The Heinz type inequality, Bloch type theorem and Lipschitz characteristic of polyharmonic mappings

TL;DR

This paper investigates polyharmonic mappings satisfying specific boundary conditions, establishing inequalities, a Bloch type theorem, and Lipschitz continuity, thereby extending classical results and solving an open problem in the field.

Contribution

It introduces a Heinz type inequality and a Bloch type theorem for polyharmonic mappings with boundary conditions, and proves Lipschitz continuity for quasiconformal solutions, addressing an open problem.

Findings

Established a Schwarz type lemma for polyharmonic mappings.

Derived a Heinz type inequality for these mappings.

Proved Lipschitz continuity with asymptotically sharp constants.

Abstract

Suppose that $f$ satisfies the following: $(1)$ the polyharmonic equation $\Delta^{m}f=\Delta(\Delta^{m-1} f)$$=\varphi_{m}$ $(\varphi_{m}\in \mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{n}))$, (2) the boundary conditions $\Delta^{0}f=\varphi_{0},\Delta^{1}f=\varphi_{1},~\ldots,~\Delta^{m-1}f=\varphi_{m-1}$ on $\mathbb{S}^{n-1}$ ($\varphi_{j}\in \mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{n})$ for $j\in\{0,1,\ldots,m-1\}$ and $\mathbb{S}^{n-1}$ denotes the boundary of the unit ball $\mathbb{B}^{n}$), and $(3)$ $f(0)=0$, where $n\geq3$ and $m\geq1$ are integers. Initially, we prove a Schwarz type lemma and use it to obtain a Heinz type inequality of mappings satisfying the polyharmonic equation with the above Dirichlet boundary value conditions. Furthermore, we establish a Bloch type theorem of mappings satisfying the above polyharmonic equation, which gives an answer to an open problem in \cite{CP-Hi}. Additionally, we show that if $f$ is a $K$-quasiconformal self-mapping of $\mathbb{B}^{n}$ satisfying the above polyharmonic equation, then $f$ is Lipschitz continuous, and the Lipschitz constant is asymptotically sharp as $K\to 1^{+}$ and $\|\varphi_{j}\|_{\infty}\to 0^{+}$ for $j\in\{1,\ldots,m\}$.

Equations

$$|x|=⟨x,x{⟩}^{1/2}={\left(\sum _{k=1}^n|x_k{|}^2\right)}^{1/2}.$$

$$D_{f}=\left(\begin{array}[]{cccc}\displaystyle D_{1}f_{1}\;\cdots\;D_{n}f_{1}\\[11.38109pt] \vdots\;\;\;\;\cdots\;\;\;\;\vdots\\[5.69054pt] \displaystyle D_{1}f_{n_{1}}\;\cdots\;D_{n}f_{n_{1}}\end{array}\right),\quad D_{j}f_{i}(x)=\frac{\partial f_{i}(x)}{\partial x_{j}}.$$

$$\Delta f=\sum _{k=1}^n{D}_{kk}f.$$

$$|A|=\underset{x\in {\mathbb{R}}^n,x\ne 0}{\sup }\frac{|Ax|}{|x|}=\max \{|A\theta |:\theta \in {\mathbb{S}}^{n-1}\},$$

$$l(A)=\inf \{|A\theta |:\theta \in {\mathbb{S}}^{n-1}\}.$$

$$[x,y]:=\left|y|x|-x^{\ast }\right|=\left|x|y|-y^{\ast }\right|.$$

$$G(x,y)=c_n\left(\frac{1}{|x-y{|}^{n-2}}-\frac{1}{[x,y{]}^{n-2}}\right),$$

$$P(x,\zeta )=\frac{1-|x{|}^2}{|x-\zeta {|}^n}.$$

$$\nabla =\left(\frac{\partial }{\partial x_1},\dots ,\frac{\partial }{\partial x_n}\right)$$

$${\Delta }^{m}f=\Delta ({\Delta }^{m-1}f)={\phi }_m\text{mbox}in{\mathbb{B}}^n$$

$${\Delta }^{0}f={\phi }_0,{\Delta }^{1}f={\phi }_1,\dots ,{\Delta }^{m-1}f={\phi }_{m-1}\text{mbox}on{\mathbb{S}}^{n-1},$$

$$\underset{r\to 1^{-}}{\lim }{\Delta }^{j-1}{f}_r(\zeta )={\phi }_{j-1}(\zeta ),$$

$$f(x)=P[{\phi }_0](x)-G_1[{\phi }_1](x),x\in {\mathbb{B}}^n,$$

$$P[{\phi }_0](x)={\int }_{{\mathbb{S}}^{n-1}}P(x,\zeta ){\phi }_0(\zeta )d\sigma (\zeta )$$

$$G_1[{\phi }_1](x)={\int }_{{\mathbb{B}}^n}G(x,y_1){\phi }_1(y_1)dV(y_1).$$

$$f(x)=P[{\phi }_0](x)+\sum _{k=1}^m(-1{)}^k{G}_k[{\phi }_k](x),x\in {\mathbb{B}}^n,$$

$$G_k[{\phi }_k](x)$$

$$G_m[{\phi }_m](x)$$

$$|f(z)|\le \frac{4}{\pi }\arctan |z|.$$

$$\left|f(z)-\frac{1-|z{|}^2}{1+|z{|}^2}f(0)\right|\le \frac{4}{\pi }\arctan |z|,$$

$$|f(x)|\le U(rN),$$

$$U(x)=P[X_{S^{+}}-{\mathcal{X}}_{S^{-}}](x).$$

$$\left|f(x)-\frac{1-|x{|}^2}{(1+|x{|}^2{)}^{\frac{n}{2}}}f(0)\right|\le U(|x|N).$$

$$\left|f(x)-\frac{1-|x{|}^2}{(1+|x{|}^2{)}^{\frac{n}{2}}}P[{\phi }_0](0)\right|$$

$$\underset{r\to 1^{-}}{\mathrm{liminf}}|D_f(r{e}^{i\theta })|\ge \frac{1}{\pi }.$$

$$\frac{n!\big{[}1+n-(n-2)F\big{(}\frac{1}{2},1;\frac{n+3}{2};-1\big{)}\big{]}}{2^{\frac{3n}{2}}\Gamma(\frac{n+1}{2})\Gamma(\frac{n+3}{2})}>\sum_{k=1}^{m}\frac{\|\varphi_{k}\|_{\infty}}{n}\left[\frac{n+4}{4n(n+2)}\right]^{k-1}\left(1+\frac{1}{2^{\frac{n}{2}}}\right).$$

$$\underset{r\to 1^{-}}{\mathrm{liminf}}\frac{|f(\zeta )-f(r\zeta )|}{1-r}$$

$${\Delta }^{1}f={\phi }_1,\dots ,{\Delta }^{m-1}f={\phi }_{m-1}\text{mbox}on{\mathbb{S}}^{n-1}.$$

$$R_1\ge \frac{3n(1+3^{n-1})}{4}{r}_{1}M,$$

$$\frac{1}{{\left(\frac{nM}{2}+\frac{n\Vert {\phi }_1{\Vert }_{\infty }}{n+1}\right)}^{n-1}}-\frac{3n(1+3^{n-1})}{2}{r}_{1}M$$

Full text

1. Preliminaries and statements of main results

For an integer $n\geq 2$, let $\mathbb{R}$ and $\mathbb{R}^{n}$ be the set of real numbers and the usual real vector space of dimension $n$, respectively. Sometimes it is convenient to identify each point $x\in\mathbb{R}^{n}$ with an $n\times 1$ column matrix so that $x=(x_{1},\ldots,x_{n})^{\prime}$, where ′ denotes the transposition of a matrix. For $y=(y_{1},\ldots,y_{n})^{\prime}$ and $x\in\mathbb{R}^{n}$, we define the Euclidean inner product $\langle\cdot,\cdot\rangle$ by $\langle x,y\rangle=\sum_{k=1}^{n}x_{k}y_{k}$ so that the Euclidean length of $x$ is defined by

[TABLE]

Denote a ball in $\mathbb{R}^{n}$ with center $x_{0}\in\mathbb{R}^{n}$ and radius $r$ by $\mathbb{B}^{n}(x_{0},r)$. In particular, let $\mathbb{B}^{n}:=\mathbb{B}^{n}(0,1)$ and $\mathbb{S}^{n-1}:=\partial\mathbb{B}^{n}$. For $n_{1}\in\mathbb{N}:=\{1,2,\ldots\}$ and $k\in\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$, we denote by $\mathcal{C}^{k}(\Omega_{1},\Omega_{2})$ the set of all $k$-times continuously differentiable functions from $\Omega_{1}$ into $\Omega_{2}$, where $\Omega_{1}$ and $\Omega_{2}$ are subsets of $\mathbb{R}^{n}$ and $\mathbb{R}^{n_{1}}$, respectively. In particular, let $\mathcal{C}(\Omega_{1},\Omega_{2}):=\mathcal{C}^{0}(\Omega_{1},\Omega_{2})$, the set of all continuous functions of $\Omega_{1}$ into $\Omega_{2}$. For $f=(f_{1},\ldots,f_{n_{1}})^{\prime}\in\mathcal{C}^{1}(\Omega_{1},\Omega_{2})$, we denote the derivative $D_{f}$ of $f$ by

[TABLE]

In particular, if $n=n_{1}$, the Jacobian of $f$ is defined by $J_{f}=\det D_{f}$ and the Laplacian of $f\in\mathcal{C}^{2}(\Omega_{1},\Omega_{2})$ is defined by

[TABLE]

For an $n_{1}\times n$ matrix $A=(a_{ij})_{n_{1}\times n}$ with $n_{1}\in\mathbb{N}$ and $n\geq 2$, the operator norm of $A$ is defined by

[TABLE]

and the matrix function $l(A)$ is defined by

[TABLE]

1.1. Polyharmonic equation

For $n\geq 3$ and $x,y\in\mathbb{R}^{n}\backslash\{0\}$, we define $x^{\ast}=x/|x|$, $y^{\ast}=y/|y|$ and let

[TABLE]

Also, for $x,y\in\mathbb{B}^{n}$ with $x\neq y$, we use $G(x,y)$ to denote the Green function:

[TABLE]

where $c_{n}=1/[(n-2)\omega_{n-1}]$ and \omega_{n-1}=2\pi^{\frac{n}{2}}/\Gamma\big{(}\frac{n}{2}\big{)} denotes the area of $\mathbb{S}^{n-1}$. The Poisson kernel $P:\,\mathbb{B}^{n}\times\mathbb{S}^{n-1}\rightarrow{\mathbb{R}}$ is defined by

[TABLE]

We use

[TABLE]

to denote the gradient.

Of particular interest for our investigation is the following polyharmonic equation:

[TABLE]

with the following associated Dirichlet boundary value condition:

[TABLE]

where $m,~{}n_{1}\in\mathbb{N}$, $\Delta^{0}f:=f$, $\Delta^{1}f:=\Delta f$, $\varphi_{m}\in\mathcal{C}(\mathbb{B}^{n},\mathbb{R}^{n_{1}})$, and $\varphi_{k}\in\mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{n_{1}})$ for $k\in\{0,1,\ldots,m-1\}$. Here the boundary condition in (1.3) are interpreted in the following distributional sense. For some fixed $r\in(0,1)$, let $f_{r}(x)=f(rx),~{}x\in\mathbb{B}^{n}$. Then for any $\zeta\in\mathbb{S}^{n-1}$,

[TABLE]

where $j\in\{1,\ldots,m\}$.

(I) If $m=1$, then all solutions to the equation (1.2) satisfying (1.3) are given by

[TABLE]

where

[TABLE]

and

[TABLE]

Here $d\sigma$ denotes the normalized Lebesgue surface measure on $\mathbb{S}^{n-1}$ and $dV$ is the Lebesgue volume measure on $\mathbb{B}^{n}$.

(II) If $m\geq 2$, then, by [20, p. 118–120] and the iterative procedure, we see that all solutions to the equation (1.2) satisfying (1.3) are given by

[TABLE]

where

[TABLE]

for $k\in\{1,\ldots,m-1\}$, and

[TABLE]

Moreover, we call $f$ a polyharmonic mapping if $f$ satisfies (1.6).

We refer the reader to [12, 17, 37] etc for more discussions in this line. In particular, if $m=1$ ($m=2$ resp.), then (1.2) is called the Poisson equation ( biharmonic equation resp.) (cf. [22, 27, 29, 32]).

1.2. Main results

Heinz in his classical paper [18] showed that the following result which is called the Schwarz Lemma of harmonic mappings: If $f$ is a harmonic mapping of the unit disk $\mathbb{D}:=\mathbb{B}^{2}$ into $\mathbb{D}$ with $f(0)=0$, then

[TABLE]

Later, Pavlović [39, Theorem 3.6.1] removed the assumption $f(0)=0$ and obtained the following sharp form

[TABLE]

where $f$ is a harmonic mapping from $\mathbb{D}$ into itself. The inequality (1.9) has been proved independently by Hethcote in [19]. For $n\geq 3$, the classical Schwarz lemma of harmonic mappings in $\mathbb{B}^{n}$ infers that if $f$ is a harmonic mapping of $\mathbb{B}^{n}$ into itself satisfying $f(0)=0,$ then

[TABLE]

where $r=|x|$, $N=(0,\ldots,0,1)^{\prime}$ and $U$ is a harmonic function of $\mathbb{B}^{n}$ into $[-1,1]$ defined by

[TABLE]

Here $\mathcal{X}$ is the indicator function, $S^{+}=\{x=(x_{1},\ldots,x_{n})^{\prime}\in\mathbb{S}^{n-1}:~{}x_{n}\geq 0\}$ and $S^{-}=\{x=(x_{1},\ldots,x_{n})^{\prime}\in\mathbb{S}^{n-1}:~{}x_{n}\leq 0\}$ (see [2]). In [25], Kalaj showed that the following result for harmonic mappings $f$ of $\mathbb{B}^{n}$ into itself:

[TABLE]

The first aim of the paper is to extend (1.10) to mappings satisfying the polyharmonic equation. More precisely, we shall prove the following.

Theorem 1.1**.**

Let $n\geq 3$, $m\geq 2,~{}n_{1}\in\mathbb{N}$, $\varphi_{m}\in\mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{n_{1}})$ and $\varphi_{k}\in\mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{n_{1}})$ for $k\in\{1,\ldots,m-1\}$. If $f\in\mathcal{C}^{2m}(\mathbb{B}^{n},\mathbb{R}^{n_{1}})\cap\mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{n_{1}})$ satisfies (1.2) with the boundary condition: $\Delta^{1}f=\varphi_{1},~{}\ldots,~{}\Delta^{m-1}f=\varphi_{m-1}$ on $\mathbb{S}^{n-1}$, then for $x\in\overline{\mathbb{B}^{n}}$,

[TABLE]

where $\varphi_{0}=f|_{\mathbb{S}^{n-1}}$, $\|\varphi_{m}\|_{\infty}=\sup_{x\in\mathbb{B}^{n}}|\varphi_{m}(x)|~{}\mbox{and}~{}\|\varphi_{k}\|_{\infty}=\sup_{\zeta\in\mathbb{S}^{n-1}}|\varphi_{k}(\zeta)|$ for $k\in\{0,1,\ldots,m-1\}$.

In particular, if we choose $f(x)=(M(1-|x|^{2(m-1)}),0,\ldots,0)\in\mathcal{C}^{2m}(\mathbb{B}^{n},\mathbb{R}^{n_{1}})\cap\mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{n_{1}})$ for $x\in\overline{\mathbb{B}^{n}}$, then the inequality (1.1) is sharp in $\mathbb{S}^{n-1}$, where $M>0$ is a constant.

Let $f$ be a harmonic homeomorphism of $\mathbb{D}$ onto itself with $f(0)=0.$ Heinz [18, Ineq. (18)] proved that, for any $\theta\in[0,2\pi]$,

[TABLE]

We refer to [25] for the extensive discussion on Heinz type inequalities for harmonic mappings in $\mathbb{R}^{n}~{}(n\geq 3)$. On the applications of the Heinz type inequalities, see [13, 26]. In the following, by using Theorem 1.1, we establish a Heinz type inequality for mappings satisfying the polyharmonic equation.

Theorem 1.2**.**

For $n\geq 3$, $m\geq 2,~{}n_{1}\in\mathbb{N}$ and $k\in\{1,\ldots,m-1\}$, suppose that $\varphi_{m}\in\mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{n_{1}})$ and $\varphi_{k}\in\mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{n_{1}})$ such that

[TABLE]

Let $f\in\mathcal{C}^{2m}(\mathbb{B}^{n},\mathbb{R}^{n_{1}})\cap\mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{n_{1}})$ satisfying (1.2) with the Dirichlet boundary value condition: $\Delta^{1}f=\varphi_{1},~{}\ldots,~{}\Delta^{m-1}f=\varphi_{m-1}$ on $\mathbb{S}^{n-1}$. If $f(0)=0$ and $\lim_{r\rightarrow 1^{-}}|f(r\zeta)|=1$ for some $\zeta\in\mathbb{S}^{n-1}$, then

[TABLE]

where $\Gamma$ and $F(\cdot,\cdot;\cdot;\cdot)$ are the Gamma function and the hypergeometric function, respectively $($see the Section 2.2$)$. In particular, if $\|\varphi_{k}\|_{\infty}=0$ for $k\in\{1,\ldots,m\}$, then this estimate (1.2) is sharp.

Next, we discuss an issue that is related to a classical result in geometric function theory: the theorem of Bloch. Recall that Bloch’s theorem says that an analytic function $f$ on the unit disk with $|f^{\prime}(0)|=1$ univalently covers a disk of radius $R$, where $R$ is a universal constant (see [42, 43]). However, for general class of functions, there is no Bloch’s Theorem. For example, consider $f_{k}(x)=(kx_{1},x_{2}/k,x_{3},\ldots,x_{n})^{\prime}$ for $k=\{1,2,\ldots\}$, where $n\geq 3$ and $x=(x_{1},\ldots,x_{n})^{\prime}\in\mathbb{B}^{n}$. It is easy to see that each $f_{k}$ is univalent and $|f_{k}(0)|=J_{f_{k}}(0)-1=0$. Furthermore, each $f_{k}(\mathbb{B}^{n})$ contains no ball with radius bigger than $1/k$. Hence, there does not exist an absolute constant $R_{0}>0$ which can work for all $k\in\{1,2,\ldots\}$, such that $\mathbb{B}(0,R_{0})$ is contained in the range $f_{k}(\mathbb{B}^{n})$. To establish analogs of the Bloch’s theorem for more general classes of functions, it is necessary to restrict our focus on certain subclasses (see [1, 4, 6, 8, 10, 14, 42, 44, 48]). In our next result, we establish a Bloch type theorem for mappings satisfying the polyharmonic equation, which gives an answer to the open problem in [10, Remark 1.2].

Definition 1.1**.**

Let $M$ be a positive constant and $n\geq 3$.

1. (1)

If $m=1$ in (1.2), then, for a given function $\varphi_{1}\in\mathcal{C}(\mathbb{B}^{n},\mathbb{R}^{n})$, we use $\mathcal{F}_{\varphi_{1}}^{M}$ to denote the set of all mappings $f\in\mathcal{C}^{2}(\mathbb{B}^{n},\mathbb{R}^{n})\cap\mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{n})$ satisfying $|f|\leq M$, $|f(0)|=J_{f}(0)-1=0$ and $\Delta f=\varphi_{1}~{}\mbox{in}~{}\mathbb{B}^{n}$. 2. (2)

If $m\geq 2$ in (1.2), then, for given functions $\varphi_{m}\in\mathcal{C}(\mathbb{B}^{n},\mathbb{R}^{n})$ and $\varphi_{1},\ldots,\varphi_{m-1}\in\mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{n})$, we denote by $\mathcal{F}_{\varphi_{1},\ldots,\varphi_{m}}^{M}$ the set of all mappings $f\in\mathcal{C}^{2m}(\mathbb{B}^{n},\mathbb{R}^{n})\cap\mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{n})$ satisfying $|f|\leq M$, $|f(0)|=J_{f}(0)-1=0$, and (1.2) with the following Dirichlet boundary value condition:

[TABLE]

Theorem 1.3**.**

Let $M$ be a positive constant and $n\geq 3$.

1. (a)

For $m\geq 2$, let $f\in\mathcal{F}_{\varphi_{1},\ldots,\varphi_{m}}^{M}$. Then there is a positive constant $R_{1}$ depending only on $M$ and $\varphi_{k}$ for $k\in\{1,\ldots,m\}$ such that $\mathbb{B}^{n}(0,R_{1})\subset f(\mathbb{B}^{n})$. 2. (b)

For $m=1$, let $f\in\mathcal{F}_{\varphi_{1}}^{M}$. Then $f(\mathbb{B}^{n})$ contains a ball $\mathbb{B}^{n}(0,R_{1})$ with the radius $R_{1}$ satisfying

[TABLE]

where $r_{1}$ is an unique solution of the equation:

[TABLE]

A homeomorphism $f:~{}\Omega\rightarrow\Omega^{\prime}$ between two open subsets $\Omega$ and $\Omega^{\prime}$ of $\mathbb{R}^{n}$ will be called a $K$-quasiconformal mapping if

1. (1)

$f$ is an absolutely continuous function in almost every segment parallel to some of the coordinate axes, and there exist the partial derivatives which are locally $L^{n}$ integrable functions on $\Omega$ (briefly, $f\in ACL^{n}$), and 2. (2)

$f$ satisfies the condition

[TABLE]

at almost every $x$ in $\Omega$.

We remark that, for a continuous mapping $f$, the condition (1) is equivalent to the condition that $f$ belongs to the Sobolev space $W_{n,loc}^{1}(\Omega)$ (cf. [45, 47]).

Given a subset $\Omega$ of $\mathbb{R}^{n}$, a function $\psi:~{}\Omega\rightarrow\mathbb{R}^{n}$ is said to be bi-Lipschitz if there is a constant $c\geq 1$ such that for all $x_{1},x_{2}\in\Omega$,

[TABLE]

Furthermore, $\psi$ is called Lipschitz if the right hand of (1.14) holds, and $\psi$ is said to be co-Lipschitz if it satisfies the left hand of (1.14).

It is well known that all sense-preserving bi-Lipschitz mappings are quasiconformal mappings (cf. [45]). But quasiconformal mappings are not necessarily bi-Lipschitz, not even Lipschitz (see [15, 23, 29]).

Pavlović [40] showed that harmonic quasiconformal mappings of the unit disk $\mathbb{D}$ onto itself are bi-Lipschitz mappings. In [38], Partyka and Sakan improved Pavlović’s corresponding result and obtained an asymptotically sharp version. By using the regularity theory of elliptic PDE’s, Kalaj and Pavlović [22] generalized the Lipschitz-property of harmonic quasiconformal mappings to the quasiconformal solutions of Poisson’s equations. The same problem in the space is much more complicated because of the lack of the techniques of complex analysis. It is well known that the harmonic extension of a homeomorphism of the unit circle is always a diffeomorphism of the unit disk $\mathbb{D}$. However, in higher dimensions, the situation is quite different. Namely, Melas [36] constructed a homeomorphism of $\mathbb{S}^{n-1}~{}(n\geq 3)$ whose harmonic extension fails to be diffeomorphic. On the discussion of the related topic, we refer to [5, 7, 8, 11, 23, 27, 29, 30, 31, 34, 35] and the references therein. By using Theorem 1.1 and Green’s potential theory, we obtain the asymptotically sharp Lipschitz constant which depends on the quasiconformal constant $K$ and the Dirichlet boundary value condition.

Theorem 1.4**.**

Let $K\geq 1$, $n\geq 3$, $m\geq 2$, $\varphi_{m}\in\mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{n})$ and $\varphi_{k}\in\mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{n})$ for $k\in\{1,\ldots,m-1\}$. Suppose that $f$ is a $K$-quasiconformal self-mapping of $\mathbb{B}^{n}$ satisfying $f(0)=0$ and (1.2) with the Dirichlet boundary value condition: $\Delta^{1}f=\varphi_{1},~{}\ldots,~{}\Delta^{m-1}f=\varphi_{m-1}$ on $\mathbb{S}^{n-1}$. Then there are nonnegative constants $N_{1}(K,\varphi_{1},\cdots,\varphi_{n})$ and $M_{1}(n,K)$ with

[TABLE]

such that for all $x_{1}$ and $x_{2}$ in $\mathbb{B}^{n}$,

[TABLE]

We will give several auxiliary results in Section 2. The proofs of Theorems 1.1 and 1.2 will be presented in Section 3, and the proof of Theorem 1.3 will be given in Section 4. Theorem 1.4 will be proved in the last Section.

2. Auxiliary results

2.1. Möbius Transformations of the Unit Ball

For $x\in\mathbb{B}^{n}$, the Möbius transformation in $\mathbb{B}^{n}$ is defined by

[TABLE]

The set of isometries of the hyperbolic unit ball is a Kleinian subgroup of all Möbius transformations of the extended spaces $\mathbb{R}^{n}\cup\{\infty\}$ onto itself. In the following, we make use of the automorphism group ${\operatorname{Aut}}(\mathbb{B}^{n})$ consisting of all Möbius transformations of the unit ball $\mathbb{B}^{n}$ onto itself. We recall the following facts from [3]: For $x\in\mathbb{B}^{n}$ and $\phi_{x}\in{\operatorname{Aut}}(\mathbb{B}^{n})$, we have $\phi_{x}(0)=x$, $\phi_{x}(x)=0$, $\phi_{x}(\phi_{x}(y))=y\in\mathbb{B}^{n}$,

[TABLE]

and

[TABLE]

2.2. Gauss Hypergeometric Functions

For $a,b,c\in\mathbb{R}$ with $c\neq 0,-1,-2,\ldots,$ the hypergeometric function is defined by the power series in the variable $x$

[TABLE]

Here $(a)_{0}=1$, $(a)_{k}=a(a+1)\cdots(a+k-1)$ for $k=1,2,\ldots$, and generally $(a)_{k}=\Gamma(a+k)/\Gamma(a)$ is the Pochhammer symbol, where $\Gamma$ is the Gamma function. In particular, for $a,b,c>0$ and $a+b<c$, we have (cf. [41])

[TABLE]

The following result is useful in showing one of our main results of the paper.

Lemma A. ([24] or [41, 2.5.16(43)])* For $\lambda_{1}>1$ and $\lambda_{2}>0$, we have*

[TABLE]

*where $\mathbf{B}(.,.)$ denotes the beta function and $r\in[0,1)$. *

2.3. The spherical coordinates

Throughout this article, by $S$ and $T$ we denote the spherical coordinates:

[TABLE]

and

[TABLE]

\big{(}S(r,\theta_{1},\cdots,\theta_{n-2},\theta_{n-1})=rT(\theta_{1},\cdots,\theta_{n-2},\theta_{n-1})\big{)}, defined by $S=(x_{1},x_{2},\cdots,x_{n-1})^{\prime},$

[TABLE]

Then we have

[TABLE]

where $\theta=(\theta_{1},\ldots,\theta_{n-1})^{\prime}$.

3. The heinz type inequalities for mappings satisfying polyharmonic equations

The following result easily follows from [10, Theorem 1].

Lemma B. * Let $G(x,y)$ be the Green function defined in (1.1). Then for $x\in\mathbb{B}^{n}$,*

[TABLE]

Lemma 3.1**.**

Let $G(x,y)$ be the Green function defined in (1.1). Then for $x\in\mathbb{B}^{n}$,

[TABLE]

Proof. Let

[TABLE]

For $x,y\in\mathbb{B}^{n}$ with $x\neq y$ and $|x|+|y|\neq 0$, let $z=\phi_{x}(y)$, where $\phi_{x}\in{\operatorname{Aut}}(\mathbb{B}^{n})$. Then $y=\phi_{x}(z)$ and

[TABLE]

It follows from (2.1) that

[TABLE]

which gives

[TABLE]

By (2.2), we have

[TABLE]

which, together with (3.2), implies that

[TABLE]

Using the spherical coordinates and Lemma A, we obtain

[TABLE]

By (3.1), (3.2), (3.3), (3) and the change of variables, we obtain

[TABLE]

The proof of this lemma is complete. ∎

Proof of Theorem 1.1

By (1.6), we have

[TABLE]

where $\varphi_{0}=f|_{\mathbb{S}^{n-1}}$, $G_{k}[\varphi_{k}]$ are defined in (1.1) for $k\in\{1,\ldots,m-1\}$, and $G_{m}[\varphi_{m}]$ is defined in (1.1). Next, we estimate $|G_{m}[\varphi_{m}]|$ and $|G_{k}[\varphi_{k}]|$ for $k\in\{1,\ldots,m-1\}$.

$\mathbf{Case~{}1.}$ $m=2$ and $k=1$.

By Lemma B, we have

[TABLE]

$\mathbf{Case~{}2.}$ $m\geq 3$ and $2\leq k\leq m-1$.

It follows from (3) and Lemma 3.1 that

[TABLE]

Now we estimate $|G_{m}[\varphi_{m}]|$, where $m\geq 2$.

By (3) and Lemma 3.1, we see that

[TABLE]

Therefore, it follows from (1.10), (3), (3) and (3) that

[TABLE]

The proof of the theorem is complete. ∎

Lemma C. ([25, Lemma 2.3])* For $r\in[0,1]$, let $\Phi(r)=\partial U(rN)/\partial r.$ Then $\Phi(r)$ is decreasing on $r\in[0,1]$ and*

[TABLE]

Proof of Theorem 1.2

By (1.6), we have

[TABLE]

where $G_{k}[\varphi_{k}]$ are defined in (1.1) for $k\in\{1,\ldots,m-1\}$, and $|G_{m}[\varphi_{m}]|$ is defined in (1.1). By the assumption, we see that

[TABLE]

which, together with (3), (3) and Theorem 1.1, implies that

[TABLE]

where $r\in[0,1)$. On the other hand, for $x\in\mathbb{B}^{n}$, there is a $\rho\in(|x|,1)$ such that

[TABLE]

where $r=|x|$. It follows from (3), (3.9) and Lemma C that

[TABLE]

At last, we prove the sharpness part. Especially, if $\|\varphi_{k}\|_{\infty}=0$ for $k\in\{1,\ldots,m\}$, then the sharpness part easily follows from [25, Theorem 2.5]. The proof of this theorem is complete. ∎

4. Bloch type theorem for mappings satisfying polyharmonic equations

The main purpose of this section is to prove Theorem 1.3. We start with some lemmas which are useful to the proof of Theorem 1.3.

Theorem D. (see [21, Theorem 2.7])* For $n\geq 3$, let $u$ be a harmonic function of $\mathbb{B}^{n}$ into itself. Then*

[TABLE]

A matrix-valued function A(x)=\big{(}a_{ij}(x)\big{)}_{n\times n}is called matrix-valued harmonic function if each of its entries $a_{ij}(x)$ is a harmonic function from an open subset $\Omega\subset\mathbb{R}^{n}$ into $\mathbb{R}$.

Lemma E. ([9, Lemma 3.1])* For $r>0$, let A(x)=\big{(}a_{ij}(x)\big{)}_{n\times n} be a matrix-valued harmonic function defined on $\mathbb{B}^{n}(0,r)$. If $A(0)=0$ and $|A(x)|\leq M$ in $\mathbb{B}^{n}(0,r),$ then*

[TABLE]

*where $M$ is a positive constant. *

Lemma F. ([29, Lemma 2.5] or [46, p. 24-26])* Let $\varrho$ be a bounded (absolutely) integrable function defined on a bounded domain $\Omega\subset\mathbb{R}^{n}$. Then the potential type integral*

[TABLE]

belongs to the space $\mathcal{C}^{k}(\mathbb{R}^{n},\mathbb{R})$, where $k+\alpha<n.$ Moreover,

[TABLE]

Lemma 4.1**.**

Suppose that $\varphi_{k}\in\mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{n})$ and $G_{k}[\varphi_{k}]$ are defined in (1.1), where $m\in\mathbb{N}\setminus\{1\}$, $k\in\{1,\ldots,m-1\}$ and $n\geq 3.$ Then, for $x\in\mathbb{B}^{n}$,

[TABLE]

where

[TABLE]

Moreover, $D_{G_{k}[\varphi_{k}]}$ has a continuous extension to the boundary and, for $\varepsilon\in\mathbb{S}^{n-1}$,

[TABLE]

Proof. We divide the proof of this lemma into two steps.

Step 4.1**.**

We first prove (4.2).

$\mathbf{Case~{}1.}$ $m\geq 3$ and $2\leq k\leq m-1$.

For $k\in\{2,\ldots,m-1\}$ and $m\geq 3$, let

[TABLE]

Then, by (3), we have

[TABLE]

where $y_{1}\in\mathbb{B}^{n}.$

Applying Lemma F to

[TABLE]

we see that, for any fixed $\xi\in\mathbb{R}^{n}$,

[TABLE]

By calculations, we obtain

[TABLE]

which gives

[TABLE]

where $x\in\mathbb{B}^{n}.$ It follows from (4.4), (4.5), (4.6), Lemma F and the Lebesgue Dominated Convergence Theorem that, for $\varepsilon\in\mathbb{S}^{n-1}$,

[TABLE]

Next, we estimate $|D_{G_{k}[\varphi_{k}]}(\varepsilon)|.$ By (4.4) and (4), we have

[TABLE]

which implies that

[TABLE]

$\mathbf{Case~{}2.}$ $m=2$ and $k=1$.

Using the spherical coordinates and Proposition 2.2, we obtain

[TABLE]

which yields that

[TABLE]

Step 4.2**.**

Next, we show that (4.1).

$\mathbf{Case~{}3.}$ $m\geq 3$ and $2\leq k\leq m-1$.

In order to estimate $|D_{G_{k}[\varphi_{k}]}(x)\xi|,$ we first show that, for $x\in\mathbb{B}^{n},$

[TABLE]

where

[TABLE]

In order to prove (4.8), we let $z=\phi_{x}(y_{1})$, where $\phi_{x}\in{\operatorname{Aut}}(\mathbb{B}^{n})$. Then, by (3.1) and (3.2), we have

[TABLE]

which, together with (2.3), gives that

[TABLE]

where

[TABLE]

and

[TABLE]

By computations, we obtain

[TABLE]

and

[TABLE]

which, together with (3), imply that

[TABLE]

where

[TABLE]

It follows from (3) and the inequality

[TABLE]

that

[TABLE]

Then combining (4.9) and (4.10) gives the estimate for (4.8). Therefore, we conclude from (4.4), (4.5) and (4.8) that

[TABLE]

which yields that

[TABLE]

$\mathbf{Case~{}4.}$ $m=2$ and $k=1$.

By [24, Theorem 2.1],

[TABLE]

which implies that

[TABLE]

The proof of this lemma is finished. ∎

Lemma 4.2**.**

Suppose that $\varphi_{m}\in\mathcal{C}(\mathbb{B}^{n},\mathbb{R}^{n})$ and $G_{m}[\varphi_{m}]$ are defined in (1.1), where $m\in\mathbb{N}\setminus\{1\}$ and $n\geq 3.$ Then, for $x\in\mathbb{B}^{n}$,

[TABLE]

where $\delta(n)$ is defined in Lemma 4.1.

Moreover, $D_{G_{m}[\varphi_{m}]}$ has a continuous extension to the boundary and, for $\varepsilon\in\mathbb{S}^{n-1}$,

[TABLE]

Proof. Let

[TABLE]

By Lemma B and (3), we have

[TABLE]

which, together with (4.6), Lemma F and the Lebesgue Dominated Convergence Theorem, implies that, for any fixed $\xi\in\mathbb{R}^{n}$,

[TABLE]

and $D_{G_{m}[\varphi_{m}]}$ has a continuous extension to the boundary.

Next we estimate $|D_{G_{m}[\varphi_{m}]}(x)|$ for $x\in\mathbb{B}^{n}$, and $|D_{G_{m}[\varphi_{m}]}(\eta)|$ for $\eta\in\mathbb{S}^{n-1}$, respectively.

It follows from (4.8), (4) and (4.13) that

[TABLE]

and

[TABLE]

The proof of this lemma is complete. ∎

Lemma G. ([33, Lemma 4])* Let $A$ be an $n\times n$ real $($or complex$)$ matrix with $|A|\neq 0$. Then for any unit vector $\theta\in\partial\mathbb{B}^{n}$, the inequality*

[TABLE]

*holds. *

Proof of Theorem 1.3

We first prove (a). By (1.6), we have

[TABLE]

where $\varphi_{0}=f|_{\mathbb{S}^{n-1}}$. Then

[TABLE]

We split the remaining proof into six steps to complete it.

Step 4.3**.**

The estimate of $|D_{P[\varphi_{0}]}(x)-D_{P[\varphi_{0}]}(0)|$ for $x\in\mathbb{B}^{n}(0,r_{0})$, where $r_{0}=1/2$.

For $x\in\mathbb{B}^{n}(0,r_{0})$, by Theorem D, we have

[TABLE]

which, together with Lemma E, implies that

[TABLE]

where

[TABLE]

Step 4.4**.**

The estimate of $\left|D_{G_{m}[\varphi_{m}]}(x)-D_{G_{m}[\varphi_{m}]}(0)\right|$ for $x\in\mathbb{B}^{n}(0,r_{0})$, where $m\geq 2$.

Claim 4.1**.**

[TABLE]

and $\lim_{x\rightarrow 0}\mathscr{L}_{m}(x)=0$, where

[TABLE]

and $M_{m}=\frac{\|\varphi_{m}\|_{\infty}}{2n}\left[\frac{n+4}{4n(n+2)}\right]^{m-2}$.

Now we prove the Claim 4.1. For $\xi\in\mathbb{S}^{n-1}$, let

[TABLE]

Then, by (4), Lemma F and the Lebesgue Dominated Convergence Theorem, we have

[TABLE]

where $\varrho_{\varphi_{m}}(y_{1})$ is defined in (4.11).

On the other hand, by Lemma F and (4.6), we see that

[TABLE]

and

[TABLE]

The proof of Claim 4.1 is complete.

Step 4.5**.**

For $m\geq 2$ and $k\in\{1,\ldots,m-1\}$, we estimate $\left|D_{G_{k}[\varphi_{k}]}(x)-D_{G_{k}[\varphi_{k}]}(0)\right|$ for $x\in\mathbb{B}^{n}(0,r_{0})$.

Claim 4.2**.**

For $m\geq 2$ and $k\in\{1,\ldots,m-1\}$, we have

[TABLE]

and $\lim_{x\rightarrow 0}\mathscr{L}_{k}(x)=0$, where

[TABLE]

and $M_{k}=\frac{\|\varphi_{k}\|_{\infty}}{2n}\left[\frac{n+4}{4n(n+2)}\right]^{k-2}$.

In order to prove Claim 4.2, we divide it into two cases. For $\xi\in\mathbb{S}^{n-1}$, let

[TABLE]

$\mathbf{Case~{}1.}$ $m\geq 3$ and $2\leq k\leq m-1$.

It follows from (4.4), Lemma F and the Lebesgue Dominated Convergence Theorem that

[TABLE]

where $\varrho_{\varphi_{k}}(y_{1})$ is defined in (4).

Next we prove $\lim_{x\rightarrow 0}\mathscr{L}_{k}(x)=0$.

By Lemma F and (4.6), we see that

[TABLE]

and

[TABLE]

$\mathbf{Case~{}2.}$ $m=2$ and $k=1$.

In this case, we have

[TABLE]

It follows from Lemma F and (4.6) that

[TABLE]

and

[TABLE]

The proof of Claim 4.2 is finished.

Step 4.6**.**

The estimate of $|D_{f}(0)\xi|$, where $\xi\in\mathbb{S}^{n-1}$.

By Theorem D, Lemmas 4.1 and 4.2, we have

[TABLE]

where $M^{\ast}=\frac{n}{2}M+\frac{n}{n+1}\|\varphi_{1}\|_{\infty}+\sum_{k=2}^{m}\frac{\|\varphi_{k}\|_{\infty}}{2n}\left[\frac{n+4}{4n(n+2)}\right]^{k-2}\delta(n)$ and $\delta(n)$ is defined in Lemma 4.1.

For any $\xi\in\mathbb{S}^{n-1}$, it follows from (4.15) and Lemma G that

[TABLE]

Step 4.7**.**

We will show that there is a constant $r_{1}\in(0,r_{0})$ such that $f$ is injective in $\mathbb{B}^{n}(0,r_{1})$.

In order to prove the injection of $f$ in $\mathbb{B}^{n}(0,r_{1})$, let

[TABLE]

$x\in\mathbb{B}^{n}(0,r_{0})$. Since $\mathscr{G}(|x|)$ is monotonous and continuous in $\mathbb{B}^{n},$

[TABLE]

we see that there is an unique constant $r_{1}\in(0,r_{0}]$ such that $\mathscr{G}(r_{1})=0$. For any $x^{\prime},x^{\prime\prime}\in\mathbb{B}^{n}(0,r_{1})$, we use $[x^{\prime},x^{\prime\prime}]$ to denote the segment from $x^{\prime}$ to $x^{\prime\prime}$ with the endpoints $x^{\prime}$ and $x^{\prime\prime}$. Hence by (4.14), Claims 4.1, 4.2 and (4.16), we have

[TABLE]

which yields that $f(x^{\prime})\neq f(x^{\prime\prime})$. Thus, from the arbitrariness of $x^{\prime}$ and $x^{\prime\prime}$, the injection of $f$ follows.

Step 4.8**.**

We will prove that the image $f(\mathbb{B}^{n}(0,r_{1}))$ contains a ball $\mathbb{B}^{n}(0,R_{1})$, where

[TABLE]

To reach this goal, let $\varsigma\in\partial\mathbb{B}^{n}(0,r_{1})$. Then we infer from (4.14), Claims 4.1, 4.2 and (4.16) that

[TABLE]

which implies that $f(\mathbb{B}^{n}(0,r_{1}))$ contains a ball $\mathbb{B}^{n}(0,R_{1})$ with

[TABLE]

Next we prove (b). If $m=1$, then, by (1.4), we have

[TABLE]

where $\varphi_{0}=f|_{\mathbb{S}^{n-1}}$ and $G_{1}[\varphi_{1}]$ is defined in (1.5). In the following, we will use the similar reasoning as in the proof of (a) to prove (b). Let $r_{1}$ be an unique solution of the following equation:

[TABLE]

Then, for any $x^{\prime},x^{\prime\prime}\in\mathbb{B}^{n}(0,r_{1})$, we have

[TABLE]

which yields that $f$ is injective in $\mathbb{B}^{n}(0,r_{1})$. Therefore, for any $\varsigma\in\partial\mathbb{B}^{n}(0,r_{1})$, we see that

[TABLE]

The proof of this theorem is complete. ∎

5. The Lipschitz continuity of quasiconformal self-mappings satisfying polyharmonic equations

Lemma 5.1**.**

Suppose that $n\geq 3$, $m\in\mathbb{N}\setminus\{1\}$, $\varphi_{m}\in\mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{n})$ and $\varphi_{k}\in\mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{n})$ for $k\in\{0,1,\ldots,m-1\}$. Let $f$ be a mapping of $\overline{\mathbb{B}^{n}}$ onto itself satisfying (1.2) and the boundary conditions $\Delta^{m-1}f=\varphi_{m-1},~{}\ldots,~{}\Delta^{1}f=\varphi_{1}$ on $\mathbb{S}^{n-1}$. In addition, let $f$ be Lipschitz continuous in $\mathbb{B}^{n}$ satisfying $|f(x)|\rightarrow 1$ as $|x|\rightarrow 1^{-}$, where $x\in\mathbb{B}^{n}$. Then, for almost every $t\in\mathbb{S}^{n-1}$, the following limits exist:

[TABLE]

Further, for $\varphi_{0}:=f|_{\mathbb{S}^{n-1}}$ and $x(\theta)=\varphi_{0}(T(\theta)):=\varphi_{0}(t)$, we have

[TABLE]

and

[TABLE]

where $M_{x}(\theta)$ and $M_{T}(\theta)$ are the square roots of Gram determinants of $D_{x}$ and $D_{T}$, respectively.

Before the proof of Lemma 5.1, let us recall the following result.

Lemma H. ([29, Lemma 2.1])* Let $u=P[f]$ be a harmonic mapping of $\mathbb{B}^{n}$ into $\mathbb{R}^{n}$, and assume that its derivative $v=D_{u}$ is bounded in $\mathbb{B}^{n}$ (or equivalently, let $u$ be Lipschitz continuous), where $u|_{\mathbb{S}^{n-1}}=f\in L^{1}(\mathbb{S}^{n-1})$. Then there exists a mapping $A\in L^{\infty}(\mathbb{S}^{n-1})$ defined in the $\mathbb{S}^{n-1}$ such that $D_{f}(x)=P[A](x)$ and for almost every $\eta\in\mathbb{S}^{n-1}$ there holds the relation*

[TABLE]

Moveover, the function $f\circ T$ is differentiable almost everywhere in $Q^{n-1}$ and there holds

[TABLE]

*where $\theta=(\theta_{1},\theta_{2},\ldots,\theta_{n-1})$, $Q^{n-1}$ and $T$ are defined in the part of 2.3. *

The proof of Lemma 5.1

We first prove the existence of the two limits in (5.1). By Lemmas 4.1 and 4.2, we see that for any $t\in\mathbb{S}^{n-1}$,

[TABLE]

and $G_{k}[\varphi_{k}]$ are Lipschitz continuous in $\mathbb{B}^{n}$, where $k\in\{1,\ldots,m\}$. Since $f$ is Lipschitz continuous in $\mathbb{B}^{n}$, we see that

[TABLE]

are also Lipschitz continuous in $\mathbb{B}^{n}$, where $\varphi_{0}=f|_{\mathbb{S}^{n-1}}$. It follows from Lemma H that, for almost every $t\in\mathbb{S}^{n-1}$,

[TABLE]

does exist, which, together with (5.4), guarantees that for almost every $t\in\mathbb{S}^{n-1}$,

[TABLE]

also exists.

By (5.5) and $J_{f}=\det D_{f},$ we conclude that

[TABLE]

exists for almost every $t\in\mathbb{S}^{n-1}$.

Next we estimate $J_{f}(t)$. It follows from (5.5) that the mapping $x$, $x(\theta)=\varphi_{0}(T(\theta))$, defines the outer normal vector field $\mathbf{n}_{x}$ almost everywhere in $\mathbb{S}^{n-1}$ at the point $x(\theta)=\varphi_{0}(T(\theta))=(x_{1},\ldots,x_{n})^{\prime}$ by the formula

[TABLE]

where $e_{1}=(1,0,\ldots,0)^{\prime}$, $\ldots$, $e_{n}=(0,0,\ldots,1)^{\prime}$ and $T$ is defined in the part of 2.3. Let $f(S(r,\theta))=y=(y_{1},\ldots,y_{n})^{\prime}$, where $S$ is defined in 2.3.

By (5.5), for $i\in\{1,\ldots,n\}$ and $j\in\{1,\ldots,n-1\}$, we have

[TABLE]

and

[TABLE]

which imply that

[TABLE]

where

[TABLE]

Since

[TABLE]

by (5), we see that

[TABLE]

In the following, we will demonstrate the estimate of $|\mathcal{X}_{k}|$ for $k\in\{1,\ldots,m\}$.

$\mathbf{Case~{}1.}$ $k=1$. Then, by (5.7), we get

[TABLE]

which implies that

[TABLE]

$\mathbf{Case~{}2.}$ $k\in\{2,\ldots,m-1\}$ and $m\geq 3$. In this case, by (5.7), we have

[TABLE]

which, together with (4.4), gives that

[TABLE]

where $\varrho_{\varphi_{k}}(y_{1})$ is defined in (4).

$\mathbf{Case~{}3.}$ $k=m$. Then it follows from (4) and (5.7) that

[TABLE]

where $\varrho_{\varphi_{m}}(y_{1})$ is defined in (4.11). Hence (5.1) and (5.1) follow from (5.8), (5), (5), (5) and

[TABLE]

The proof of this lemma is complete. ∎

Lemma I. ([29, Lemma 2.2])* Let $u$ be a harmonic Lipschitz continuous mapping defined in $\mathbb{B}^{n}$. Suppose that $D_{u}$ exists almost everywhere in $\mathbb{S}^{n-1}$. Then for $x\in\mathbb{B}^{n}$,*

[TABLE]

Lemma J. ([27, Lemma 4.8])* Let $\mathcal{A}:~{}\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a linear operator such that $\mathcal{A}=[a_{ij}]_{i,j=1,\ldots,n}$. If $\mathcal{A}$ is $K$-quasiconformal, then the following sharp inequalities hold:*

[TABLE]

Lemma K. ([27, Corollary 3.7])* Assume that $u:~{}\overline{\mathbb{B}^{n}}\rightarrow\mathbb{R}^{n}$ is a $K$-quasiregular, twice differentiable mapping, continuous on $\overline{\mathbb{B}^{n}}$, and that $u|_{\mathbb{S}^{n-1}}\in\mathcal{C}^{1,\alpha}$. If, in addition, $u$ satisfies the differential inequality*

[TABLE]

*for some positive constants $a$ and $b$, then $|D_{u}|$ is bounded and $u$ is Lipschitz continuous. *

The following is the so-called Mori’s Theorem of quasiconformal mappings defined in $\mathbb{B}^{n}$ (see [15]).

Theorem L. * If $u$ is a $K$-quasiconformal self-mapping of $\mathbb{B}^{n}$ with $u(0)=0$, then there exists a constant $q(n,K)$, satisfying the condition $q(n,K)\rightarrow 1$ as $K\rightarrow 1$, such that, for any $x,y\in\mathbb{B}^{n}$,*

[TABLE]

*Moreover, the mapping $u(x)=|x|^{-1+K^{1/(1-n)}}x$ shows that the exponent $K^{1/(1-n)}$ is optimal in the class of arbitrary $K$-quasiconformal homeomorphism from $\mathbb{B}^{n}$ onto itself. *

The proof of Theorem 1.4

Let’s begin the proof of this theorem with the following claim.

Claim 5.1**.**

The limits

[TABLE]

exist almost everywhere in $\mathbb{S}^{n-1}$.

In order to prove the existence of these two limits, we need to obtain the upper bound of $|\Delta f(x)|$ in $\mathbb{B}^{n}$, and we divide it into two cases to estimate.

$\mathbf{Case~{}1.}$ $m=2.$

By [20, pp. 118-120] (see also [27, 28]), we have that for $x\in\mathbb{B}^{n}$,

[TABLE]

It follows from Lemma B that

[TABLE]

$\mathbf{Case~{}2.}$ $m\geq 3.$

By (1.6), we have that for $x\in\mathbb{B}^{n}$,

[TABLE]

where

[TABLE]

for $k\in\{1,\ldots,m-2\}$, and

[TABLE]

For $x\in\mathbb{B}^{n}$ and $k\in\{1,\ldots,m-2\}$, by (3), we obtain

[TABLE]

and, by (3), we have

[TABLE]

which imply that

[TABLE]

Since $f$ is a $K$-quasiconformal self-mapping of $\mathbb{B}^{n}$, we see that $f$ can be extended to the homeomorphism of $\overline{\mathbb{B}^{n}}$ onto itself. Hence Claim 5.1 follows from (5), Lemmas K and 5.1.

In the following, for convenience, let

[TABLE]

Since for almost all $x_{1}$ and $x_{2}\in\mathbb{B}^{n}$,

[TABLE]

we see that, to prove the Lipschitz continuity of $f$, it suffices to estimate the quantity $C_{2}(K,\varphi_{1},\cdots,\varphi_{n})$. To reach this goal, we first show that the quantity $C_{2}(K,\varphi_{1},\cdots,\varphi_{n})$ satisfies an inequality which is stated in the following claim.

Claim 5.2**.**

C_{2}(K,\varphi_{1},\cdots,\varphi_{n})\leq\big{(}C_{2}(K,\varphi_{1},\cdots,\varphi_{n})\big{)}^{1-K^{1/(1-n)}}\mu_{1}+\mu_{2},* where*

[TABLE]

$q(n,K)$* is from Theorem L, $\mu_{2}=\mu_{3}+\mu_{4},$*

[TABLE]

and

[TABLE]

To prove the claim, we need the following preparation. Firstly, we prove that

[TABLE]

almost everywhere in $Q^{n-1}$, where $\varphi_{0}=f|_{\mathbb{S}^{n-1}}$.

Since $f$ is $K$-quasiconformal mapping, by Claim 5.1, we see that

[TABLE]

exists almost everywhere in $Q^{n-1}$. By Lemma H, we obtain

[TABLE]

exists almost everywhere in $Q^{n-1}$. It follows from (5.17), Lemma J and

[TABLE]

that

[TABLE]

where $M_{x}(\theta)$ and $M_{T}(\theta)$ are defined in Lemma 5.1. From (5.1) in Lemma 5.1, (5.16) and (5.18), we infer that

[TABLE]

almost everywhere in $Q^{n-1}$, which yields that (5.15).

Secondly, we show that for any $\epsilon>0$, there exists $\theta_{\epsilon}\in Q^{n-1}$ such that

[TABLE]

Since

[TABLE]

is harmonic, by Lemma I, we see that

[TABLE]

which, together with Lemmas 4.1 and 4.2, gives that

[TABLE]

Hence (5.19) follows from (5) and Claim 5.1.

For $\theta\in Q^{n-1}$, let

[TABLE]

Then by Theorem L, we have

[TABLE]

which, together with (5.15) and (5.19), gives that

[TABLE]

By letting $\epsilon\rightarrow 0^{+}$, we get from (5) that

[TABLE]

which yields that

[TABLE]

Claim 5.3**.**

If \big{(}1-K^{1/(1-n)}\big{)}\mu_{1}<1, then

[TABLE]

where \mu_{5}=\frac{K^{1/(1-n)}\mu_{1}+\mu_{2}}{1-\big{(}1-K^{1/(1-n)}\big{)}\mu_{1}}.

The proof of this claim easily follows from [24, Lemma 2.9].

In the following, an upper bound of $C_{2}(K,\varphi_{1},\cdots,\varphi_{n})$ will be established. By Claims 5.2 and 5.3, we obtain that

[TABLE]

where

[TABLE]

In the following, we will break $C_{3}$ down into the form we need. By (5.23), we have

[TABLE]

where $M^{\ast}_{1}=M_{1}^{{}^{\prime}}(n,K)+N_{1}^{{}^{\prime}}(K,\varphi_{1},\cdots,\varphi_{n})$, $M^{\ast}_{2}=M_{1}^{{}^{\prime\prime}}(n,K)+N_{1}^{{}^{\prime\prime}}(K,\varphi_{1},\cdots,\varphi_{n})$, $M_{1}^{{}^{\prime}}(n,K)=(K\mu_{1})^{K^{1/(n-1)}}$, M_{1}^{{}^{\prime\prime}}(n,K)=\frac{K^{1/(1-n)}\mu_{1}}{1-\big{(}1-K^{1/(1-n)}\big{)}\mu_{1}}, $N_{1}^{{}^{\prime}}(K,\varphi_{1},\cdots,\varphi_{n})=(K\mu_{1}+\mu_{2})^{K^{1/(n-1)}}-(K\mu_{1})^{K^{1/(n-1)}}$, and N_{1}^{{}^{\prime\prime}}(K,\varphi_{1},\cdots,\varphi_{n})=\frac{\mu_{2}}{1-\big{(}1-K^{1/(1-n)}\big{)}\mu_{1}}.

Let

[TABLE]

and

[TABLE]

It follows from the facts

[TABLE]

that these two constants are what we need. The proof of this theorem is complete. ∎

6. Acknowledgments

This research was partly supported by the National Science Foundation of China (grant no. 12071116), the Hunan Provincial Natural Science Foundation of China (No. 2022JJ10001), the Key Projects of Hunan Provincial Department of Education (grant no. 21A0429); the Double First-Class University Project of Hunan Province (Xiangjiaotong [2018]469), the Science and Technology Plan Project of Hunan Province (2016TP1020), and the Discipline Special Research Projects of Hengyang Normal University (XKZX21002)