# Pointwise convergence of the heat and subordinates of the heat semigroups
associated with the Laplace operator on homogeneous trees and two weighted
$L^{p}$ maximal inequalities.

I. Alvarez-Romero †,1 ,  B. Barrios‡,2 and  J. J. Betancor‡,3 Departamento de
Matemáticas, Universidad de Las Palmas de Gran Canaria, Edificio de
Informática y Matemáticas, Campus de Tafira, 35017, Tafira Baja, Las Palmas de
Gran Canaria, Spain.  Departamento de Análisis Matemático, Universidad de La
Laguna C/. Astrofísico Francisco Sánchez s/n, 38200 - La Laguna, Spain.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper we consider the heat semigroup $\\{W_{t}\\}_{t>0}$ defined by
the combinatorial Laplacian and two subordinated families of
$\\{W_{t}\\}_{t>0}$ on homogeneous trees $X$. We characterize the weights $u$
on $X$ for which the pointwise convergence to initial data of the above
families holds for every $f\in L^{p}(X,\mu,u)$ with $1\leq p<\infty$, where
$\mu$ represents the counting measure in $X$ . We prove that this convergence
property in $X$ is equivalent to the fact that the maximal operator on
$t\in(0,R)$, for some $R>0$, defined by the semigroup is bounded from
$L^{p}(X,\mu,u)$ into $L^{p}(X,\mu,v)$ for some weight $v$ on $X$.

##  1\. Introduction

Along this work we will denote by $X$, a homogeneous tree of degree $q+1$,
that is, a connected graph without loops, in which every vertex has $q+1$
neighbors, with $q\in\mathbb{N}$, $q\geq 1$. We consider on $X$ the natural
distance $d$, that is, if $x,\,y\in X$, $x\neq y$, $d(x,y)$ is the number of
edges between the vertexes $x$ and $y$ and $d(x,x)=0$, $x\in X$. By $\mu$ we
represent the counting measure on $X$. Note that $\mu$ is not doubling because
$q\geq 1$. The canonical, also called combinatorial, Laplace operator on $X$
is defined by

(1.1) $\mathcal{L}(f)(x)=f(x)-\frac{1}{q+1}\sum_{y\in X,\,d(x,y)=1}f(y),\quad
x\in X,$

where $f$ is a complex function defined on $X$. When $q=1$ we clearly get that
the homogeneous tree $X$ is equal to $\mathbb{Z}$ and $(\mathbb{Z},d,\mu)$ is
a homogeneous type space in the sense of Coifman and Weiss ([11]). This fact
leads to use all the tools of the harmonic analysis in this framework, in
particular the standard Calderón-Zygmund theory works in this space. We also
observe that in this case

$\mathcal{L}(f)(x)\left(=-\Delta_{1}(f)(x)\right)=-\frac{1}{2}(f(x+1)-2f(x)+f(x-1)),\quad
x\in\mathbb{Z},$

and the harmonic analysis associated with $\Delta_{1}$ on $\mathbb{Z}$ has
been developed in the last years by several authors (see, for instance [1, 2,
9, 10, 19, 27] and the references therein). The operator $\mathcal{L}$ is
bounded in $L^{p}(X,\mu)$ for every $1\leq p\leq\infty$ and it is selfadjoint
in $L^{2}(X,\mu)$. Hence $-\mathcal{L}$ generates a $C_{0}$\- semigroup
$\\{W_{t}\\}_{t\geq 0}:=\\{e^{-t\mathcal{L}}\\}_{t\geq 0}$ of operators in
$L^{p}(X,\mu)$ for every $1\leq p<\infty$. This semigroup $\\{W_{t}\\}_{t\geq
0}$ is usually called the heat semigroup associated with the operator
$\mathcal{L}$ because for every $f\in L^{p}(X,\mu)$, $1\leq p<\infty$, the
function $u(t,x):=W_{t}(f)(x)$, $t>0$ and $x\in X$, solves the initial value
problem

(1.2) $\left\\{\begin{array}[]{ll}\left(\frac{\partial}{\partial
t}+\mathcal{L}\right)u(t,x)=0,\quad t>0,\,x\in X,\\\ u(x,0)=f(x),\quad x\in
X.\end{array}\right.$

We also observe that, for every $t>0$, the operator $W_{t}$ is invariant under
the action of the group of isometries of $(X,d)$, so that, $W_{t}$ is a
convolution operator given by

(1.3) $W_{t}(f)(x)=\int_{X}h_{t}(x,y)f(y)d\mu(y)=\sum_{y\in
X}h_{t}(x,y)f(y),\quad x\in X.$

The function $h_{t}$, $t>0$ is usually called the heat kernel associated to
$\mathcal{L}$ and we have that $h_{t}(x,y)=H_{t}(d(x,y))$, that is,
$h_{t}(x,y)$ depends actually on $d(x,y)$, $x,\,y\in X$. According to [17] we
have that, for every $t>0$,

$H_{t}(k)=\frac{2e^{-(q+1)t}}{\pi
q^{k/2-1}}\int_{0}^{\pi}\frac{e^{2t\sqrt{q}cos(u)}sin(u)(qsin((k+1)u)-sin((k-1)u))}{(q+1)^{2}-4qcos^{2}(u)}du,\quad
k\in\mathbb{N}\setminus\\{0\\},$

and

$H_{t}(0)=\frac{2q(q+1)e^{-(q+1)t}}{\pi}\int_{0}^{\pi}\frac{e^{2t\sqrt{q}cos(u)}sin^{2}(u)}{(q+1)^{2}-4qcos^{2}(u)}du.$

We also consider other semigroups of operators defined subordinating with
respect to $\\{W_{t}\\}_{t>0}.$ For that let us take $0<\alpha<2$ and we
define, for every $t>0$,

$f_{\alpha,t}(s):=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{2\pi\,i}\int_{a-i\infty}^{a+i\infty}e^{zs-
tz^{\frac{\alpha}{2}}}\,dz,\quad s\geq 0,\,a>0,\\\ 0,\quad
s<0.\end{array}\right.$

Note that the last integral does not depend on $a>0$. As in [32, Chapter
IX,11], we also define, for every $t>0$ and $f\in L^{p}(X,\mu),\,1\leq
p\leq\infty$,

(1.4) $\displaystyle P_{t}^{\alpha}(f)(x)$
$\displaystyle:=\int_{0}^{\infty}f_{\alpha,t}(s)W_{s}(f)(x)\,ds$
$\displaystyle=\sum_{y\in X}P_{t}^{\alpha}(d(x,y))f(y),\quad x\in X,$

where

$P_{t}^{\alpha}(k)=\int_{0}^{\infty}f_{\alpha,t}(s)H_{s}(k)ds,\quad
k\in\mathbb{N}.$

Thus, for every $1\leq p<\infty$, $\\{P_{t}^{\alpha}\\}_{t\geq 0}$ is a
$C_{0}$-semigroup of operators on $L^{p}(X,\mu)$ whose infinitesimal generator
is $\mathcal{L}^{\alpha/2}$ ([32, Theorem 2]). Then, for every $f\in
L^{p}(X,\mu)$, the function $v(t,x):=P_{t}^{\alpha}(f)(x)$, $t>0$, $x\in X$,
solves the initial value problem

(1.5) $\left\\{\begin{array}[]{ll}\left(\frac{\partial}{\partial
t}+\mathcal{L}^{\frac{\alpha}{2}}\right)v(t,x)=0,\quad t>0,\,x\in X,\\\
v(x,0)=f(x),\quad x\in X.\end{array}\right.$

Here $\mathcal{L}^{\frac{\alpha}{2}}$ denotes the $\frac{\alpha}{2}$ power of
the operator $\mathcal{L}$ that can be defined, for every $x\in X$, by (see
[32, (5), p. 260])

$\mathcal{L}^{\frac{\alpha}{2}}f(x):=\frac{1}{\Gamma(\frac{\alpha}{2})}\int_{0}^{\infty}{\frac{W_{t}(f)(x)-f(x)}{t^{1+\frac{\alpha}{2}}}}\,dt,\quad
f\in L^{p}(X,\mu),$

for every $1\leq p<\infty$. Since we have that

$f_{1,t}(s)=\frac{t}{2\sqrt{\pi}}s^{-3/2}e^{-t^{2}/4s},\,t,\,s\in(0,\infty),$

then $\\{P_{t}^{1}\\}_{t\geq 0}$ is the so called Poisson semigroup associated
to $\mathcal{L}$. When $\alpha\neq 1$ we do not know a non integral form for
the function $f_{\alpha,t}$.

The last subordinate family that we will consider is the following one. Let
$\nu>0$. For every $t>0$ and $f\in L^{p}(X,\mu)$, $1\leq p\leq\infty$, we
define

$\displaystyle T_{t}^{\nu}(f)(x)$
$\displaystyle:=\frac{t^{2\nu}}{4^{\nu}\Gamma(\nu)}\int_{0}^{\infty}e^{-t^{2}/4s}W_{s}(f)(x)\frac{ds}{s^{1+\nu}}$
(1.6) $\displaystyle=\sum_{y\in X}T_{t}^{\nu}(d(x,y))f(y),\quad x\in X,$

where

$T_{t}^{\nu}(k)=\frac{t^{2\nu}}{4^{\nu}\Gamma(\nu)}\int_{0}^{\infty}e^{-t^{2}/4s}H_{s}(k)\frac{ds}{s^{1+\nu}},\quad
k\in\mathbb{N}.$

As before the family $\\{T_{t}^{\nu}\\}_{t\geq 0}$ is a $C_{0}$\- semigroup in
$L^{p}(X,\mu)$ and, for every $f\in L^{p}(X,\mu)$ the function
$w(t,x):=T_{t}^{\nu}(f)(x)$, $t>0$, $x\in X$, solves the initial value problem

(1.7) $\left\\{\begin{array}[]{ll}\left(\frac{\partial^{2}}{\partial
t^{2}}+\frac{1-2\nu}{t}\frac{\partial}{\partial
t}+\mathcal{L}\right)w(t,x)=0,\quad t>0,\,x\in X,\\\ w(x,0)=f(x),\quad x\in
X.\end{array}\right.$

We note that $T_{t}^{1/2}=P_{t}^{1}$ for every $t>0$.

We can consider [13] as the seminal paper in the development of the harmonic
analysis associated with the operator $\mathcal{L}$ in homogeneous trees. In
this work Cowling, Meda and Setti established some $L^{p}-L^{q}$ mapping
properties for the heat semigroup defined by $\mathcal{L}$. They also proved
$L^{p}$ boundedness properties for the maximal and Littlewood-Paley functions
associated with $\\{W_{t}\\}_{t>0}$. The corresponding properties for the
Poisson semigroup $\\{P_{t}^{1/2}\\}_{t>0}$ were established in [30]. In the
last years the study of harmonic analysis in the homogeneous trees has turned
to take great interest. Heat and Poisson semigroups ([24] and [31]), Poincaré
and Hardy inequalities [5], Hardy and BMO spaces ([3, 4, 7]), nondoubling flow
measures ([25, 26]), uncertainty principles ([15]), Carleson measures ([12]),
special multipliers ([8]) and maximal functions ([14, 18, 28, 29]) are some of
the topics that have being recently studied in this setting.

For now on we denote $\\{S_{t}\\}_{t>0}$ one of the three uniparametric
families that we have introduced, that is, $\\{W_{t}\\}_{t>0}$,
$\\{P_{t}^{\alpha}\\}_{t>0}$, $\alpha\in(0,2)$, and $\\{T^{\nu}_{t}\\}_{t>0}$,
$\nu>0$. It is well known that if $f\in L^{p}(X,\mu)$, $1\leq p<\infty$ then

(1.8) $\lim_{t\to 0^{+}}S_{t}(f)(x)=f(x),\,x\in X.$

Motivated by the results in [16, 21] the main objective in this work is to
find an optimal weighted Lebesgue space for which (1.8) still holds. That is,
we want to find the weights $w\in X$, that is, functions $w:X\to(0,\infty)$,
such that (1.8) is satisfied for every $f\in L^{p}(X,\mu,w)$, $1\leq
p<\infty$. Here $L^{p}(X,\mu,w)$ denotes the usual weighted Lebesgue space.
The characterization of these weights are related with the maximal function
associated to each of the families $\\{W_{t}\\}_{t>0}$,
$\\{P_{t}^{\alpha}\\}_{t>0}$, $\alpha\in(0,2)$ and $\\{T^{\nu}_{t}\\}_{t>0}$,
$\nu>0$. In fact we have the next result.

######  Theorem 1.1.

Let $1\leq p<\infty$, $\alpha\in(0,2)$, and $q\in\mathbb{N}$, $q\geq 1$.
Assume that $u$ is a weight on $X$. The following assertions are equivalent:

  * a)

There exist $R>0$ and a weight $v$ on $X$ such that the operator

$P_{*,R}^{\alpha}(f):=\sup_{0<t<R}|P_{t}^{\alpha}(f)|,$

is bounded from $l^{p}(X,\mu,u)$ into $l^{p}(X,\mu,v)$.

  * b)

There exist $R>0$ and a weight $v$ on $X$ such that the operator
$P_{\star,R}^{\alpha}$ is bounded from $l^{p}(X,\mu,u)$ into
$l^{p,\infty}(X,\mu,v)$.

  * c)

For every $f\in l^{p}(X,\mu,u)$, $\lim_{t\to 0^{+}}P_{t}^{\alpha}(f)(x)=f(x)$,
$x\in X$.

  * d)

For every $f\in l^{p}(X,\mu,u)$, there exists $x\in X$ such that $\lim_{t\to
0^{+}}P_{t}^{\alpha}(f)(x)=f(x)$.

  * e)

There exists $R>0$ such that $P_{*,R}^{\alpha}(f)(x)<\infty,\,x\in X$ for
every $f\in l^{p}(X,\mu,u)$.

  * f)

There exists $R>0$ such that $P_{R}^{\alpha}(f)(x)<\infty,\,x\in X$ for every
$f\in l^{p}(X,\mu,u)$.

  * g)

There exists $R>0$ such that, when $1<p<\infty$,

$\sum_{y\in
X}u(y)^{-p^{\prime}/p}P_{R}^{\alpha}(d(x,y))^{p^{\prime}}<\infty,\,\,\,x\in
X,$

and, when $p=1$,

$\sup_{y\in X}P_{R}^{\alpha}(d(x,y))u(y)^{-1}<\infty,\quad x\in X.$

  * h)

There exist $R>0$ and $x\in X$ satisfying, when $1<p<\infty$,

$\sum_{y\in X}u(y)^{-p^{\prime}/p}P_{R}^{\alpha}(d(x,y))^{p^{\prime}}<\infty,$

and, when $p=1$,

$\sup_{y\in X}P_{R}^{\alpha}(d(x,y))u(y)^{-1}<\infty.$

  * i)

There exists $x\in X$ such that, when $1<p<\infty$,

$\sum_{y\in
X}\left(q^{d(x,y)}(1+d(x,y))^{1+\frac{\alpha}{2}}\right)^{-p^{\prime}}u(y)^{-p^{\prime}/p}<\infty,$

and, when $p=1$,

$\sup_{y\in X}(q^{d(x,y)}(1+d(x,y))^{1+\frac{\alpha}{2}}u(y))^{-1}<\infty.$

######  Theorem 1.2.

Let $1<p<\infty$, $\nu>0$ and $q\in\mathbb{N}$, $q\geq 1$. Assume that $u$ is
a weight on $X$. The following assertions are equivalent:
a) to h) as in Theorem 1.1 when the semigroup $\\{P_{t}^{\alpha}\\}_{t>0}$ is
replaced by $\\{T_{t}^{\nu}\\}_{t>0}$, $\nu>0$.

  * i)

There exists $x\in X$ such that, when $1<p<\infty$,

$\sum_{y\in
X}\left(q^{d(x,y)}(1+d(x,y))^{\nu+1}\right)^{-p^{\prime}}u(y)^{-p^{\prime}/p}<\infty,$

and, when $p=1$,

$\sup_{y\in X}(q^{d(x,y)}(1+d(x,y))^{1+\nu}u(y))^{-1}<\infty.$

######  Theorem 1.3.

Let $1\leq p<\infty$ and $q\in\mathbb{N}$, $q\geq 1$. Assume that $u$ is a
weight on $X$. The following assertions are equivalent:
a), b), e), f) and g) as in Theorem 1.1 with the semigroup $\\{W_{t}\\}_{t>0}$
instead of $\\{P_{t}^{\alpha}\\}_{t>0}$.

  * c)

For a certain $R_{1}>0$ there exists $\lim_{t\to R_{1}}W_{t}(f)(x)$ for every
$f\in l^{p}(X,\mu,u)$ and $x\in X$.

  * d)

For a certain $R_{1}>0$ and for every $f\in l^{p}(X,\mu,u)$ there exists $x\in
X$ for which there exists $\lim_{t\to R_{1}}W_{t}(f)(x)$.

Each of the above conditions a)-g) implies that for every $f\in
l^{p}(X,\mu,u)$ and $x\in X$, $\lim_{t\to 0^{+}}W_{t}(f)(x)=f(x).$

To conclude this introduction we want to mention that harmonic analysis
related to flow Laplacian on $X$ has been developed recently in [3, 4, 5, 25,
26]. Main definitions and results about trees with nondoubling flow measures
can be found in [26]. On $X$ the canonical flow measure $\lambda$ is defined
by

$\lambda(x):=q^{l(x)},\,x\in X,$

where $l$ is the level function on $X$ defined after fixing a reference point
and a mythical ancestor. The flow measure $\lambda$ is constant only when
$q=1$ and the flow Laplacian (also studied in [22]), denoted by $\mathbb{L}$
on $X$ and defined by $\mu$ is given by

$\mathbb{L}(f)(x):=f(x)-\frac{1}{2\sqrt{q}}\sum_{y\sim
x}\frac{\lambda(y)}{\lambda(x)}f(y),\,x\in X.$

We have that

$\mathbb{L}:=\frac{1}{1-b}\lambda^{-1/2}(\mathcal{L}-bI)\lambda^{1/2},\,\mbox{where
}b:=\frac{(\sqrt{q}-1)^{2}}{q+1}.$

We denote by $\\{\mathbb{W}_{t}\\}_{t>0}$ the heat semigroup defined by
$\mathbb{L}$ in $L^{2}(X,\lambda)$. Thus, for every $t>0$, the semigroups
$W_{t}$ and $\mathbb{W}_{t}$ are connected through the expression

$\mathbb{W}_{t}=\lambda^{-1/2}e^{\frac{bt}{1-b}}W_{\frac{t}{1-b}}\lambda^{1/2}.$

By using the last equality we can deduce from Theorems 1.1-1.3 the
corresponding proper–ties for $\\{\mathbb{W}_{t}\\}_{t>0}$ and the
subordinates semigroups associated to it.

Organization of the work: In the next three sections we will prove Theorems
1.1-1.3 and for that some estimations for the heat kernel associated to the
Laplacian $\mathcal{L}$ established in [13] will be fundamental. The results
when $q=1$ can be proved in a similar way that the corresponding ones for
$q\geq 2$. However, since some estimates used in those are not valid for the
case $X=\mathbb{Z}$, for convenience of the reader, the proofs for $q=1$ are
sketched.

Throughout this paper by $C$ and $c$ we always denote positive constants that
can change from one line to another.

##  2\. Proof of Theorem 1.1 for $q\geq 2$

We consider $q\geq 2$. The properties $a)\Rightarrow b)$, $c)\Rightarrow d)$,
$e)\Rightarrow f)$ and $g)\Rightarrow h)$ are clear. We denote by

$\mathcal{F}=\\{f:X\to\mathbb{C}\mbox{ the set $\\{x\in X:\,f(x)\neq 0\\}$ is
finite}\\}$

a dense subspace of $l^{q}(X,\mu,u)$. Then, since for every $f\in\mathcal{F}$,
$\displaystyle\lim_{t\to 0^{+}}P_{t}^{\alpha}f(x)=f(x)$, by using a standard
argument we can see that $b)\Rightarrow c)$.

Suppose now that $d)$ holds. Let $0\leq f\in l^{p}(X,\mu,u)$. We assume that
$\displaystyle\lim_{t\to 0^{+}}P_{t}^{\alpha}(f)(x_{0})=f(x_{0})$, for some
$x_{0}\in X$. Then, there exists $R_{0}\in(0,1)$ such that

(2.1) $\sup_{0<t<R_{0}}P_{t}^{\alpha}(f)(x_{0})<\infty.$

According to [31, Theorem 3.1] the following property holds for the
$\alpha$-stable kernel $P_{t}^{\alpha}(k)$, $t>0$ and $k\in\mathbb{N}$,

$P_{t}^{\alpha}(k)\sim\phi_{0}(k)tk^{-2-\frac{\alpha}{2}}q^{-k/2},\,t>0,\,k\in\mathbb{N},\,k>t^{2/\alpha}.$

By using the fact that $q\geq 2$ (see [13, p. 4282] or [31, (1)]),

$\phi_{0}(k)\sim(k+1)q^{-k/2},\,k\in\mathbb{N},$

with $\phi_{0}$ being the spherical function ([13, (1.1)]), we get

(2.2) $P_{t}^{\alpha}(k)\sim
k^{-1-\frac{\alpha}{2}}tq^{-k},\,t>0,\,k\in\mathbb{N},\,k>t^{2/\alpha}.$

Let us now consider $x_{1}\in X\setminus\\{x_{0}\\}$ and the function
$g:X\to\mathbb{R}$ such that $g(x_{1})=1$ and $g(x)=0$, $x\in
X\setminus\\{x_{1}\\}$. We have that

$P_{t}^{\alpha}(g)(x)=\sum_{y\in
X}P_{t}^{\alpha}(d(x,y))g(y)=P_{t}^{\alpha}(d(x,x_{1})),\,x\in X,\,t>0.$

Then, it is clear that

(2.3) $\lim_{t\to
0^{+}}P_{t}^{\alpha}(k)=0,\,k\in\mathbb{N}\setminus\\{0\\},\,\lim_{t\to
0^{+}}P_{t}^{\alpha}(0)=1.$

By (2.3) there exists $R_{1}\in(0,R_{0})$ such that $P_{t}^{\alpha}(0)\sim 1$
and $P_{t}^{\alpha}(d(x_{1},x_{0}))\leq C$, for every $t\in(0,R_{1}]$.

For every $t>0$, it follows that

(2.4) $\begin{split}P_{t}^{\alpha}(f)(x_{1})&=\sum_{y\in
X}P_{t}^{\alpha}(d(x_{1},y))f(y)=P_{t}^{\alpha}(d(x_{1},x_{0}))f(x_{0})\\\
&+P_{t}^{\alpha}(0)f(x_{1})+\sum_{y\in
X\setminus\\{x_{0},\,x_{1}\\}}\frac{P_{t}^{\alpha}(d(x_{1},y))}{P_{t}^{\alpha}(d(x_{0},y))}P_{t}^{\alpha}(d(x_{0},y))f(y).\end{split}$

Therefore, since without loss of generality we can assume that for
$0<t<R_{1}<1$, $\min\\{d(x_{0},y),\,d(x_{1},y)\\}>t^{2/\alpha}$, $y\in
X\setminus\\{x_{0},\,x_{1}\\}$, according to (2.2), we get that

$\begin{split}\frac{P_{t}^{\alpha}(d(x_{1},y))}{P_{t}^{\alpha}(d(x_{0},y))}&\sim
q^{d(x_{0},y)-d(x_{1},y)}\left(\frac{d(x_{0},y)+1}{d(x_{1},y)+1}\right)^{1+\frac{\alpha}{2}}\\\
&\leq
Cq^{d(x_{0},x_{1})}\left(\frac{d(x_{0},x_{1})+1+d(x_{1},y)}{d(x_{1},y)+1}\right)^{1+\frac{\alpha}{2}}\\\
&\leq C,\,t\in(0,R_{1}).\end{split}$

Thus, by (2.4) and (2.1) we obtain that

$P_{t}^{\alpha}(f)(x_{1})\leq
C+\sup_{0<t<R_{1}}|P_{t}(f)(x_{0})|<\infty,\,0<t\leq R_{1}.$

Once we have obtained the previous inequality our next objective is to get

(2.5) $\sup_{R_{1}\leq t<1}|P_{t}(f)(x_{1})|\leq C|P_{R_{1}}(f)(x_{1})|$

that clearly implies $e)$. It is known that ([20, p. 89])

(2.6)
$\eta_{t}^{\alpha}(u)\sim\left\\{\begin{array}[]{ll}\displaystyle{t^{\frac{1}{2-\alpha}}u^{\frac{-(4-\alpha)}{4-2\alpha}}e^{-c_{1}t^{\frac{2}{2-\alpha}}u^{\frac{-\alpha}{2-\alpha}}}},\,u\leq
t^{2/\alpha},\\\
\displaystyle{tu^{-1-\frac{\alpha}{2}}e^{-c_{1}t^{\frac{2}{2-\alpha}}u^{\frac{-\alpha}{2-\alpha}}}},\,u\geq
t^{2/\alpha}.\end{array}\right.$

where
$c_{1}:=\frac{2-\alpha}{2}\Big{(}\frac{\alpha}{2}\Big{)}^{\alpha/(2-\alpha)}$.
We fix now $0<a<b<\infty$. We observe that (2.5) will be proved if we see
that, for some $C>0$,

$\eta_{t}^{\alpha}(u)\leq
C\eta^{\alpha}_{{a}}(u),\,u\in(0,\infty),\,t\in[a,b].$

To prove the previous inequality let us consider $t\in[a,b]$. By (2.6) we have
that

  * •

If $u\in(0,a^{2/\alpha})$, then $u\in(0,t^{2/\alpha})$ and

$\begin{split}\eta_{t}^{\alpha}(u)&\leq
C\displaystyle{b^{\frac{1}{2-\alpha}}u^{\frac{-(4-\alpha)}{4-2\alpha}}e^{-c_{1}a^{\frac{2}{2-\alpha}}u^{\frac{-\alpha}{2-\alpha}}}}\\\
&\leq
C\left(\frac{b}{a}\right)^{\frac{1}{2-\alpha}}\displaystyle{a^{\frac{1}{2-\alpha}}u^{\frac{-(4-\alpha)}{4-2\alpha}}e^{-c_{1}a^{\frac{2}{2-\alpha}}u^{\frac{-\alpha}{2-\alpha}}}}\\\
&\leq C\eta_{a}^{\alpha}(u).\end{split}$

  * •

If $u\in(b^{2/\alpha},\infty)$ then
$u\in(t^{2/\alpha},\infty)\subseteq(a^{2/\alpha},\infty)$ and

$\eta_{t}^{\alpha}(u)\leq
C\displaystyle{bu^{-1-\frac{\alpha}{2}}e^{-c_{1}a^{\frac{2}{2-\alpha}}u^{\frac{-\alpha}{2-\alpha}}}}\leq
C\eta^{\alpha}_{a}(u).$

  * •

In the case $u\in[a^{2/\alpha},t^{2/\alpha}]$ it follows

$\begin{split}\eta_{t}^{\alpha}(u)&\leq
C\displaystyle{b^{\frac{1}{2-\alpha}}a^{\frac{-(4-\alpha)}{4-2\alpha}}e^{-c_{1}a^{\frac{2}{2-\alpha}}u^{\frac{-\alpha}{2-\alpha}}}}\\\
&\leq
C\displaystyle{b^{\frac{1}{2-\alpha}+\frac{2}{\alpha}(1+\frac{\alpha}{2})}a^{\frac{-(4-\alpha)}{4-2\alpha}-1}au^{-1-\frac{\alpha}{2}}e^{-c_{1}a^{\frac{2}{2-\alpha}}u^{\frac{-\alpha}{2-\alpha}}}}\\\
&\leq C\eta^{\alpha}_{a}(u).\end{split}$

  * •

Finally when
$u\in[t^{2/\alpha},b^{2/\alpha}]\subseteq[a^{2/\alpha},b^{2/\alpha}]$ we also
obtain

$\eta_{t}^{\alpha}(u)\leq
C\displaystyle{bu^{-1-\frac{\alpha}{2}}e^{-c_{1}a^{\frac{2}{2-\alpha}}u^{\frac{-\alpha}{2-\alpha}}}}\leq
C\eta^{\alpha}_{a}(u).$

Suppose now that $f)$ holds. Let $x\in X$. We consider the mapping
$Q_{x}:l^{p}(X,\mu,u)\to l^{1}(X,\mu)$ where

$Q_{x}(f)(y):=P_{R}^{\alpha}(d(x,y))f(y),\,y\in X.$

By using the closed graph Theorem we deduce that $Q_{x}$ is bounded from
$l^{p}(X,\mu,u)$ into $l^{1}(X,\mu)$ and by duality arguments this implies
that

$\sum_{y\in X}P_{R}^{\alpha}(d(x,y))^{p^{\prime}}u(y)^{-p^{\prime}/p}<\infty,$

when $1<p<\infty$, and

$\sup_{y\in X}\frac{P_{R}^{\alpha}(d(x,y))}{u(y)}<\infty,$

when $p=1$, as wanted.

If $h)$ holds, then from (2.2) it follows that $i)$ is true.

Let us finally assume that $i)$ is true and let us prove that $a)$ is
satisfied. By using (2.2), for a fixed and arbitrary $x\in X$, we obtain

$\begin{split}\sup_{t\in(0,R_{1})}|P_{t}^{\alpha}(f)(x)|&\leq\sup_{t\in(0,R_{1})}P_{t}^{\alpha}(0)|f(x)|+\sup_{t\in(0,R_{1})}\sum_{y\in
X\setminus\\{x\\}}P_{t}^{\alpha}(d(x,y))|f(y)|\\\ &\leq
C\left(|f(x)|+\sum_{y\in
X\setminus\\{x\\}}(d(x,y)+1)^{-1-\alpha/2}q^{-d(x,y)}|f(y)|\right),\end{split}$

for $0<t<R_{1}$ where $R_{1}$ is given in the proof of $d)$ implies $e)$.
Since for every $x,\,y\in X$,

$\left(\frac{d(x_{0},y)+1}{d(x,y)+1}\right)^{1+\alpha/2}\leq\left(\frac{d(x,y)+d(x_{0},x)+1}{d(x,y)+1}\right)^{1+\alpha/2}\leq(d(x_{0},x)+1)^{1+\alpha/2},$

where $x_{0}\in X$ is the fixed element for which the condition $i)$ is true,
then

(2.7) $\begin{split}&\sup_{t\in(0,R_{1})}|P_{t}^{\alpha}(f)(x)|\\\ &\leq
C\left(|f(x)|+\left(d(x_{0},x)+1\right)^{1+\frac{\alpha}{2}}\sum_{y\in
X\setminus\\{x\\}}\frac{|f(y)|}{(d(x_{0},y)+1)^{1+\frac{\alpha}{2}}q^{d(x,y)}}\right)\\\
&\leq
C\left(|f(x)|+\left(d(x_{0},x)+1\right)^{1+\frac{\alpha}{2}}q^{d(x_{0},x)}\sum_{y\in
X\setminus\\{x\\}}\frac{|f(y)|}{(d(x_{0},y)+1)^{1+\frac{\alpha}{2}}q^{d(x_{0},y)}}\right).\end{split}$

Since by Hölder inequality we get when $1<p<\infty$

$\begin{split}S&:=\sum_{y\in
X\setminus\\{x\\}}\frac{|f(y)|}{(d(x_{0},y)+1)^{1+\frac{\alpha}{2}}q^{d(x_{0},y)}}\\\
&\leq\left(\sum_{y\in
X\setminus\\{x\\}}\left(\frac{1}{(d(x_{0},y)+1)^{1+\frac{\alpha}{2}}q^{d(x_{0},y)}}\right)^{p^{\prime}}u^{-p^{\prime}/p}(y)\right)^{1/p^{\prime}}\left(\sum_{y\in
X\setminus\\{x\\}}|f(y)|^{p}u(y)\right)^{1/p},\end{split}$

and when $p=1$

$\begin{split}S&:=\sum_{y\in
X\setminus\\{x\\}}\frac{|f(y)|}{(d(x_{0},y)+1)^{1+\frac{\alpha}{2}}q^{d(x_{0},y)}}\\\
&\leq\sup_{y\in
X\setminus\\{x\\}}\frac{1}{(d(x_{0},y)+1)^{1+\frac{\alpha}{2}}q^{d(x_{0},y)}u(y)}\sum_{y\in
X\setminus\\{x\\}}|f(y)|u(y),\end{split}$

by the fact that $f\in l^{p}(X,\mu,u)$ and the hypothesis $i)$ it follows that

(2.8) $S<\infty.$

On the other hand, since we can find $w$ a weight on $X$ such that

$\sum_{y\in
X\setminus\\{x\\}}\left((d(x_{0},y)+1)^{1+\frac{\alpha}{2}}q^{d(x_{0},y)}\right)^{p}w(y)<\infty,$

then $P_{*,R_{1}}$ is bounded from $l^{p}(X,\mu,u)$ into $l^{p}(X,\mu,v)$ as
wanted where $v:=\min\\{w,u\\}$. Indeed given $f\in l^{p}(X,\mu,u)$ by (2.7)
and (2.8) it follows that

$\begin{split}&\left(\sum_{x\in
X}\left(\sup_{t\in(0,R_{1})}|P_{t}^{\alpha}(f)(x)|\right)^{p}v(x)\right)^{1/p}\\\
&\leq\left(\sum_{y\in X}|f(y)|^{p}v(y)\right)^{1/p}+S\left(\sum_{y\in
X\setminus\\{x\\}}\left((d(x_{0},y)+1)^{1+\frac{\alpha}{2}}q^{d(x_{0},y)}\right)^{p}v(y)\right)^{1/p}\\\
&\leq C\left(\sum_{y\in X}|f(y)|^{p}u(y)\right)^{1/p}.\end{split}$

Thus, the proof of Theorem 1.1 is finished when $q\geq 2$.

##  3\. Proof of Theorem 1.2 for $q\geq 2$

We consider $q\geq 2$. Before proving Theorem 1.2 we establish the following
auxiliary result that plays for the family $\\{T_{t}^{\nu}\\}_{t>0}$ a similar
role as (2.2) for the semigroup $\\{W_{t}^{\alpha}\\}_{t>0}$, $\nu>0$ given in
(1).

######  Proposition 3.1.

Let $\nu>0$ and $q\in\mathbb{N}$, $q\geq 2$. We have that

  * a)

For every $t>0$, $k\in\mathbb{N}$ with $k<\nu$, there exists $C=C({q})>0$ such
that

$0\leq T_{t}^{\nu}(k)\leq
C\frac{t^{2k}}{(k+1)^{k+\frac{1}{2}}}\Big{(}\frac{2e}{q+1}\Big{)}^{k+1}.$

  * b)

For every $t\in(0,1)$, $k\in\mathbb{N}$ with $k\geq\nu$, there exists $C>0$
such that

$0\leq T_{t}^{\nu}(k)\leq C\frac{t^{2\nu}}{k^{\nu+1}q^{k}}.$

  * c)

For every $t\in(0,1)$, $k\in\mathbb{N}\setminus\\{0\\}$ there exists $C>0$
such that

$T_{t}^{\nu}(k)\geq C\frac{t^{2\nu}}{k^{\nu+1}q^{k}}.$

  * d)

For every $t\in(0,1)$, $T_{t}^{\nu}(0)\sim 1.$

###### Proof.

We adapt some ideas developed in [31, Theorem 3.1]. We recall that

(3.1)
$T_{t}^{\nu}(k):=\frac{t^{2\nu}}{4^{\nu}\Gamma(\nu)}\int_{0}^{\infty}e^{-t^{2}/4u}H_{u}(k)\frac{du}{u^{1+\nu}},\quad
k\in\mathbb{N},\,t>0.$

For every $k\in\mathbb{N}\setminus\\{0\\}$ and $t>0$ by [13, Proposition 2.5
and p. 4275] and [25, Proposition 3.1] it follows that

$T_{t}^{\nu}(k)\sim
t^{2\nu}\int_{0}^{\infty}\frac{e^{-\frac{t^{2}}{4u}-b_{2}u}}{u^{\nu+2}}\phi_{0}(k)H_{u\frac{2\sqrt{q}}{q+1}}^{\mathbb{Z}}(k+1)\,du,\quad
t>0,\quad k\in\mathbb{N},$

with $b_{2}:=1-\frac{2\sqrt{q}}{q+1}\in(0,1)$. Here $H_{t}^{\mathbb{Z}}(k)$,
$t>0$ and $k\in\mathbb{N}$, denotes the heat kernel associated with
$X=\mathbb{Z}$ given by

$H_{t}^{\mathbb{Z}}(k)=e^{-t}I_{k}(t),\quad t>0,\quad k\in\mathbb{N},$

where $I_{\nu}$ is the modified Bessel function of the first kind and order
$\nu$. Therefore

$\begin{split}T_{t}^{\nu}(k)&\leq
Ct^{2\nu}\int_{0}^{\infty}e^{-\frac{t^{2}}{4u}}(k+1)q^{-k/2}\frac{e^{-\left(b_{2}+\frac{2\sqrt{q}}{q+1}\right)u+\sqrt{(k+1)^{2}+\frac{4q}{(q+1)^{2}}u^{2}}}}{\left(1+(k+1)^{2}+\frac{4qu^{2}}{(q+1)^{2}}\right)^{1/4}}\\\
&\cdot\left(\frac{\frac{2\sqrt{q}u}{q+1}}{k+1+\sqrt{(k+1)^{2}+\frac{4q^{2}u^{2}}{(q+1)^{2}}}}\right)^{k+1}\frac{du}{u^{\nu+2}}\\\
&\leq
C\frac{t^{2\nu}}{(k+1)^{k+\frac{1}{2}}}\Big{(}\frac{2e}{q+1}\Big{)}^{k+1}\int_{0}^{\infty}e^{-b_{2}u-\frac{t^{2}}{4u}}u^{k-1-\nu}\,du\\\
&\leq
C\frac{t^{2\nu}}{(k+1)^{k+\frac{1}{2}}}\Big{(}\frac{2e}{q+1}\Big{)}^{k+1}\int_{0}^{\infty}e^{-\frac{t^{2}}{4u}}u^{k-1-\nu}\,du\\\
&\leq
C\frac{t^{2k}}{(k+1)^{k+\frac{1}{2}}}\Big{(}\frac{2e}{q+1}\Big{)}^{k+1}\int_{0}^{\infty}e^{-z}z^{\nu-k-1}dz\\\
&\leq
C\Gamma(\nu-k)\frac{t^{2k}}{(k+1)^{k+\frac{1}{2}}}\Big{(}\frac{2e}{q+1}\Big{)}^{k+1},\quad
t>0,\quad k\in\mathbb{N},\quad k<\nu,\end{split}$

where we have used the fact that $b_{2}>0$ and in the fourth inequality we
have done the change of variable $z=t^{2}/4u$. Thus the conclusion given in
$a)$ follows. Let us prove the estimates given in $c)$ and $d)$. Indeed, doing
similar computations as before, by using the fact that
$b_{2}+\frac{2\sqrt{q}}{q+1}=1$, for every $0<t<1$, we get that

(3.2) $\begin{split}T_{t}^{\nu}(k)&\geq
Ct^{2\nu}(k+1)q^{-k/2}\int_{0}^{\infty}\frac{e^{-u+\sqrt{(k+1)^{2}+\frac{4q}{(q+1)^{2}}u^{2}}}}{\left(1+(k+1)^{2}+\frac{4qu^{2}}{(q+1)^{2}}\right)^{1/4}}\\\
&\cdot\left(\frac{\frac{2\sqrt{q}u}{q+1}}{k+1+\sqrt{(k+1)^{2}+\frac{4q^{2}u^{2}}{(q+1)^{2}}}}\right)^{k+1}\frac{du}{u^{\nu+2}}\\\
&\geq
Ct^{2\nu}k^{-\nu-\frac{1}{2}}\left(\frac{2}{q+1}\right)^{k}\int_{0}^{\infty}\frac{e^{(k+1)\left(-u+\sqrt{1+\frac{4q}{(q+1)^{2}}u^{2}}\right)}u^{k-\nu-1}}{\left(1+\frac{2\sqrt{q}u}{q+1}\right)^{1/2}\left(1+\sqrt{1+\frac{4qu^{2}}{(q+1)^{2}}}\right)^{k+1}}\,du\\\
&\geq
Ct^{2\nu}k^{-\nu-\frac{1}{2}}\left(\frac{2}{q+1}\right)^{k}\int_{0}^{\infty}\frac{e^{(k+1)g(u)}}{u^{\nu+2}\left(1+\frac{2\sqrt{q}u}{q+1}\right)^{1/2}}\,du\\\
&:=Ct^{2\nu}k^{-\nu-\frac{1}{2}}\left(\frac{2}{q+1}\right)^{k}I(\nu,q,k).\end{split}$

where

$g(u):=\sqrt{1+\frac{4q}{(q+1)^{2}}u^{2}}-u+\ln
u-\ln\left(1+\sqrt{1+\frac{4qu^{2}}{(q+1)^{2}}}\right),\,u>0,$

is a real function that attains its maximum in $u=(q+1)/(q-1)$ (see [31, p.
1443]). By using the Laplace method we notice that

(3.3)
$I(\nu,q,k)\sim\frac{1}{\sqrt{k}}e^{-k\ln\left(\frac{2q}{q+1}\right)},\quad
k\to\infty.$

Therefore, since by similar computations as the ones done in (3.2) we can also
obtain that

$T_{t}^{\nu}(k)\leq
Ct^{2\nu}k^{-\nu-\frac{1}{2}}\left(\frac{2}{q+1}\right)^{k}I(\nu,q,k),\quad
k\geq\nu,$

by (3.3) we conclude that

$T_{t}^{\nu}(k)\geq C\frac{t^{2\nu}}{k^{\nu+1}q^{k}},\quad
k\to\infty,\,\mbox{uniformly in }\,t\in(0,1),$

and

$T_{t}^{\nu}(k)\leq C\frac{t^{2\nu}}{k^{\nu+1}q^{k}},\quad
k\to\infty,\,k>\nu,\,\mbox{ uniformly in }\,t\in(0,1),$

as wanted.

Let us finally prove the assertion $d)$. Note that, since from (3.1)

$T_{t}^{\nu}(0):=\frac{1}{\Gamma(\nu)}\int_{0}^{\infty}e^{-v}v^{\nu-1}H_{\frac{t^{2}}{4v}}(0)\,dv,\quad
t>0,$

by using again [13, Proposition 2.5 and p. 4275] and [25, Proposition 3.1], it
follows that

$T_{t}^{\nu}(0)\sim\int_{0}^{\infty}e^{-v}v^{\nu-1}e^{-\frac{t^{2}}{4v}}\frac{\sqrt{q}}{2(q+1)\left(1+\sqrt{1+\left(\frac{t^{2}\sqrt{q}}{2v(q+1)}\right)^{2}}\right)}\,dv,\quad
t>0.$

Then

$T_{t}^{\nu}(0)\leq C\int_{0}^{\infty}e^{-v}v^{\nu-1}\,dv\leq
C(\Gamma(\nu)),\,\mbox{if }t>0,$

and

$T_{t}^{\nu}(0)\geq
C\int_{0}^{\infty}\frac{e^{-v-\frac{1}{4v}}v^{\nu}}{v+\sqrt{1+v^{2}}}\,dv\geq
C,\,\mbox{when }0<t<1.$

Thus, we conclude that $T_{t}^{\nu}(0)\sim 1$ when $0<t<1$. ∎

Now we can obtain the

Proof of Theorem 1.2.
To prove $a)\Rightarrow b)\Rightarrow c)\Rightarrow d)$ we can proceed as in
the proof of Theorem 1.1. Let us suppose now that $d)$ holds. We consider
$0\leq f\in l^{p}(X,\mu,u)$ and $x_{0}\in X$ such that $lim_{t\to
0^{+}}T_{t}^{\nu}(f)(x_{0})=f(x_{0})$. Then there exists $R_{1}\in(0,1)$
satisfying

(3.4) $sup_{0<t\leq R_{1}}T_{t}^{\nu}(f)(x_{0})<\infty.$

Let us consider $x_{1}\in X$ and we are going to prove that

(3.5) $sup_{0<t\leq 1}T_{t}^{\nu}(f)(x_{1})<\infty.$

Indeed, first of all we notice that, for every $t\in[R_{1},1]$, by (1), it
follows that

$T_{t}^{\nu}(f)(x_{1})\leq\frac{1}{4^{\nu}\Gamma(\nu)}\int_{0}^{\infty}\frac{e^{-\frac{R_{1}^{2}}{4s}}}{s^{1+\nu}}W_{s}(f)(x_{1})ds\leq
R_{1}^{-2\nu}T_{R_{1}}^{\nu}(f)(x_{1}).$

Secondly, for every $t>0$, we get

$\displaystyle T_{t}^{\nu}(f)(x_{1})$ $\displaystyle=\sum_{y\in
X}T_{t}^{\nu}(d(x_{1},y))f(y)$ $\displaystyle=\sum_{y\in
X,\,min\\{d(x_{0},y),d(x_{1},y)\\}<\nu}T_{t}^{\nu}(d(x_{1},y))f(y)$ (3.6)
$\displaystyle\hskip 42.67912pt+\sum_{y\in
X,\,min\\{d(x_{0},y),d(x_{1},y)\\}\geq\nu}\frac{T_{t}^{\nu}(d(x_{1},y))}{T_{t}^{\nu}(d(x_{0},y))}T_{t}^{\nu}(d(x_{0},y))f(y).$

By Proposition 3.1 b), it is clear that

(3.7)
$\begin{split}\frac{T_{t}^{\nu}(d(x_{1},y))}{T_{t}^{\nu}(d(x_{0},y))}&\leq
C\left(\frac{d(x_{0},y)}{d(x_{1},y)}\right)^{\nu+1}q^{d(x_{0},y)-d(x_{1},y)}\\\
&\leq
C\left(\frac{d(x_{0},x_{1})+d(x_{1},y)}{d(x_{1},y)}\right)^{\nu+1}q^{d(x_{0},x_{1})}\\\
&\leq C,\,y\in X,\quad min\\{d(x_{0},y),d(x_{1},y)\\}\geq\nu\quad
t\in(0,1).\end{split}$

By Proposition 3.1 (a) and (b), (3.4) and (3) we deduce that there exists
$C>0$ such that

$\displaystyle T_{t}^{\nu}(f)(x_{1})$ $\displaystyle\leq\sum_{y\in
X,\,min\\{d(x_{0},y),d(x_{1},y)\\}<\nu}T_{t}^{\nu}(d(x_{1},y))f(y)$
$\displaystyle\hskip 42.67912pt+C\sum_{y\in
X,\,min\\{d(x_{0},y),d(x_{1},y)\\}\geq\nu}T_{R_{1}}^{\nu}(d(x_{0},y))f(y)$
(3.8) $\displaystyle\leq\sum_{y\in
X,\,min\\{d(x_{0},y),d(x_{1},y)\\}<\nu}T_{t}^{\nu}(d(x_{1},y))f(y)+CT_{t}^{\nu}(f)(x_{0})\leq
C,\quad t\in(0,R_{1}].$

We have used that the same sum has a finite number of terms. So that, (3.5)
follows and, consequently also the assertion $e)$ and $f)$.

Let us suppose that $f)$ holds. By proceeding as in the proof of
$f)\Rightarrow g)$ in the previous theorem we deduce that, when $1<p<\infty$,

$\sum_{y\in
X}T_{R}^{\nu}(d(x,y))^{p^{\prime}}v(y)^{-p^{\prime}/p}<\infty,\quad x\in X,$

and, when $p=1$,

$\sup_{y\in X}\frac{T_{R}^{\nu}(d(x,y))}{u(y)}<\infty,\quad x\in X,$

for some $R>0$, and $g)$ is just proved.

The property $g)\Rightarrow h)$ is clear.

Suppose now that $h)$ holds. By using Proposition 3.1, $c)$, we obtain, when
$1<p<\infty$,

$\sum_{y\in
X\setminus\\{x\\}}\left(q^{d(x,y)}d(x,y)^{\nu+1}\right)^{-p^{\prime}}u(y)^{-p^{\prime}/p}\leq
R^{-2\nu p^{\prime}}\sum_{y\in
X\setminus\\{0\\}}(T_{R}^{\nu}(d(0,y)))^{p^{\prime}}u(y)^{-p^{\prime}/p}<\infty,$

and, when $p=1$,

$\sup_{y\in X\setminus\\{x\\}}(q^{d(x,y)}d(x,y)^{\nu+1}u(y))^{-1}\leq
R^{-2\nu}\sup_{y\in
X\setminus\\{x\\}}\frac{T_{R}^{\nu}(d(x,y))}{u(y)}<\infty,$

where $R>0$ and $x\in X$ are given in the assumption $h)$. That is, $i)$ is
proved.

Let us assume that $i)$ is satisfied. There exists $x_{0}\in X$ such that,
when $1<p<\infty$,

$\sum_{y\in
X}\left(q^{d(x_{0},y)}(1+d(x_{0},y))^{1+\frac{\alpha}{2}}\right)^{-p^{\prime}}u(y)^{-p^{\prime}/p}<\infty,$

and, when $p=1$,

$\sup_{y\in
X}(q^{d(x_{0},y)}(1+d(x_{0},y))^{1+\frac{\alpha}{2}}u(y))^{-1}<\infty.$

Let us fix $R>0$. Then, for every $0<t<R$ and $y\in X$, $y\neq x$ for a fixed
but arbitrary $x\in X$, by (3.7) we know that

(3.9) $\left(\frac{d(x_{0},y)}{d(x,y)}\right)^{\nu+1}q^{d(x_{0},y)-d(x,y)}\leq
C(d(x_{0},x)+1)^{\nu+1}q^{d(x_{0},x)}.$

Thus, by Proposition 3.1 a), for every $0<t<R$, we get

$\begin{split}|T_{t}^{\nu}(f)(x)|&\leq|T_{t}^{\nu}(|f|)(x)|\leq
C\left(\sum_{y\in
X,\,d(x,y)<\nu}\frac{R^{2d(x,y)}}{(d(x,y)+1)^{d(x,y)+\frac{1}{2}}}\left(\frac{2e}{q+1}\right)^{d(x,y)+1}|f(y)|\right.\\\
&+\left.\sum_{y\in
X,\,d(x,y)\geq\nu}R^{\nu}d(x,y)^{-\nu-1}q^{-d(x,y)}|f(y)|\right)\\\ &\leq
C\left(\sum_{y\in X,\,d(x,y)<\nu}|f(y)|+\sum_{y\in
X,\,d(x,y)\geq\nu}d(x,y)^{-\nu-1}q^{-d(x,y)}|f(y)|\right)\\\ &\leq
C\left(|f(x)|+\sum_{y\in
X\setminus\\{x\\}}d(x,y)^{-\nu-1}q^{-d(x,y)}|f(y)|\right),\quad x\in
X.\end{split}$

We notice that Hölder inequality and (3.9) lead, when $1<p<\infty$, to

$\begin{split}&\sum_{y\in
X\setminus\\{x\\}}d(x,y)^{-\nu-1}q^{-d(x,y)}|f(y)|\\\
&\leq(d(x_{0},x)+1)^{\nu+1}q^{d(x_{0},x)}\sum_{y\in
X\setminus\\{x\\}}d(x_{0},y)^{-\nu-1}q^{-d(x_{0},y)}|f(y)|\\\
&\leq(d(x_{0},x)+1)^{\nu+1}q^{d(x_{0},x)}\left(\sum_{y\in
X\setminus\\{x\\}}(d(x_{0},y)^{\nu+1}q^{d(x_{0},y)})^{-p^{\prime}}u(y)\right)^{\frac{1}{p^{\prime}}}\left(\sum_{y\in
X}|f(y)|^{p}\,u(y)\right)^{\frac{1}{p}},\end{split}$

and, when $p=1$, to

$\begin{split}&\sum_{y\in
X\setminus\\{x\\}}d(x,y)^{-\nu-1}q^{-d(x,y)}|f(y)|\\\
&\leq(d(x_{0},x)+1)^{\nu+1}q^{d(x_{0},x)}\sum_{y\in
X\setminus\\{x\\}}d(x_{0},y)^{-\nu-1}q^{-d(x_{0},y)}|f(y)|\\\
&\leq(d(x_{0},x)+1)^{\nu+1}q^{d(x_{0},x)}\sup_{y\in
X\setminus\\{x\\}}(d(x_{0},y)^{\nu+1}q^{d(x_{0},y)}u(y))^{-1}\sum_{y\in
X}|f(y)|\,u(y).\end{split}$

Since we can guarantee that there exists $w$ a weight in $X$ such that

$\sum_{x\in
X}\left((d(x_{0},x)+1)^{\nu+1}q^{d(x_{0},x)}\right)^{p}\,w(x)<\infty,$

by the previous computation we get that $T_{\ast,R}^{\nu}$ is bounded from
$l^{p}(X,\mu,u)$ into $l^{p}(X,\mu,v)$ where $v=\min\\{u,w\\}$ as wanted.

##  4\. Proof of Theorem 1.3 for $q\geq 2$

We consider $q\geq 2$. Properties $a)\Rightarrow b)\Rightarrow c)\Rightarrow
d)$ can be proved as in Theorem 1.1.

Assume that $d)$ holds. There exists $R_{1}>0$ such that, for every $f\in
l^{p}(X,\mu,u)$, the limit

$\lim_{t\to R_{1}}H_{t}(f)(x_{0})$

exists, for some $x_{0}\in X$.

Let $0\leq f\in l^{p}(X,\mu,u)$ and $x_{0}\in X$ such that $\lim_{t\to
R_{1}}W_{t}(f)(x_{0})$ exists. Then, there is a $t_{1}\in(0,R_{1})$, such that

$\sup_{t\in[t_{1},R_{1})}W_{t}(f)(x_{0})=A<\infty.$

Suppose that $R>0$. We have that

$\displaystyle\frac{1}{t}H_{\frac{t\sqrt{2q}}{q+1}}^{\mathbb{Z}}(1+d(x,y))$
$\displaystyle=\frac{1}{t}e^{-t\frac{\sqrt{2q}}{q+1}}I_{1+d(x,y)}(t\frac{\sqrt{2q}}{q+1})$
$\displaystyle=\frac{1}{t}e^{-t\frac{\sqrt{2q}}{q+1}}\sum_{k=0}^{\infty}\frac{(t\sqrt{2q}/(q+1))^{2k+1+d(x,y)}}{\Gamma(k+1)\Gamma(k+d(x,y)+2)2^{1+d(x,y)+2k}}$
$\displaystyle=e^{-t\frac{\sqrt{2q}}{q+1}}\Big{(}\frac{\sqrt{2q}}{q+1}\Big{)}^{1+d(x,y)}t^{d(x,y)}\sum_{k=0}^{\infty}\frac{(t\sqrt{2q}/(q+1))^{2k}}{\Gamma(k+1)\Gamma(k+d(x,y)+2)2^{1+d(x,y)+2k}}$
$\displaystyle\leq
R^{d(x,y)}\Big{(}\frac{\sqrt{2q}}{q+1}\Big{)}^{1+d(x,y)}\sum_{k=0}^{\infty}\frac{(R\sqrt{2q}/(q+1))^{2k}}{\Gamma(k+1)\Gamma(k+d(x,y)+2)2^{1+d(x,y)+2k}}$
$\displaystyle\leq\frac{1}{R}e^{R\sqrt{2q}/(q+1)}H_{\frac{R\sqrt{2q}}{q+1}}^{\mathbb{Z}}(1+d(x,y)),\quad
x,y\in X,\quad 0<t<R.$

Then, according to [13, Proposition 2.5], we get

(4.1) $\begin{split}H_{t}(d(x,y))&\leq
C\frac{(q+1)^{3}\sqrt{q}}{(q-1)^{3}}\frac{e^{-t}}{t}\phi_{0}(d(x,y))H_{\frac{t\sqrt{2q}}{q+1}}^{\mathbb{Z}}(1+d(x,y))\\\
&\leq
C\frac{(q+1)^{3}\sqrt{q}}{(q-1)^{3}}\phi_{0}(d(x,y))\frac{e^{2R\sqrt{2q}/(q+1)}}{R}H_{\frac{R\sqrt{2q}}{q+1}}^{\mathbb{Z}}(1+d(x,y))\\\
&\leq C\frac{(q+1)^{3}q}{(q-1)^{3}}e^{2R(1+\sqrt{2q}/(q+1))}H_{R}(d(x,y))\\\
&=CH_{R}(d(x,y)),\quad x,y\in X,\quad 0<t<R.\end{split}$

Thus

$W_{t}(f)(x_{0})\leq C\max\\{A,W_{t_{1}}(f)(x_{0})\\},\quad 0<t<R_{1}.$

We are going to see that

$W_{R_{0}/4}(f)(x)<\infty,\quad x\in X,$

where $R_{0}=R_{1}/2$.

Let $x\in X$, by using [13, Proposition 2.5 and p. 4282] and [13, Proposition
3.1] we get

$\begin{split}\frac{h_{R_{0}/4}(d(x,y))}{h_{R_{0}}(d(x_{0},y))}&\leq
C\frac{(d(x,y)+1)q^{-d(x,y)/2}}{(d(x_{0},y)+1)q^{-d(x_{0},y)/2}}\left(\frac{1+(1+d(x_{0},y))^{2}+4R_{0}^{2}q/(q+1)^{2}}{1+(1+d(x,y))^{2}+R_{0}^{2}q/4(q+1)^{2}}\right)^{1/4}\\\
&\times\exp\left\\{\sqrt{(d(x,y)+1)^{2}+\frac{R_{0}^{2}q}{4(q+1)^{2}}}-\sqrt{(d(x_{0},y)+1)^{2}+\frac{4R_{0}^{2}q}{(q+1)^{2}}}\right\\}\\\
&\times\left(\frac{R_{0}\sqrt{q}/(2(q+1))}{d(x,y)+1+\sqrt{(d(x,y)+1)^{2}+\frac{R_{0}^{2}q}{4(q+1)^{2}}}}\right)^{1+d(x,y)}\\\
&\times\left(\frac{d(x_{0},y)+1+\sqrt{(d(x_{0},y)+1)^{2}+4R_{0}^{2}q/(q+1)^{2}}}{2R_{0}\sqrt{q}/(q+1)}\right)^{1+d(x_{0},y)},\quad
y\in X.\end{split}$

We divide the estimate in several parts:

  * (4.2) $\begin{split}\frac{d(x,y)+1}{d(x_{0},y)+1}&\leq\frac{d(x,x_{0})+d(x_{0},y)+1}{d(x_{0},y)+1}\leq d(x,x_{0})+1,\quad y\in X.\end{split}$

  * (4.3) $\begin{split}\frac{q^{-d(x,y)/2}}{q^{-d(x_{0},y)/2}}&\leq\frac{q^{-(d(x_{0},y)-d(x,x_{0}))/2}}{q^{-d(x_{0},y)/2}}\leq q^{-d(x,x_{0})/2},\quad y\in X.\end{split}$

  * (4.4) $\begin{split}\frac{1+(1+d(x_{0},y))^{2}+\frac{4R_{0}^{2}q}{(q+1)^{2}}}{1+(1+d(x,y))^{2}+\frac{R_{0}^{2}q}{4(q+1)^{2}}}&\leq\frac{1+(1+d(x_{0},x)+d(x,y))^{2}+\frac{4R_{0}^{2}q}{(q+1)^{2}}}{1+(1+d(x,y))^{2}+\frac{R_{0}^{2}q}{4(q+1)^{2}}}\\\ &\leq C\frac{1+(1+d(x,y))^{2}+d(x_{0},x)^{2}+\frac{4R_{0}^{2}q}{(q+1)^{2}}}{1+(1+d(x,y))^{2}+\frac{R_{0}^{2}q}{4(q+1)^{2}}}\\\ &\leq C(1+d(x_{0},x)^{2}),\quad y\in X.\end{split}$

  * (4.5) $\begin{split}&\sqrt{(d(x,y)+1)^{2}+\frac{R_{0}^{2}q}{4(q+1)^{2}}}-\sqrt{(d(x_{0},y)+1)^{2}+\frac{4R_{0}^{2}q}{(q+1)^{2}}}\\\ &\leq\frac{(d(x,x_{0})+d(x_{0},y)+1)^{2}-(d(x_{0},y)+1)^{2}-\frac{15R_{0}^{2}q}{4(q+1)^{2}}}{\sqrt{(d(x,y)+1)^{2}+\frac{R_{0}^{2}q}{4(q+1)^{2}}}+\sqrt{(d(x_{0},y)+1)^{2}+\frac{4R_{0}^{2}q}{(q+1)^{2}}}}\\\ &=\frac{d(x,x_{0})^{2}+2d(x,x_{0})(d(x_{0},y)+1)-\frac{15R_{0}^{2}q}{4(q+1)^{2}}}{\sqrt{(d(x,y)+1)^{2}+\frac{R_{0}^{2}q}{4(q+1)^{2}}}+\sqrt{(d(x_{0},y)+1)^{2}+\frac{4R_{0}^{2}q}{(q+1)^{2}}}}\\\ &\leq 2d(x,x_{0})+\frac{d(x,x_{0})^{2}(q+1)}{2R_{0}\sqrt{q}},\quad y\in X.\end{split}$

  * We put

$F(y)=A^{1+d(x,y)}B^{1+d(x_{0},y)},\quad y\in X,$

where

$\begin{split}A&=\frac{R_{0}\sqrt{q}/(2(q+1))}{d(x,y)+1+\sqrt{(d(x,y)+1)^{2}+\frac{R_{0}^{2}q}{4(q+1)^{2}}}}\quad\quad\text{
and }\\\
B&=\frac{d(x_{0},y)+1+\sqrt{(d(x_{0},y)+1)^{2}+4R_{0}^{2}q/(q+1)^{2}}}{2R_{0}\sqrt{q}/(q+1)}.\end{split}$

We have that

(4.6) $\begin{split}&F(y)\leq A^{1+d(x_{0},y)-d(x_{0},x)}B^{1+d(x_{0},y)}\\\
&\leq C4^{-(1+d(x_{0},y))}A^{-d(x_{0},x)}\\\
&\times\left(\frac{d(x_{0},y)+1+\sqrt{(d(x_{0},y)+1)^{2}+4\frac{R_{0}^{2}q}{(q+1)^{2}}}}{d(x,y)+1+\sqrt{(d(x,y)+1)^{2}+\frac{R_{0}^{2}}{2(q+1)}}}\right)^{1+d(x_{0},y)}.\end{split}$

On one hand, we have that $F(y)\leq C$ for $y\in X,$ $d(x_{0},y)\leq
4d(x_{0},x)$.

On the other hand we obtain

(4.7) $\begin{split}F(y)&\leq
C4^{-(1+d(x_{0},y))}\left(\frac{2d(x_{0},y)+2+2\frac{R_{0}\sqrt{q}}{q+1}}{\frac{3}{4}d(x_{0},y)+1}\right)^{1+d(x_{0},y)}\\\
&\times\left(\frac{2(1+\frac{5}{4}d(x_{0},y))+\frac{R_{0}\sqrt{q}}{q+1}}{R_{0}\sqrt{q}/(2(q+1))}\right)^{d(x_{0},x)}\\\
&\leq
C\Big{(}\frac{3}{2}\Big{)}^{-(1+d(x_{0},y))}\left(\frac{d(x_{0},y)+1+\frac{R_{0}\sqrt{q}}{q+1}}{d(x_{0},y)+\frac{4}{3}}\right)^{1+d(x_{0},y)}\\\
&\times\left(\frac{2(1+\frac{5}{4}d(x_{0},y))+\frac{R_{0}\sqrt{q}}{q+1}}{R_{0}\sqrt{q}/(2(q+1))}\right)^{d(x_{0},x)}\leq
C,\end{split}$

for $y\in X$ and $d(x_{0},y)\geq 4d(x_{0},x)$

We conclude that

$H_{R_{0}/4}(d(x,y))\leq CH_{R_{0}}(d(x_{0},y)),\quad y\in X.$

It follows that

$W_{R_{0}/4}(f)(x)\leq CW_{R_{0}}(f)(x_{0}),\quad x\in X,$

and $e)$ is established by using the property (4.1).

It is clear that $e)\Rightarrow f)$.

Assume that $f)$ holds. By using the closed graph theorem and duality as in
Theorem 1.1 to prove property $g)$, we deduce that $g)$ is proved.

The property $g)\Rightarrow h)$ is clear.

Suppose now that $h)$ holds, that is, for a certain $x_{0}\in X$ and
$R_{0}>0$, when $1<p<\infty$,

$\sum_{y\in
X}P_{R_{0}}^{\alpha}(d(x_{0},y))^{p^{\prime}}u(y)^{-p^{\prime}/p}<\infty,$

and, when $p=1$,

$\sup_{y\in X}\frac{P_{R_{0}}^{\alpha}(d(x_{0},y)}{u(y)}<\infty.$

From the estimations in the proof of $d)\Rightarrow e)$, we deduce that there
exists a function $G:\mathbb{N}\to(0,\infty)$ such that

$H_{R_{0}/4}(d(x,y))\leq G(d(x_{0},x))H_{R_{0}}(d(x_{0},y)),\quad x,y\in X.$

By using the Hölder’s inequality and (4.1) we get, when $1<p<\infty$,

$\begin{split}&\sup_{0<t<R_{0}/4}\Big{|}\sum_{y\in
X}H_{t}(d(x,y))f(y)\Big{|}\leq\sum_{y\in X}H_{R_{0}/4}(d(x,y))|f(y)|\\\ &\leq
G(d(x_{0},x))\sum_{y\in X}H_{R_{0}}(d(0,y))|f(y)|\\\ &\leq
G(d(x_{0},x))\Big{(}\sum_{y\in
X}H_{R_{0}}(d(x_{0},y))^{p^{\prime}}u(y)^{-p^{\prime}/p}\Big{)}^{1/p^{\prime}}\Big{(}\sum_{y\in
X}|f(y)|^{p}u(y)\Big{)}^{1/p},\quad x\in X,\end{split}$

and, when $p=1$,

$\sup_{0<t<R_{0}/4}\Big{|}\sum_{y\in X}H_{t}(d(x,y))f(y)\Big{|}\leq
G(d(x_{0},x))\sup_{y\in X}\frac{P_{R_{0}}^{\alpha}(d(x_{0},y)}{u(y)}\sum_{y\in
X}|f(y)|u(y),\quad x\in X.$

Hence, if $v$ is a weight on $X$ such that

$\sum_{x\in X}G(d(x,x_{0}))^{p}v(x)<\infty$

then $W_{*,R_{0}/4}$ is bounded from $l^{p}(X,\mu,u)$ into $l^{p}(X,\mu,v)$
and $a)$ is proved

##  5\. Proof of Theorems 1.1, 1.2 and 1.3 for $q=1$

In this section we prove the results in the case $q=1$. The homogeneous tree
$X$ coincides with $\mathbb{Z}$ and $(\mathbb{Z},d,\mu)$ is a space of
homogeneous type in the sense of Coifman and Weiss [11].
According to [25, Proposition 3.1] we observe

(5.1) $H_{t}^{Z}(k)\sim\frac{1}{(1+|k|+t)^{1/2}}e^{-t(1+\Phi(|k|/t))},\quad
k\in\mathbb{Z}.$

where $\Phi(z)=z\log(z+\sqrt{1+z^{2}})-\sqrt{1+z^{2}}$, $z\geq 0$. In addition
we note the following behaviours of $\Phi$, which will be useful in the
sequel:

(5.2) $\Phi(z)\sim\frac{z^{2}}{2},\quad\text{ as }z\to 0^{+},\quad\text{ and
}\quad\Phi(z)\sim z\log z,\quad\text{ as }z\to+\infty.$

######  Proposition 5.1.

We define

$\mathbb{P}_{t}^{\alpha}(k)=\begin{cases}&t|k|^{-1-\alpha},\quad
k\in\mathbb{Z}\setminus\\{0\\}\\\ &1,\quad k=0.\end{cases}$

Then, $\mathbb{P}_{t}^{\alpha}\sim P_{t}^{\alpha}$, $0<t<1$.

###### Proof.

We choose $\beta>1$ and $\alpha/2<\gamma<1$ such that $\Phi(z)>\gamma z\log
z,$ $z>\beta$. We can write

$\begin{split}P_{t}^{\alpha}(k)&=\int_{0}^{|k|/\beta}\eta_{t}^{\alpha}(s)H_{s}^{Z}(k)ds+\int_{|k|/\beta}^{\infty}\eta_{t}^{\alpha}(s)H^{Z}_{s}(k)ds\\\
&=I_{1}^{\alpha}(t,k)+I_{2}^{\alpha}(t,k),\quad t>0,\quad\text{ and }\quad
k\in\mathbb{Z}.\end{split}$

By using (5.1) and (5.2) and by applying [(9) [6]] we obtain

$\begin{split}I_{2}^{\alpha}(t,k)&\leq
C\int_{|k|/\beta}^{\infty}\eta_{t}^{\alpha}(s)\frac{e^{ck^{2}/2s}}{\sqrt{s}}ds\leq
Ct\int_{|k|/\beta}^{\infty}\frac{e^{-ck^{2}/2s}}{s^{(\alpha+3)/2}}ds\\\ &\leq
C\frac{t}{|k|^{\alpha+1}},\quad t>0,\quad
k\in\mathbb{Z}\setminus\\{0\\}.\end{split}$

On the other hand using once again (5.1), (5.2) and [6, (9)] and by writing
$\gamma=a+b$, where $a>\alpha/2$ and $b>0,$ we get

$\begin{split}I_{1}^{\alpha}(t,k)&\leq
C\int_{0}^{|k|/\beta}\eta_{t}^{\alpha}(s)\frac{e^{-\gamma|k|\log(|k|/s)}}{\sqrt{|k|}}ds\\\
&\leq
Ce^{-b|k|\log\beta}\frac{t}{\sqrt{|k|}}\int_{0}^{|k|/\beta}\frac{1}{s^{1+\alpha/2}}\left(\frac{|k|}{s}\right)^{-a|k|}ds\\\
&\leq
Ce^{-b|k|\log\beta}\frac{t}{|k|^{1/2+a|k|}}\int_{0}^{|k|/\beta}s^{a|k|-1-\alpha/2}ds\\\
&\leq
Ce^{-b|k|\log\beta}\frac{t}{|k|^{1/2+a|k|}}\left(\frac{|k|}{\beta}\right)^{a|k|-\alpha/2}\leq
C\frac{t}{|k|^{1+\alpha}},\quad t>0,\quad
k\in\mathbb{Z}\setminus\\{0\\}.\end{split}$

Thus we have proved $P_{t}^{\alpha}(k)\leq C\mathbb{P}_{t}^{\alpha}(k)$,
$k\in\mathbb{Z}\setminus\\{0\\}$. To see $P_{t}^{\alpha}\geq
C\mathbb{P}_{t}^{\alpha}$, we have that there exists $s_{0}=s_{0}(\alpha)>0$
such that the following expression holds (see [6, (10)])

(5.3) $\eta_{t}^{\alpha}(s)\geq Cts^{-1-\alpha/2},\quad t>0,\quad\text{ and
}\quad s>s_{0}t^{2/\alpha},$

and by [31, (7)], for every $a\in(0,1)$ there exists $C>0$ such that

(5.4) $I_{\nu}(z)\geq Ce^{z}/z^{1/2},\quad z>\max\\{1,a\nu^{2}\\},\quad\nu\geq
1,$

where as above $I_{\nu}$ is the modified Bessel function of the first kind and
order $\nu$.

By combining (5.3) and (5.4) we deduce:

$\begin{split}P_{t}^{\alpha}(k)&\geq
C\int_{k^{2}/2}^{\infty}ts^{-1-\alpha/2}e^{-s}\frac{e^{s}}{s^{1/2}}ds\\\
&=Ct\int_{k^{2}/2}^{\infty}s^{-(\alpha+3)/2}ds=C\frac{t}{|k|^{\alpha+1}},\quad
t\in(0,1),\quad
k\in\mathbb{Z},\quad|k|\geq\max\\{\sqrt{2s_{0}},\sqrt{2}\\}:=k_{0}.\end{split}$

Using [31, (6)] we get

$\begin{split}P_{t}^{\alpha}(k)&\geq
C\int_{s_{0}}^{\infty}ts^{-1-\alpha/2}e^{-s}\frac{e^{\sqrt{s^{2}+k^{2}}}}{\sqrt{s+|k|}}\left(\frac{s}{|k|+\sqrt{k^{2}+s^{2}}}\right)^{|}k|ds\\\
&\geq
Ct\int_{s_{0}}^{\infty}s^{-(\alpha+3)/2}\left(\frac{s}{k_{0}+\sqrt{k_{0}^{2}+s^{2}}}\right)^{k_{0}}ds\\\
&\geq Ct\int_{s_{0}}^{\infty}s^{-(\alpha+3)/2}ds=Ct\\\ &\geq
C\frac{t}{|k|^{\alpha+1}},\quad t\in(0,1),\quad k\in\mathbb{Z},\quad 0<|k|\leq
k_{0}.\end{split}$

Thus we have $P_{t}^{\alpha}(k)\sim\mathbb{P}_{t}^{\alpha}(k)$, $t\in(0,1)$
and $k\in\mathbb{Z}\setminus\\{0\\}$.
Finally for $k=0$, on one hand, by using [6, (8)] we obtain

$\begin{split}P_{t}^{\alpha}(0)&=\int_{0}^{\infty}\eta_{t}(s)h^{Z}_{s}(0)ds=t^{-2/\alpha}\int_{0}^{\infty}\eta_{1}(t^{-2/\alpha}s)e^{-2s}I_{0}(2s)ds\\\
&\geq
Ct^{-\alpha/2}\int_{0}^{1}\eta_{1}(t^{-2/\alpha}s)ds=C\int_{0}^{t^{-\alpha/2}}\eta_{1}(u)du\\\
&\geq C\int_{0}^{1}\eta_{1}(u)du,\quad t\in(0,1),\end{split}$

and, on the other hand, from (5.1) it follows that

$\begin{split}I_{2}^{\alpha}(t,0)\leq
C\int_{0}^{\infty}\eta_{t}^{\alpha}(s)ds\leq C,\quad t\in(0,1).\end{split}$

We conclude that $P_{t}^{\alpha}(0)\sim 1$, $t\in(0,1)$.

∎

###  5.1. Proof of Theorem 1.1 for $q=1$.

Properties $a)\Rightarrow b)\Rightarrow c)\Rightarrow d)$ can be proved as in
Theorem 1.1.

Suppose that $d)$ holds. Let $0\leq f\in l^{p}(\mathbb{Z},\mu,u)$. We choose
$n_{0}\in\mathbb{Z}$ such that there exists

$\lim_{t\to 0^{+}}P^{\alpha}_{t}(f)(n_{0}).$

Then, we can find $R_{1}\in(0,1)$ such that

$\sup_{0<t\leq R_{1}}P^{\alpha}_{t}(f)(n_{0})<\infty.$

We define

$\mathbb{P}_{t}^{\alpha}(f)(k)=\sum_{n\in\mathbb{Z}}\mathbb{P}_{t}^{\alpha}(k-n)f(n),\quad
k\in\mathbb{Z}.$

Since by Proposition 5.1 $\mathbb{P}_{t}^{\alpha}(k)\sim P^{\alpha}_{t}(k)$,
$k\in\mathbb{Z}$ and $t\in(0,1)$, we have that
$\mathbb{P}_{R_{1}}^{\alpha}(f)(n_{0})<\infty.$ We also have that

$P^{\alpha}_{t}(f)(n_{0})\leq
C\mathbb{P}_{t}^{\alpha}(f)(n_{0})\leq\frac{C}{R_{1}}\mathbb{P}_{R_{1}}^{\alpha}(f)(n_{0}),\quad
t\in(R_{1},1).$

Hence, $\sup_{0<t<1}P_{t}^{\alpha}(f)(n_{0})<\infty$. To see that $e)$ is
true, we need to prove that for any $n\in\mathbb{Z}\setminus\\{n_{0}\\}$,
$\sup_{0<t<1}P_{t}^{\alpha}(f)(n)<\infty$. Thus, let
$n_{1}\in\mathbb{Z}\setminus\\{n_{0}\\}$. We can write

$\Big{|}\frac{n_{0}-k}{n_{1}-k}\Big{|}\leq\frac{|n_{0}-n_{1}|+|n_{1}-k|}{|n_{1}-k|}\leq|n_{0}-n_{1}|+1,\quad
k\neq n_{1}.$

Then,

$\begin{split}P_{t}^{\alpha}(f)(n_{1})&\leq
C\mathbb{P}_{t}^{\alpha}(f)(n_{1})\\\
&=C\left(f(n_{1})+tf(n_{0})|n_{0}-n_{1}|^{-1-\alpha}+\sum_{k\in\mathbb{Z}\setminus\\{n_{0},n_{1}\\}}\Big{(}\frac{|n_{0}-k|}{|n_{1}-k|}\Big{)}^{1+\alpha}\mathbb{P}_{t}^{\alpha}(n_{0}-k)f(k)\right)\\\
&\leq
C\Big{(}f(n_{1})+t\frac{f(n_{0})}{|n_{0}-n_{1}|^{1+\alpha}}+(|n_{0}-n_{1}|+1)^{1+\alpha}\mathbb{P}_{t}^{\alpha}(f)(n_{0})\Big{)}\\\
&\leq
C\Big{(}f(n_{1})+f(n_{0})+(|n_{0}-n_{1}|+1)^{1+\alpha}P_{t}^{\alpha}(f)(n_{0})\Big{)},\quad
t\in(0,1).\end{split}$

Hence $\sup_{t\in(0,1)}P_{t}^{\alpha}(f)(n_{1})<\infty$ and the property $e)$
is proved.

The property $e)\Rightarrow f)$ is clear.

Suppose that $f)$ holds with $R\in(0,1)$. Then, by using Hölder’s inequality
and duality arguments we obtain when $1<p<\infty$ that

$\sum_{k\in\mathbb{Z}}P_{R}^{\alpha}(|n-k|)^{p^{\prime}}u(k)^{-p^{\prime}/p}<\infty,\quad
n\in\mathbb{Z},$

and, when $p=1$, that

$\sup_{k\in\mathbb{Z}}\frac{P_{R}^{\alpha}(|n-k|)}{u(k)}<\infty,\quad
n\in\mathbb{Z}.$

Then, when $1<p<\infty$,

$\sum_{k\in\mathbb{Z}}(1+|n-k|)^{-(1+\alpha)p^{\prime}}u(k)^{-p^{\prime}/p}<\infty,\quad
n\in\mathbb{Z},$

and, when $p=1$,

$\sup_{k\in\mathbb{Z}}\frac{1}{(1+|n-k|)^{1+\alpha}u(k)}<\infty,\quad
n\in\mathbb{Z}.$

Thus $g)$ is proved.

The property $g)\Rightarrow h)$ is clear.

Assume now that $h)$ is true. We observe that

$\begin{split}\frac{1}{1+|n-k|}&=\frac{1+|k|}{1+|n-k|}\frac{1}{1+|k|}\leq\frac{1+|n-k|+|n|}{1+|n-k|}\frac{1}{1+|k|}\\\
&\leq\frac{1+|n|}{1+|k|},\quad n,k\in\mathbb{Z}.\end{split}$

Let $f:\mathbb{Z}\to[0,\infty)$. By proceeding as above it follows that

$\begin{split}P_{t}^{\alpha}(f)(n)&\leq C\mathbb{P}_{t}^{\alpha}(f)(n)\leq
C\sum_{k\in\mathbb{Z}}(1+|n-k|)^{-(1+\alpha)}f(k)\\\ &\leq
C(1+|n|)^{1+\alpha}\left(\sum_{k\in\mathbb{Z}}(1+|k|)^{-(1+\alpha)p^{\prime}}u(k)^{-p^{\prime}/p}\right)^{1/p^{\prime}}\left(\sum_{k\in\mathbb{Z}}f(k)^{p}u(k)\right)^{1/p},\end{split}$

with $t\in(0,1)$. Then, if $v$ is a weight on $\mathbb{Z}$ such that

$\sum_{k\in\mathbb{Z}}(1+|k|)^{-(1+\alpha)p}v(k)<\infty,$

$P_{*,t}^{\alpha}$ is bounded from $l^{p}(\mathbb{Z},u)$ into
$l^{p}(\mathbb{Z},v)$.

###  5.2. Proof of Theorem 1.2 for $q=1$.

We observe that

$\begin{split}t^{2\nu}\frac{e^{-t^{2}/4u}}{u^{\nu+1}}&=\frac{t^{2\nu-1}}{u^{\nu-1/2}}\frac{te^{-t^{2}/4u}}{u^{3/2}}=2\sqrt{\pi}\frac{t^{2\nu-1}}{u^{\nu-1/2}}\eta^{1/2}_{t}(u),\quad
t,u\in(0,\infty).\end{split}$

Thus, by arguing as in the proof of Proposition 5.1, we get

$T_{t}^{\nu}(k)\sim\frac{t^{2\nu}}{|k|^{2\nu+1}},\quad t\in(0,1)\quad\text{
and }\quad k\in\mathbb{Z}\setminus\\{0\\}.$

By using the last equivalence, Theorem 1.2 can be proved as Theorem 1.1

###  5.3. Proof of Theorem 1.3 for $q=1$.

By using [13, Proposition 5.1] and by taking into account that

$\sup_{t\in(0,1]}W_{t}^{Z}(f)\sim W_{1}^{Z}(f),$

provided that $f:\mathbb{Z}\to[0,+\infty)$, we deduce that

$W_{*,1}^{Z}(f)=\sup_{t\in(0,1]}|W_{t}^{Z}(f)|\leq CM(f).$

Here $M(f)$ denotes the Hardy-Littlewood maximal function. Since
$(\mathbb{Z},d,\mu)$ is a space of homogeneous type we have that
$W_{*,1}^{Z}(f)$ is bounded from $l^{p}(\mathbb{Z},\mu,w)$ into itself,
provided that $w$ is in the Muckenhoupt class $A^{p}(\mathbb{Z})$, see [23].
The class of weights in Theorem 1.3 is wider than $A^{p}(\mathbb{Z})$. The
proof of Theorem 1.3 follows in a similar way than the one of Theorem 1.1 for
$q\geq 2$.

Acknowledgements B. B. was partially supported by Ramón y Cajal fellowship
RYC2018-026098-I (Spain) and by grant PID2019-110712GB-I00 from the Spanish
Go–vernment. J. B. was partially supported by grant PID2019-106093GB-I00 from
the Spanish Government. The third author wishes to thank to Professor J. L.
Torrea (UAM, Madrid) for sharing with him some works that show the connection
between convergence of the initial data for evolution equations and the weight
$L^{p}$ inequalities for the associated maximal semigroup operators.

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Silhouettes of wormholes traversed for radiation

S.V. Repin1, M.A. Bugaev2, I.D. Novikov1,3,4, I.D. Novikov jr.1

1AstroSpace Center of P.N. Lebedev Physical Institute,
84/32, Profsoyusnaya str., Moscow, 117997, Russia

2Moscow Institute of Physics and Technology,
9, Institutskiy per., town Dolgoprudnyi, Moscow region, 141700, Russia

3The Niels Bohr International Academy, The Niels Bohr Institute,
Blegdamsvej 17, DK-2100, Copenhagen, Denmark

4National Research Center Kurchatov Institute,
1, Akademika Kurchatova pl., Moscow, 123182, Russia

###### Abstract

The problem of the passage of light through the mouth of a zero-mass wormhole
and the possibility of observing the objects from another asymptotically flat
space-time through the mouth of a wormhole are considered. It is shown that an
individual star can have several images and the fact that the image of a flat
Lambertian screen has a complex brightness distribution for an observer
located on the opposite side of the throat. Images of two such screens visible
inside the silhouette of a massless wormhole and the distribution of radiation
intensity in their images are constructed.

Keywords: wormhole, black hole, General relativity.

##  1 Introduction

In papers [1, 2, 3, 4] we started a program of systematic comparison of
silhouette111In numerous works on the bending and scattering of light rays by
black holes and wormholes, the resulting dark formations are called either
“shadows” [5, 6, 7], or “silhouettes” [8, 9]. We consider the term
“silhouette” more appropriate and will continue to use it. features — the dark
spots arising from the observation of radiation interacting with black holes
(BHs) and wormholes (WHs). To elucidate and interpret these features, it is
advisable to start with the simplest models of radiation sources and the
simplest WH models. In this approach, the features associated with the
different nature of these objects, and not with additional effects caused by
the complexities of more realistic models, are most clearly manifested.

In this work, we will continue to use this approach. In [1] we used the
Ellis–Bronnikov–Morris–Thorne WH model [10, 11, 12, 13], assuming that the WH
is filled with an opaque substance and considering only the rays moving in our
space around the entrance of a wormhole located in the space of the observer.
In this work, we take into account the light passing through the WH from
another space, i.e. we consider the WH to be freely traversable for light. See
links to previous works in [1]. The equations of motion are also derived there
(see [14] for a method) and the description of our numerical methods is given.

The wormhole metric we are considering can be written as:

$ds^{2}=c^{2}dt^{2}-dR^{2}-\left(R^{2}+q^{2}\right)\left(d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}\right)\,.$
(1)

Here $c$ is the speed of light. In what follows, we set $c=1$. We further set
$c=1$.

##  2 Formulation of the problem

First of all, we will look at the images and silhouettes (shadows) that are
created (formed) by the rays that have passed through the wormhole.

Figure 1: Section of observer space (spacetime-1) with $\theta=90^{\circ}$
(equator), $0<R<\infty$, $0<\varphi<2\pi$. The trajectories of null geodesics
(rays of light) in this space are given. These trajectories extend from the
observer to the right at infinity $R\to\infty$, $\varphi=0$ to the wormhole
entrance area. Trajectories have the impact parameters $b=0$, 0.2, 0.4, 0.6,
0.8, 0.9, 1.1, 1.2, 1.4, 1.6 and 1.8. The circumference is the throat of a
wormhole. The system coordinates correspond to the formula (1). Distances
along the vertical and horizontal axes are plotted in units of the size of the
throat radius.

Figure 2: Spacetime-2. Continuation of trajectories with impact parameters: 0,
0.2, 0.4, 0.6, 0.8 and 0.9 in space-2 after crossing the throat. Distances
along the coordinate axes are measured in units of the throat radius.

Figure 3: Trajectories of zero geodesics in space-2, strongly curving around
the throat. The impact parameter is indicated next to the trajectory.The
horizontal trajectory next to the arrow is the incoming trajectories in
space-1 that merged at a given resolution for all the trajectories shown in
the figure.

We emphasize that in the metric of a spherical wormhole, as well as in the
metric of a spherical black hole, the spatial trajectories of light rays are
always flat. Therefore, it will suffice for us to consider only one plane
section of the spatial metric, for example, the equatorial one. For an
observer who is far enough away from the entrance of the wormhole, the rays
coming to him from the entrance and its surroundings are almost parallel. Note
that the trajectories of null geodesics in the metric (1) for $|R|\gg 1$
quickly become almost straight lines. Figure 1 shows the outer space of the
observer in the equatorial section $\theta=90^{\circ}$ (in the coordinates of
the formula (1)). The circle is the throat of the wormhole, where $R=0$. The
observer is at $\varphi=0$, $R\to\infty$. We will call the space $0<R<+\infty$
as space-1. A parallel beam of null geodesics leaves the wormhole region
towards the observer. Let’s call it the sheaf $A$.

Figure 1 shows the trajectories of null geodesics with different impact
parameter $b$ with respect to the observer. The parameter $b$ is measured in
units of the throat radius. Zero geodesics with $b>1$ are the rays in the
space-1 of the observer. Trajectories with $b<1$ reach the WH throat and go to
another space. We will call it space-2, where $R$ changes from $R=0$ to
$R\to-\infty$.

Figure 1 shows the trajectories with impact parameter $b$ above the horizontal
line $b=0$, i.e. moving counterclockwise. Of course, a similar picture should
be repeated for the impact parameter $b$ below the line $b=0$, i.e. moving
clockwise.

Figure 2 shows space-2. Shown here are null geodesics that arrive from space-1
crossing the throat. The corresponding impact parameters $b<1$ are indicated
next to the trajectories. For values $b<1$, but very close to 1 (i.e.
$(1-b)\ll 1$) the trajectories turn many times around the $\varphi$ coordinate
before going to $R\to-\infty$. This is shown in fig. 3.

Trajectories with $b<1$ are the object of our attention in this article. They
are used to build the images and silhouettes that are visible to the observer
through the wormhole.

##  3 Images of individual stars seen through a wormhole

Let us now proceed to our main task - to construct the images of individual
objects and silhouettes visible to the observer through the wormhole. We start
with the image of individual stars.

Figure 4: Multiple images of the same star in space-2. The image marked with
an arrow can be conventionally called the “main” image. In the right panel,
the same images are shown with the horizontal coordinate $\log(1-|b|)$.

Let there be a distant point source (star) in space-2 located in our
$R-\theta$ section at the coordinate $\theta$. How will it appear to the
observer? Rays of light from a distant star (zero geodesic) fall on the
entrance of the wormhole in space-2 in a parallel beam. Let’s call it the
sheaf $B$. A ray of light, the asymptotics of which for $R\to-\infty$
corresponds to the coordinate $\theta$, will reach the observer. Its impact
parameter $b_{1}$ will correspond to the position of the star image for the
observer. But the image created will not be a single one. A ray from beam $B$
going near the throat and making a full turn near it can get into the beam $A$
and come to the observer. This is how the second image of the star $b_{2}$
appears closer to the throat circumference, and so on. There will be an
infinite number of images, getting closer and closer to the circle
corresponding to the throat. The rays from beam $B$ can also make full turns
in the opposite direction. Such images will also be observed near the throat
circumference, but near its opposite side. Multiple images of a single star
are shown in Fig. 4 on the left. The same images are shown on the right, but
the value $\log(1-|b|)$ is used as the horizontal coordinate. This is done in
order to distinguish individual merging images near the edge of the throat.

Figure 5: Multiple images of the same star located in space-2 and arranged
symmetrically in the observer’s field of view. For the same images in the
right panel, the $\log(1-|b|)$ coordinate is used.

Figure 6: Multiple images of the same star located in space-2 and arranged
symmetrically in the observer’s field of view. The “main” image is in the
center of the field of view. For the same images in the right panel, the
$\log(1-|b|)$ coordinate is used.

The nature of the image of the star itself and its repetition depend, of
course, on its position in the sky of space-2.

We will assume that the meridian of the section $\varphi=90^{\circ}$, along
which we consider the different positions of the stars, corresponds to a
horizontal line in the field of view of the wormhole image observer. Then the
pattern with repeated images is shown schematically in Fig. 5. In the region
of the pole, the first image of the star is absent. The rays from this star do
not follow a straight path directly to the observer. At the same time, the
outermost points in the field of view in the left panel consist of two merged
images that cannot be resolved at this scale. On the right panel with the
horizontal coordinate $\lg(1-|b|)$ they are quite clearly visible.

Fig. 6 shows another symmetrical situation, when the star from space-2 is
visible in the center of the field of view.

##  4 Inner silhouettes (shadows) of a wormhole

Let us now consider the images created for the observer in our space-1 by
extended luminous screens located in space-2. As in the papers [1]–[3] we will
assume that the screen glow intensity is the same in all directions
(Lambertian source). First of all, consider a screen located in space-2 in the
direction of the pole $\varphi=180^{\circ}$ perpendicular to the radial
direction and located far enough $R\to-\infty$. Fig. 7 schematically shows the
position of the observer in space-1 and two screens located in space-2. It is
clear from the Figure that the light beam coming from the observer exactly to
the center of the wormhole (i.e. with zero impact parameter) arrives at
screen-2 perpendicular to it and therefore does not hit screen-1. And in order
to hit the screen-1, the light ray must move with a non-zero impact parameter
$b$ and, therefore, turn in space-2 (near the throat) by an angle greater than
$90^{\circ}$. Looking again at Fig. 2 one can see that the impact parameter of
such a beam is approximately $b\approx 0.5$.

Figure 7: Layout of the observer in space-1 and extended Lambertian screens in
space-2.

Figure 8: The image of a flat Lambertian screen-1, which the observer sees
through the mouth of a wormhole. The silhouette of this screen is shown on the
left, and the image emission intensity distribution on a logarithmic scale is
shown on the right.

Figure 9: Radiation intensity distribution along the diameter of the wormhole
throat for the first screen.

First of all, we will consider how the image of screen-1 looks like from the
point of view of an observer located in space-1. We consider the screen
dimensions to be infinitely large The methods of constructing the images are
similar to those described in the paper [1]. The image seen by an observer in
space-1 is shown in Fig. 8. The left panel shows the shape of the image as
seen by the observer, while the right panel shows the distribution of
radiation intensity in the same image, shown on a logarithmic scale. The inner
radius of the light region, as mentioned above, corresponds to a rotation of
the ray in space-2 by the angle of $90^{\circ}$, and the outer boundary
corresponds to a rotation by $270^{\circ}$. The exact values of the impact
parameters for these values of the rotation angle can only be obtained by
numerical integration of the zero geodesic equations. These values are:

$b_{90}=0.4704q,\qquad\qquad b_{270}=0.94008q\,.$ (2)

The beam, however, can reach the screen in a more complicated way: it can make
several revolutions near the throat of the wormhole and only then hit the
screen. So, when turning the ray through the angles from $450^{\circ}$ to
$630^{\circ}$, the ray will reach the screen after making one full turn, and
when turning through the angles from $810^{\circ}$ to $990^{\circ}$ — two full
turns, etc. These rays should pass through the mouth of the wormhole with
impact parameters very close to unity and form an infinite number of thin
rings. For the first such ring, the impact parameters of the inner and outer
boundaries refer respectively to:

$b_{450}=0.9969445q,\qquad\qquad b_{630}=0.9998713q,\qquad\qquad
b_{630}-b_{450}\approx 0.003q\,.$ (3)

The accuracy of these values is four to five units of the last decimal place.
For the second and subsequent rings, the integration with extremely high
accuracy is required to determine these parameters, since the system of
equations becomes too stiff for standard numerical methods.

The intensity distribution along the image diameter is shown as a graph in
Fig. 9. The local intensity minimum falls at the value of the impact parameter
$b=0.793$, which corresponds to the beam rotation by $180^{\circ}$. The
appearance of this minimum can also be easily understood from general physical
considerations. The point is that the rays emitted from an observer with an
impact parameter $b=0.793$ hit the same point of screen-1 in space-2 and,
consequently, the observer sees this point when registering any ray with an
impact parameter $b=$0.793.

Consider now the screen-2, in Fig. 7, located in space-2 at
$\varphi=0^{\circ}$, $R\to-\infty$. In this case, the beam coming with a zero
impact parameter hits the center of the screen and, therefore, the center of
the screen should be visible bright. Moreover, a ray that hits the screen-2
cannot hit the screen-1 and vice versa. Conventionally, we can say that the
screen-2 image should be “complementary” to the screen-1 image shown in Fig.
8.

Figure 10: The image of a flat Lambert screen-2, which the observer sees
through the mouth of a wormhole. The silhouette of this screen is shown on the
left, and the image emission intensity distribution on a logarithmic scale is
shown on the right.

Figure 11: Details of the radiation intensity distribution in the first ring,
i.e. in a ring formed by quanta that have made one complete revolution in the
throat of a wormhole.

The image of screen-2 for an observer in space-1 is shown in Fig. 10.
Similarly to Fig. 8, the left panel shows only the silhouette of the image
that the observer sees, while the right panel shows the distribution of the
radiation intensity in the same image on a logarithmic scale. As expected, the
center of the screen is bright, but the distribution of brightness differs
from Fig. 8. In the center of the screen, the intensity has a minimum and
increases towards the edge of the bright disk. In addition, we see a bright
ring formed by the rays that have made a complete revolution in the mouth of a
wormhole. In this case, the width of the ring is such that we can plot the
radiation intensity distribution inside it along the radial coordinate. On an
enlarged scale, the intensity distribution in the ring is shown in Fig. 11,
where one can see its details. For example, the intensity minimum is clearly
visible, and this minimum is located asymmetrically and strongly shifted to
the outer edge of the ring. The radiation intensity increases towards the
edges of the ring. As in the previous case, the minimum arises due to the fact
that all rays with this impact parameter, having made a revolution, fall into
the same point of the screen. The value of the impact parameter for the
intensity minimum in the ring is:

$b=0.985q.$ (4)

The graph of radiation intensity distribution along the equatorial section of
the wormhole silhouette is shown in Fig. 12.

Figure 12: Radiation intensity distribution along the diameter of the wormhole
throat for the second screen.

Of course, an observer in space-1 will see both the rays that have passed in
his space around the entrance to the wormhole, as well as the rays that have
passed through the wormhole. We considered the first category of rays in [1,
2, 3]. The complete picture for the observer is formed by merging the
depictions considered in [1, 2, 3] and considered in this paper.

##  5 Conclusion

In the considered model of the silhouette of a traversable wormhole, there are
characteristic image details that can be used to identify these objects in
interferometric observations [7]. Thus, in contrast to the black holes, fairly
complex ring structures with varying image brightness can be observed inside
the silhouette of a wormhole. Wormholes themselves, like supermassive black
holes, may possibly exist in the centers of galaxies [15]. In order to search
for and observe such objects, one can use the catalog of supermassive black
holes [16], in which one can find a sufficient number of candidates for
observations using the ground-based interferometers.

Even wider opportunities for observing the wormholes are provided by a ground-
space interferometer, in which one of the receiving antennas is located on a
satellite located in space at a distance from units to hundreds of Earth
radii. One of the space objects for installing such an antenna could be the
Moon. Another possibility is provided by the Russian project Millimetron [17,
18] with a ten-meter cooled antenna. The resolution of the Millimetron in the
interferometer mode can reach tens of nano-arcseconds. Such a high resolution
is enough to study in detail the silhouettes of black holes and, possibly,
wormholes in a large number of galaxies. The discovery of such objects will
undoubtedly give a new impetus to the development of Astrophysics.

##  6 Acknowledgments

S.R. expresses his gratitude to O.N. Sumenkova, R.E. Beresneva and O.A.
Kosareva for the opportunity to successfully work on this problem.

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