# Rigidity transitions in zero-temperature polygons

M. C. Gandikota1,2, Amanda Parker1,3, J. M. Schwarz1,4 1Department of Physics
and BioInspired Institute, Syracuse University, Syracuse, NY USA
2Department of Chemistry, Columbia University, New York, NY USA
3SymBioSys, Chicago, Il USA
4Indian Creek Farm, Ithaca, NY USA

###### Abstract

We study geometrical clues of a rigidity transition due to the emergence of a
system-spanning state of self stress in under-constrained systems of
individual polygons and spring networks constructed from such polygons. When a
polygon with harmonic bond edges and an area spring constraint is subject to
an expansive strain, we observe that convexity of the polygon is a necessary
condition for such a self stress. We prove that the cyclic configuration of
the polygon is a sufficient condition for the self stress. This correspondence
of geometry and rigidity is akin to the straightening of a one dimensional
chain of springs to rigidify it. We predict the onset of the rigidity
transition using a purely geometrical method. We also estimate the transition
strain for a given initial configuration by approximating irregular polygons
as regular polygons. These findings help determine the rigidity of an area-
preserving polygon just by looking at it. Since two-dimensional spring
networks can be considered as a network of polygons, we look for similar
geometric features in under-constrained spring networks under isotropic
expansive strain. In particular, we observe that all polygons attain convexity
at the rigidity transition such that the fraction of convex, but not cyclic,
polygons predicts the onset of the rigidity transition. Interestingly, acyclic
polygons in the network correlate with larger tensions, thus, forming
effective force chains.

##  I Introduction

Determining the rigidity of a disordered network is a nontrivial problem that
depends on both the topology and geometry of the network. Specifically, there
are two, independent manners in which an under-constrained disordered network
can be rigidified. The first is to add sufficient number of bonds to remove
all nontrivial floppy modes in the system. Here, a floppy mode is a
deformation of the network that preserves the bond lengths to first order in
the displacement of the vertices at either end of the bond. While nontrivial
floppy modes change the shape/geometry of the network, trivial floppy modes
are just the global translations and rotations of the network. When all
nontrivial floppy modes are removed, the network is said to be first-order, or
infinitesimally, rigid. This is a microscopic definition of rigidity Connelly
and Guest (2022). By adding bonds, or constraints, one is changing the
topology of the network such that it is rigidified when the number of degrees
of freedom is equal to the number of independent constraints in the system
Maxwell (1864). A canonical example of this phenomenon is that the
infinitesimal rigidity of a two-dimensional spring network with central-force
spring interactions can be determined solely by using a combinatorial theorem
that identifies the number of independent constraints Laman (1970). In other
words, no geometry need be invoked.

Another way of rigidifying an under-constrained network is through a system
spanning state of self stress Alexander (1998a); Connelly and Guest (2022).
Here, a state of self stress is a distribution of stresses in the network that
adds up to zero net force on every vertex Lubensky  _et al._ (2015). For
example, a stretched out one-dimensional chain with equal but non-zero
tensions in its springs satisfies force balance at every point on the chain.
The tension in the chain imposes a bending energy cost on all its transverse
fluctuations Alexander (1998a). Straining such under-constrained central-force
spring networks can reposition the vertices such that a state of self stress
can be established. The difference between the number of nontrivial floppy
modes $N_{0}$ and the number of states of self stress $N_{S}$ for a given
configuration is determined by Maxwell-Calladine theorem Calladine (1978);
Lubensky  _et al._ (2015),

$N_{0}-N_{S}=dN-N_{b}-\frac{d(d+1)}{2},$ (1)

where $N_{b}$ is the number of constraints in the spring-network embedded in a
$d$ dimensional space. This is a topological condition that does not involve
any geometrical details of the network. However, the existence of a state of
self stress is a question of geometry where we need to identify potential
shapes that can satisfy force balance on the vertices. A canonical example for
this latter type of rigidification is diluted two-dimensional spring networks
that are under isotropic expansive strain. Even though the numerical data
could not decidedly find the order of the transition, a continuous phase
transition in the bulk modulus appears to agree better with the numerical
results Sheinman  _et al._ (2012). In contrast, for disordered spring networks
derived from the contact network of jammed particles, strain in the form of
isotropic expansion establishes a discontinuous rigidity transition in the
bulk modulus Merkel  _et al._ (2019). Strain-induced rigidity transitions,
such as these, are being studied via several approaches Vermeulen  _et al._
(2017); Arzash  _et al._ (2019); Damavandi  _et al._ (2022a, b); Zhang  _et
al._ (2021); Damavandi  _et al._ (2022b); Lee and Merkel (2022).

In an attempt to gain some novel perspective on the phase transition of
underconstrained spring-networks induced by strain, we here study the rigidity
transition in its constituent loops under an expansive strain. While a non-
zero macroscopic elastic constant typically determines the rigidity of a loop,
we see that we can predict the rigidity of these underconstrained systems
using its geometry given its trivial topology. Since a one-dimensional chain
of edges when straightened out rigidifies via a state of self stress, a
natural two-dimensional equivalent of this system is a polygon under expansive
strain. By focusing on this minimal structure, we prove that such polygons can
sustain a system spanning state of self stress when they attain a cyclic
configuration. Such self stresses can rigidify the polygon. The correspondence
of shapes in judging the presence of a system-spanning state of self stress
can be compared to an important result of Cauchy which guarantees the rigidity
of three dimensional convex polyhedrons as long as they are convex Cauchy
(1813); Aigner  _et al._ (2010). While Cauchy’s theorem does not apply to
polygons, an extension of this theorem by Alexandrov proves that all convex
polytopes in $\mathbb{R}^{d}$ with $d\geq 3$ are rigid Pak (2010).

In the context of rigidity theory, Connelly proved that for a closed loop of
edges, there exists an expanding motion that conserves the edge lengths. Such
a motion convexifies the loop and increases its area Connelly  _et al._
(2000). The proof involves the addition of struts 111Struts are elastic rods
which are in compression and hence want to expand. connecting the edges in the
polygon. This is reminiscent of a ‘pressure’ within the loop. In another work,
such an expansive motion inspired by electrostatic charges has also been
studied Cantarella  _et al._ (2004). Our work does not involve the
incorporation of struts.

Figure 1: Correspondence of geometry and rigidity. a) A straightened out chain
of springs rigidifies with additional extensional strain. b) A polygon under
expansive strain is rigid when it attains a cyclic configuration. c) Convex
polyhedrons are rigid. While (a) and (b) are rigid due to the virtue of a
system spanning state of self stress, (c) is rigid solely due to the shape-
preserving constraints of the polyhedron faces.

The statistical mechanics of shape (not rigidity) transitions in two-
dimensional random polymer rings with a pressure energy term has been
carefully studied. These were studied as idealized models for membrane
vesicles Rudnick and Gaspari (1991); Leibler  _et al._ (1987); Haleva and
Diamant (2006); Mitra  _et al._ (2012). Specifically, the polymer rings are
constructed as $N$ number of linear springs and a pressure $p$ coupled with
the enclosed area $A$. While the energetic pressure term $-pA$ favors
inflation of the ring, entropic effects in these thermal models favor the
crumpling of the ring 222There are more possible configurations for a crumpled
polygon as compared to the inflated configurations.. Thus, entropy counters
the outward pressure force and serves as an ‘inward’ pressure. In these shape-
transition studies, the mean area $\left<A\right>$ is used as the order
parameter and pressure is the tuning parameter. For self-intersecting rings,
at the critical pressure $p_{c}\sim N^{-1}$, the ring blows up achieving
infinite area at the transition point Rudnick and Gaspari (1991); Gaspari  _et
al._ (1993). By replacing stretchable rods with rigid rods, a continuous phase
transition between the crumpled and the inflated phases was found Haleva and
Diamant (2006). This behavior is different for self-avoiding rods where smooth
crossovers between three distinct scaling regimes were recognized Leibler  _et
al._ (1987). In these studies, the critical pressure of transition is
$p_{c}=4\pi\;k_{B}Tl^{2}/N$, where $l$ is the length of each edge of the loop.
Near zero temperature, the critical pressure for the polygon to blow up into a
regular, cyclic polygon will be $p_{c}=0^{+}$. However, the zero-temperature
spring networks, which we are motivated by, have a rigidity transition at
finite strain Wyart  _et al._ (2008). To have a nontrivial transition in an
isolated polygon at zero temperature, we add higher order area terms to the
energy. By adding an energy term $k_{A}A^{2}$ to the energy with the constant
$k_{A}>0$ and completing the square as $k_{A}(A-p/(2k_{A}))^{2}$, we can write
the new term in the Hamiltonian as $k_{A}(A-A_{0})^{2}$. We will show that an
isolated polygon with such an energy has a nontrivial strain-induced rigidity
transition at zero temperature.

The outline of the manuscript is as follows. We introduce the model in Sec. II
and present the numerical results in Sec. III. We observe that there is no
singularity associated with this transition in the thermodynamic limit. In
Sec. V, we prove that the cyclic polygon is the unique configuration
sustaining a state of self stress and is the geometrical signature of rigidity
in the loop. In Sec. VI.1, we identify the transition using a purely
geometrical method by employing nontrivial floppy modes. In Sec. VI.2, we
estimate the rigidity transition strain by approximating the initial random
polygon as a regular polygon. In Sec. VII, we discuss the geometry of the
constituent loops in a spring network under isotropic expansive strain.
Lastly, in Sec. VIII, we discuss the relevance of the individual polygon
rigidity transition with regards to the rigidity of disordered spring networks
more generally.

Notation: We represent vectors by lower case bold letters as in $\bm{r}$ and
matrices by upper case bold letters as in $\bm{R}$.

##  II Random polygon model

To generate a random polygon with $N$ vertices ($N-$gon), we use Graham’s
algorithm O’Rourke (1994). We randomly pick $N$ points in the $xy$ plane
within a two dimensional box of dimensions
$\widetilde{L}_{B}\times\widetilde{L}_{B}$. The nearest of these points to the
‘center of mass’ of this set is chosen to be the first vertex of the polygon
333Graham’s algorithm picks a point within the convex hull of these set of
points. We are assuming here that the ‘center of mass’ of the points is within
the convex hull of the points. We perform this approximation for numerical
simplicity.. The rest of the points are sorted according to the angle of the
vector joining the first vertex with each of the other $N-1$ points. This
sorting assigns the set of points with vertex indices ${2,3,...,N}$. The
polygon is constructed by joining these vertices in the order of the assigned
indices. The area enclosed by this random polygon is $\widetilde{a}_{0}$ and
the $i^{\text{th}}$ edge length is $\widetilde{l}_{0_{i}}$. While we construct
an 8-gon constrained to a box of lengths
$\widetilde{L}_{B}\times\widetilde{L}_{B}$, all $N$-gons with $N>8$, are
constrained to a box of edge lengths $\sqrt{N/8}\;\widetilde{L}_{B}$. This
ensures the average length of edges to be the same for all $N\geq 8$.

Once we have the random polygon, the Hamiltonian of the system is defined as,

$\widetilde{H}=\frac{1}{2}\widetilde{K}_{\ell}\sum_{i=1}^{N}(\widetilde{\ell}_{i}-\widetilde{\ell}_{i_{0}})^{2}+\frac{1}{2}\widetilde{K}_{A}(\widetilde{A}-\widetilde{A}_{0})^{2},$

where $\widetilde{K}_{\ell}$ and $\widetilde{K}_{A}$ are the spring constants
of two-body springs associated with the edges and the area spring,
respectively. Variables $\widetilde{\ell}_{i}$ and $\widetilde{A}$ are the
lengths of the $i^{\text{th}}$ spring and the area of the enclosed loop
respectively while $\widetilde{\ell}_{i_{0}}$’s and $\widetilde{A}_{0}$ as the
rest lengths of the two-body spring and the rest area of the “area spring”,
respectively. At zero strain, the rest lengths and rest area match those of
the initial configuration of the random polygon such that $\widetilde{H}=0$.
In this work, we measure lengths in units of $\widetilde{L}_{B}$ and energies
in units of $\widetilde{K}_{A}\;\widetilde{L}_{B}^{4}$. In non-dimensionalized
units, the Hamiltonian is,

$H=\frac{1}{2}K_{\ell}\sum_{i=1}^{N}(\ell_{i}-\ell_{i_{0}})^{2}+\frac{1}{2}(A-A_{0})^{2},$
(2)

where $K_{\ell}=\widetilde{K}_{\ell}/\widetilde{K}_{A}\widetilde{L}_{B}^{2}$.
For a polygon embedded in $\mathbb{R}^{2}$ and $\hat{z}$ being the direction
perpendicular to this plane, the area of the loop is taken to be the algebraic
area Rudnick and Gaspari (1991); Gaspari  _et al._ (1993); Haleva and Diamant
(2006),

$A({\bm{r}})=\sum_{i=1}^{N}(\bm{r}_{i}\times\bm{r}_{i+1})\cdot\hat{z}.$ (3)

Note that we imply cyclicity of indices in $i+1$ and $N+1$ index is taken to
be index 1.

##  III Numerical results

We can equivalently impose expansive strain on the system by decreasing
$\ell_{i_{0}}$ of each spring proportionally or by increasing $A_{0}$ in Eq.
(2), i.e., strain can be imposed by tuning down the dimensionless ratios
$l_{i_{0}}/\sqrt{A_{0}}$. We choose to strain the polygon by increasing
$A_{0}$. The strain $\gamma$ is defined as,

$\gamma=\frac{A_{0}-a_{0}}{a_{0}},$ (4)

where $a_{0}$ is the initial area of the random polygon. Energy minimization
for every imposed strain is performed numerically in C++ using the BFGS2
method in the multimin package of GNU scientific library Galassi  _et al._
(2002). While Graham’s algorithm O’Rourke (1994) ensures a non-intersecting
polygon at zero strain, we allow self-intersections of floppy polygons at non-
zero strains. This retains the simplicity of the system for the purpose of
energy minimization. We denote the minimized energy as $E$. Stress $\sigma$
and stiffness $K$ are defined as derivatives of the energy density as,

$\sigma=\frac{1}{A}\;\frac{dE}{d\gamma},$ (5)

and,

$K=\frac{d\sigma}{d\gamma}.$ (6)

For reporting numerical results, we choose the spring constants $K_{\ell}=1$.
This choice does not change the critical transition strain. The energy
minimized configuration for regular polygons can be easily solved analytically
due to the symmetry of the polygon (see Appendix. IX). For irregular polygons,
we observe a floppy to rigid transition in the system. Stress is seen to be
continuous across the transition point $\gamma_{c}$ and the stiffness jumps.
The size of this jump decreases with $N$. The finite size analysis for
stiffness is reported in Fig. (2). Judging from the nature of the decrease in
step-size of the stiffness with increasing $N$, there is no true phase
transition, i.e., a transition where derivatives of the energy show
singularities in the thermodynamic limit of $N\rightarrow\infty$. We, instead,
have a smooth crossover between the floppy and rigid regimes. The critical
strain scales with polygon size as $\gamma_{c}\sim N^{3/2}$ (see Fig. 3).

Figure 2:  A floppy to rigid transition is seen for varying $N$. Stiffness
jumps at $\gamma=\gamma_{c}$, however the step-size of the jump decreases with
polygon size. Note that larger $N$ have a smaller step-size at transition.
Every curve is averaged over 100 runs.  Figure 3: Average critical strain of
rigidity transition scales as $\sim N^{3/2}$. Transition strains are obtained
from the numerical stiffness-strain curves. The data points are obtained by
averaging over 100 runs and error bars are standard deviation about the
average.

##  IV Convexity is necessary for a system spanning state of self stress

We now present two short arguments to show that convexity is necessary for the
loop to sustain a state of self stress where non-zero forces due to the area
spring and the tensions in the spring add up to a zero net force on each
vertex of the polygon.

The force on the $i^{\text{th}}$ vertex due to the area spring is given by the
negative gradient of the corresponding energy term in Eq. (2),

$\bm{f}_{A_{i}}=(A_{0}-A)\;\left(y_{i+1}-y_{i-1},x_{i-1}-x_{i+1}\right).$ (7)

This vector is perpendicular to the line joining the adjoining vertices of the
$i^{\text{th}}$ vertex. This is easily seen by noticing that the dot product
of this force with $\left(x_{i+1}-x_{i-1},y_{i+1}-y_{i-1}\right)$ is zero.
Thus, the direction of force due to the area spring at any given vertex can be
easily constructed geometrically. See (Fig. 5).

An $i^{\text{th}}$ vertex is called an ear/mouth if the line joining
$i-1^{\text{th}}$ and $i+1^{\text{th}}$ vertices lie inside/outside the
polygon Devadoss and O’Rourke (2011). When the tuning parameter $A_{0}$ is
increased (see Eq. 2), a non-convex polygon attempts to increase its area
without changing edge lengths. This is achieved by flipping the two dashed
edges in Fig. (4a) into the two solid edges. A non-convex polygon can always
increase its enclosed area without changing edge lengths via this discrete
floppy mode motion which transforms a mouth to an ear. Thus non-convex
polygons cannot be rigid under expansive strain expansion.

Another independent argument can be given as follows. When there is an outward
force on the vertices due to the area spring, the edges should necessarily be
in tension (as opposed to being in compression) to satisfy force-balance at
all the ears. Since a mouth is flanked by ears, this implies that the edges at
the mouth are in tension. However, at the mouth, edges in tension cannot
sustain a force-balance with the outward area spring force (See Fig. 4 b).
Since, force balance cannot be satisfied by non-convex loops, they cannot
sustain a rigidifying state of self stress. The necessity of convexity is
demonstrated in Fig. (7 a) where convexity transition precedes the rigidity
transition.

##  V Cyclicity is sufficient for a system spanning state of self stress

In three dimensional spaces, Cauchy established an important connection
between geometry and rigidity, assuming a trivial topology Cauchy (1813);
Aigner  _et al._ (2010). First, the congruence theorem of Cauchy proves that
two convex polyhedrons are congruent if corresponding faces of polyhedrons are
congruent. A corollary of this theorem shows that a convex polyhedron with
elastic faces is rigid i.e. the shape of the polyhedron cannot be changed
without changing the shape of at least one of its faces. The geometry of the
polyhedron is thus sufficient to judge the rigidity of the polyhedron. For the
polygons with an area spring constrained to two dimensions, the transition
point is again purely determined by the geometry of the polygon approaching
the cyclic polygon. A cyclic polygon is a polygon whose vertices can be
inscribed on a circle which is referred to as the circumcircle.

For two-dimensional polygons, a fundamental result in Euclidean geometry shows
that the maximum area of a flexible polygon is the polygon whose vertices lie
on a common circle (Theorem 12.5a of Niven (1981)). It is very intuitive to
expect that when the loop cannot expand any more, it will become rigid. The
subtlety here is displayed when we ask the question - how can the polygon
shape which cannot increase area any more without changing edge lengths be the
same shape that can support a state of self stress? We present analytical and
numerical arguments to show that the configuration of a cyclic polygon is a
sufficient condition to support a state of self stress.

Figure 4: Convexity is necessary for rigidity. a) Upon an expansive strain,
the initial non-convex configuration represented by dashed-line edges can flip
themselves over into the solid-line edges. Thus the polygon achieves an area
increase without changing edge lengths. b) The pressure force of the area
spring pointing outwards (blue arrow) and tensions in the two body springs
(green arrows) cannot satisfy force balance at the mouth of the polygon.

A cyclic configuration of a $N$-gon reduces $2N$ degrees of freedom of $N$
vertices to $N+1$ degrees of freedom which are the $N$ number of angular
coordinates of the vertices and the radius $r$. Without loss of generality we
ignore the radius since it only serves as a scale factor and write the
coordinates of the polygon vertices as,

$(x_{i},y_{i})=(\text{cos}\;\phi_{i},\text{sin}\;\phi_{i}).$ (8)

The angles between the tangent at the $i^{\text{th}}$ vertex and the
$j^{\text{th}}$ edge is denoted by $\theta_{i,j}$ as shown in Figure 5. We
decompose the forces at each vertex along the direction of force of the area
spring and the direction tangent to this. At each vertex, there are two force
balance conditions along each of these directions. This system of simultaneous
linear equations in the spring tensions $t_{i}$ and the forces due to the area
spring $n_{i}$ can be neatly represented as a matrix equation
$\bm{A}\bm{F}=\bm{0}$,

$\begin{bmatrix}\text{cos}\;\theta_{1,2}&0&0&\cdot\cdot\cdot&-\text{cos}\;\theta_{1,N}&0&0&\cdot\cdot\cdot&\cdot\cdot\cdot&0\\\
-\text{cos}\;\theta_{2,1}&\text{cos}\;\theta_{2,3}&0&\cdot\cdot\cdot&0&0&0&\cdot\cdot\cdot&\cdot\cdot\cdot&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&\cdot\cdot\cdot&-\text{cos}\;\theta_{N,N-1}&\text{cos}\;\theta_{N,1}&0&0&\cdot\cdot\cdot&\cdot\cdot\cdot&0\\\
\text{sin}\;\theta_{1,2}&0&0&...&\text{sin}\;\theta_{1,N}&-1&0&0&\cdot\cdot\cdot&0\\\
\text{sin}\;\theta_{2,1}&\text{sin}\;\theta_{2,3}&0&0&\cdot\cdot\cdot&0&-1&\cdot\cdot\cdot&\cdot\cdot\cdot&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&\cdot\cdot\cdot&\text{sin}\;\theta_{N,N-1}&\text{sin}\;\theta_{N,1}&0&0&\cdot\cdot\cdot&\cdot\cdot\cdot&-1\end{bmatrix}\begin{bmatrix}t_{1}\\\
t_{2}\\\ \vdots\\\ \vdots\\\ t_{N}\\\ n_{1}\\\ n_{2}\\\ \vdots\\\ \vdots\\\
n_{N}\end{bmatrix}=\begin{bmatrix}0\\\ 0\\\ \vdots\\\ \vdots\\\ 0\\\ 0\\\ 0\\\
\vdots\\\ \vdots\\\ 0\\\ \end{bmatrix},$ (9)

where $\bm{A}$ is a purely geometrical $2N\times 2N$ matrix and $F$ is the
column matrix of forces. We show that the rank of the matrix
$\text{Rank}(\bm{A})=2N-1$ by showing that the rank of the tangent components
of the force, i.e., the first $N$ rows of matrix $\bm{A}$ is less than $N$.
For this purpose, let us consider matrix $\bm{B}$ constructed with the top $N$
rows and the first $N$ columns of matrix $\bm{A}$. We write the cosines of
angles $\theta_{i,j}$ in the matrix as a dot product of unit tangent vector
$\hat{p}_{i}$ and the tension unit vector $\hat{t}_{j}$ in the spring
connecting the $i^{\text{th}}$ and the $j^{\text{th}}$ vertices. These unit
vectors are defined as $\hat{p}_{i}=\bm{p}_{i}/p_{i}$ and
$\hat{t}_{i}=\bm{t}_{i}/t_{i}$. Here,

$\begin{split}\bm{t}_{i}&=(\text{cos}\phi_{i+1}-\text{cos}\phi_{i},\;\text{sin}\phi_{i+1}-\text{sin}\phi_{i}),\\\
\bm{p}_{i}&=(\text{cos}\phi_{i+1}-\text{cos}\phi_{i-1},\;\text{sin}\phi_{i+1}-\text{sin}\phi_{i-1}),\end{split}$
(10)

Figure 5: Schematic of a cyclic pentagon in a state of self stress. Purple
arrows indicate the inwards tension forces in the springs. Green arrows
indicate the outward non-radial forces of the area spring. The thin solid
black line segment at each vertex is the line perpendicular to the outward
force of the area spring. $\theta_{i,j}$ is the angle between the ‘tangent’ at
vertex $i$ and the vertex $j$. Note that the ‘tangent’ lines are not tangent
to the circumcircle. The black-dashed line is the circumcircle.

and $t_{i}$, $p_{i}$ are their respective magnitudes. In this section,
$(i+1)^{\text{th}}$ index is considered as $(i+1)$ mod $N$. We now decompose
$\bm{B}$ as $\bm{P}^{-1}\bm{C}\bm{T}^{-1}$ where $\bm{P}^{-1}$ and
$\bm{T}^{-1}$ are full rank diagonal matrices with elements
$p_{i}^{-1}\delta_{i,j}$ and $t_{i}^{-1}\delta_{i,j}$ respectively. Since the
rank of a matrix does not change on multiplication with a full rank matrix,
$\text{Rank}(\bm{B})=\text{Rank}(\bm{C})$. The simplified matrix $\bm{C}$ for
the tangential (Tangential directions are perpendicular to the force of the
area spring at the vertex and are not tangential to the circumcircle) force
balance at the vertices is then

$\begin{bmatrix}|\bm{p}_{1}\cdot\bm{t}_{1}|&0&0&...&-|\bm{p}_{1}\cdot\bm{t}_{N}|\\\
-|\bm{p}_{2}\cdot\bm{t}_{1}|&|\bm{p}_{2}\cdot\bm{t}_{2}|&0&...&0\\\
0&-|\bm{p}_{3}\cdot\bm{t}_{2}|&|\bm{p}_{3}\cdot\bm{t}_{3}|&...&0\\\
0&0&...&...&...\\\
0&0&...&-|\bm{p}_{N}\cdot\bm{t}_{N-1}|&|\bm{p}_{N}\cdot\bm{t}_{N}|\\\
\end{bmatrix}$ (11)

We take the modulus of the dot product since the angles between the tension
vectors and the tangential vectors are always acute and, hence, the cosine of
the angle must always be positive.

The rank of a matrix is the dimension of the space spanned by its row or
column vectors equivalently. To find the rank and nullity of $\bm{C}$, we
attempt to find solutions (if they exist) to a set of $N$ number of equations
given by

$\sum_{i=1}^{N}\beta_{i}\;\bm{C}_{i}=\bm{0},$ (12)

where $\bm{C}_{i}$ is the $i^{\text{th}}$ row of matrix $\bm{C}$. While the
trivial solution to $\beta_{i}$’s always exists, we seek nontrivial solutions.
The coefficients $\beta_{i}$ can be solved for iteratively as

$\beta_{i+1}=\beta_{i}\;\frac{|\bm{p}_{i}\cdot\bm{t}_{i}|}{|\bm{p}_{i+1}\cdot\bm{t}_{i}|}~{}~{}~{}~{}~{}~{}\text{for
}i=1,2,...,N.$ (13)

Using the coordinate representation in Eq. (10), and taking advantage of
trigonometric identities, we obtain

$\begin{split}\beta_{i+1}=\beta_{i}\;\bigg{|}&\text{sin}\;\left(\frac{\phi_{i+1}-\phi_{i-1}}{2}\right)\text{cosec}\;\left(\frac{\phi_{i+2}-\phi_{i}}{2}\right)\\\
\times&\text{cos}\;\left(\frac{\phi_{i-1}-\phi_{i}}{2}\right)\text{sec}\;\left(\frac{\phi_{i+2}-\phi_{i+1}}{2}\right)\bigg{|}.\\\
\end{split}$ (14)

By iterating through the first $N-1$ equations, and cancellation of the (sine
and cosine)/(cos and secant) terms of subsequent/next-subsequent iterations we
can express $\beta_{N}$ in terms of $\beta_{1}$ as

$\begin{split}\beta_{N}=\beta_{1}\;\bigg{|}&\text{cosec}\;\left(\frac{\phi_{1}-\phi_{N-1}}{2}\right)\text{sin}\;\left(\frac{\phi_{2}-\phi_{N}}{2}\right)\\\
\times&\text{sec}\;\left(\frac{\phi_{N-1}-\phi_{N}}{2}\right)\text{cos}\;\left(\frac{\phi_{2}-\phi_{1}}{2}\right)\bigg{|}.\\\
\end{split}$ (15)

This relation is the same as the one implied by the $N^{\text{th}}$ equation
of Eq. (15). Thus, the conditions on the $\beta_{i}$’s are not independent and
a nontrivial solution to $\beta_{i}$’s can be written in terms of $\beta_{1}$
which serves as the chosen parameter for the set of solutions. The nullity of
this matrix is one and the rank of the matrix is $N-1$. $\bm{C}$ not being
full rank implies that $\bm{B}$ is not full rank. Since the rest of the row
vectors in $\bm{A}$ are clearly independent among themselves and with the row
vectors of $\bm{B}$ due to the $-1$ terms, rank of $\bm{A}$ is $2N-1$ and its
nullity is one.

Figure 6:  The irregular polygon rigidifies when a cyclic polygon
configuration is achieved (shown in solid red lines). The red dashed line is
the circumcircle.

While nullity of the geometric matrix $\bm{A}$ would remain zero in the floppy
phase of the polygon allowing only trivial zero solutions for tensions in the
polygon, post-rigidity transition, the nullity becomes one. This is akin to
the emergence of a system spanning state of self stress under shear strain of
spring networks Vermeulen  _et al._ (2017). Physically, for the random
polygon, this means that for a given cyclic configuration of rigid rods, at
every imposed expansion strain on the loop, there exists a set of tensions
parametrized by $t_{1}$ that satisfies force balance at all vertices. Note
that not only at the critical strain, but at all strains post transition, the
configuration of the polygon is cyclic for increasing radii. The
configurations of cyclic polygons are a sufficient condition to sustain a
system-spanning state of self stress. These self stresses can then rigidity
the loop. Given that the polygon which maximizes area for a given set of edges
is unique Pinelis (2005), this further implies that at the rigidity
transition, the cyclic polygon supports a unique state of self stress. A
unique state of self stress supporting the network’s rigidity is seen in other
strain-induced transitions as well Alexander (1998b); Merkel  _et al._ (2019);
Vermeulen  _et al._ (2017).

##  VI Rigidity transition

The critical strain of the rigidity transition in stretching a chain of
springs or inflating random polygons is determined solely by the geometry of
the systems. For a chain of floppy springs, the total contour length of the
springs is sufficient to predict the critical strain of transition where the
chain straightens out. In this section, we discuss the approach to the
transition and approximating the transition strain. The simplest method would
be to calculate the asphericity of the polygon which should fall to zero at
the point of transition. Alternatively, in Sec. (VI.1) we delineate a method
that explicitly shows the insufficiency of the nontrivial floppy modes in
increasing the area of the polygon as it approaches the point of rigidity
transition. In Sec. (VI.2), to approximate the transition strain in irregular
polygons, we approximate the area of the cyclic irregular polygon at the
transition point by the area of the corresponding regular polygon. Unlike the
measurement of critical strain reported in Fig. (3), both these methods employ
only the configurations of the polygons and do not require energy measurements
to predict the rigidity transition.

###  VI.1 Identifying the approach to transition

In this section, we predict the critical strain by observing the shape of the
loop. At every strain, we use the energy minimized configuration to construct
the rigidity matrix $\bm{R}$ for the two-body spring system whose zero modes
are the floppy modes. We then evaluate the gradient of area $A$ expressed as a
function of the nontrivial floppy variables of the polygon. The modulus of
this gradient continuously decreases and falls to zero at the rigidity
transition. This serves as a measure to indicate the transition.

For the purpose of formally defining the rigidity matrix $\bm{R}$, let
$\bm{p}(\tau)$ be a continuous, analytic deformation of a network with $N$
vertices with $\tau$ denoting time such that $\bm{p}(0)=\bm{r}$ with

$\bm{r}:=(\bm{r}_{1},\bm{r}_{2},...\bm{r}_{n}).$ (16)

The constraints of preserving the rest lengths of the two-body springs are

$|\bm{r}_{i}-\bm{r}_{j}|^{2}=d^{2}_{ij},$ (17)

where $i,j$ are the vertices which flank the constraining edge of length
$d_{ij}$, and $\bm{r}_{i}$ is the position vector of the $i^{\text{th}}$
vertex. Taking the derivative of Eq. (17) with respect to $\tau$ at $\tau=0$,
we have,

$(\bm{r}_{i}-\bm{r}_{j})\cdot(\bm{p}_{i}^{\prime}-\bm{p}_{j}^{\prime})=\bm{0}.$
(18)

$\bm{p}_{i}^{\prime}=\bm{p}_{i}^{\prime}(0)$ is the velocity of the
$i^{\text{th}}$ vertex which deforms the network infinitesimally. Instead of
preserving the constraints exactly as in Eq. (17), the above set of equations
preserve the constraints only to first order in displacements. The system of
equations that represents this first-order theory may be written in terms of a
matrix equation

$\bm{R}(\bm{r})\bm{p}^{\prime}=\bm{0},$ (19)

where $\bm{R}(\bm{r})$ is the rigidity matrix. A network is first-order rigid
if there are no nontrivial solutions to Eq. (19) with the trivial solutions
being rigid body displacements of the entire network. A nonzero solution to
this equation is a nontrivial floppy mode. Floppy modes are vectors that span
the null space of the transformation. The motion of the bonds in a floppy mode
is such that the velocity of each directed bond
$\bm{p}^{\prime}_{i}-\bm{p}^{\prime}_{j}$ is perpendicular to the direction of
the bond $\bm{r}_{i}-\bm{r}_{j}$, thus satisfying Eq. (18).

When expansive strain is imposed on the polygon by changing the tuning
parameters, the polygon responds by moving to a configuration that minimizes
the elastic energy (see Eq. 2). Until the rigidity transition, this can be
achieved via floppy motion: motion that preserves the lengths of the springs
while also keeping the area constant. For $N_{S}=0$ and $d=2$, we know from
Eq. (1) that the number of floppy modes $N_{0}$ is $N-3$. We seek to write the
change in position vector of the network $\Delta\bm{r}$ in terms of $N_{0}$
number of independent floppy variables,

$\bm{u}:=(u_{1},u_{2},...u_{N_{0}}).$ (20)

We choose independent definitions for nontrivial floppy variables which we
encode in the matrix equation $\bm{F}\Delta\bm{r}=\bm{u}$. Three more
constraining equations are provided by fixing the two translations and
rotation of the system. The associated matrix equation is denoted by
$\bm{G}\Delta\bm{r}=\bm{0}$. Finally Eq. (19) provides a set of $N$ equations
as $\bm{R}(\bm{r})\Delta\bm{r}=\bm{0}$.

Dimensions | Matrix equations
---|---
$N_{b}\times 2N$ | $\bm{R}\;\Delta\bm{r}=\bm{0}$
$3\times 2N$ | $\bm{G}\;\Delta\bm{r}=\bm{0}$
$N_{0}\times 2N$ | $\bm{F}\;\Delta\bm{r}=\bm{u}$
$2N\times 2N$ |  $\bm{R}^{\prime}\;\Delta\bm{r}=\bm{u^{\prime}}$,

In the last row, we have vertically stacked these equations. In block notation
of matrices the same operation may be written as,

$\bm{R}^{\prime}=\begin{bmatrix}\bm{R}\\\ \bm{G}\\\ \bm{F}\\\
\end{bmatrix},~{}~{}~{}\bm{u}^{\prime}=\begin{bmatrix}\bm{0}\\\ \bm{0}\\\
\bm{u}\\\ \end{bmatrix}.$ (21)

Note that the dimensions of the zero vectors differ. Assuming its
invertibility, we invert the square matrix $\bm{R}^{\prime}$ to express the
coordinate displacements of the system in terms of the nontrivial floppy
variables,

$\Delta\bm{r}(\bm{u})=(\bm{R}^{\prime})^{-1}\bm{u^{\prime}}.$

On imposition of strain, the displacements of vertices which obey all the
spring constraints to first-order and that exclude trivial translations and
rotation is a function of the independent nontrivial floppy variables.

The area of the polygon can be found using Green’s theorem/shoelace formula,
Eq. (3). Upon strain imposition, we have,

$\bm{r}^{\prime}(\bm{u})=\bm{r}+\Delta\bm{r}(\bm{u}).$ (22)

The geometric measure for the rigidity transition is,

$\bm{\nabla}A(\bm{r}^{\prime})\bigg{|}_{\bm{u}=\bm{0}}=\left.\frac{dA(\bm{r}+\Delta\bm{r}(\bm{u}))}{d\bm{u}}\right|_{\bm{u}=\bm{0}}=\bm{0},$
(23)

i.e. the gradient of the area function with respect to the independent
nontrivial floppy variables is uniformly zero. If an expansive area strain is
imposed at this configuration, it cannot be achieved via the motion of
nontrivial floppy variables and an energy cost needs to be paid.

Figure 7: The approach to the rigidity transition can be followed by measuring
the capacity of the nontrivial floppy modes to increase the area of the
polygon. The irregular octagon used here has five nontrivial floppy modes. a)
Convexity being a necessary condition for transition precedes the rigidity
transition. The rigidity transition coincides with the normalized gradient
square of area, $1/N_{0}(dA/d\bm{u})^{2}$ approaching zero. b) All the five
components of the gradient of the area goes to zero at the point of
transition.

Inversion of the rigidity matrix was done using the SymPy package of python
Meurer  _et al._ (2017). As an example, we use a two dimensional irregular
octagon which has $N_{0}=5$ number of floppy modes (see Eq. (1) with
$N_{S}=0$). The choice of nontrivial floppy variables $\bm{u}$ is,

$\begin{split}u_{1}&=\Delta x_{6}+\Delta y_{6},\\\ u_{2}&=\Delta x_{7}+\Delta
y_{5},\\\ u_{3}&=\Delta x_{5}+\Delta y_{7},\\\ u_{4}&=\Delta x_{8}+\Delta
y_{4},\\\ u_{5}&=\Delta x_{4}+\Delta y_{8}.\end{split}$ (24)

The results are reported in Fig. (7). We see that the gradient of the area
goes to zero at the critical strain. The gradient of the area function
approaching zero identifies the critical strain in a similar fashion as the
singular value of the equilibrium matrix approaching zero identifies the
rigidity transition in a shear strained network Vermeulen  _et al._ (2017). We
also conclude from the loss of all nontrivial zero modes at the transition
that the system is microscopically rigid.

###  VI.2 Approximation of transition strain

Figure 8: Regular polygons approximately predict critical strain of irregular
polygons. $A_{c}$ is the numerically determined area of a cyclic polygon at
critical strain and $A_{\text{reg}}$ is the area of the approximated regular
polygon. The approximation is seen to work well in the $N\rightarrow\infty$
limit. Error bars are standard deviations about the average of 100 runs. The
solid line is a guide to the eye.

To predict the transition strain, we need to know the area of the cyclic
polygon corresponding to the initial edge length distribution of the polygon.
Finding the area of general cyclic polygons given the edges is an area of
ongoing research in mathematics. For $N=4$, we can find the area of irregular
cyclic polygons using Brahmagupta’s formula,

$\text{Area}_{\text{ cyclic quadrilateral}}=\sqrt{(s-a)(s-b)(s-c)(s-d)},$ (25)

where $a,b,c,d$ are the lengths of edges and $s$ is the semi-perimeter. The
area of cyclic pentagons and hexagons have been found in more recent work
Robbins (1994, 1995). It is a nontrivial problem to find an exact expression
for the maximum area of the polygon. However, the area of any regular polygon
(Polygons with equal edge lengths) is easily expressible in terms of its edge
length. Here, we make a simplifying approximation by considering an irregular
polygon as a regular polygon whose edge length is the average edge length of
the irregular polygon. This approximation holds well in the limit of
$N\rightarrow\infty$ (see Fig. 8) simply because the area of both irregular
polygons and regular polygons approach the area of the circumcircle with
radius $R$ which satisfies the implicit equation,

$\sum_{i=1}^{N}\text{sin}^{-1}\left(\frac{\ell_{i_{0}}}{2R}\right)=\pi$ (26)

In the limit of large $N$, we have, $l_{i_{0}}/2R\rightarrow 0$. The above
implicit equation, under this limit, simplifies to $\sum_{i=1}^{N}l_{i0}=2\pi
R$, which is the perimeter of a circle of radius $R$.

Figure 9: Isotropic strain-induced rigidity transition in under-constrained
periodic spring networks. a) A Floppy spring network at $l_{0}=0.8$ and a
strain-rigidified spring network at $l_{0}=0.6182$. Note that lower the
$l_{0}$, more is the strain. b) Left: A floppy network does not have any
spring tensions and constituent loops in the network need not be convex.
Right: A rigidified network has non-zero spring tensions and all polygons are
convex. Note the scale bar difference in color map of tensions in springs
pre/post transition. c) Top/Bottom: Rigidity/Convexity transition as a
function of isotropic strain. Convexity transition in the spring network
coincides with the rigidity transition. The blue dashed line marks the point
of rigidity transition. Finite size effects are weak in the system sizes
studied.

##  VII Spring networks

For random polygons, we established that convexity is necessary for a system
spanning state of self stress in Sec. IV and that cyclicity is sufficient to
guarantee a system spanning state of self stress in Sec. V. How do such
geometric signatures translate to a system that is built out of random
polygons? At the rigidity transition, do all polygons become cyclic polygons?
These are the two main questions we address in this section.

We consider a two-dimensional spring network with the energy

$H_{SN}=\sum_{i=1}^{N}(\ell_{i}-\ell_{0})^{2},$ (27)

where $l_{i}$ is the spring length of the $i^{\text{th}}$ spring and $l_{0}$
is their rest length. The network is three-fold coordinated and, therefore,
under-constrained via Maxwell constraint counting. We implement periodic
boundary conditions in both directions such that the network is constrained to
a torus. With this construction, we no longer require the area spring
constraint. An isotropic strain is then imposed by decreasing the rest length
$l_{0}$. Note that this choice is different from the one made in Merkel  _et
al._ (2019) where isotropic strain is imposed by proportionally decreasing the
rest lengths $l_{0_{i}}$ of all springs.

As $l_{0}$ is decreased, the energy of the system is minimized using FIRE
minimization for a modified version of cellGPU for two-body springs Sussman
(2017). Once the average tension in the spring network is non-zero, we
characterize it as rigid. In our simulations (see Fig. 9 b), we see that at
the rigidity transition, all constituent loops in the network become convex.
We also see in Fig. 9 c that the convexity transition in the spring network -
the strain at which all the loops in the network become convex - is the same
as the point of rigidity transition. Additionally, the approach to transition
can be judged by measuring the fraction of convex polygons since this fraction
increases linearly with strain (see Fig. 9 c). As for cyclicity, Fig. (9 a)
clearly shows that post-transition strain, all polygons are not cyclic.
Interestingly, the polygons supporting above-average tensions in the network
appear less cyclic than the polygons supporting less than average tensions.

The important geometric signature is not cyclicity but convexity.
Interestingly, prior work studying ordered spring networks consisting of
polygons of equal edge lengths with periodic boundary conditions demonstrates
that all polygons are cyclic at the rigidity transition Guiducci  _et al._
(2015). In the presence of disorder, not all polygons are equal in terms of
its cyclicity. As loops of polygons form, those polygons composed of higher-
tension springs are more stressed in a particular direction. Being more
stressed, or tensioned in a particular direction, causes them to be acyclic.
However, the necessary condition of convexity still prevails, even in the
disordered network. With convexity as the geometric signature, perhaps
understanding of the onset of rigidity may be enhanced with a focus on loops
of polygons as opposed to focus on stretching out lines of springs, which does
not, in of itself, relate directly to convexity. Coincidentally, convex and
nonconvex can map to a Boolean variable such that a mapping to an Ising-like
description is possible, provided one quantifies the interaction between
convex polygons, etc. One can also imagine that including area-conservation
constraints for each polygon may modify the interaction such that an Ising
spin glass-like description is possible given the additional constraints.

##  VIII Discussion

For a random polygon made up of harmonic springs and including an additional
area spring, we demonstrate that while a convex configuration is necessary,
the cyclic configuration is sufficient for sustaining a system spanning state
of self stress. Such a self stress can then rigidify the polygon with
expansion strain. An isolated loop attaining a cyclic polygon configuration at
the critical strain of rigidity transition, is independent of the details of
the nonlinearities in the Hamiltonian. While we have explicitly proved that
the cyclic configuration of the polygon can sustain a state of self stress,
our numerical simulations show that such a self stress rigidifies the polygon
as evidenced in the non-zero stiffness moduli. We, therefore, have arrived at
a two-dimensional version of Cauchy’s geometry-rigidity correspondence for
isolated structures that are topologically trivial. It would be interesting to
study the nature of this transition for a thermal system with a quadratic
area-energy term and investigate the existence of a phase transition as
previously done in the context of polymer rings with pressure coupled to area
Rudnick and Gaspari (1991); Leibler  _et al._ (1987); Haleva and Diamant
(2006); Mitra  _et al._ (2012).

Given our result for an individual polygon, a natural extension would be to
investigate the geometry-rigidity correspondence for two polygons sharing an
edge. There, we anticipate that the cyclicity-rigidity correspondence for each
polygon still holds. One can then naturally extend the investigation further
by considering multiple polygons with shared edges to form a network of
polygons formed with harmonic springs to arrive at a spring network. We
numerically study this particular limit by imposing periodic boundary
conditions such that the area springs are no longer needed as the network is
now constrained to a toroidal surface. We find that while the cyclic
configuration is important for the case of the isolated random polygon to
attain rigidity, all polygons are not cyclic for a strain-rigidified, under-
constrained spring-network. However, the data suggests that all polygons are
convex at the transition. Intriguingly, earlier work on an ordered, under-
constrained spring network reports that the network rigidity correlates with
the cyclicity of the polygons making up the network Guiducci  _et al._ (2015).
How such a correlation is modified for an infinitesimal amount of disorder is
an open question.

For the disordered, under-constrained spring networks studied here, it would
be useful to study the correlation of the cyclicity of the constituent
polygons in the network and the average tensions in such a loop. We expect the
two quantities to be anti-correlated. More specifically, polygons that are
less cyclic may become part of the ‘force chains’ in the network that sustain
the system spanning state of self stress. Of course, force chains have long
been studied in granular systems Mueth  _et al._ (1998); Majmudar and
Behringer (2005). Moreover, a correspondence between geometry and rigidity has
been established in force tilings of particle packings to, again, highlight
the importance of the correspondence Sarkar  _et al._ (2013). Specifically,
for frictionless, particle packings, all force tilings are convex in the rigid
phase Sarkar  _et al._ (2013). As we work to understand the correspondence
between geometry and rigidity in general disordered systems, hopefully we can
adopt a “rigid or not by looking” approach, at least in some limits.

Figure 10:  Energy of regular polygon under isotropic expansive strain imposed
via the area spring. Analytical solution (solid-line) is possible due to the
symmetry of regular polygons. Note that we use $\gamma^{\prime}$ strain where
$A_{0}$ spans very large values.

## Acknowledgments

MCG acknowledges useful discussions with Manu Mannattil. JMS acknowledges
financial support from NSF-DMR-1832002 and an Isaac Newton Award from the DoD.

##  IX Appendix: Regular polygons

For a regular polygon, we can analytically solve for the minimized energy
configuration under expansive strain. A regular polygon has only one degree of
freedom - the edge length $l$. The area of the regular polygon is
$N\;l^{2}/(4\;\text{tan}(\pi/N))$. An exact expression can be found by
minimizing Eq. (2) with respect to $l$ at each strain. This gives a cubic
equation in $l$ which was solved in MATHEMATICA. Instead of using, strain
$\gamma$ as defined in Eq. (4), we increase $A_{0}$ as
$A_{0}=a_{0}/(1-\gamma^{\prime\;2})$ with $0\leq\gamma^{\prime}<1$ to capture
the complete range of possible $A_{0}$’s. See Fig. (10).

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# Positive Toeplitz Operators from
a Weighted Harmonic Bloch Space
into Another

Ömer Faruk Doğan  Department of Mathematics, Tek$\dot{\hbox{\i}}$rdağ
Namik Kemal University, 59030
Tek$\dot{\hbox{\i}}$rdağ, Turkey<EMAIL_ADDRESS>
(Date: January 1, 2004)

###### Abstract.

We define positive Toeplitz operators between weighted harmonic Bloch spaces
$b^{\infty}_{\alpha}$ on the unit ball of $\mathbb{R}^{n}$ for the full range
of parameter $\alpha\in\mathbb{R}$. We give characterizations of bounded and
compact Toeplitz operators taking one weighted harmonic Bloch space into
another in terms of Carleson and vanishing Carleson measures.

###### Key words and phrases:

Toeplitz operator, Harmonic Bergman-Besov space, Weighted Harmonic Bloch
Space, Carleson measure

###### 1991 Mathematics Subject Classification:

Primary 47B35; Secondary 31B05

##  1\. Introduction

Let $n\geq 2$ be an integer and $\mathbb{B}=\mathbb{B}_{n}$ be the open unit
ball in $\mathbb{R}^{n}$. Let $\nu$ be the Lebesgue volume measure on
$\mathbb{B}$ normalized so that $\nu(\mathbb{B})=1$. For
$\alpha\in\mathbb{R}$, we define the weighted volume measures $\nu_{\alpha}$
on $\mathbb{B}$ by

$d\nu_{\alpha}(x)=\frac{1}{V_{\alpha}}(1-|x|^{2})^{\alpha}d\nu(x).$

These measures are finite when $\alpha>-1$ and in this case we choose
$V_{\alpha}$ so that $\nu_{\alpha}(\mathbb{B})=1$. Naturally $V_{0}=1$. For
$\alpha\leq-1$, we set $V_{\alpha}=1$. We denote the Lebesgue classes with
respect to $\nu_{\alpha}$ by $L^{p}_{\alpha}$, $0<p<\infty$, and the
corresponding norms by $\|\cdot\|_{L^{p}_{\alpha}}$.

Let $h(\mathbb{B})$ be the space of all complex-valued harmonic functions on
$\mathbb{B}$ with the topology of uniform convergence on compact subsets. For
$0<p<\infty$ and $\alpha>-1$, the harmonic weighted Bergman space
$b^{p}_{\alpha}$ is defined by $b^{p}_{\alpha}=L^{p}_{\alpha}\cap
h(\mathbb{B})$ endowed with the norm $\|\cdot\|_{L^{p}_{\alpha}}$. The
subfamily $b^{2}_{\alpha}$ is a reproducing kernel Hilbert space with respect
to the inner product
$[f,g]_{b^{2}_{\alpha}}=\int_{\mathbb{B}}f\overline{g}\,d\nu_{\alpha}(x)$ and
with the reproducing kernel $R_{\alpha}(x,y)$ such that
$f(x)=[f,R_{\alpha}(x,\cdot)]_{b^{2}_{\alpha}}$ for every $f\in
b^{2}_{\alpha}$ and $x\in\mathbb{B}$. It is well-known that $R_{\alpha}$ is
real-valued and $R_{\alpha}(x,y)=R_{\alpha}(y,x)$. The homogeneous expansion
of $R_{\alpha}(x,y)$ is given in the $\alpha>-1$ part of the formulas (1.2)
and (1.3) below (see [13], [15]).

The well-known harmonic Bloch space $b$ is the space of all $f\in
h(\mathbb{B})$ such that

$\sup_{x\in\mathbb{B}}(1-|x|^{2})|\nabla f(x)|<\infty.$

We denote by $L^{\infty}$ the Lebesgue class of essentially bounded functions
on $\mathbb{B}$, and for $\alpha\in\mathbb{R}$ we define

$L^{\infty}_{\alpha}=\\{\varphi:(1-|x|^{2})^{\alpha}\varphi(x)\in
L^{\infty}\\},$

so that $L^{\infty}_{0}=L^{\infty}$. The norm on $L^{\infty}_{\alpha}$ is

$\|\varphi\|_{L^{\infty}_{\alpha}}=\|(1-|x|^{2})^{\alpha}\varphi(x)\|_{L^{\infty}}.$

For $\alpha>0$, the weighted harmonic Bloch space $b_{\alpha}$ is
$h(\mathbb{B})\cap L^{\infty}_{\alpha}$ endowed with the norm
$\|\cdot\|_{L^{\infty}_{\alpha}}$.

For $\alpha>0$, the weighted harmonic Bloch space $b^{\infty}_{\alpha}$ is
naturally imbedded in $L^{\infty}_{\alpha}$ by the inclusion $i$. For
$\alpha>0$, the harmonic Bergman projection $Q_{\alpha}:L^{\infty}_{\alpha}\to
b^{\infty}_{\alpha}$ is given by the integral operator

$Q_{\alpha}f(x)=\frac{1}{V_{\alpha}}\int_{\mathbb{B}}R_{\alpha}(x,y)f(y)(1-|y|^{2})^{\alpha}d\nu(y)\quad(f\in
L^{\infty}_{\alpha}).$ (1.1)

This integral operator plays a major role in the theory of weighted harmonic
Bloch spaces and the question when the projection
$Q_{\alpha}:L^{\infty}_{\beta}\to b^{\infty}_{\beta}$ is bounded is studied in
many sources such as [3, 16, 17] for $\beta=0$ and [23] with a different
integral representation valid for $\beta>-1$. Then one defines the Toeplitz
operator ${{}_{\alpha}}T_{\phi}:b^{\infty}_{\alpha}\to b^{\infty}_{\alpha}$
with symbol $\phi\in L^{1}$ by ${{}_{\alpha}}T_{\phi}=Q_{\alpha}M_{\phi}i$,
where $M_{\phi}$ is the operator of multiplication by $\phi$. Let $\mu$ be a
finite complex Borel measure on $\mathbb{B}$. The Toeplitz operator
${{}_{\alpha}}T_{\mu}$ with symbol $\mu$ is defined by

${{}_{\alpha}}T_{\mu}f(x)=\frac{1}{V_{\alpha}}\int_{\mathbb{B}}R_{\alpha}(x,y)f(y)(1-|y|^{2})^{\alpha}d\mu(y)$

for $f\in L^{\infty}_{\alpha}$. The operator ${{}_{\alpha}}T_{\mu}$ is more
general and reduces to ${{}_{\alpha}}T_{\phi}$ when $d\mu=\phi d\nu$. Toeplitz
operators have been studied extensively on the harmonic Bergman spaces by many
authors. Especially, positive symbols of bounded and compact Toeplitz
operators are completely characterized in term of Carleson measures as in
[19],[20] on the ball and in [6] on smoothly bounded domains. Toeplitz
operators from a harmonic Bergman space into another are considered and
positive symbols of bounded and compact Toeplitz operators are characterized
in [7] on smoothly bounded domains and in [4] on the half space.

The weighted harmonic Bergman $b^{p}_{\alpha}$ and Bloch $b^{\infty}_{\alpha}$
spaces can be extended to the whole range $\alpha\in\mathbb{R}$. These are
studied in detail in [15] and [12], respectively. We call the extended family
$b^{p}_{\alpha}$ $(\alpha\in\mathbb{R})$ harmonic Bergman-Besov spaces and the
corresponding reproducing kernels $R_{\alpha}(x,y)$ $(\alpha\in\mathbb{R})$
harmonic Bergman-Besov kernels. The homogeneous expansion of $R_{\alpha}(x,y)$
can be expressed in terms of zonal harmonics

$R_{\alpha}(x,y)=\sum_{k=0}^{\infty}\gamma_{k}(\alpha)Z_{k}(x,y)\quad(\alpha\in\mathbb{R},\,x,y\in\mathbb{B}),$
(1.2)

where (see [14, Theorem 3.7], [15, Theorem 1.3])

$\gamma_{k}(\alpha):=\begin{cases}\dfrac{(1+n/2+\alpha)_{k}}{(n/2)_{k}},&\text{if
$\,\alpha>-(1+n/2)$};\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\dfrac{(k!)^{2}}{(1-(n/2+\alpha))_{k}(n/2)_{k}},&\text{if
$\,\alpha\leq-(1+n/2)$},\end{cases}$ (1.3)

and $(a)_{b}$ is the Pochhammer symbol. For definition and details about
$Z_{k}(x,y)$, see [2, Chapter 5].

The spaces $b^{\infty}_{\alpha}$ can be defined by using the radial
differential operators $D^{t}_{s}$ $(s,t\in\mathbb{R})$ introduced in [14] and
[15]. These operators are defined in terms of reproducing kernels of harmonic
Besov spaces and still mapping $h(\mathbb{B})$ onto itself. The properties of
$D^{t}_{s}$ will be reviewed in Section 2. Consider the linear transformation
$I_{s}^{t}$ defined for $f\in h(\mathbb{B})$ by

$I^{t}_{s}f(x):=(1-|x|^{2})^{t}D^{t}_{s}f(x).$

######  Definition 1.1.

For $\alpha\in\mathbb{R}$, we define the weighted harmonic Bloch space
$b^{\infty}_{\alpha}$ to consist of all $f\in h(\mathbb{B})$ for which
$I^{t}_{s}f$ belongs to $L^{\infty}_{\alpha}$ for some $s,t$ satisfying (see
[12] )

$\alpha+t>0.$ (1.4)

The quantity

$\|f\|_{b^{\infty}_{\alpha}}=\|I^{t}_{s}f\|_{L^{\infty}_{\alpha}}=\sup_{x\in\mathbb{B}}\,(1-|x|^{2})^{\alpha+t}|D^{t}_{s}f(x)|<\infty,$

defines a norm on $b^{\infty}_{\alpha}$ for any such $s,t\in\mathbb{R}$.

It is well-known that the above definition is independent of $s,t$ under
(1.4), and the norms on a given space are all equivalent. Thus for a given
pair $s,t$, $I^{t}_{s}$ isometrically imbeds $b^{\infty}_{\alpha}$ into
$L^{\infty}_{\alpha}$ if and only if (1.4) holds.

Harmonic Bergman-Besov projections $Q_{s}$ that map Lebesgue classes boundedly
onto weighted Bloch spaces $b^{\infty}_{\alpha}$ can be precisely identified
as in the case of $\alpha>0$ by

$s>\alpha-1.$ (1.5)

Then $I_{s}^{t}$ is a right inverse to $Q_{s}$. This is all done in [12].

Now let $\alpha\in\mathbb{R}$, s and t satisfing (1.5) and (1.4), and a
measurable function $\phi$ on $\mathbb{B}$ be given. Harmonic Bergman-Besov
projections $Q_{s}$ forces us to define Toeplitz operators on all
$b^{\infty}_{\alpha}$ as follows. We define the Toeplitz operator
${{}_{s,t}}T_{\phi}:b^{\infty}_{\alpha}\to b^{\infty}_{\alpha}$ with symbol
$\phi$ by ${{}_{s,t}}T_{\phi}=Q_{s}M_{\phi}I_{s}^{t}$. Explicitly,

${{}_{s,t}}T_{\phi}f(x)=\int_{\mathbb{B}}R_{s}(x,y)\phi(y)I^{t}_{s}f(y)d\nu_{s}(y)\quad(f\in
b^{\infty}_{\alpha}).$

We see that ${{}_{s,t}}T_{\phi}$ makes sense if $\phi\in L^{1}_{s-\alpha}$.
When $\alpha>0$, one can choose $t=0$ and a value of $s$ satisfying (1.5) is
$s=\alpha$. Then $I_{\alpha}^{0}$ is inclusion, and ${{}_{s,t}}T_{\phi}$
reduces to the classical Toeplitz operator
${{}_{\alpha}}T_{\phi}=Q_{\alpha}M_{\phi}i$ on the harmonic weighted Bloch
spaces $b^{\infty}_{\alpha}$, $\alpha>0$. We use the term classical to mean a
Toeplitz operator with $i=I_{\alpha}^{0}$. It is possible to take
$s\neq\alpha$ also when $\alpha>-1$. So we have more general Toeplitz
operators defined via $I_{s}^{t}$ strictly on harmonic Bloch spaces too. It
turns out that the properties of Toeplitz operators studied in this paper are
independent of $s,t$ under (1.5) and (1.4).

Having obtained the integral form for ${{}_{s,t}}T_{\phi}$, we can now define
Toeplitz operators on $b^{\infty}_{\alpha}$ with symbol $\mu$. Let $\alpha$,
and $s$ and $t$ satisfing (1.5) and (1.4) be given. We let

$d\kappa(y)=(1-|y|^{2})^{s+t}d\mu(y)$

and define

$\displaystyle{{}_{s,t}}T_{\mu}f(x)$
$\displaystyle=\frac{1}{V_{s}}\int_{\mathbb{B}}R_{s}(x,y)I^{t}_{s}f(y)(1-|y|^{2})^{s}d\mu(y)$
$\displaystyle=\frac{1}{V_{s}}\int_{\mathbb{B}}R_{s}(x,y)D^{t}_{s}f(y)d\kappa(y).$

The operator ${{}_{s,t}}T_{\mu}$ is more general and reduces to
${{}_{s,t}}T_{\phi}$ when $d\mu=\phi d\nu$. It makes sense when

$d\psi(y)=(1-|y|^{2})^{-(\alpha+t)}d\kappa(y)=(1-|y|^{2})^{s-\alpha}d\mu(y)$

is finite. Note that $\mu$ need not be finite in conformity with that $\alpha$
is unrestricted.

The holomorphic analogues of our characterizations for Dirichlet spaces have
been obtained in [1]; these results are all concerned with Toeplitz operators
from a Dirichlet space into itself. Recently, in [11], positive symbols of
bounded and compact general Toeplitz operators between harmonic Bergman-Besov
spaces are completely characterized in term of Carleson measures. In this
paper, we consider the Toeplitz operator ${{}_{s,t}}T_{\mu}$ with positive
symbol and characterize those that are bounded and compact from a weighted
harmonic Bloch space $b^{\infty}_{\alpha_{1}}$ into another
$b^{\infty}_{\alpha_{2}}$ for $\alpha_{1},\alpha_{2}\in\mathbb{R}$. Our main
tool is Carleson measure.

Let $\mu$ be a positive Borel measure on $\mathbb{B}$. For $\alpha>-1$, we say
that $\mu$ is a $\alpha$-Carleson measure if the inclusion
$i:b^{p}_{\alpha}\to L^{p}(\mu)$ is bounded, that is, if

$\left(\int_{\mathbb{B}}|f(x)|^{p}\,d\mu(x)\right)^{1/p}\lesssim\|f\|_{b^{p}_{\alpha}},\qquad(f\in
b^{p}_{\alpha}).$

We also denote

$\|\mu\|_{\alpha}=\sup_{f\in b^{p}_{\alpha},\|f\|_{b^{p}_{\alpha}}\leq
1}\int_{\mathbb{B}}|f(x)|^{p}\,d\mu(x).$

As is common with Carleson measure theorems, the property of being an
$\alpha$-Carleson measure is independent of $p$, because Theorem 3.1 is true
for any $p$. However, it depends on $\alpha>-1$. So for a fixed $\alpha$, an
$\alpha$-Carleson measure for one $b^{p}_{\alpha}$ is a Carleson measure for
all $b^{p}_{\alpha}$ with the same $\alpha$. We can now state our main result.

######  Theorem 1.2.

Let $\alpha_{1},\alpha_{2}\in\mathbb{R}$. Suppose that $\alpha_{1}+t>0$,
$\alpha_{2}+t>0$ and

$s>\alpha_{i}-1,\qquad i=1,2.$ (1.6)

Let

$\gamma=s+t+\alpha_{1}-\alpha_{2}.$

Let $\mu$ be a positive Borel measure on $\mathbb{B}$ and
$d\kappa(y)=(1-|x|^{2})^{s+t}d\mu(y)$. Then the following statements are
equivalent:

  1. (i)

${{}_{s,t}}T_{\mu}$ is bounded from $b^{\infty}_{\alpha_{1}}$ to
$b^{\infty}_{\alpha_{2}}$.

  2. (ii)

$\kappa$ is a $\gamma$-Carleson measure.

In order to characterize compact positive Toeplitz operators
${{}_{s,t}}T_{\mu}$ from weighted harmonic Bloch spaces
$b^{\infty}_{\alpha_{1}}$ into another $b^{\infty}_{\alpha_{2}}$ for all
$\alpha_{1},\alpha_{2}\in\mathbb{R}$, we introduce the notion of vanishing
$\alpha$-Carleson measures. We say that $\mu\geq 0$ is a vanishing
$\alpha$-Carleson measure if for any sequence $\\{f_{k}\\}$ in
$b^{p}_{\alpha}$ with $f_{k}\to 0$ uniformly on each compact subset of
$\mathbb{B}$ and $\|f_{k}\|_{b^{p}_{\alpha}}\leq 1$,

$\lim_{k\to\infty}\int_{\mathbb{B}}|f_{k}(x)|^{p}\,d\mu(x)=0.$

One can see from Theorem 3.3 that the notion of vanishing $\alpha$-Carleson
measures on $b^{p}_{\alpha}$ is also independent of $p$.

######  Theorem 1.3.

Let $\alpha_{1},\alpha_{2}\in\mathbb{R}$. Let $s,t$, and $\gamma$ be as in
Theorem 1.2. Let $\mu$ be a positive Borel measure on $\mathbb{B}$ and
$d\kappa(y)=(1-|x|^{2})^{s+t}d\mu(y)$. Then the following statements are
equivalent:

  1. (i)

${{}_{s,t}}T_{\mu}$ is compact from $b^{\infty}_{\alpha_{1}}$ to
$b^{\infty}_{\alpha_{2}}$.

  2. (ii)

$\kappa$ is a vanishing $\gamma$-Carleson measure.

The proofs of our results are inspired by the work of Pau and Zhao in [21],
where bounded and compact classical Toeplitz operators between holomorphic
weighted Bergman spaces are characterized.

The paper is organized as follows. The notation and some preliminary results
are summarized in Section 2. We will recall various characterizations of
(vanishing) $\alpha$-Carleson measures for weighted harmonic Bergman spaces in
Section 3. Section 4 is devoted to the proof of our main results, Theorem 1.2
and 1.3.

In the following for two positive expressions $X$ and $Y$ we write $X\lesssim
Y$ if there exists a positive constant $C$, whose exact value is inessential,
such that $X\leq CY$. If both $X\lesssim Y$ and $Y\lesssim X$, we write $X\sim
Y$.

##  2\. Preliminaries

In this section we collect some known facts that will be used throughout the
paper.

The Pochhammer symbol $(a)_{b}$ is defined by

$(a)_{b}=\frac{\Gamma(a+b)}{\Gamma(a)},$

when $a$ and $a+b$ are off the pole set $-\mathbb{N}$ of the gamma function.
By Stirling formula,

$\frac{(a)_{c}}{(b)_{c}}\sim c^{a-b}\quad(c\to\infty).$ (2.1)

A harmonic function $f$ on $\mathbb{B}$ has a homogeneous expansion, that is,
there exist homogeneous harmonic polynomials $f_{k}$ of degree $k$ such that
$f(x)=\sum_{k=0}^{\infty}f_{k}(x)$. The series uniformly and absolutely
converges on compact subsets of $\mathbb{B}$.

###  2.1. Pseudohyperbolic metric

The canonical Möbius transformation on $\mathbb{B}$ that exchanges $a$ and $0$
is

$\varphi_{a}(x)=\frac{(1-|a|^{2})(a-x)+|a-x|^{2}a}{[x,a]^{2}}.$

Here the bracket $[x,a]$ is defined by

$[x,a]=\sqrt{1-2x\cdot a+|x|^{2}|a|^{2}},$

where $x\cdot a$ denotes the inner product of $x$ and $a$ in $\mathbb{R}^{n}.$
The pseudohyperbolic distance between $x,y\in\mathbb{B}$ is

$\rho(x,y)=|\varphi_{x}(y)|=\frac{|x-y|}{[x,y]}.$

For a proof of the following lemma see [5, Lemma 2.2].

######  Lemma 2.1.

Let $a,x,y\in\mathbb{B}$. Then

$\frac{1-\rho(x,y)}{1+\rho(x,y)}\leq\frac{[x,a]}{[y,a]}\leq\frac{1+\rho(x,y)}{1-\rho(x,y)}.$

The following two lemmas show that if $x,y\in\mathbb{B}$ are close in the
pseudohyperbolic metric, then certain quantities are comparable. Both of them
easily follow from Lemma 2.1 (note that $[x,x]=1-|x|^{2}$).

######  Lemma 2.2.

Let $0<\delta<1$. Then

$[x,y]\sim 1-|x|^{2}\sim 1-|y|^{2},$

for all $x,y\in\mathbb{B}$ with $\rho(x,y)<\delta$.

######  Lemma 2.3.

Let $0<\delta<1$. Then

$[x,a]\sim[y,a],$

for all $a,x,y\in\mathbb{B}$ with $\rho(x,y)<\delta$.

For $0<\delta<1$ and $x\in\mathbb{B}$ we denote the pseudohyperbolic ball with
center $x$ and radius $\delta$ by $E_{\delta}(x)$. The pseudohyperbolic ball
$E_{\delta}(x)$ is also a Euclidean ball with center $c$ and radius $r$, where

$c=\frac{(1-\delta^{2})x}{1-\delta^{2}|x|^{2}}\qquad\text{and}\qquad
r=\frac{(1-|x|^{2})\delta}{1-\delta^{2}|x|^{2}}.$

It follows that for fixed $0<\delta<1$, we have
$\nu(E_{\delta}(x))\sim(1-|x|^{2})^{n}$. More generally, for
$\alpha\in\mathbb{R}$, by Lemma 2.2

$\nu_{\alpha}(E_{\delta}(x))=\frac{1}{V_{\alpha}}\int_{E_{\delta}(x)}(1-|y|^{2})^{\alpha}\,d\nu(y)\sim(1-|x|^{2})^{\alpha}\nu(E_{\delta}(x))\sim(1-|x|^{2})^{\alpha+n}.$
(2.2)

Let $\\{a_{k}\\}$ be a sequence of points in $\mathbb{B}$ and $0<\delta<1$. We
say that $\\{a_{k}\\}$ is $\delta$-separated if $\rho(a_{j},a_{k})\geq\delta$
for all $j\neq k$. For a proof of the following lemma see, for example, [18].

######  Lemma 2.4.

Let $0<\delta<1$. There exists a sequence of points $\\{a_{k}\\}$ in
$\mathbb{B}$ satisfying the following properties:

  1. (i)

$\\{a_{k}\\}$ is $\delta$-separated.

  2. (ii)

$\displaystyle\bigcup_{k=1}^{\infty}E_{\delta}(a_{k})=\mathbb{B}$.

  3. (iii)

There exists a positive integer $N$ such that every $x\in\mathbb{B}$ belongs
to at most $N$ of the balls $E_{\delta}(a_{k})$.

In what follows whenever we use expressions like
$\widehat{\mu}_{\alpha,\delta}(a_{k})$, the sequence
$\\{a_{k}\\}=\\{a_{k}(\delta)\\}$ will always refer to the sequence chosen in
Lemma 2.4.

We also need to the following pointwise estimate. See [10, Lemma 3.1] for a
proof.

######  Lemma 2.5.

Let $0<p<\infty$ and $\alpha>-1$. Then

$|u(x)|\lesssim\frac{\|u\|_{b^{p}_{\alpha}}}{(1-|x|^{2})^{(n+\alpha)/p}}$

for all $u\in b^{p}_{\alpha}$ and $x\in\mathbb{B}$.

###  2.2. Reproducing Kernels and the Operators $D^{t}_{s}$

For every $\alpha\in\mathbb{R}$ we have $\gamma_{0}(\alpha)=1$, and therefore

$R_{\alpha}(x,0)=R_{\alpha}(0,y)=1,\quad(x,y\in\mathbb{B},\alpha\in\mathbb{R}).$

Checking the two cases in (1.3), we have by (2.1)

$\gamma_{k}(\alpha)\sim k^{1+\alpha}\quad(k\to\infty).$ (2.3)

$R_{\alpha}(x,y)$ is harmonic as a function of either of its variables on
$\overline{\mathbb{B}}$. Using the coefficients in the extended kernels we
define the radial differential operators $D^{t}_{s}$.

######  Definition 2.6.

Let $f=\sum_{k=0}^{\infty}f_{k}\in h(\mathbb{B})$ be given by its homogeneous
expansion. For $s,t\in\mathbb{R}$ we define $D_{s}^{t}$ on $h(\mathbb{B})$ by

$D_{s}^{t}f:=\sum_{k=0}^{\infty}\frac{\gamma_{k}(s+t)}{\gamma_{k}(s)}\,f_{k}.$

By (2.3), $\gamma_{k}(s+t)/\gamma_{k}(s)\sim k^{t}$ for any $s,t$ and, roughly
speaking, $D_{s}^{t}$ multiplies the $k$th homogeneous part of $f$ by $k^{t}$.
For every $s\in\mathbb{R}$, $D_{s}^{0}=I$, the identity. An important property
of $D^{t}_{s}$ is that it is invertible with two-sided inverse $D_{s+t}^{-t}$:

$D^{-t}_{s+t}D^{t}_{s}=D^{t}_{s}D^{-t}_{s+t}=I,$ (2.4)

which follows from the additive property $D_{s+t}^{z}D_{s}^{t}=D_{s}^{z+t}$.

For every $s,t\in\mathbb{R}$, the map $D^{t}_{s}:h(\mathbb{B})\to
h(\mathbb{B})$ is continuous in the topology of uniform convergence on compact
subsets (see [15, Theorem 3.2]). The parameter $s$ plays a minor role and is
used to have the precise relation

$D_{s}^{t}R_{s}(x,y)=R_{s+t}(x,y)$ (2.5)

One of the most important properties about the operator $D^{t}_{s}$ is that it
allows us to pass from one Bloch space to another. More precisely, we have the
following isomorphism. For a proof see [12, Proposition 4.6].

######  Lemma 2.7.

Let $\alpha,s,t\in\mathbb{R}$. The map $D^{t}_{s}:b^{\infty}_{\alpha}\to
b^{\infty}_{\alpha+t}$ is an isomorphism.

The following duality result is [12, Theorem 5.4].

######  Theorem 2.8.

Let $q\in\mathbb{R}$. Pick $s^{\prime},t^{\prime}$ such that

$\displaystyle s^{\prime}$ $\displaystyle>q,$ $\displaystyle q+t^{\prime}$
$\displaystyle>-1.$

The dual of $b^{1}_{q}$ can be identified with $b_{\alpha}$ (for any
$\alpha\in\mathbb{R}$) under the pairing

$\langle
f,g\rangle=\int_{\mathbb{B}}I^{t^{\prime}}_{s^{\prime}}f\,\overline{I^{s^{\prime}-q-\alpha}_{t^{\prime}+q+\alpha}g}\
d\nu_{q+\alpha},\qquad(f\in b^{1}_{q},\ g\in b_{\alpha}).$

###  2.3. Estimates of Reproducing Kernels

In case $\alpha>-1$, the reproducing kernels $R_{\alpha}(x,y)$ are well-
studied by various authors. We recall some of their properties below. For
extension of these properties to $\alpha\in\mathbb{R}$ we refer to [15].

For a proof of the following pointwise estimate see [8, 22] when $\alpha>-1$
and [15] when $\alpha\in\mathbb{R}$.

######  Lemma 2.9.

Let $\alpha\in\mathbb{R}$. For all $x,y\in\mathbb{B}$,

$|R_{\alpha}(x,y)|\lesssim\begin{cases}\dfrac{1}{[x,y]^{\alpha+n}},&\text{if
$\,\alpha>-n$};\\\ 1+\log\dfrac{1}{[x,y]},&\text{if $\,\alpha=-n$};\\\
1,&\text{if $\,\alpha<-n$}.\end{cases}$

The next lemma shows that the first part of the above estimate continues to
hold when $x$ and $y$ are close enough in the pseudohyperbolic metric. It can
be proved along the same lines as [19, Proposition 5].

######  Lemma 2.10.

Let $\alpha>-n$. There exists $0<\delta<1$ such that for every
$x\in\mathbb{B}$ and $y\in E_{\delta}(x)$,

$R_{\alpha}(x,y)\sim\frac{1}{(1-|x|^{2})^{\alpha+n}}.$

##  3\.  Carleson Measures

Characterizations of (vanishing) $\alpha$-Carleson measures for weighted
harmonic Bergman spaces $b^{p}_{\alpha}$ ($\alpha>-1$) are established by
various authors in more general settings. In this subsection we will recall
these results.

For $0<\delta<1$ the averaging function $\widehat{\mu}_{\delta}$ is defined by

$\widehat{\mu}_{\delta}(x)=\frac{\mu(E_{\delta}(x))}{\nu(E_{\delta}(x))}\qquad(x\in\mathbb{B}).$

More generally, for $\alpha\in\mathbb{R}$ we define

$\widehat{\mu}_{\alpha,\delta}(x):=\frac{\mu(E_{\delta}(x))}{\nu_{\alpha}(E_{\delta}(x))}\qquad(x\in\mathbb{B}).$

By (2.2),
$\widehat{\mu}_{\alpha,\delta}(x)\sim\mu(E_{\delta}(x))/(1-|x|^{2})^{\alpha+n}$.

Now, we cite the following characterization of Carleson measures in terms of
averaging functions which justify the fact that the notion of
$\alpha$-Carleson measures on $b^{p}_{\alpha}$ depend only on $\alpha$.

######  Theorem 3.1.

Let $\mu$ be a positive Borel measure on $\mathbb{B}$, $0<p<\infty$ and
$\alpha>-1$. The following are equivalent:

  1. (a)

$\mu$ is a $\alpha$-Carleson measure.

  2. (b)

$\widehat{\mu}_{\alpha,\delta}\lesssim 1$ for some (every) $0<\delta<1$.

Note that (b) is equivalent to

$\mu(E_{\delta}(x))\lesssim(1-|x|^{2})^{(\alpha+n)}\quad\text{for some (every)
$0<\delta<1$.}$

###### Proof.

Equivalence of (a) and (b) for $\alpha=0$ is proved in [6, Theorem 3.5] for
bounded smooth domains. The proof works equally well for other $\alpha$ too. ∎

We also need the following proposition. The proof is similar to [11,
Proposition 3.6], but for the sake of completeness,we give the details of it.

######  Proposition 3.2.

Let $\mu$ be a positive Borel measure on $\mathbb{B}$. Let $\alpha_{1}>0$ and
$-1<\alpha_{2}<\infty$ and let

$\varrho=\alpha_{1}+\alpha_{2}.$

If $\mu$ is a $\varrho$-Carleson measure, then

$\int_{\mathbb{B}}|f(x)||g(x)|\,d\mu(x)\lesssim\|f\|_{b^{\infty}_{\alpha_{1}}}\|g\|_{b^{1}_{\alpha_{2}}}\qquad(f\in
b^{\infty}_{\alpha_{1}},g\in b^{1}_{\alpha_{2}}).$

###### Proof.

Let $f\in b^{\infty}_{\alpha_{1}},g\in b^{1}_{\alpha_{2}}$. By definitions of
$b^{\infty}_{\alpha_{1}}$ and $b^{1}_{\alpha_{2}}$,

$\displaystyle\|fg\|_{L^{1}_{\varrho}}$
$\displaystyle=\frac{1}{V_{\varrho}}\int_{\mathbb{B}}|f(x)g(x)|(1-|x|^{2})^{\varrho}d\nu(x)$
$\displaystyle\lesssim\|f\|_{b^{\infty}_{\alpha_{1}}}\int_{\mathbb{B}}|g(x)|(1-|x|^{2})^{\alpha_{2}}d\nu(x)$
$\displaystyle\lesssim\|f\|_{b^{\infty}_{\alpha_{1}}}\|g\|_{b^{1}_{\alpha_{2}}}.$
(3.1)

Thus, $fg\in L^{1}_{\varrho}$. Let $0<\delta<1$. Because $E_{\delta/2}(x)$ is
an Euclidean ball with center
$c=(1-(\delta/2)^{2})x/(1-(\delta/2)^{2}|x|^{2})$ and the radius behaves like
$1-|x|^{2}$ when $\delta/2$ is fixed, it follows from [10, Lemma 3.3] that

$|f(x)g(x)|\lesssim\frac{1}{r^{n}}\int_{B(x,r)}|f(y)g(y)|d\nu(y),$

whenever $B(x,r)=\\{y:|y-x|<r\\}\subset E_{\delta/2}(x)$ for all
$x\in\mathbb{B}$. This immediately leads to the pointwise estimate

$|f(x)g(x)|\lesssim\frac{1}{(1-|x|^{2})^{n+\varrho}}\int_{E_{\delta/2}(x)}|f(y)g(y)|(1-|y|^{2})^{\varrho}d\nu(y)$

for all $x\in\mathbb{B}$. For $a\in\mathbb{B}$ and $x\in E_{\delta/2}(a)$, we
note that $E_{\delta/2}(x)\subset E_{\delta}(a)$. Let $E_{\delta/2}(a_{k})$ be
the associated sets to the sequence $\\{a_{k}\\}=\\{a_{k}(\delta/2)\\}$ in
Lemma 2.4. Thus we have

$\displaystyle|f(x)g(x)|$
$\displaystyle\lesssim\frac{1}{(1-|x|^{2})^{n+\varrho}}\int_{E_{\delta/2}(x)}|f(y)g(y)|(1-|y|^{2})^{\varrho}d\nu(y)$
$\displaystyle\lesssim\frac{1}{(1-|x|^{2})^{n+\varrho}}\int_{E_{\delta}(a_{k})}|f(y)g(y)|(1-|y|^{2})^{\varrho}d\nu(y),\quad
x\in E_{\delta/2}(a_{k})$

for $k=1,2,\dots$. Then by Lemma 2.4 and Lemma 2.2, we have

$\displaystyle\int_{\mathbb{B}}|f(x)g(x)|\,d\mu(x)$
$\displaystyle\lesssim\sum_{k=1}^{\infty}\int_{E_{\delta/2}(a_{k})}|f(x)g(x)|\,d\mu(x)$
$\displaystyle\lesssim\sum_{k=1}^{\infty}\int_{E_{\delta}(a_{k})}|f(y)g(y)|(1-|y|^{2})^{\varrho}d\nu(y)$
$\displaystyle\times\int_{E_{\delta/2}(a_{k})}\frac{d\mu(x)}{(1-|x|^{2})^{n+\varrho}}$
$\displaystyle\lesssim\sum_{k=1}^{\infty}\frac{\mu(E_{\delta/2}(a_{k}))}{(1-|a_{k}|^{2})^{n+\varrho}}\int_{E_{\delta}(a_{k})}|f(y)g(y)|(1-|y|^{2})^{\varrho}d\nu(y).$
(3.2)

Since $\mu$ is a $(1,\varrho)$-Bergman-Carleson measure, by Theorem 3.1 we
have

$\mu(E_{\delta/2}(a_{k}))\lesssim(1-|a_{k}|^{2})^{n+\varrho}.$

Then it follows from this together with (3.2) and Lemma 2.4 that

$\displaystyle\int_{\mathbb{B}}|f(x)g(x)|\,d\mu(x)$
$\displaystyle\lesssim\sum_{k=1}^{\infty}\int_{E_{\delta}(a_{k})}|f(y)g(y)|(1-|y|^{2})^{\varrho}d\nu(y)$
$\displaystyle\leq N\|fg\|_{L^{1}_{\varrho}}\lesssim\|fg\|_{L^{1}_{\varrho}},$
(3.3)

where $N$ is the number provided by Lemma 2.4. Combining (3.1) and (3.3)
concludes the proof. ∎

Now we will characterize vanishing $\alpha$-Carleson measures.

######  Theorem 3.3.

Let $\mu$ be a positive Borel measure on $\mathbb{B}$, $0<p<\infty$ and
$\alpha>-1$. The following are equivalent:

  1. (a)

$\mu$ is a vanishing $\alpha$-Carleson measure.

  2. (b)

$\lim_{|x|\to
1^{-}}(1-|x|^{2})^{(\alpha+n)(1-\lambda)}\,\widehat{\mu}_{\alpha,\varepsilon}(x)=0$
for some (every) $0<\varepsilon<1$.

  3. (c)

$\lim_{k\to\infty}(1-|a_{k}|^{2})^{(n+\alpha)(1-\lambda)}\widehat{\mu}_{\alpha,\delta}(a_{k})=0$
for some (every) $0<\delta<1$.

###### Proof.

Equivalence of (a), (b) and (c) for $\alpha=0$ is proved in [7, Theorem 3.5]
for bounded smooth domains. The proof works equally well for other $\alpha$
too. ∎

##  4\. Proofs of Theorems 1.2 and 1.3

In this section we will prove Theorems 1.2 and 1.3. Before that we present a
very useful intertwining relation for transforming certain problems for
Toeplitz operators between weighted Bloch spaces $b^{\infty}_{\alpha}$,
$\alpha\in\mathbb{R}^{n}$ to similar problems for classical Toeplitz operators
between weighted Bloch spaces when $\alpha>0$ . The harmonic and holomorphic
Bergman-Besov space versions are in [11] and [1], respectively.

######  Theorem 4.1.

We have $D^{t}_{s}({{}_{s,t}}T_{\mu})=({{}_{s+t}}T_{\kappa})D^{t}_{s}$, where

${{}_{s+t}}T_{\kappa}f(x)=\frac{1}{V_{s}}\int_{\mathbb{B}}R_{s+t}(x,y)f(y)d\kappa(y)$

is classical Toeplitz operator from $b^{\infty}_{\alpha_{1}+t}$ to
$b^{\infty}_{\alpha_{2}+t}$. Consequently,

$({{}_{s,t}}T_{\mu})=D^{-t}_{s+t}({{}_{s+t}}T_{\kappa})D^{t}_{s},\qquad({{}_{s+t}}T_{\kappa})=D^{t}_{s}({{}_{s,t}}T_{\mu})D^{-t}_{s+t}.$

###### Proof.

By differentiation under the integral sign and (2.5), we have

$\displaystyle D^{t}_{s}({{}_{s,t}}T_{\mu}f)(x)$
$\displaystyle=\frac{1}{V_{s}}\int_{\mathbb{B}}R_{s+t}(x,y)D^{t}_{s}f(y)d\kappa(y)$
$\displaystyle=({{}_{s+t}}T_{\kappa})(D^{t}_{s}f)(x)\qquad(f\in
b^{\infty}_{\alpha_{1}}).$

For the second and third assertions, we note that
$(D^{t}_{s})^{-1}=D^{-t}_{s+t}$ by (2.4). ∎

By Theorem 4.1, ${{}_{s,t}}T_{\mu}$ is bounded from $b^{\infty}_{\alpha_{1}}$
to $b^{\infty}_{\alpha_{2}}$ if and only if ${{}_{s+t}}T_{\kappa}$ is bounded
from $b^{\infty}_{\alpha_{1}+t}$ to $b^{\infty}_{\alpha_{2}+t}$. With all the
preparation done in earlier sections, now we are ready to prove the main
result.

###  4.1. Proof of Theorem 1.2

(i) Implies (ii). Let ${{}_{s,t}}T_{\mu}:b^{\infty}_{\alpha_{1}}\to
b^{\infty}_{\alpha_{2}}$ is bounded. First note that $[x,y]\gtrsim(1-|x|^{2})$
and $[x,y]\gtrsim(1-|y|^{2})$ for $x,y\in\mathbb{B}$. Then fix
$x\in\mathbb{B}$ and consider $R_{s+t}(x,.)$. Under the condition
$n+s+t>\alpha_{1}+t$ provided by (1.6), it is easy to check using Lemma 2.9
that $R_{s+t}(x,.)\in b^{\infty}_{\alpha_{1}+t}$ with

$\|R_{s+t}(x,.)\|b^{\infty}_{\alpha_{1}+t}\lesssim(1-|x|^{2})^{\alpha_{1}-(n+s)}.$

Take $\delta=\delta_{0}$ where $\delta_{0}$ is the number provided by Lemma
2.10. By Lemma 2.2 and Lemma 2.10, we have

$\displaystyle\kappa(E_{\delta}(x))$
$\displaystyle\lesssim\frac{V_{\alpha_{1}}}{V_{s}}(1-|x|^{2})^{2(n+s+t)}\int_{E_{\delta}(x)}|R_{s+t}(x,y)|^{2}d\kappa(y)$
$\displaystyle\lesssim\frac{V_{\alpha_{1}}}{V_{s}}(1-|x|^{2})^{2(n+s+t)}\int_{\mathbb{B}}|R_{s+t}(x,y)|^{2}d\kappa(y)$
$\displaystyle=(1-|x|^{2})^{2(n+s+t)}{{}_{s+t}}T_{\kappa}[R_{s+t}(x,.)](x),$

and therefore

$\displaystyle\widehat{\kappa}_{\gamma,\delta}(x)$
$\displaystyle=\frac{\kappa(E_{\delta}(x))}{\nu_{\gamma}(E_{\delta}(x))}$
$\displaystyle\lesssim(1-|x|^{2})^{2(n+s+t))-(n+\gamma)}{{}_{s+t}}T_{\kappa}[R_{s+t}(x,.)](x).$

On the other hand, by the definition of $b^{\infty}_{\alpha},\,\alpha>0$
together with the boundedness of the Toeplitz operator ${{}_{s+t}}T_{\kappa}$
and an inequality above, we get

$\displaystyle{{}_{s+t}}T_{\kappa}[R_{s+t}(x,.)](x)$
$\displaystyle=|{{}_{s+t}}T_{\kappa}[R_{s+t}(x,.)](x)|$
$\displaystyle\lesssim(1-|x|^{2})^{-t-\alpha_{2}}\|{{}_{s+t}}T_{\kappa}[R_{s+t}(x,.)]\|_{b^{\infty}_{\alpha_{2}+t}}$
$\displaystyle\lesssim(1-|x|^{2})^{-t-\alpha_{2}}\|{{}_{s+t}}T_{\kappa}\|\|R_{s+t}(x,.)\|_{b^{\infty}_{\alpha_{1}+t}}$
$\displaystyle\lesssim(1-|x|^{2})^{-t-\alpha_{2}-n-s+\alpha_{1}}\|{{}_{s+t}}T_{\kappa}\|,$

where $\|{{}_{s+t}}T_{\kappa}\|$ denote the operator norm of
${{}_{s+t}}T_{\kappa}:b^{\infty}_{\alpha_{1}+t}\to b^{\infty}_{\alpha_{2}+t}$.
Combining these estimates we have

$\widehat{\kappa}_{\gamma,\delta}(x)\lesssim\|{{}_{s+t}}T_{\kappa}\|.$

By Theorem 3.1 this means that $\kappa$ is a $\gamma$-Carleson measure.

(ii) Implies (i). Now suppose $(ii)$ holds, that is, $\kappa$ is a
$\gamma$-Carleson measure. Let $\alpha^{\prime}_{2}$ be a number satisfying

$\alpha^{\prime}_{2}=s-\alpha_{2}>-1.$ (4.1)

Since $\alpha^{\prime}_{2}>-1$ and $\alpha_{2}+t>0$, applying Theorem 2.8
(with $s^{\prime}=\alpha^{\prime}_{2}+\alpha_{2}+t$ and $t^{\prime}=0$), we
get that the dual of $b^{1}_{\alpha^{\prime}_{2}}$ can be identified with
$b^{\infty}_{\alpha_{2}+t}$ under the pairing

$[f,g]_{b^{2}_{s+t}}=\int_{\mathbb{B}}f(x)\overline{g(x)}\,d\nu_{s+t}(x).$

Let $f\in b^{\infty}_{\alpha_{1}+t}$ and $h\in b^{1}_{\alpha^{\prime}_{2}}$.
By using Fubini’s theorem and the reproducing formula (1.5) of [15], since
$\alpha^{\prime}_{2}>-1$ and $\alpha^{\prime}_{2}<s+t$ by the
$\alpha_{2}+t>0$, we get that

$\displaystyle[h,{{}_{s+t}}T_{\kappa}f]_{b^{2}_{s+t}}$
$\displaystyle=\frac{1}{V_{s}}\int_{\mathbb{B}}h(y)\int_{\mathbb{B}}R_{s+t}(x,y)\overline{f(x)}d\kappa(x)\,d\nu_{s+t}(y)$
$\displaystyle=\frac{1}{V_{s}}\int_{\mathbb{B}}\left(\int_{\mathbb{B}}R_{s+t}(x,y)h(y)d\nu_{s+t}(y)\right)\overline{f(x)}\,d\kappa(x)$
$\displaystyle=\frac{1}{V_{s}}\int_{\mathbb{B}}h(x)\overline{f(x)}\,d\kappa(x).$

The condition for $\gamma$ in the theorem is equivalent to

$\gamma=\alpha_{1}+t+\alpha^{\prime}_{2}.$

Thus, by Proposition 3.2,

$|[h,{{}_{s+t}}T_{\kappa}f]_{b^{2}_{s+t}}|\lesssim\int_{\mathbb{B}}|h(x)||f(x)|\,d\kappa(x)\lesssim\|f\|_{b^{\infty}_{\alpha_{1}+t}}\|h\|_{b^{1}_{\alpha^{\prime}_{2}}}.$

Hence ${{}_{s+t}}T_{\kappa}$ is bounded from $b^{\infty}_{\alpha_{1}+t}$ to
$b^{\infty}_{\alpha_{2}+t}$.

###  4.2. Proof of Theorem 1.3

In this subsection we will prove Theorem 1.3. Note again that by Theorem 4.1,
${{}_{s,t}}T_{\mu}$ is compact from $b^{\infty}_{\alpha_{1}}$ to
$b^{\infty}_{\alpha_{2}}$ if and only if ${{}_{s+t}}T_{\kappa}$ is compact
from $b^{\infty}_{\alpha_{1}+t}$ to $b^{\infty}_{\alpha_{2}+t}$.

(i) Implies (ii). Since ${{}_{s+t}}T_{\kappa}$ is compact, then
$\|{{}_{s+t}}T_{\kappa}f_{k}\|_{b^{\infty}_{\alpha_{2}+t}}\to 0$ for any
bounded sequence $\\{f_{k}\\}$ in $b^{\infty}_{\alpha_{1}+t}$ converging to
zero uniformly on compact subsets of $\mathbb{B}$. Let
$\\{a_{k}\\}\subset\mathbb{B}$ with $|a_{k}|\to 1^{-}$ and consider the
functions

$f_{k}(x)=(1-|a_{k}|^{2})^{n+s-\alpha_{1}}R_{s+t}(x,a_{k}).$

Due to the conditions on $s$ and Lemma 2.9, since $[x,y]\gtrsim(1-|x|^{2})$
and $[x,y]\gtrsim(1-|y|^{2})$ for $x,y\in\mathbb{B}$, we have
$\sup_{k}\|f_{k}\|_{b^{\infty}_{\alpha_{1}+t}}<\infty$, and it is obvious that
$f_{k}$ converges to zero uniformly on compact subsets of $\mathbb{B}$. Hence
$\|{{}_{s+t}}T_{\kappa}f_{k}\|_{b^{\infty}_{\alpha_{2}+t}}\to 0$. Therefore,
proceeding as (i) Implies (ii) in Theorem 1.2, for any $\delta>0$, we get

$\displaystyle\widehat{\kappa}_{\gamma,\delta}(a_{k})$
$\displaystyle\lesssim(1-|a_{k}|^{2})^{2(n+s+t)-(n+\gamma)-(n+s-\alpha_{1})}{{}_{s+t}}T_{\kappa}f_{k}(a_{k})$
$\displaystyle=(1-|a_{k}|^{2})^{\alpha_{2}+t}{{}_{s+t}}T_{\kappa}f_{k}(a_{k})$
$\displaystyle\lesssim\|{{}_{s+t}}T_{\kappa}f_{k}\|_{b^{\infty}_{\alpha_{2}+t}}\to
0.$

Thus, by Theorem 3.3, the measure $\kappa$ be a vanishing $\delta$-Carleson
measure.

(ii) Implies (i). Now suppose $(ii)$ holds, that is, $\kappa$ is a vanishing
$\gamma$-Carleson measure. To prove that the operator ${{}_{s+t}}T_{\kappa}$
is compact, we must show that
$\|{{}_{s+t}}T_{\kappa}f_{k}\|_{b^{\infty}_{\alpha_{2}+t}}\to 0$ for any
bounded sequence $\\{f_{k}\\}$ in $b^{\infty}_{\alpha_{1}+t}$ converging to
zero uniformly on compact subsets of $\mathbb{B}$. Similarly, as in the proof
of Theorem 1.2, by duality we have (the number $\alpha^{\prime}_{2}$ is the
one defined by (4.1)

$\displaystyle\|{{}_{s+t}}T_{\kappa}f_{k}\|_{b^{\infty}_{\alpha_{2}+p_{2}t}}$
$\displaystyle\lesssim\sup_{\|h\|_{b^{1}_{\alpha^{\prime}_{2}}}\leq
1}|[h,{{}_{s+t}}T_{\kappa}f_{k}]_{b^{2}_{s+t}}|$
$\displaystyle\leq\sup_{\|h\|_{b^{1}_{\alpha^{\prime}_{2}}}\leq
1}\int_{\mathbb{B}}|h(x)||f_{k}(x)|d\kappa(x).$

Let $\kappa_{r}=\kappa|_{\mathbb{B}\backslash\mathbb{\overline{B}}_{r}}$,
where $0<r<1$ and $\mathbb{B}_{r}=\\{x\in\mathbb{B}:|x|<r\\}$. Then
$\kappa_{r}$ is also a $\gamma$-Carleson measure, and

$\lim_{r\to 1}\|\kappa_{r}\|_{\gamma}=0$

(see, p.$130$ of [9]). Let $\vartheta_{r}$ be the measure defined by

$d\vartheta_{r}=\frac{|f_{k}|d\kappa_{r}(x)}{\|f_{k}\|_{b^{\infty}_{\alpha_{1}+t}}}.$

Using Proposition 3.2, it is easy to check that, $\vartheta_{r}$ is a
$\alpha^{\prime}_{2}$-Carleson measure on $b^{1}_{\alpha^{\prime}_{2}}$, and

$\lim_{r\to 1}\|\vartheta_{r}\|_{\alpha^{\prime}_{2}}=0.$

Hence,

$\displaystyle\int_{\mathbb{B}\backslash\mathbb{\overline{B}}_{r}}|h(x)||f_{k}(x)|d\kappa(x)$
$\displaystyle\leq\int_{\mathbb{B}}|h(x)||f_{k}(x)|\kappa_{r}(x)$
$\displaystyle=\|f_{k}\|_{b^{\infty}_{\alpha_{1}+t}}\int_{\mathbb{B}}|h(x)|d\vartheta_{r}$
$\displaystyle\lesssim\|\vartheta_{r}\|_{\alpha^{\prime}_{2}}\|f_{k}\|_{b^{\infty}_{\alpha_{1}+t}}\|h\|_{b^{1}_{\alpha^{\prime}_{2}}}$
$\displaystyle\lesssim\epsilon$ (4.2)

as $r$ sufficiently close to $1$. Fix such an $r$. Since $\\{f_{k}\\}$
converges to $0$ uniformly in compact subsets of $\mathbb{B}$, there is a
constant $K>0$ such that for any $k>K$, $|f_{k}(x)|<\epsilon$ for any
$x\in\mathbb{\overline{B}}_{r}$. Therefore, using Proposition 3.2, we have

$\displaystyle\int_{\mathbb{\overline{B}}_{r}}|h(x)||f_{k}(x)|d\kappa(x)$
$\displaystyle\leq\epsilon\int_{\mathbb{B}}|1||h(x)|\kappa(x)$
$\displaystyle\lesssim\epsilon\|1\|_{b^{\infty}_{\alpha_{1}+t}}\|h\|_{b^{1}_{\alpha^{\prime}_{2}}}\lesssim\epsilon$
(4.3)

for any $x\in\mathbb{\overline{B}}_{r}$. Combining (4.2) and (4.2), we get
that

$\sup_{\|h\|_{b^{1}_{\alpha^{\prime}_{2}}}\leq
1}\int_{\mathbb{B}}|h(x)||f_{k}(x)|d\kappa(x)\to 0.$

Thus, $\|{{}_{s+t}}T_{\kappa}f_{k}\|_{b^{\infty}_{\alpha_{2}+p_{2}t}}\to 0$,
finishing the proof.

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