{n-1}\}.$ By (4.6), (4.7) and (4.8), we obtain that there is a constant $C(s)>0$ such that

[TABLE]

for all $n\in\mathbb{N}$ and $k_{n}\in\{1,...,2^{n-1}\}.$

Define

[TABLE]

For any $x\in R$ and any $n\geq n^{s}_{0},$ there is $b_{n}\in\partial I_{n}$ such that $|b_{n}-x|=\inf\limits_{b\in\partial I_{n}}|b-x|.$ By (4.4) and (4.10), we have that

[TABLE]

It follows that $\lim_{n\rightarrow+\infty}b_{n}=x$ and $\{f^{1}(b_{n})\}$ is a Cauchy sequence. Therefore

[TABLE]

is a well-defined function on $[0,1].$

Proposition 4.1**.**

Let $f_{s}$ be as in (4.11) with $s\in(0,+\infty).$ Then $f_{s}(0)=0$, $f_{s}(1)=1$ and $f_{s}$ is increasing on $[0,1].$ Moreover there is a constant $C(s)>0$ such that

[TABLE]

for all $x,y\in[0,1]$ with $x\neq y.$

Proof.

By (4.11), we have that $f_{s}(0)=\lim_{n\rightarrow\infty}f^{1}_{s}(2^{-j_{n}})=\lim_{n\rightarrow\infty}2^{-n}=0.$ Analogously $f_{s}(1)=1.$

We next prove the monotonicity of $f_{s}$. Let $x_{1}\in[0,1],\ x_{2}\in[0,1]$ with $x_{1}\leq x_{2}.$ If $x_{1}\in I_{n,k^{\prime}_{n}}$ and $x_{2}\in I_{n,k^{\prime\prime}_{n}}$ with $k^{\prime}_{n}\leq k^{\prime\prime}_{n},$ from (4.11) we have that

[TABLE]

Assume $x_{1}\in I_{n_{1},k_{n_{1}}}$ and $x_{2}\in I_{n_{2},k_{n_{2}}}$ with $n_{1}\neq n_{2}.$ Let $q=|n_{2}-n_{1}|.$ If $n_{1}<n_{2},$ from the construction of $\{I_{n,k_{n}}\}$ we have that $k_{n_{2}}\geq 2^{q}(k_{n_{1}}-1)+2^{q-1}+1.$ It then follows from (4.10) that

[TABLE]

If $n_{2}<n_{1},$ from the construction of $\{I_{n,k_{n}}\}$ we have that

[TABLE]

It follows that

[TABLE]

and

[TABLE]

By combining (4.15) with (4.16), we deduce from (4.10) that

[TABLE]

Assume $x_{1}\in R$ and $x_{2}\in I.$ By (4.11), there is $\{b_{n}\}\subset\partial I$ such that $\lim\limits_{n\rightarrow\infty}b_{n}=x_{1}.$ Together with $x_{1}<x_{2},$ it follows that $b_{n}<x_{2}$ whenever $n\gg 1.$ Via the arguments for (4.13), (4.14) and (4.17), we have that

[TABLE]

By taking limit for (4.18), we have that

[TABLE]

Assume either $x_{1}\in I$ and $x_{2}\in R,$ or $x_{1}\in R$ and $x_{2}\in R.$ Via analogous arguments as for (4.19), we can also prove $f_{s}(x_{1})\leq f_{s}(x_{2})$ at these two cases. By preceding arguments, we conclude that $f_{s}$ is increasing on $[0,1]$.

We next prove (4.12). Let $T_{n}=\{b_{n,i_{n}}:i_{n}=1,...,2^{n+1}\}$ with $n\in\mathbb{N}$ and $f^{1}_{i,s}$ be as in (4.10). For a given $n\in\mathbb{N},$ define

[TABLE]

[TABLE]

Then $f_{n,s}$ is piecewise affine, increasing and continuous on $[0,1].$ Furthermore we claim:

1. (i)

$\lim_{n\rightarrow\infty}f_{n,s}(x_{0})=f_{s}(x_{0})$ for all $x_{0}\in[0,1],$ 2. (ii)

there are constant $C(s)>0$ and $N(s)>0$ such that

[TABLE]

for all $n\geq N(s).$

If both (i) and (ii) hold, we can prove (4.12).

We first prove (i). Let $x_{0}\in[0,1].$ If $x_{0}\in I,$ without loss of generality we assume that $x_{0}\in I_{n_{0},k_{n_{0}}}.$ From (4.11) and (4.20), we have that $f_{n}(x_{0})=f(x_{0})$ for all $n\geq n_{0}.$ Therefore (i) holds. If $x_{0}\in R$, from (4.11) there is $\{b_{n}\}\subset\partial I$ such that $\lim_{n\rightarrow\infty}b_{n}=x_{0}$ and $\lim_{n\rightarrow\infty}f^{1}_{s}(b_{n})=f_{s}(x_{0}).$ Moreover by (4.20), we have that

[TABLE]

Together with $|f_{n,s}(x_{0})-f_{s}(x_{0})|\leq|f_{n,s}(x_{0})-f^{1}_{s}(b_{n})|+|f^{1}_{s}(b_{n})-f_{s}(x_{0})|,$ we have that (i) also holds at this case.

We next prove (ii). Given $n\geq 1,\ x\in[0,1]$ and $y\in[0,1]$ with $x<y,$ set

[TABLE]

Then $0\leq k_{n}(x,y)\leq 2^{n}-1.$

Assume $k_{n}(x,y)=0$. If $x\in\cup_{m=1}^{n}I_{m},$ there are $m\in\{1,...,n\}$ and $k_{m}\in\{1,...,2^{m-1}\}$ such that $x\in I_{m,k_{m}}.$ For the location of $y,$ possibly we have that

[TABLE]

If $y\in I_{m,m_{k}}$, by (4.20) we have that

[TABLE]

If $y\in I_{m,m_{k}+1},$ then $|x-y|\geq 2^{-j_{n}}.$ It follows from (4.20) that

[TABLE]

If $y\in[0,1]\setminus(\cup_{m=1}^{n}I_{m})$, there is $x_{0}\in[x,y)\cap T_{n}$ such that

[TABLE]

Since there is $n^{s}_{1}>0$ such that $\log(2^{j^{s}_{n^{s}_{1}}})-s>0,$ we have that

[TABLE]

for all $n\geq n^{s}_{1}$ and every $t\in(0,2^{-j^{s}_{n}}].$ By (4.20), (4.23) and (4.24), we then have that

[TABLE]

whenever $n\geq N(s):=\max\{n^{s}_{0},n^{s}_{1}\}.$ If $x\in[0,1]\setminus(\cup_{m=1}^{n}I_{m}),$ for the location of $y$ we possibly have that

[TABLE]

If $y\in[0,1]\setminus(\cup_{m=1}^{n}I_{m}),$ then $0<y-x<2^{-j_{n}}.$ By (4.20) and (4.24) we have that

[TABLE]

for all $n\geq N(s).$ If $y\in\cup_{m=1}^{n}I_{m},$ by analogous arguments as for (4) we have that

[TABLE]

for all $n\geq N(s).$ By (4), (4.22), (4), (4.26) and (4.27), we conclude that

[TABLE]

for all $n\geq N(s)\mbox{ and }k_{n}(x,y)=0.$

Assume $k_{n}(x,y)\in\{1,...,2^{n}-1\}.$ Define

[TABLE]

and

[TABLE]

If $k_{n}(x,y)=1,$ by (4.20) we have that

[TABLE]

If $2^{m}\leq k_{n}(x,y)\leq 2^{m+1}-1$ for $m=1,...,n-1,$ by (4.5), (4.9) and (4.20) we have that

[TABLE]

and

[TABLE]

Whenever $n\geq n^{s}_{0}+1,$ it follows from (4.1) that

[TABLE]

Notice that there are two cases for the location of $x$

[TABLE]

If $x\in(x^{\prime}-2^{-j_{n}},x^{\prime}],$ by analogous arguments as for (4.26) we have that

[TABLE]

If $x\in\cup_{m=1}^{n}I_{m},$ same arguments as (4) imply (4.31). Analogously, we have that

[TABLE]

Since

[TABLE]

by (4.29), (4), (4.31) and (4.32) there is a constant $C(s)>0$ such that

[TABLE]

whenever $n\geq N(s)\mbox{ and }k_{n}(x,y)\in\{1,...,2^{n}-1\}.$ By (4.28) and (4.33), we finish the proof of (ii). ∎

Let $\varphi:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ be a homeomorphism. In the following we denote by $P[\varphi]:{\mathbb{D}}\rightarrow{\mathbb{D}}$ the harmonic extension of $\varphi.$

Example 4.2**.**

For a given $p\in(1,2),$ there is a homeomorphism $\varphi:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ such that $\mathcal{V}(p,p-2,0,\varphi)<\infty$, $I_{1}(p,p-2,0,P[\varphi])=\infty$ and $I_{2}(p,p-2,0,P[\varphi])=\infty.$

Proof.

We first introduce a class of self-homeomorphisms on ${\mathbb{S}}^{1}$ and their properties. Let $f_{s}$ be as in (4.11) with $s\in(0,+\infty).$ Define

[TABLE]

Then $g_{s}:[0,1]\rightarrow[0,1]$ is strictly increasing and continuous, i.e. $g_{s}$ is homeomorphic. Moreover by (4.1), there is a constant $C(s)>0$ such that

[TABLE]

for all $x,y\in[0,1]\mbox{ with }x\neq y.$ Let $\mbox{arg}z\in(-\pi,\pi]$ be the principal value of the argument $z.$ Define

[TABLE]

Then $\varphi_{s}:{\mathbb{S}}^{1}\rightarrow{\mathbb{S}}^{1}$ is homeomorphic and $\varphi(e^{i\pi})=e^{i\pi}.$ Next we prove that

[TABLE]

for all $z_{1},z_{2}\in\mathbb{S}^{1}$ with $z_{1}\neq z_{2}.$ Let $\Gamma(z_{1},z_{2})$ be the arc in $\mathbb{S}^{1}$ joining $z_{1}$ to $z_{2}$ with smaller length. Denote by $\ell(\Gamma(z_{1},z_{2}))$ the length of $\Gamma(z_{1},z_{2}).$ In order to prove (4.37), it is enough to consider the case $\ell(\Gamma(z_{1},z_{2}))\ll 1.$ If $e^{i\pi}\notin\Gamma(z_{1},z_{2}),$ we have that

[TABLE]

whenever $\ell(\Gamma(z_{1},z_{2}))\ll 1.$ Together with (4.35), we then have that

[TABLE]

If $e^{i\pi}\in\Gamma(z_{1},z_{2})$ and $\ell(\Gamma(\varphi(z_{1}),\varphi(e^{i\pi})))>\ell(\Gamma(\varphi(e^{i\pi}),\varphi(z_{2}))),$ there is $z_{0}\in\Gamma(z_{1},e^{i\pi})$ such that

[TABLE]

Same arguments as for (4) imply that

[TABLE]

Combining (4.39) with (4.40) therefore implies that (4.37) holds when $e^{i\pi}\in\Gamma(z_{1},z_{2})$ and $\ell(\Gamma(\varphi_{s}(z_{1}),\varphi_{s}(e^{i\pi})))>\ell(\Gamma(\varphi(e^{i\pi}).$ Analogously, we can prove that (4.37) holds when $e^{i\pi}\in\Gamma(z_{1},z_{2})$ and $\ell(\Gamma(\varphi_{s}(z_{1}),\varphi_{s}(e^{i\pi})))\leq\ell(\Gamma(\varphi_{s}(e^{i\pi}),\varphi_{s}(z_{2}))).$

Let $p\in(1,2).$ There is $s\in(1,+\infty)$ such that $p-1<1/s<1.$ Based on this $s,$ we obtain a homeomorphism $\varphi=\varphi_{s}:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1},$ where $\varphi_{s}$ is from (4.36). By Jensen’s inequality and (4.37), we have that

[TABLE]

Let $n^{s}_{0}$ be as in (4.2) with $s$ chosen above. For any $n\geq n^{s}_{0}$ and any $j_{n}<j\leq j_{n+1},$ by (4.34) and (4.11) we have that

[TABLE]

Notice that $j_{n+1}-j_{n}\approx 2^{n/s}$ whenever $n\geq n_{0}.$ We then derive from (4) that

[TABLE]

By Lemma 3.2, Lemma 3.3 and Lemma 3.4, it follows that $I_{1}(p,p-2,0,P[\varphi])=\infty$ and $I_{2}(p,p-2,0,P[\varphi])=\infty.$ ∎

Example 4.3**.**

For a given $p\in(2,+\infty),$ there is a homeomorphism $\varphi:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ such that $\mathcal{V}(p,p-2,0,\varphi)=\infty$, $I_{1}(p,p-2,0,P[\varphi])<\infty$ and $I_{2}(p,p-2,0,P[\varphi])<\infty.$

Proof.

Since $p\in(2,+\infty),$ there is $s\in(0,1)$ such that $p-1>1/s>1.$ Based on this chosen $s,$ we obtain a homeomorphism $\varphi=\varphi_{s}:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1},$ where $\varphi_{s}$ is from (4.36). In order to prove $\mathcal{V}(p,p-2,0,\varphi)=\infty$, by Jensen’s inequality it suffices to prove that

[TABLE]

For any $\sigma\in{\mathbb{S}}^{1}$ and $\tau\in{\mathbb{S}}^{1},$ let $\ell(\sigma,\tau)$ be the arc length of the shorter arc in ${\mathbb{S}}^{1}$ joining $\sigma$ and $\tau.$ Let $n^{s}_{0}$ be from (4.2) with $s$ chosen above. For any $n\geq n^{s}_{0},$ set

[TABLE]

We have that

[TABLE]

Given $n\geq n^{s}_{0}$ and $k=1,...,2^{n},$ let

[TABLE]

[TABLE]

For any $\xi\in\varphi(\Gamma^{{}^{\prime}}_{n,k})$ and $\eta\in\varphi(\Gamma^{{}^{\prime\prime}}_{n,k}),$ we have that

[TABLE]

Notice that by (4.2) we have that $2^{-j_{n+1}}<2^{-j_{n}}-2^{1-j_{n+1}}$ whenever $n\geq n^{s}_{0}$. It then follows from (4.44) that

[TABLE]

for all $n\geq n^{s}_{0}$ and all $k=1,...,2^{n}.$ Moreover from (4.36), (4.34) and (4.11), it follows that

[TABLE]

for all $n\geq n^{s}_{0}$ and all $k=1,...,2^{n}.$ Similarly

[TABLE]

Since $(\varphi(\Gamma^{{}^{\prime}}_{n,k})\times\varphi(\Gamma^{{}^{\prime\prime}}_{n,k}))\cap(\varphi(\Gamma^{{}^{\prime}}_{n,j})\times\varphi(\Gamma^{{}^{\prime\prime}}_{n,j}))=\emptyset$ for all $n\geq n^{s}_{0}$ and $k,j\in\{1,...,2^{n}\}$ with $k\neq j,$ it follows (4.45), (4) and (4.47) that

[TABLE]

for all $n\geq n^{s}_{0}.$ Combining (4) with (4.48) hence implies that

[TABLE]

Therefore (4.42) is complete.

For any $n\geq n_{0}$ and $j_{n}<j\leq j_{n+1},$ by (4.36), (4.34), (4.11) and Jensen’s inequality we have that

[TABLE]

Notice $j_{n+1}-j_{n}\approx 2^{n/s}$ whenever $n\geq n^{s}_{0}.$ We then derive from (4) that

[TABLE]

Since $\sum_{j=1}^{j_{n_{0}}}\sum_{k=1}^{2^{j}}\ell(\varphi(\Gamma_{j,k}))^{p}$ is finite, it follows from (4) that $\mathcal{E}_{1}(p,p-2,0,\varphi)<+\infty.$ Moreover by Lemma 3.3 and Lemma 3.4 we have that $I_{1}(p,p-2,0,P[\varphi])<+\infty$ and $I_{2}(p,p-2,0,P[\varphi])<+\infty.$ ∎

Example 4.4**.**

There is a homeomorphism $\varphi:{\mathbb{S}}^{1}\rightarrow{\mathbb{S}}^{1}$ such that both $I_{1}(p,\alpha,\lambda,P[\varphi])<+\infty$ and $I_{2}(p,\alpha,\lambda,P[\varphi])<+\infty$ hold for all $p>1,$ $\alpha\in(-1,p-1)$ and $\lambda\in\mathbb{R}.$ Moreover for any $p>1,$ there is a homeomorphism $\varphi:{\mathbb{S}}^{1}\rightarrow{\mathbb{S}}^{1}$ such that $I_{1}(p,\alpha,\lambda,p[\varphi])=\infty$ and $I_{2}(p,\alpha,\lambda,P[\varphi])=\infty$ whenever either $\alpha\in(-1,p-2)$ and $\lambda\in\mathbb{R}$ or $\alpha=p-2$ and $\lambda\in[-1,+\infty).$

Proof.

Take $\varphi:{\mathbb{S}}^{1}\rightarrow{\mathbb{S}}^{1}$ as the identity mapping. We have that

[TABLE]

whenever $p>1$, $\alpha\in(-1,p-1)$ and $\lambda\in\mathbb{R}.$ Therefore by Lemma 3.3 and Lemma 3.4 both $I_{1}(p,\alpha,\lambda,P[\varphi])$ and $I_{2}(p,\alpha,\lambda,P[\varphi])$ are finite now.

For a given $p>1,$ set $j_{n}$ in (4.1) as $[e^{2^{n(p-1)}}].$ There is $n_{0}\geq 1$ such that (4.2) holds for all $n\geq n_{0}-1.$ By following the arguments for (4.5), we have $f$ as in (4.11). Moreover by same arguments as in the proof of Proposition 4.1, there is a constant $C>0$ depending only on $p$ such that

[TABLE]

for all $x,y\in[0,1]$ with $x\neq y.$ As in (4.36), we finally obtain a homeomorphism $\varphi:{\mathbb{S}}^{1}\rightarrow{\mathbb{S}}^{1}.$ For any $n\geq n_{0}$ and $j_{n}<j\leq j_{n+1},$ by analogous arguments for (4) we have that

[TABLE]

Notice that $\sum_{j_{n}<j\leq j_{n+1}}j^{-1}\approx\log j_{n+1}-\log j_{n}\gtrsim 2^{n(p-1)}$ for all $n\geq n_{0}.$ For any $\lambda\in[-1,+\infty)$ it then follows from (4.51) that

[TABLE]

For any $\alpha\in(-1,p-2)$ and $\lambda\in\mathbb{R},$ we have that $2^{j(p-2-\alpha)}j^{\lambda}\gtrsim j^{-1}$ whenever $j\gg 1.$ Without loss of generality, we assume that $2^{j(p-2-\alpha)}j^{\lambda}\gtrsim j^{-1}$ for all $n\geq n_{0}$ and $j_{n}<j\leq j_{n+1}.$ Hence from (4) we have that

[TABLE]

for all $\alpha\in(-1,p-2)$ and $\lambda\in\mathbb{R}.$ By Lemma 3.2, Lemma 3.3 and Lemma 3.4, we conclude from (4) and (4) that for any $p>1$ there is a homeomorphism $\varphi:\mathbb{S}^{1}\rightarrow\mathbb{S}^{1}$ such that $I_{1}(p,\alpha,\lambda,P[\varphi])=\infty$ and $I_{2}(p,\alpha,\lambda,P[\varphi])=\infty$ whenever either $\alpha\in(-1,p-2)$ and $\lambda\in\mathbb{R}$ or $\alpha=p-2$ and $\lambda\in[-1,+\infty).$ ∎

Acknowledgment

The author has been supported by China Scholarship Council (project No. 201706340060). This paper is a part of the author’s doctoral thesis. The author thanks his advisor Professor Pekka Koskela for posing this question and for valuable discussions. The author thanks Aleksis Koski and Zhuang Wang for comments on the earlier draft.