# A simple random matrix model for the vibrational spectrum of jammed packings

E. Stanifer, P.K. Morse, A.A. Middleton, M.L. Manning  Syracuse University
Department of Physics

###### Abstract

To better understand the surprising low-frequency vibrational modes in
structural glasses, we study the spectra of a large ensemble of sparse random
matrices where disorder is controlled by the distribution of bond weights and
network coordination. We find $D(\omega)$ has three regimes: a very-low
frequency regime that can be predicted analytically using extremal statistics,
an intermediate regime with quasi-localized modes, and a plateau with
$D(\omega)\sim\omega^{0}$. In the special case of uniform bond weights, the
intermediate regime displays $D(\omega)\sim\omega^{4}$, independent of network
coordination and system size, just as recently discovered in simulations of
structural glasses.

The vibrational spectra of disordered glassy materials exhibit universal
features. Although these features govern the mechanical response and provide
insight into mechanisms for material failure, their origin remains poorly
understood.

Perhaps the most well-studied feature of the density of vibrational states
$D(\omega)$ is the boson peak, which is an excess of vibrational modes above
the Debye prediction, $D(\omega)\propto\omega^{d-1}$ (Manning and Liu, 2011;
O’Hern  _et al._ , 2003; DeGiuli  _et al._ , 2014). In jammed packings the
frequency at which the peak occurs, $\omega^{*}$, scales linearly with the
average excess number of contacts $\delta z$ above the isostatic point where
the number of constraints equals the degrees of freedom (Wyart  _et al._ ,
2005; Silbert  _et al._ , 2005; O’Hern  _et al._ , 2003). Additionally, the
eigenvector statistics of modes in the boson peak follow a universal
distribution (Manning and Liu, 2015).

Recently, another universal feature has been identified in simulations of low-
dimensional jammed systems: $D(\omega)\sim\omega^{4}$ below $\omega^{*}$
(Lerner  _et al._ , 2016; Mizuno  _et al._ , 2017; Kapteijns  _et al._ ,
2018), which deviates from recent mean-field calculations for the spectra in
infinite dimensions that predict $D(\omega)\sim\omega^{2}$ (Charbonneau  _et
al._ , 2016; Parisi and Zamponi, 2010). This interesting behavior has also
been found in Heisenberg spin glass systems (Baity-Jesi  _et al._ , 2015).
Understanding this regime is important, as the vibrational modes are
quasilocalized and help govern flow and failure in disordered solids (Manning
and Liu, 2011; Wijtmans and Manning, 2017; Tanguy  _et al._ , 2010; Tsamados
_et al._ , 2008; Ashton and Garrahan, 2009; Brito and Wyart, 2007; Baity-Jesi
_et al._ , 2015).

Given the success of random matrix theory in predicting universal features in
other physical systems (Mehta, 2004),it is natural to wonder if a random
matrix model may also explain the $\omega^{4}$ scaling in jammed packings.
Other features, including the boson peak, have already been understood in
terms of Euclidean random matrices, which are dynamical matrices for a set of
points that are randomly and uniformly distributed in space (Parisi, 2002).

Although there are generic arguments that the global minima of random
functions should have a spectrum that scales as $\omega^{4}$ (Gurarie and
Chalker, 2003), we would like to construct a random matrix model to provide
insight into how features of the $\omega^{4}$ region, such as the prefactor,
or the location of the scaling regime, change with parameters such as the
excess coordination $\delta z$. Such an understanding is important for
predicting how material preparation protocols alter the mechanical response of
glassy materials.

We study matrices that share three important features with the dynamical
matrix: they are symmetric, positive semidefinite, and force balancing. In
higher dimensions, force balance corresponds to $d$ sum rules on partial sums
of entries in each row of a matrix, while in 1D, the force balancing
restriction simply requires the sum over all the entries in a row must be zero
(Manning and Liu, 2011). This rule is also obeyed by standard or weighted
Laplacians, $L_{ij}$, which are also symmetric and positive semi-definite.
They are defined by

$L_{ij}=\begin{cases}-k_{ij}&i\text{ and }j\text{ are connected},\\\
\sum_{l\neq i}k_{ik}&i=j,\\\ 0&\text{Otherwise},\end{cases}$ (1)

where $k_{ij}$ is the independently chosen random weight of the edge between
particles $i$ and $j$ and in the special case of the standard Laplacian,
$k_{ij}=1$ (Merris, 1994). Standard Laplacian matrices are well-studied and
possess distinctive vibrational spectra (Aspelmeier and Zippelius, 2011), so
we focus on weighted Laplacians for the remainder of this Letter.

In order to calculate the Laplacian we must specify the topology of the
underlying graph. Although recent advances have been made in analytically
characterizing the spectra of Laplacians on an Erdős-Rényi graph (Cicuta  _et
al._ , 2018), Erdős-Rényi networks are not locally isostatic, as a significant
fraction of nodes are under-coordinated (fewer than isostatic coordination
$z_{c}=2d$), which leads to highly localized excitations that are not seen in
jammed packings.

Instead, we consider the weighted Laplacian on a $z_{c}$-regular graph with a
small number of additional edges, or crossbonds. Since weighted Laplacians
only obey one sum rule, they are effectively 1D and $z_{c}=2$. The number of
additional bonds is $\delta zN$ where $N$ is the number of points and $\delta
z$ is the excess coordination.

Another important control parameter is the distribution of the edge weights
and, in particular, the weight of this distribution near zero. We choose to
parameterize this distribution as a power law with exponent $\alpha$,
normalized so that the mean is 1, $\rho(k)\propto k^{\alpha}$ on
$\left[0,\frac{\alpha+2}{\alpha+1}\right]$. A uniform distribution corresponds
to $\alpha=0$ and we only consider normalizable distributions, $\alpha>-1$.

_Finite size scaling for the weighted ring_ : We first study the finite size
scaling of the low frequency excitations at isostaticity, when $\delta z=0$
and the underlying network topology is simply a ring of size $N$. Although
this is a well-studied model, we believe its finite-size scaling can provide
insight into the case with $\delta z>0$.

Figure 1: The rescaled density of states, $D(\omega^{\prime})$, for the two-
regular graph with $N$=16, 64, 256, 1024, and 4096 and $\alpha=0$, normalized
by system size, $N$, averaged over at least $10^{6}$ matrices. The analytic
prediction for the low-frequency scaling is shown as the black dashed line. In
the upper-left we have a sketch of a 1d chain with periodic boundary
conditions (the open circles are the same node) Inset: Unscaled density of
states, $D(\omega)$.

The inset to Fig. 1 shows the sample averaged density of states for
$\alpha=0$, calculated via diagonalization of the matrix, as a function of
system size $N$, averaged over $2\times 10^{6}$ matrices. The main panel shows
the sample averaged density of states as a function of the normalized
frequency, $\omega^{\prime}=\omega N$, highlighting a region of power-law
scaling at the lowest frequencies that disappears in the thermodynamic limit.

We hypothesize that the lowest-energy mode on a weighted ring is well
approximated by a stretching of the two weakest bonds, with all other bond
lengths relatively fixed. We expect this to be the case when $\alpha\leq 0$,
so that the weight of the lowest two bonds are well separated from bonds with
larger values of $k_{ij}$, especially in the limit of low $\omega$,
$\omega<N^{-\frac{2\alpha+3}{4\alpha+3}}$.

If the two weakest bonds have strengths $k_{1}$ and $k_{2}$ and are separated
by $m$ nodes, the frequency of this mode is
$\sqrt{\frac{N(k_{1}+k_{2})}{m(N-m)}}$. As we show in the supplement, one can
use extremal statistics to find the exact distribution of the weakest bonds on
the ring to predict that the low-frequency density of states scales as:

$D(\omega)\propto N^{2\alpha+3}\omega^{4\alpha+3}$ (2)

For a uniform distribution of bond weights ($\alpha=0$), the contribution of
these modes to the density of states scales as $(N\omega)^{3}$. The scaling of
Eq. 2, using $\alpha=0$, is shown as the black dashed line in Fig. 1.

_Crossbonded ring with uniform bond weights:_ We hypothesize that adding a
small number of crossbonds alters the low-frequency behavior by reducing the
effective distance between the two weakest bonds. In the case of $\delta z=0$,
the two weakest bonds separate the ring into two segments that can move
relative to one another at nearly zero cost, but if a crossbond connects those
two segments it will significantly increase the energy of that mode.
Therefore, the weak bonds that contribute to low-frequency modes must both be
in a segment between crossbonds. Because there are $N\delta z$ such segments,
we expect that crossbonds give rise to an extensive number of low-energy
modes, so that the scaling regime described in the previous section persists
in the thermodynamic limit.

We search for very low-weight edges that generate a two-cut of the network:
two edges that, if removed, disconnect the network. In the supplement, we show
the low-frequency density of states scales as

$D_{\alpha}(\omega)\propto\frac{\omega^{4\alpha+3}}{\delta z^{2\alpha+3}},$
(3)

independent of system size.

Figure 2:  The density of states for fixed system size (N=1000) and changing
$\delta z=0.1,0.168,0.282,0.476,0.8$ In the upper-left we have a sketch of a
1d chain with periodic boundary conditions (the open circles are the same
node) with additional bonds. Inset: The density of states, $D(\omega)$, for
fixed $\delta z=0.1$ and changing system size $N=20$, $60$, $120$, $240$,
$500$, $1000$, $2000$, and $4000$.

To test the universal form predicted by Eq. 3, we computed the spectrum
$D(\omega)$ for rings with crossbonds and uniform bond weights ($\alpha=0$).
For each value of $\delta z$ and $N$ we generated between $10^{5}$ and
$2\times 10^{6}$ matrices samples 111 For $\delta z=0.1$ and $N=500$ and
$1000$, we calculate $2\times 10^{6}$ matrices and for $N=2000$ and $4000$, we
calculate $522240$ and $261120$ matrices. For all other values, we calculate
$10^{6}$ matrices., with independently chosen weights and uniformly random
placements of the endpoints of the $N\delta z/2$ crossbonds. The inset to Fig.
2 displays plots of the sample-averaged density of states $D(\omega)$ for
fixed $\delta z=0.1$ as $N$ increases. This example plot supports the
convergence of $D(\omega)$ to a gapless distribution as $N\rightarrow\infty$.
The main panel of Fig. 2 displays the computed density of states (solid lines)
for large $N$ ($N=1000$) and varying $\delta z$. The dashed lines in Fig. 2
show fits of the form $D(\omega)\propto\omega^{3}$ to the low frequency
region, as predicted by Eq. 3. These fits are in good agreement with the
computed spectra.

Based on Eq. 3 and the more complete form of the density of states derived in
Appendix B, we expect a collapse of $D(\omega)$ when frequencies are scaled by
$\delta z$. Fig. 3(a) shows the density of states for the scaled frequency,
$\omega’=\omega/\delta z$. For $\delta z=0.168$ we numerically identify a
frequency $\omega_{e}$ that best separates the $\omega^{3}$ scaling regime
from the remaining spectrum. Eq. 3 then predicts that all other cutoff
frequencies should scale linearly with $\delta z$, which is in good agreement
with the data as shown by the open squares in Fig 2 and 3(a).

In addition to the crossover at $\omega_{e}$, there is a second crossover
where $D(\omega)$ flattens to a plateau. In jammed packings at zero
temperature, where the boson peak occurs at the onset of the plateau,
$\omega^{*}$ is often defined as the frequency at which the density of states
attains a fixed fraction $f$ (typically 25 %) of its value in the plateau (Xu
_et al._ , 2010). We use that same definition here with $f=0.25$.

Figure 3: a)The density of states, $D(\omega)$, rescaled by $\delta z$. The
blue dashed line indicates the transition from the $\omega^{3}$ regime to the
$\omega^{4}$ regime while the black dashed line indicates the transition to
the plateau. The inset shows the scaling of $\omega^{*}$ and $\omega_{e}$ with
$\delta z$ is linear. b)The inverse participation ratio, IPR, rescaled by
$\delta z$. The IPR approaches a quasilocalized plateau in the $\omega^{3}$
region.

In many disordered solids, numerical evidence suggests
$\omega^{*}\propto\delta z$ (O’Hern  _et al._ , 2003; Silbert  _et al._ ,
2005). To check whether this is also true for our matrices, we plot the
density of states as a function of the rescaled frequency
$\omega^{\prime}=\omega/\delta z$, for various values of $\delta z$, shown in
Fig 3(a). We see a good collapse of the three regions, suggesting that both
crossovers are linear in $\delta z$, which is also highlighted by the inset to
Fig 3(a).

Importantly, this confirms that although the intermediate region between the
two crossover frequencies spans less than a decade in frequency, it is well-
defined and does not change as a function of excess coordination or system
size. Specifically, these results mandate the following functional form for
the density of states in our random matrix model with $\alpha=0$:

$D(\omega)=\begin{cases}\frac{4}{L^{2}}\left(\frac{\omega}{\delta
z}\right)^{3}&\omega\leq\omega_{e}\\\
\propto\omega^{\psi}&\omega_{e}\leq\omega\leq\omega^{*}\\\
\propto\omega^{0}&\omega^{*}\leq\omega\end{cases}$ (4)

To extract the scaling of $D(\omega)$ below the boson peak, we fit $D(\omega)$
to this functional form and extract the best-fit $\psi$ for each value of
$\delta z$ (See table in supplemental materials). We find that all curves are
consistent with $\psi=4.0\pm 0.05$ for frequencies
$\omega_{e}\leq\omega\leq\omega^{*}$. This suggests
$D(\omega)\propto\omega^{4}$, just as seen below the plateau in simulations of
jammed packings.

Given the striking similarities between the density of states in this simple
model and jammed packings, we would also like to know if the eigenvector
statistics are similar. In jammed systems, many modes at frequencies below the
boson peak are quasilocalized (Xu  _et al._ , 2010). This is quantified by the
inverse participation ratio (IPR),
$IPR(\omega)=\sum_{i}v_{i}^{4}/(\sum_{i}v_{i}^{2})^{2}$, where $v$ is the
vector associated with the eigenfrequency $\omega$. In Fig 2(b), the very low-
frequency regime of the IPR plateaus, and the value of this plateau scales
with $\delta z$, indicating that only about $\frac{1}{\delta z}$ nodes are
participating in the vibration.

Interestingly, the intermediate region exhibits values of IPR that are
typically associated with quasilocalized excitations. Moreover, the size of
those excitations seems to decrease as $\delta z$ increases. In jammed solids,
an outstanding open question is how the size of localized excitations changes
as one approaches the jamming transition.

__Crossbonded ring with power-law bond weights__ : Having a simple
constructive model that reproduces many features of the vibrational modes in
jammed packings is useful, because we can vary the model and ask what features
are necessary to generate the $\omega^{4}$ scaling in the density of states.
One natural choice is to perturb the distribution of bond strengths away from
the uniform distribution by changing the power-law exponent $\alpha$.

Figure 4: The density of states for $\alpha=-0.4$, $-0.2$, $0$, $0.25$, $0.5$,
$1$, and $2$, with $\delta z=0.1$. Inset:
$D(\omega^{\prime}=A\omega^{4\alpha+3})$ for the same values of $\alpha$ as in
the main figure, where $A$ is the coefficient predicted in Appendix B. The
black dashed line is the predicted scaling for the low frequency regime.

For $\alpha>0$, very weak bonds become rare and the assumptions that lead to
Eq. 3 break down. Numerically, we observe that a gap appears to open up in the
spectrum as $\alpha$ increases, as seen in Fig 4. For $\alpha<0$, we expect
Eq. 3 should still hold, as shown by the numerical data in the inset of Fig 4.
In this case, however, the crossover frequency no longer scales linearly with
$\delta z$, and so the power-law scaling between $\omega_{e}$ and $\omega^{*}$
– the exponent $\psi$ in Eq. 4 – is no longer independent of $\delta z$. In
other words, an intermediate regime consistent with
$D(\omega)\propto\omega^{4}$, independent of $\delta z$, is only possible for
$\alpha=0$.

Discussion: In this Letter, we propose a simple random matrix model that is
locally nearly isostatic and captures features of the vibrational states of
disordered packings that are typically associated with marginality.
Specifically, the model recapitulates a plateau in the density of states above
$\omega^{*}$, and a regime consistent with $\omega^{4}$ scaling immediately
below that. Our model also has a second crossover frequency $\omega_{e}$,
below which $D(\omega)$ scales as $\omega^{3}$.

The modes in this extremely low frequency regime are governed by extremal
statistics, and so we can calculate their properties analytically. This allows
us to demonstrate that $\omega_{e}$ scales linearly with excess coordination
$\delta z$ if and only if the weak bonds are uniformly distributed, suggesting
that $\omega^{4}$ seen in jammed packings arises due to a special, self-
organized distribution of the weakest bonds.

Of course, jammed packings only exist in dimensions greater than unity. Above
one dimension, the bond between particles is described by a tensor and not a
scalar weight. The $d$ by $d$ interaction block that corresponds to a single
bond in the Hessian matrix can be written as
$H_{ij\alpha\beta}=-V^{\prime\prime}|u_{\parallel}|^{2}-\frac{V^{\prime}}{r_{ij}}|u_{\perp}|^{2}$.
The first term is often referred to as the stiffness while the second term is
called the prestress term (Ellenbroek, 2007).

Interestingly, observations in 3D jammed packings suggest that the
$\omega^{4}$ regime only exists when the $V^{\prime}$ term is unperturbed;
even very small perturbations to the prestress open up a gap in the density of
states (Lerner and Bouchbinder, 2018). This suggests that a self-organized
balance between the stiffness and prestress must occur in systems near
isostaticity. Moreover, the stiffness is always positive and the prestress
always decreases the entries in the Hessian, so it is plausible that the
prestress term is driving some interactions to be very weak near isostaticity,
similar to our simple model.

While suggestive, a more concrete connection will require us to extend our
analysis to higher dimensions. We see an $\omega^{4}$ regime when bond
strengths are uniform, but it is unclear what quantity would be analogous to a
uniform bond weight in a $d\times d$ sub-block. Concurrent work by Benetti et
al focused on $d$-dimensional Laplacian matrices where the magnitude of each
bond is unity, but the geometry of the bond is randomly distributed, and these
also generate scaling consistent with $\omega^{4}$ at low frequencies (Benetti
_et al._ , 2018). To better understand the connections between these models
and why both generate $\omega^{4}$ scaling, one could study systems with
random bond weights and ordered geometries, or have both be disordered.

Furthermore, although $\omega^{4}$ scaling as been observed in several glass
forming systems (Lerner  _et al._ , 2016), the $\omega^{3}$ regime may be
unique to 1D systems, as it has not been reported in simulations or in the
random matrices with $3\times 3$ sub-blocks Benetti  _et al._ (2018). In
addition, we see about half a decade of frequency consistent with $\omega^{4}$
scaling, while the most recent data from Lerner and collaborators (Lerner  _et
al._ , 2016; Lerner and Bouchbinder, 2018; Kapteijns  _et al._ , 2018) finds
almost a full decade.

Nevertheless, the $\omega^{3}$ scaling regime is interesting. Disordered rings
are well-studied, but major results focus on localization caused by disorder
(Dyson, 1953; Dean, 1964). To our knowledge, the finite-size scaling effects
of the vibrational spectrum have not been discussed previously. Our model
demonstrates that finite size effects in the disordered ring, such as this
gapless low-frequency scaling, can be promoted into properties that are
maintained in the thermodynamic limit by network disorder.

Although we have excellent understanding of the $\omega^{3}$ regime in this
simple model, and convincing numerical evidence demonstrating $D(\omega)$
scaling as $\omega^{4}$ over a window of about half of a decade in $\omega$,
we have not identified a mechanism for the $\omega^{4}$ regime, where we know
the assumption of two weak bonds and two rigid arms breaks down. There are
many higher order modes that may contribute, and visual inspection of the
eigenvector structure suggests that no single one dominates, so there is no
obvious simple extension of our argument for $\omega^{3}$.

One possible avenue for understanding this regime is suggested by recent
numerical work that shows universality in the eigenvector statistics
associated with the boson peak. Specifically, eigenstatistics in jammed
packings match those from both the random matrix model described here, as well
as the dense limit of this model where all nodes are connected to one another
(Manning and Liu, 2015). Interestingly, the eigenvector statistics are also
identical in a much simpler model which is just the sum of a diagonal matrix
and a Gaussian orthogonal matrix. Very recent analytic work suggests that such
matrices are marginal; they are on the edge of a non-ergodic localized phase
(Facoetti  _et al._ , 2016). It would therefore be very interesting to extend
this analytic work to sparse matrices and study the tail of the density of
states.

Another way to extend our model is to alter the loop structure of the
underlying graph. In our random matrix model, the loop structure is
uncontrolled since we add crossbonds with uniform probability across the
graph. This is different from jammed systems where neighbors of one particle
are more likely to be neighbors of each other and loops are small. It is
fairly straightforward to extend our analytic analysis of the $\omega^{3}$
regime to random matrix models with smaller loops, and we expect that the
prefactor and the onset of the scaling $\omega_{e}$ will change, but the
$\omega^{3}$ scaling will not. However, this change could impact the behavior
of the $\omega^{4}$ regime.

Acknowledgements We thank Fernanda Benetti, Gabriele Sicuro, and Giorgio
Parisi for discussions. This work was partially supported by the Simons
Foundation grant number 454947 (ES, PM, MLM), and by NSF-DMR-1352184 (ES,
MLM). Computational resources were provided by support from Syracuse
University and NSF ACI-1541396.

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On harmonic quasiconformal quasi-isometries

M. Mateljević and M. Vuorinen

FILE: main.tex, printed on: 2024-8-27, 16.07

Abstract. The purpose of this paper is to explore conditions which guarantee
Lipschitz-continuity of harmonic maps w.r.t. quasihyperbolic metrics. For
instance, we prove that harmonic quasiconformal maps are Lipschitz w.r.t.
quasihyperbolic metrics.

2000 Mathematics Subject Classification. Primary 30C85. Secondary 30C65.

Keywords. harmonic mapping, Lipschitz mapping, quasiconformal mapping,
quasihyperbolic metric.

##  1 Introduction

Let $G\subset{\mathbb{R}}^{2}$ be a domain and let
$f:G\to{\mathbb{R}}^{2},f=(f_{1},f_{2}),$ be a harmonic mapping. This means
that $f$ is a map from $G$ into ${\mathbb{R}}^{2}$ and both $f_{1}$ and
$f_{2}$ are harmonic functions, i.e. solutions of the two-dimensional Laplace
equation

$\Delta u=0\,.$ (1.1)

The Cauchy-Riemann equations, which characterize analytic functions, no longer
hold for harmonic mappings and therefore these mappings are not analytic.
Intensive studies during the past two decades show that much of the classical
function theory can be generalized to harmonic mappings (see the recent book
of Duren [9] and the survey of Bshouty and Hengartner [7]). The purpose of
this paper is to continue the study of the subclass of quasiconformal and
harmonic mappings, introduced by Martio in [31] and further studied for
example in [32, 33, 34, 37, 38, 16, 2, 3, 19, 20, 21, 17]. The above
definition of a harmonic mapping extends in a natural way to the case of
vector-valued mappings $f:G\to{\mathbb{R}}^{n},f=(f_{1},\dots,f_{n}),$ defined
on a domain $G\subset{\mathbb{R}}^{n},n\geq 2.$

We first recall the classical Schwarz lemma for the unit disk
${\mathbb{D}}=\\{z\in{\mathbb{C}}:|z|<1\\}\,$:

1.2. Lemma.  Let $f:{\mathbb{D}}\to{\mathbb{D}}$ be an analytic function with
$f(0)=0\,.$ Then $|f(z)|\leq|z|$, $z\in{\mathbb{D}}\,.$

For the case of harmonic mappings this lemma has the following counterpart.

1.3. Lemma.  ([15], [9, p. 77]) Let $f:{\mathbb{D}}\to{\mathbb{D}}$ be a
harmonic mapping with $f(0)=0\,.$ Then
$|f(z)|\leq\frac{4}{\pi}{\mathrm{tan}}^{-1}|z|$ and this inequality is sharp
for each point $z\in{\mathbb{D}}\,.$

The classical Schwarz lemma is one of the cornerstones of geometric function
theory and it also has a counterpart for quasiconformal maps ([1, 26, 41,
45]). Both for analytic functions and for quasiconformal mappings it has a
form that is conformally invariant under conformal automorphisms of
${\mathbb{D}}\,.$

In the case of harmonic mappings this invariance is no longer true. In
general, if $\varphi:{\mathbb{D}}\to{\mathbb{D}}$ is a conformal automorphism
and $f:{\mathbb{D}}\to{\mathbb{D}}$ is harmonic, then $\varphi\circ f$ is
harmonic only in exceptional cases. Therefore one expects that harmonic
mappings from the disk into a strip domain behave quite differently from
harmonic mappings from the disk into a half-plane and that new phenomena will
be discovered in the study of harmonic maps. For instance, it follows from
Lemma 1 that holomorphic functions in plane do not increase hyperbolic
distances. In general, planar harmonic mappings do not enjoy this property. On
the other hand, we shall give here an additional hypothesis under which the
situation will change, in the plane as well as in higher dimensions. It turns
out that the local uniform boundedness property, which we are going to define,
has an important role in our study.

For a domain $G\subset{\mathbb{R}}^{n},n\geq 2,x,y\in G,$ let

$r_{G}(x,y)=\frac{|x-y|}{\mathrm{min}\\{{d}(x),{d}(y)\\}}\,\textrm{where}\,d(x)=d(x,\partial
G)\equiv\inf\\{|z-x|:z\in\partial G\\}\,.$

If the domain $G$ is understood from the context, we write $r$ instead
$r_{G}$. This quantity is used, for instance, in the study of quasiconformal
and quasiregular mappings, cf. [45]. It is a basic fact that [43, Theorem
18.1] for $n\geq 2,K\geq 1,c_{2}>0$ there exists $c_{1}\in(0,1)$ such that
whenever $f:G\to fG$ is a $K$-quasiconformal mapping with
$G,fG\subset{\mathbb{R}}^{n}$ then $x,y\in G$ and $r_{G}(x,y)\leq c_{1}$ imply
$r_{fG}(f(x),f(y))\leq c_{2}.$ We call this property the local uniform
boundedness of $f$ with respect to $r_{G}\,.$ Note that quasiconformal
mappings satisfy the local uniform boundedness property and so do quasiregular
mappings under appropriate conditions; it is known that one to one mappings
satisfying the local uniform boundedness property may not be quasiconformal.
We also consider a weaker form of this property and say that $f:G\to fG$ with
$G,fG\subset{\mathbb{R}}^{n}$ satisfies the weak uniform boundedness property
on $G$ (with respect to $r_{G}\,$) if there is a constant $c>0$ such that
$r_{G}(x,y)\leq 1/2$ implies $r_{fG}(f(x),f(y))\leq c\,.$ Univalent harmonic
mappings fail to satisfy the weak uniform boundedness property as a rule, see
Example 2.3 below.

We show that if $f:G\to fG$ is harmonic then $f$ is Lipschitz w.r.t.
quasihyperbolic metrics on $G$ and $fG$ if and only if it satisfies the weak
uniform boundedness property; see Theorem 2.5. The proof is based on a higher
dimensional version of the Schwarz lemma: harmonic maps satisfy the inequality
(2.15) below. An inspection of the proof of Theorem 2.5 shows that the class
of harmonic mappings can be replaced by $OC^{1}$ class defined by (3.1) (see
Section 3 below) and it leads to generalizations of the result; see Theorem 3.

Another interesting application is Theorem 2.5 which shows that if $f$ is a
harmonic $K$-quasiregular map such that the boundary of the image is a
continuum containing at least two points, then it is Lipschitz. In Subsection
2.5, we study conditions under which a qc mapping is quasi isometry with
respect to the corresponding quasihyperbolic metrics; see Theorems 2.6 and
2.6. In particular, using a quasiconformal analogue of Koebe’s theorem, cf.
[4], we give a simple proof of the following result, cf. [30, 33]: if $D$ and
$D^{\prime}$ are proper domains in $\mathbb{R}^{2}$ and $h:D\rightarrow
D^{\prime}$ is $K$-qc and harmonic, then it is bi-Lipschitz with respect to
quasihyperbolic metrics on $D$ and $D^{\prime}$.

The results in this paper may be generalized into various directions. One
direction is to consider weak continuous solutions of the $p$-Laplace equation

$div(|\nabla u|^{p-2}\nabla u)=0,\quad 1<p<\infty,$

so called $p$-harmonic functions. Note that $2$-harmonic functions in the
above sense are harmonic in the usual sense.

It seems that the case of the upper half space is of particular interest, cf.
[37, 33, 16, 3]. In Subsection 2.6, using Theorem 3.1 [23] we prove that if
$h$ is a quasiconformal $p$-harmonic mapping of the upper half space
$\mathbb{H}^{n}$ onto itself and $h(\infty)=\infty$, then $h$ is quasi-
isometry with respect to both the Euclidean and the Poincaré distance.

##  2  Lipschitz property of harmonic maps w.r.t. quasihyperbolic metrics

###  2.1 Hyperbolic type metrics

Let $B^{n}(x,r)=\\{z\in\mathbb{R}^{n}:|z-x|<r\\},S^{n-1}(x,r)=\partial
B^{n}(x,r)$ and let $\mathbb{B}^{n},S^{n-1}$ stand for the unit ball and the
unit sphere in $\mathbb{R}^{n}$, respectively. Sometimes we write $\mathbb{D}$
instead of $\mathbb{B}^{2}$. For a domain $G\subset\mathbb{R}^{n}$ let
$\rho:G\to(0,\infty)$ be a continuous function. We say that $\rho$ is a weight
function or a metric density if for every locally rectifiable curve $\gamma$
in $G,$ the integral

$l_{\rho}(\gamma)=\int_{\gamma}\rho(x)ds$

exists. In this case we call $l_{\rho}(\gamma)$ the $\rho$-length of $\gamma.$
A metric density defines a metric $d_{\rho}:G\times G\to(0,\infty)$ as
follows. For $a,b\in G,$ let

$d_{\rho}(a,b)=\inf_{\gamma}l_{\rho}(\gamma)$

where the infimum is taken over all locally rectifiable curves in $G$ joining
$a$ and $b.$ For a fixed $a,b\in G\,,$ suppose that there exists a
$d_{\rho}$-length minimizing curve $\gamma:[0,1]\to G$ with
$\gamma(0)=a,\gamma(1)=b$ such that

$d_{\rho}(a,b)=l_{\rho}(\gamma|[0,t])+l_{\rho}(\gamma|[t,1])$

for all $t\in[0,1]\,.$ Then $\gamma$ is called a geodesic segment joining $a$
and $b\,.$ It is an easy exercise to check that $d_{\rho}$ satisfies the
axioms of a metric. For instance, the hyperbolic (or Poincaré) metric of the
unit ball $\mathbb{B}^{n}$ and the upper half space
$\mathbb{H}^{n}=\\{x\in\mathbb{R}^{n}:x_{n}>0\\}$ are defined in terms of the
densities $\rho(x)=2/(1-|x|^{2})$ and $\rho(x)=1/x_{n}\,,$ respectively. It is
a classical fact that in both cases the length-minimizing curves, geodesics,
exist and that they are circular arcs orthogonal to the boundary [6]. In both
cases we have even explicit formulas for the distances:

$\sinh\frac{\rho_{\mathbb{B}^{n}}(x,y)}{2}=\frac{|x-y|}{\sqrt{(1-|x|^{2})(1-|y|^{2})}}\,,\quad
x,y\in\mathbb{B}^{n}\,,$ (2.1)

and

$\cosh{\rho_{\mathbb{H}^{n}}(x,y)}=1+\frac{|x-y|^{2}}{2x_{n}y_{n}}\,,\quad
x,y\in\mathbb{H}^{n}\,.$ (2.2)

Because the hyperbolic metric is invariant under conformal mappings, we may
define the hyperbolic metric in any simply connected plane domain by using the
Riemann mapping theorem, see for example [24]. The Schwarz lemma may now be
formulated by stating that an analytic function from a simply connected domain
into another simply connected domain is a contraction mapping, i.e. the
hyperbolic distance between the images of two points is at most the hyperbolic
distance between the points. The hyperbolic metric is often the natural metric
in classical function theory. For the modern mapping theory, which also
considers dimensions $n\geq 3\,,$ we do not have a Riemann mapping theorem and
therefore it is natural to look for counterparts of the hyperbolic metric. So
called hyperbolic type metrics have been the subject of many recent papers.
Perhaps the most important of these metrics are the quasihyperbolic metric
$k_{G}$ and the distance ratio metric $j_{G}$ of a domain
$G\subset{\mathbb{R}^{n}}\,.$ They are defined as follows.

2.3. The quasihyperbolic and distance ratio metrics .  Let
$G\subset\mathbb{R}^{n}$ be a domain. The quasihyperbolic metric $k_{G}$ is a
particular case of the metric $d_{\rho}$ when $\rho(x)=\frac{1}{d(x,\partial
G)}$ (see [13, 12, 45]). It was proved in [12] that for given $x,y\in G\,,$
there exists a geodesic segment of length $k_{G}(x,y)$ joining them. The
distance ratio metric is defined for $x,y\in G$ by setting

$j_{G}(x,y)=\log(1+r_{G}(x,y))=\log(1+\frac{|x-y|}{\mathrm{min}\\{{d}(x),{d}(y)\\}})\,$

where $r_{G}$ is as in the Introduction. It is clear that

$j_{G}(x,y)\leq r_{G}(x,y)\,.$

Some applications of these metrics are reviewed in [46]. The recent PhD theses
[27], [24], [29] study the quasihyperbolic geometry or use it as a tool.

2.4. Lemma.  ([13], [45, (3.4), Lemma 3.7]) Let $G$ be a proper subdomain of
$\mathbb{R}^{n}\,.$

  1. (a)

If $x,y\in G$ and $|y-x|\leq d(x)/2$, then $k_{G}(x,y)\leq 2j_{G}(x,y)\,.$

  2. (b)

For $x,y\in G$ we have $k_{G}(x,y)\geq
j_{G}(x,y)\geq\log\left(1+\frac{|y-x|}{d(x)}\right)$.

###  2.2 Quasiconformal and quasiregular maps

2.5. Maps of class ACL and ACLn .  For each integer $k=1,...,n$ we denote
$R_{k}^{n-1}=\\{x\in R^{n}:x_{k}=0\\}$. The orthogonal projection
$P_{k}:\mathbb{R}^{n}\to\mathbb{R}_{k}^{n-1}$, is given by
$P_{k}x=x-x_{k}\,e_{k}\,.$

Let $I=\\{x\in\mathbb{R}^{n}:a_{k}\leq x_{k}\leq b_{k}\\}$ be a closed
$n$-interval. A mapping $f:I\rightarrow\mathbb{R}^{m}$ is said to be
absolutely continuous on lines (ACL) if $f$ is continuous and if $f$ is
absolutely continuous on almost every line segment in $I$, parallel to the
coordinate axes. More precisely, if $E_{k}$ is the set of all $x\in P_{k}I$
such that the function $t\mapsto u(x+te_{k})$ is not absolutely continuous on
$[a_{k},b_{k}]$, then $m_{n-1}(E_{k})=0$ for all $1\leq k\leq n$.

If $\Omega$ is an open set in $\mathbb{R}^{n}$, a mapping
$f:\Omega\rightarrow\mathbb{R}^{m}$ is absolutely continuous if $f|I$ is ACL
for every closed interval $I\subset\Omega$. If $\Omega$ and $\Omega^{\prime}$
are domains in $\mathbb{\overline{R}}^{n}$, a homeomorphism
$f:\Omega\rightarrow\Omega^{\prime}$ is called ACL if
$f|\Omega\setminus\\{\infty,f^{-1}(\infty)\\}$ is ACL.

If $f:\Omega\rightarrow\mathbb{R}^{m}$ is ACL, then the partial derivatives of
$f$ exist a.e. in $\Omega$, and they are Borel functions. We say that $f$ is
ACLn if the partials are locally integrable.

2.6. Quasiregular mappings .  Let $G\subset\mathbb{R}^{n}$ be a domain. A
mapping $f:G\to\mathbb{R}^{n}$ is said to be quasiregular (qr) if $f$ is ACLn
and if there exists a constant $K\geq 1$ such that

$|f^{\prime}(x)|^{n}\leq
KJ_{f}(x)\;,\;\;|f^{\prime}(x)|=\max_{|h|=1}|f^{\prime}(x)h|\;,$

a.e. in $G$. Here $f^{\prime}(x)$ denotes the formal derivative of $f$ at
$x\,,$ The smallest $K\geq 1$ for which this inequality is true is called the
outer dilatation of $f$ and denoted by $K\,_{O}(f)$. If $f$ is quasiregular,
then the smallest $K\geq 1$ for which the inequality

$J_{f}(x)\leq
K\,l(f^{\prime}(x))^{n}\;,\;\;l(f^{\prime}(x))=\min_{|h|=1}|f^{\prime}(x)h|\;,$

holds a.e. in $G$ is called the inner dilatation of $f$ and denoted by
$K\,_{I}(f)$. The maximal dilatation of $f$ is the number
$K(f)=\max\\{\,K\,_{I}(f),\,K\,_{O}(f)\,\\}$. If $K(f)\leq K$, then $f$ is
said to be $K$–quasiregular ($K$–qr). If $f$ is not quasiregular, we set
$K\,_{O}(f)=K\,_{I}(f)=K(f)=\infty\,.$

Let $\Omega_{1}$ and $\Omega_{2}$ be domains in $\mathbb{R}^{n}$ and fix
$K\geq 1\,.$ We say that a homeomorphism $f:\Omega_{1}\rightarrow\Omega_{2}$
is a $K$-quasiconformal (qc) mapping if it is $K$-qr and injective. Some of
the standard references for qc and qr mappings are [11], [26], [43], and [45].
These mappings generalize the classes of conformal maps and analytic functions
to Euclidean spaces. The Kühnau handbook [25] contains several reviews dealing
with qc maps. It should be noted that various definitions for qc maps are
studied in [43]. The above definition of $K$-quasiconformality is equivalent
to the definition based on moduli of curve families in [43, p. 42]. It is
well-known that qr maps are differentiable a.e., satisfy condition (N) i.e.
map sets of measure zero (w.r.t. Lebesgue’s $n$-dimensional measure) onto sets
of measure zero. The inverse mapping of a $K$-qc mapping is also $K$-qc. The
composition of a $K_{1}$-qc and of a $K_{2}$-qc map is a $K_{1}K_{2}$-qc map
if it is defined.

###  2.3 Examples

We first show that, as a rule, univalent harmonic mappings fail to satisfy the
local uniform boundedness property.

2.7. Example .  The univalent harmonic mapping $f:{\mathbb{H}}^{2}\to
f({\mathbb{H}}^{2})\,,$ $f(z)=\mathrm{arg}\,z+i\,\mathrm{Im}z,$ fails to
satisfy the local uniform boundedness property with respect to
$r_{{\mathbb{H}}^{2}}\,.$

Let $z_{1}=\rho e^{i\pi/4}$, $z_{2}=\rho e^{i3\pi/4}$, $w_{1}=f(z_{1})$ and
$w_{2}=f(z_{2})$. Then $r_{{\mathbb{H}}^{2}}(z_{1},z_{2})=2$ and
$r_{f{\mathbb{H}}^{2}}(w_{1},w_{2})=\frac{\pi}{\sqrt{2}\,\rho}$ if $\rho$ is
small enough and we see that $f$ does not satisfy the local uniform
boundedness property.

2.8. Example.  The univalent harmonic mapping
$f:{\mathbb{H}}^{2}\to{\mathbb{H}}^{2}\,,$
$f(z)=\mathrm{Re}\,z\,\mathrm{Im}\,z+i\,\mathrm{Im}z,$ fails to satisfy the
local uniform boundedness property with respect to $r_{{\mathbb{H}}^{2}}\,.$

For a harmonic mapping $f(z)=h(z)+\overline{g(z)}\,,$ we introduce the
following notation

$\lambda_{f}(z)=|h^{\prime}(z)|-|g^{\prime}(z)|\,,\quad\Lambda_{f}(z)=|h^{\prime}(z)|+|g^{\prime}(z)|\,\,\,{\rm
and}\quad\nu(z)=g^{\prime}(z)/f^{\prime}(z).$

The following Proposition shows that a one to one harmonic function satisfying
the local uniform boundedness property need not be quasiconformal.

2.9. Proposition .  The function $f(z)={\mathrm{log}}(|z|^{2})+2iy$ is a
univalent harmonic mapping and satisfies the local uniform boundedness
property, but $f$ is not quasiconformal on $V=\\{z:x>1,0<y<1\\}$.

Proof.  It is clear that $f$ is harmonic in $\Pi^{+}=\\{z:\,{\rm Re\,}z>0\\}$.
Next $f(z)=h(z)+\overline{g(z)}$, where $h(z)={\mathrm{log}}z+z$ and
$g(z)={\mathrm{log}}z-z$. Since $h^{\prime}(z)=1+1/z$ and
$g^{\prime}(z)=-1+1/z$, we have $|\nu(z)|<1$ for $z\in\Pi^{+}$.

Moreover, $f$ is quasiconformal on every compact subset $D\subset\Pi^{+}$ and
$\lambda_{f}$, $\Lambda_{f}$ are bounded from above and below on $D$.
Therefore $f$ is a quasi-isometry on $D$ and by Theorem 2.5 below, $f$
satisfies the local uniform boundedness property on $D$.

From now on we consider the restriction of $f$ to $V=\\{z=x+iy:x>1,0<y<1\\}$.
Then $fV=\\{w=(u,v):u>\log(1+v^{2}/4),0<v<2\\}\,.$

We are going to show that:

  1. $\bullet$

$f$ satisfies the local uniform boundedness property, but $f$ is not
quasiconformal on $V$ .

We see that $f$ is not quasiconformal on $V\,,$ because $|\nu(z)|\rightarrow
1$ as $z\rightarrow\infty,z\in V\,.$

For $s>1$, define $V_{s}=\\{z:1<x<s,0<y<1\\}$. Note that $f$ is qc on $V_{s}$
and therefore $f$ satisfies the property of local uniform boundedness on
$V_{s}$ for every $s>1$.

We consider separately two cases.

Case A. $z\in V_{4}\,.$ If $r>1$ is big enough, then $d(z,\partial
V_{r})=d(z,\partial V)$ and $d\big{(}f(z),\partial
f(V_{r})\big{)}=d(f(z),\partial f(V))$ and therefore $f$ satisfies the
property of local uniform boundedness on $V_{4}$ with respect to $r_{V}$.

Case B. It remains to prove that $f$ satisfies the property of local uniform
boundedness on $V\setminus V_{4}$ with respect to $r_{V}$.

Observe first that for $z,z_{1}\in V$ and $|z_{1}|\geq|z|\geq 1\,,$ we have
the estimate

$\log\left(\frac{|z_{1}|}{|z|}\right)\leq\frac{|z_{1}|}{|z|}-1\leq|z_{1}-z|,$

and therefore for $z,z_{1}\in V$

$|f(z_{1})-f(z)|\leq 4|z_{1}-z|.$ (2.10)

We write

$\partial V=[1,1+i]\cup A\cup B\,;A=\\{(x,0):x\geq 1\\}\,,B=\\{(x,1):x\geq
1\\}\,.$

Then

$\partial(fV)=f(\partial V)\subset f[1,1+i]\cup(fA)\cup(fB)$

and by the definition of $f$ we see that

$fA=\\{(x,0):x\geq 0\\}\,,\quad fB=\\{(x,2):x\geq\log 2\\},\quad
f[1,1+i]\subset[0,\log 2]\times[0,2]\,.$

Clearly for $w\in fV$

$d(w)=\min\\{d(w,fA),d(w,fB),d(w,f[1,1+i])\\}\,,$

and for ${\rm Re}w>1+\log{2}$ and $w\in fV\,,$ we find

$d(w)=\min\\{d(w,fA),d(w,fB)\\}\,.$ (2.11)

For $z\in V\setminus V_{4}$ we have ${\rm Re}f(z)\geq\log(16)>1+\log{2}$ and
therefore, in view of the definition of $f$, (2.11) yields $d(f(z))=2d(z)$.
This together with (2.10) shows that $f$ satisfies the property of local
uniform boundedness on $V\setminus V_{4}$. $\Box$

###  2.4 Higher dimensional version of Schwarz lemma

Before giving a proof of the higher dimensional version of the Schwarz lemma
for harmonic maps we first establish some notation.

Suppose that $h:\overline{B}^{n}(a,r)\rightarrow\mathbb{R}^{n}$ is a
continuous vector-valued function, harmonic on $B^{n}(a,r)$, and let

$M_{a}^{*}=\sup\\{|h(y)-h(a)|:y\in S^{n-1}(a,r)\\}.$

Let $h=(h^{1},h^{2},\dots,h^{n})$. A modification of the estimate in [14,
Equation (2.31)] gives

$r|\nabla h^{k}(a)|\leq nM_{a}^{*}\,,\quad k=1,\dots,n.$

We next extend this result to the case of vector valued functions. See also
[8] and [5, Theorem 6.16].

2.12. Lemma.  Suppose that
$h:{\overline{B}^{n}(a,r)}\rightarrow\mathbb{R}^{n}$ is a continuous mapping,
harmonic in ${B}^{n}(a,r)$. Then

$r|h^{\prime}(a)|\leq nM_{a}^{*}\,.$ (2.13)

Proof.  Without loss of generality, we may suppose that $a=0$ and $h(0)=0$.
Let

$K(x,y)=K_{y}(x)=\frac{r^{2}-|x|^{2}}{n\omega_{n}\,r|x-y|^{n}},$

where $\omega_{n}$ is the volume of the unit ball $\mathbb{B}^{n}$ in
${\mathbb{R}}^{n}$.

Then

$h(x)=\int_{S^{n-1}(0,r)}K(x,t)h(t)d\sigma,\quad x\in{B}^{n}(0,r),$

where $d\sigma$ is the $(n-1)$-dimensional surface measure on $S^{n-1}(0,r)$.

A simple calculation yields

$\frac{\partial}{\partial
x_{j}}K(x,\xi)=\frac{1}{n\omega_{n}\,r}\left(\frac{-2x_{j}}{|x-\xi|^{n}}-n(r^{2}-|x|^{2})\frac{x_{j}-\xi_{j}}{|x-\xi|^{n+2}}\right).$

Hence, for $1\leq j\leq n$, we have

$\frac{\partial}{\partial
x_{j}}K(0,\xi)=\frac{\xi_{j}}{\omega_{n}\,r^{n+1}}\,.$

Let $\eta\in S^{n-1}$ be a unit vector and $|\xi|=r$. For given $\xi$, it is
convenient to write $K_{\xi}(x)=K(x,\xi)$ and consider $K_{\xi}$ as a function
of $x$.

Then

$K_{\xi}^{\prime}(0)\eta=\frac{1}{\omega_{n}\,r^{n+1}}(\xi,\eta)\,.$

Since $|(\xi,\eta)|\leq|\xi||\eta|=r$, we see that

$|K_{\xi}^{\prime}(0)\eta|\leq\frac{1}{\omega_{n}\,r^{n}}\,,\textrm{ and
therefore\,}\quad|\nabla K^{\xi}(0)|\leq\frac{1}{\omega_{n}r^{n}}\,.$

This last inequality yields

$|h^{\prime}(0)({\eta})|\leq\int_{{S^{n-1}(a,r)}}|\nabla
K^{y}(0)|\,|h(y)|\,d\sigma(y)\leq\frac{M_{0}^{*}\,n\omega_{n}r^{n-1}}{\omega_{n}r^{n}}=\frac{M_{0}^{*}n}{r}$

and the proof is complete. $\Box$

Let $G\subset\mathbb{R}^{n},$ be a domain, let $h:G\to{\mathbb{R}}^{n}$ be
continuous. For $x\in G\,$ let $B_{x}=B^{n}(x,\frac{1}{4}d(x))$ and

$M_{x}=\omega_{h}(x)=\sup\\{|h(y)-h(x)|:y\in B_{x}\\}.$ (2.14)

If $h$ is a harmonic mapping, then the inequality (2.13) yields

$\frac{1}{4}d(x)|h^{\prime}(x)|\leq n\,\omega_{h}(x),\quad x\in G\,.$ (2.15)

We also refer to (2.15) as the inner gradient estimate.

###  2.5 Harmonic quasiconformal quasi-isometries

For our purpose it is convenient to have the following lemma.

2.16. Lemma.  Let $G$ and $G^{\prime}$ be two domains in $\mathbb{R}^{n}$, and
let $\sigma$ and $\rho$ be two continuous metric densities on $G$ and
$G^{\prime}$, respectively, which define the elements of length
$ds=\sigma(z)|dz|$ and $ds=\rho(w)|dw|$, respectively; and suppose that
$f:G\rightarrow G^{\prime}$, is a $C^{1}$-mapping.
a) If there is a positive constant $c_{1}$ such that
$\rho(f(z))\,|f^{\prime}(z)|\leq c_{1}\,\sigma(z),$ $z\in G\,,$ then
$d_{\rho}(f(z_{2}),f(z_{1}))\leq c_{1}\,d_{\sigma}(z_{2},z_{1})$,
$z_{1},z_{2}\in G$.
b) If $f(G)=G^{\prime}$ and there is a positive constant $c_{2}$ such that
$\rho(f(z))\,l(f^{\prime}(z))\geq c_{2}\,\sigma(z),$ $z\in G\,,$ then
$d_{\rho}(f(z_{2}),f(z_{1}))\geq c_{2}\,d_{\sigma}(z_{2},z_{1})$,
$z_{1},z_{2}\in G$

The proof of this result is straightforward and it is left to the reader as an
exercise.

2.17. Pseudo-isometry and a quasi-isometry .  Let $f$ be a map from a metric
space $(M,d_{M})$ into another metric space $(N,d_{N})$.

  * •

We say that $f$ is a pseudo-isometry if there exist two positive constants $a$
and $b$ such that for all $x,y\in M$,

$a^{-1}d_{M}(x,y)-b\leq d_{N}(f(x),f(y))\leq ad_{M}(x,y).$

  * •

We say that $f$ is a quasi-isometry or a bi-Lipschitz mapping if there exists
a positive constant $a\geq 1$ such that for all $x,y\in M$,

$a^{-1}d_{M}(x,y)\leq d_{N}(f(x),f(y))\leq ad_{M}(x,y).$

For the convenience of the reader we begin our discusssion for the unit disk
case.

2.18. Theorem. Suppose that $h:{\mathbb{D}}\to{\mathbb{R}}^{2}$ is harmonic
and satisfies the weak uniform boundedness property.

  1. (c)

Then $h:({\mathbb{D}},k_{\mathbb{D}})\to(h({\mathbb{D}}),k_{h({\mathbb{D}})})$
is Lipschitz.

  2. (d)

If, in addition, $h$ is a qc mapping, then
$h:({\mathbb{D}},k_{\mathbb{D}})\to(h({\mathbb{D}}),k_{h({\mathbb{D}})})$ is a
quasi-isometry.

Proof.  The part (d) is proved in [33].

For the proof of part (c) fix $x\in\mathbb{D}$ and $y\in
B_{x}=B(x,\frac{1}{4}d(x))$. Then $d(y)\geq\frac{3}{4}d(x)$ and therefore
$r(x,y)<1/2$. By the hypotheses $|h(y)-h(x)|\leq c\,d(h(x))$. The Schwarz
lemma, applied to $B_{x}\,,$ yields in view of (2.14)

$\frac{1}{4}d(x)|h^{\prime}(x)|\leq 2M_{x}\leq 2c\,d(h(x))$

The proof of part (c) follows from Lemma 2.5. $\Box$

A similar proof applies for higher dimensions; the following result is a
generalization of the part $(c)$ of Theorem 2.5 .

2.19. Theorem. Suppose that $G$ is a proper subdomain of $\mathbb{R}^{n}$ and
$h:G\to\mathbb{R}^{n}$ is a harmonic mapping. Then the following conditions
are equivalent

  1. (1)

$h$ satisfies the weak uniform boundedness property.

  2. (2)

$h:({G},k_{G})\to(h({G}),k_{h({G})})$ is Lipschitz.

Proof.  Let us prove that $(1)$ implies $(2)$.

By the hypothesis $(1)$ $f$ satisfies the weak uniform boundedness property:
for every $x\in G$ and $t\in B_{x}$

$|f(t)-f(x)|\leq c_{2}\,d(f(x))\,.$ (2.20)

This inequality together with Lemma 2.4 gives $d(x)|f^{\prime}(x)|\leq
c_{3}\,d(f(x))$ for every $x\in G$. Now an application of Lemma 2.5 shows that
$(1)$ implies $(2)$.

It remains to prove that $(2)$ implies $(1)$.

Suppose that $f$ is Lipschitz with the multiplicative constant $c_{2}$. Fix
$x,y\in G$ with $r_{G}(x,y)\leq 1/2$. Then $|y-x|\leq d(x)/2$ and therefore by
Lemma 2.1

$k_{G}(x,y)\leq 2j_{G}(x,y)\leq 2\,r_{G}(x,y)\leq 1.$

Hence $k_{G^{\prime}}(fx,fy)\leq c_{2}$. Since $j_{G^{\prime}}(fx,fy)\leq
k_{G^{\prime}}(fx,fy)\leq c_{2}$, we find
$j_{G^{\prime}}(fx,fy)=\log(1+r_{G^{\prime}}(fx,fy))\leq c_{2}$ and therefore
$r_{G^{\prime}}(fx,fy)\leq e^{c_{2}}-1$. $\Box$

Since $f^{-1}$ is qc, an application of [12, Theorem 3] to $f^{-1}$ and
Theorem 2.5 give the following corollary:

2.21. Corollary.  Suppose that $G$ is a proper subdomain of $\mathbb{R}^{n}$,
$h:G\rightarrow hG$ is harmonic and $K$-qc. Then
$h:({G},k_{G})\to(h({G}),k_{h({G})})$ is a pseudo-isometry.

In [45, Example 11.4] (see also [44, Example 3.10]), it is shown that the
analytic function $f:{\mathbb{D}}\to
G,G={\mathbb{D}}\setminus\\{0\\},f(z)=\exp((z+1)/(z-1))\,,$
$f({\mathbb{D}})=G,$ fails to be uniformly continuous as a map

$f:({\mathbb{D}},k_{{\mathbb{D}}})\to(G,k_{G})\,.$

Therefore bounded analytic functions do not satisfy the weak uniform
boundedness property in general. The situation will be different for instance
if the boundary of the image domain is a continuum containing at least two
points. Note that if $k_{G}$ is replaced by the hyperbolic metric
$\lambda_{G}$ of $G,$ then
$f:({\mathbb{D}},k_{{\mathbb{D}}})\to(G,\lambda_{G})\,$ is Lipschitz.

2.22. Theorem. Suppose that $G\subset\mathbb{R}^{n}$,
$f:G\rightarrow\mathbb{R}^{n}$ is $K$-qr and $G^{\prime}=f(G)$. Let $\partial
G^{\prime}$ be a continuum containing at least two distinct points. If $f$ is
a harmonic mapping, then $f:({G},k_{G})\to({G^{\prime}},k_{G^{\prime}})$ is
Lipschitz.

Proof.  Fix $x\in G$ and let $B_{x}=B^{n}(x,d(x)/4)$. If $|y-x|\leq d(x)/4$,
then $d(y)\geq 3d(x)/4$ and therefore,

$r_{G}(y,x)\leq\frac{4}{3}\frac{|y-x|}{d(x)}.$

Because $j_{G}(x,y)=\log(1+r_{G}(x,y))\leq r_{G}(x,y)$, using Lemma 2.1(a), we
find

$k_{G}(y,x)\leq 2\,j_{G}(y,x)\leq 2/3<1.$

By [45, Theorem 12.21] there exists a constant $c_{2}>0$ depending only on $n$
and $K$ such that

$k_{G^{\prime}}(fy,fx)\leq
c_{2}\max\\{k_{G}(y,x)^{\alpha},k_{G}(y,x)\\},\alpha=K^{1/(1-n)},$

and hence, using Lemma 2.1(b) and $k_{G}(y,x)\leq 1$, we see that

$|fy-fx|\leq e^{c_{2}}d(fx),\quad\mbox{i.e.}\quad M_{x}=\omega_{f}(x)\leq
e^{c_{2}}\,d(fx).$ (2.23)

By (2.15) applied to $B_{x}=B^{n}(x,d(x)/4)$, we have

$\frac{1}{4}d(x)|f^{\prime}(x)|\leq 2M_{x}$

and therefore using the inequality (2.23), we have

$\frac{1}{4}d(x)|f^{\prime}(x)|\leq 2\,c\,d(f(x)),$

where $c=e^{c_{2}}$; and the proof follows from Lemma 2.5. $\Box$

The first author has asked the following Question (cf. [33]: Suppose that
$G\subset\mathbb{R}^{n}$ is a proper subdomain, $f:G\rightarrow\mathbb{R}^{n}$
is harmonic $K$-qc and $G^{\prime}=f(G)$. Determine whether $f$ is a quasi-
isometry w.r.t. quasihyperbolic metrics on $G$ and $G^{\prime}$. This is true
for $n=2$ (see Theorem 2.6 below). It seems that one can modify the proof of
Proposition 4.6 in [42] and show that this is true for the unit ball if $n\geq
3$ and $K<2^{n-1}$, cf. also [20].

###  2.6 Quasi-isometry in planar case

Astala and Gehring [4] proved a quasiconformal analogue of Koebe’s theorem,
stated here as Theorem 2.6. These concern the quantity

$a_{f}(x)=a_{f,G}(x):={\rm
exp}\left(\frac{1}{n|B_{x}|}\int_{B_{x}}{\textrm{log}}J_{f}(z)dz\right),x\in
G,$

associated with a quasiconformal mapping $f:G\rightarrow
f(G)\subset\mathbb{R}^{n}$; here $J_{f}$ is the Jacobian of f ; while $B_{x}$
stands for the ball $B(x;d(x,\partial G)$; and $|B_{x}|$ for its volume.

2.24. Theorem[4]. Suppose that G and $G^{\prime}$ are domains in $R^{n}$: If
$f:G\rightarrow G^{\prime}$ is $K$\- quasiconformal, then

$\frac{1}{c}\frac{d(fx,\partial G^{\prime})}{d(x,\partial G)}\leq
a_{f,G}(x)\leq c\frac{d(fx,\partial G^{\prime})}{d(x,\partial G)},\quad x\in
G,$

where $c$ is a constant which depends only on $K$ and $n$.

Let $\Omega\in\mathbb{R}^{n}$ and $\mathbb{R}^{+}=[0,\infty)$. If
$f,g:\Omega\rightarrow\mathbb{R}^{+}$ and there is a positive constant $c$
such that

$\frac{1}{c}\,g(x)\leq f(x)\leq c\,g(x)\,,\quad x\in\Omega\,,$

we write $f\approx g$ on $\Omega$.

Our next result concerns the quantity

$E_{f,G}(x):=\frac{1}{|B_{x}|}\int_{B_{x}}J_{f}(z)dz\,,x\in G,$

associated with a quasiconformal mapping $f:G\rightarrow
f(G)\subset\mathbb{R}^{n}$; here $J_{f}$ is the Jacobian of $f$ ; while
$B_{x}$ stands for the ball $B(x,d(x,\partial G)/2$ and $|B_{x}|$ for its
volume.

Define

$A_{f,G}=\sqrt[n]{E_{f,G}}\,.$

2.25. Theorem. Suppose $f:\Omega\rightarrow\Omega^{\prime}$ is a $C^{1}$ qc
homeomorphism. The following conditions are equivalent:

a)$f$ is bi-Lipschitz with respect to quasihyperbolic metrics on $\Omega$ and
$\Omega^{\prime}~\,,$

b) $\sqrt[n]{J_{f}}\approx d_{*}/d\,,$

c) $\sqrt[n]{J_{f}}\approx a_{f}\,,$

d) $\sqrt[n]{J_{f}}\approx A_{f}$,

where $d(x)=d(x,\partial\Omega)$ and
$d_{*}(x)=d(f(x),\partial\Omega^{\prime})$.

Proof.  It is known that a) is equivalent to b) (see for example [36]).

In [36], using Gehring’s result on the distortion property of qc maps (see
[10], p.383; [43], p.63), the first author gives short proofs of a new version
of quasiconformal analogue of Koebe’s theorem; it is proved that $A_{f}\approx
d_{*}/d$.

By Theorem 2.6, $a_{f}\approx d_{*}/d$ and therefore b) is equivalent to c).
The rest of the proof is straightforward. $\Box$

If $\Omega$ is planar domain and $f$ a harmonic qc map, then we proved that
the condition d) holds.

The next theorem is a short proof of a recent result of V. Manojlovic [30],see
also [33].

2.26. Theorem. Suppose D and $D^{\prime}$ are proper domains in
$\mathbb{R}^{2}$. If $h:D\rightarrow D^{\prime}$ is $K$\- qc and harmonic,
then it is bi-Lipschitz with respect to quasihyperbolic metrics on D and
$D^{\prime}$.

Proof.  Without loss of generality, we may suppose that $h$ is preserving
orientation. Let $z\in D$ and $h=f+\overline{g}$ be a local representation of
$h$ on $B_{z}$, where $f$ and $g$ are analytic functions on $B_{z}$,
$\Lambda_{h}(z)=|f^{\prime}(z)|+|g^{\prime}(z)|$,
$\lambda_{h}(z)=|f^{\prime}(z)|-|g^{\prime}(z)|$ and $k=\frac{K-1}{K+1}$.

Since $h$ is $K$\- qc, we see that

$(1-k^{2})|f^{\prime}|^{2}\leq J_{h}\leq K|f^{\prime}|^{2}$ (2.27)

on $B_{z}$ and since ${\textrm{log}}\,|f^{\prime}(\zeta)|$ is harmonic,

${\textrm{ log}}\,|f^{\prime}(z)|=\frac{1}{2|B_{z}|}\int_{B_{z}}\textrm{
log}\,|f^{\prime}(\zeta)|^{2}d\xi\,d\eta\,.$

Hence, using the right hand side of (2.27), we find

$\displaystyle{\textrm{log}}\,a_{h,D}(z)\leq\frac{1}{2}\,{\textrm{
log}}K+\frac{1}{2|B_{z}|}\int_{B_{z}}{\textrm{
log}}\,|f^{\prime}(\zeta)|^{2}d\xi\,d\eta$ (2.28) $\displaystyle={\textrm{
log}}\,\sqrt{K}\,|f^{\prime}(z)|\,.$ (2.29)

Hence,

$a_{h,D}(z)\leq\sqrt{K}\,|f^{\prime}(z)|$

and in a similar way using the left hand side of (2.27), we have

$\sqrt{1-k^{2}}\,|f^{\prime}(z)|\leq a_{h,D}(z)\,.$

Now, an application of the Astala-Gehring result gives

$\Lambda_{h}(z)\asymp\frac{d(hz,\partial D^{\prime})}{d(z,\partial
D}\asymp\lambda_{h}(z)\,.$

This pointwise result, combined with Lemma 2.5 (integration along curves),
easily gives

$k_{D^{\prime}}(h(z_{1}),h(z_{2}))\asymp k_{D}(z_{1},z_{2}),\quad
z_{1},z_{2}\in D\,.$

$\Box$

Note that in [30] the proof makes use of the interesting fact that
${\log}\frac{1}{J_{h}}$ is a subharmonic function; but we do not use it here.

Define $m_{f}(x,r)=\min\\{|f(x^{\prime})-f(x)|:|x^{\prime}-x|=r\\}$.

Suppose that $G$ and $G^{\prime}$ are domains in $\mathbb{R}^{n}$: If
$f:G\rightarrow G^{\prime}$ is $K$\- quasiconformal; by the distortion
property we find $m_{f}(x,r)\geq a(x)r^{1/\alpha}$. Hence, as in [20] and [36]
, we get:

2.30. Lemma.  If $f\in C^{1,1}$ is a $K-$ quasiconformal mapping defined in a
domain $\Omega\subset\mathbb{R}^{n}$ $(n\geq 3)$, then

$J_{f}(x)>0,\quad x\in\Omega$

provided that $K<2^{n-1}$. The constant $2^{n-1}$ is sharp.

2.31. Theorem. Under the hypothesis of the lemma, if
$\overline{G}\subset\Omega$, then $f$ is bi-Lipschitz with respect to
Euclidean and quasihyperbolic metrics on $G$ and $G^{\prime}=f(G).$

Proof.  Since $\overline{G}$ is compact $J_{f}$ attains minimum on
$\overline{G}$ at a point $x_{0}\in\overline{G}$. By Lemma 2.6,
$m_{0}=J_{f}>0$ and therefore since $f\in C^{1,1}$ is a $K-$ quasiconformal,
we conclude that functions $|f_{x_{k}}|$, $1\leq k\leq n$ are bounded from
above and below on $\overline{G}$; hence $f$ is bi-Lipschitz with respect to
Euclidean metric on $G$.

By Theorem 2.6, we find $a_{f,G}\approx d_{*}/d$, where $d(x)=d(x,\partial G)$
and $d_{*}(x)=d(f(x),\partial G^{\prime})$. Since we have here
$\sqrt[n]{J_{f}}\approx a_{f}$, we find $\sqrt[n]{J_{f}}\approx d_{*}/d$ on
$G$. An application of Theorem 2.6 completes the proof. $\Box$

###  2.7 The upper half space ${\mathbb{H}}^{n}.$

Let $\mathbb{H}^{n}$ denote the half-space in $\mathbb{R}^{n}$. If $D$ is a
domain in $\mathbb{R}^{n}$, by $QCH(D)$ we denote the set of Euclidean
harmonic quasiconformal mappings of $D$ onto itself.

In particular if $x\in\mathbb{R}^{3}$, we use notation $x=(x_{1},x_{2},x_{3})$
and we denote by $\partial_{x_{k}}f=f^{\prime}_{x_{k}}$ the partial derivative
of $f$ with respect to $x_{k}$ .

A fundamental solution in space $\mathbb{R}^{3}$ of the Laplace equation is
$\frac{1}{|x|}$. Let $U_{0}=\frac{1}{|x+e_{3}|}$, where $e_{3}=(0,0,1)$.
Define $h(x)=(x_{1}+\varepsilon_{1}U_{0},x_{2}+\varepsilon_{2}U_{0},x_{3}).$
It is easy to verify that $h\in QCH(\mathbb{H}^{3})$ for small values of
$\varepsilon_{1}$ and $\varepsilon_{2}$.

Using the Herglotz representation of a nonnegative harmonic function $u$ (see
Theorem 7.24 and Corollary 6.36 [5]), one can get:

Lemma A. If $u$ is a nonnegative harmonic function on a half space
$\mathbb{H}^{n}$, continuous up to the boundary with $u=0$ on
$\mathbb{H}^{n}$, then $u$ is (affine) linear.

In [33], the first author has outlined a proof of the following result:

Theorem A. If $h$ is a quasiconformal harmonic mapping of the upper half space
$\mathbb{H}^{n}$ onto itself and $h(\infty)=\infty$, then $h$ is quasi-
isometry with respect to both the Euclidean and the Poincaré distance.

Note that the outline of proof in [33] can be justified by Lemma A.

We show that the analog statement of this result holds for $p$-harmonic vector
functions (solutions of $p$-Laplacian equations) using the mentioned result
obtained in the paper [23], stated here as:

Theorem B. If $u$ is a nonnegative $p$-harmonic function on a half space
$\mathbb{H}^{n}$, continuous up to the boundary with $u=0$ on
$\mathbb{H}^{n}$, then $u$ is (affine) linear.

2.32. Theorem. If $h$ is a quasiconformal $p$-harmonic mapping of the upper
half space $\mathbb{H}^{n}$ onto itself and $h(\infty)=\infty$, then both
$h:(\mathbb{H}^{n},|\cdot|)\to(\mathbb{H}^{n},|\cdot|)$ and
$h:(\mathbb{H}^{n},\rho_{\mathbb{H}^{n}})\to(\mathbb{H}^{n},\rho_{\mathbb{H}^{n}})$
are bi-Lipschitz where $\rho=\rho_{\mathbb{H}^{n}}$ is the $Poincar\acute{e}$
distance.

Since $2$-harmonic mapping are Euclidean harmonic this result includes Theorem
A.

Proof.  It suffices to deal with the case $n=3$ as the proof for the general
case is similar. Let $h=(h_{1},h_{2},h_{3})$.

By Theorem B, we get $h_{3}(x)=ax_{3}$, where $a$ is a positive constant.
Without loss of generality we may suppose that $a=1$.

Since $h_{3}(x)=x_{3}$, we have $\partial_{x_{3}}h_{3}(x)=1$, and therefore
$|h^{\prime}_{x_{3}}(x)|\geq 1$. In a similar way,
$|g^{\prime}_{x_{3}}(x)|\geq 1$, where $g=h^{-1}$. Hence, there exists a
constant $c=c(K)$,

$|h^{\prime}(x)|\leq c\quad{\rm and}\quad 1/c\leq l(h^{\prime}(x))\,.$

Therefore partial derivatives of $h$ and $h^{-1}$ are bounded from above; and,
in particular, $h$ is Euclidean bi-Lipschitz.

Since $h_{3}(x)=x_{3}$,

$\frac{|h^{\prime}(x)|}{h_{3}(x)}\leq\frac{c}{x_{3}};$

and hence, by Lemma 2.5, $\rho(h(a),h(b))\leq c\rho(a,b)\,.$ $\Box$

##  3 Pseudo-isometry and $OC^{1}(G)$

In this section, we give a sufficient condition for a qc mapping
$f:G\rightarrow f(G)$ to be a pseudo-isometry w.r.t. quasihyperbolic metrics
on $G$ and $f(G)$. First we adopt the following notation.

If $V$ is a subset of $\mathbb{R}^{n}$ and $u:V\rightarrow\mathbb{R}^{m}$, we
define

${\rm osc}_{V}u=\sup\\{|u(x)-u(y)|\,:x,y\in V\\}\,.$

Suppose that $G\subset\mathbb{R}^{n}$ and $B_{x}=B(x,d(x)/2)$. Let $OC^{1}(G)$
denote the class of $f\in C^{1}(G)$ such that

$d(x)|f^{\prime}(x)|\leq c_{1}\,{\rm osc}_{B_{x}}f$ (3.1)

for every $x\in G$. Similarly, let $SC^{1}(G)$ be the class of functions $f\in
C^{1}(G)$ such that

$|f^{\prime}(x)|\leq ar^{-1}\,\omega_{f}(x,r)\quad{\rm
for\,all}\,\,B^{n}(x,r)\subset G,$ (3.2)

where $\omega_{f}(x,r)=\sup\\{|f(y)-f(x)|:\,y\in B^{n}(x,r)\\}$.

The proof of Theorem 2.5 gives the following more general result:

3.3. Theorem. Suppose that $G\subset\mathbb{R}^{n}$, $f:G\rightarrow
G^{\prime}$, $f\in OC^{1}(G)$ and it satisfies the weak property of uniform
boundedness with a constant $c$ on $G$. Then

  1. (e)

$f:({G},k_{G})\to({G^{\prime}},k_{G^{\prime}})$ is Lipschitz.

  2. (f)

In addition, if $f$ is $K$-qc, then $f$ is pseudo-isometry w.r.t.
quasihyperbolic metrics on $G$ and $f(G)$.

Proof.  By the hypothesis $f$ satisfies the weak property of uniform
boundedness: $|f(t)-f(x)|\leq c_{2}\,d(f(x)$ for every $t\in B_{x}$, that is

${\rm osc}_{B_{x}}f\leq c_{2}\,d(f(x))$ (3.4)

for every $x\in G$. This inequality together with (3.1) gives
$d(x)|f^{\prime}(x)|\leq c_{3}\,d(f(x))$. Now an application of Lemma 2.5
gives part (e). Since $f^{-1}$ is qc, an application of [12, Theorem 3] on
$f^{-1}$ gives part (f). $\Box$

In order to apply the above method we introduce subclasses of $OC^{1}(G)$
(see, for example, below (3.5)).

Let $f:G\rightarrow G^{\prime}$ be a $C^{2}$ function and $B_{x}=B(x,d(x)/2).$
We denote by $OC^{2}(G)$ the class of functions which satisfy the following
condition:

$\sup_{B_{x}}d^{2}(x)|\Delta f(x)|\leq c\,\,{\rm osc}_{B_{x}}f$ (3.5)

for every $x\in G$.

If $f\in OC^{2}(G)$, then by Theorem 3.9 in [14], applied to $\Omega=B_{x}$,

$\sup_{t\in B_{x}}d(t)|f^{\prime}(t)|\leq C(\sup_{t\in
B_{x}}|f(t)-f(x)|+\sup_{t\in B_{x}}d^{2}(t)|\Delta f(t)|)$

and hence by (3.5)

$d(x)|f^{\prime}(x)|\leq c_{1}\,{\rm osc}_{B_{x}}f$ (3.6)

for every $x\in G$ and therefore $OC^{2}(G)\subset OC^{1}(G)$.

Now the following result follows from the previous theorem.

3.7. Corollary.  Suppose that $G\subset\mathbb{R}^{n}$ is a proper subdomain,
$f:G\rightarrow G^{\prime}$ is $K$-qc and $f$ satisfies the condition (3.5).
Then $f:({G},k_{G})\to({G^{\prime}},k_{G^{\prime}})$ is Lipschitz.

We will now give some examples of classes of functions to which Theorem 3 is
applicable. Let $SC^{2}(G)$ denote the class of $f\in C^{2}(G)$ such that

$|\Delta f(x)|\leq ar^{-1}\sup\\{|f^{\prime}(y)|:\,y\in B^{n}(x,r)\\},$

for all $B^{n}(x,r)\subset G$, where $a$ is a positive constant. Note that the
class $SC^{2}(G)$ contains every function for which $d(x)|\Delta f(x)|\leq
a|f^{\prime}(x)|$, $x\in G$. It is clear that $SC^{1}(G)\subset OC^{1}(G)$ and
by the mean value theorem, $OC^{2}(G)\subset SC^{2}(G)$. For example, in [39]
it is proved that $SC^{2}(G)\subset SC^{1}(G)$ and that the class $SC^{2}(G)$
contains harmonic functions, eigenfunctions of the ordinary Laplacian if $G$
is bounded, eigenfunctions of the hyperbolic Laplacian if $G={\mathbb{B}}^{n}$
and thus our results are applicable for instance to these classes.

Acknowledgements. This work was initiated during the visit of the first author
to the Universities of Helsinki and Turku in October 2005 and continued during
the visit of the second author to Belgrade in December 2006. The authors’
research was supported by the grant no. 8118525 the Academy of Finland. The
first author was partially supported by MNTRS, Serbia, Grant No. 144 020. The
authors are indebted to the referee for valuable comments and to O. Martio and
S. Ponnusamy for interesting discussions on this paper.

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M. Mateljević
Faculty of Mathematics
University of Belgrade
Studentski trg 16
11000 Beograd
Serbia
miodrag’at’matf.bg.ac.yu

M. Vuorinen
Department of Mathematics
FIN-20014 University of Turku
FINLAND
vuorinen’at’utu.fi