Here we also need to query the neighbors of $u_{j}$. Since $u_{j}$’s are
sampled from $S^{\prime}\Delta C(r)$ which also has a size $\Omega(d(r))$, we
can use the same way to sample them as we previously sampling $u_{i}$’s.

We will prove the following lemma. Then, according to union bound, all the
costs are estimated well within a small error. By changing the constant in the
definition of $\Delta$ (in Section 5), the same algorithm could also work.

######  Lemma 55.

Let
$\zeta<\left(s^{\eta^{5}}\right)^{\eta}\left(1+\log_{1+\epsilon^{\prime}}(4/\epsilon)\right)^{\eta}=2^{\epsilon^{-O(1)}}$
denote the number of functions calls of GenerateCluster in each round. Let
$\eta^{\prime}>38220595200\epsilon^{-54}W^{2}\zeta$ be a sufficiently large
constant. With probability at least $1-2\exp(-\eta)$,

$\mathrm{EstImprovement}_{w}(C,S^{\prime},r)\in\textsc{Cost}_{w}(\mathcal{C})-\textsc{Cost}_{w}(\mathcal{C}+S^{\prime})+\frac{\epsilon^{13}}{4}\cdot\frac{\epsilon^{8}}{576}|D(r)|\cdot|N(r)|.$
(25)

###### Proof.

Let $\textsc{Cost}(\mathcal{C},uv)$ be the cost by the clustering scheme
$\mathcal{C}$ for the edge or nonedge $uv$. For each $u\in S^{\prime}$, let

$\displaystyle X_{u}=$ $\displaystyle\frac{1}{2}\sum_{v\in
S^{\prime}}(\textsc{Cost}_{w}(\mathcal{C},uv)-\textsc{Cost}_{w}(\mathcal{C}+S^{\prime},uv))+$
$\displaystyle\sum_{v\in C(u)\setminus
S^{\prime}}(\textsc{Cost}_{w}(\mathcal{C},uv)-\textsc{Cost}_{w}(\mathcal{C}+S^{\prime},uv))$

Observe that

$\displaystyle X_{u}=$ $\displaystyle\frac{1}{2}\sum_{v\in
S^{\prime}}(\textsc{Cost}_{w}(\mathcal{C},uv)-\textsc{Cost}_{w}(\mathcal{C}+S^{\prime},uv))+$
$\displaystyle\sum_{v\in C(u)\setminus
S^{\prime}}(\textsc{Cost}_{w}(\mathcal{C},uv)-\textsc{Cost}_{w}(\mathcal{C}+S^{\prime},uv))$
$\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{v\in
S^{\prime}}(\textsc{Cost}_{w}(\mathcal{C},uv)-\textsc{Cost}_{w}(\mathcal{C}+S^{\prime},uv))+$
$\displaystyle\sum_{v\in V\setminus
S^{\prime}}(\textsc{Cost}_{w}(\mathcal{C},uv)-\textsc{Cost}_{w}(\mathcal{C}+S^{\prime},uv))$
$\displaystyle=$ $\displaystyle\sum_{v\in
V}(\textsc{Cost}_{w}(\mathcal{C},uv)-\textsc{Cost}_{w}(\mathcal{C}+S^{\prime},uv))-$
$\displaystyle\frac{1}{2}\sum_{v\in
S^{\prime}}(\textsc{Cost}_{w}(\mathcal{C},uv)-\textsc{Cost}_{w}(\mathcal{C}+S^{\prime},uv))$
$\displaystyle=$
$\displaystyle\mathrm{CostStays}_{w}(S^{\prime},u)-\mathrm{CostMoves}_{w}(S^{\prime},u)-\frac{1}{2}\sum_{v\in
S^{\prime}}(\textsc{Cost}_{w}(\mathcal{C},uv)-\textsc{Cost}_{w}(\mathcal{C}+S^{\prime},uv))$

Note that $X_{u}$ can be computed in $O(d(u)+|S^{\prime}|)$.

$\textsc{Cost}_{w}(\mathcal{C})-\textsc{Cost}_{w}(\mathcal{C}+S^{\prime})=\sum_{u\in
S^{\prime}\oplus C(r)}X_{u}$ (26)

Since $|X_{u}|\leq W(|S^{\prime}|+|C(u)|)\leq W(|N(r)|+|C(u)|)$, by Corollary
33 and lemma 29, we have

$\displaystyle|X_{u}|$ $\displaystyle\leq W(|N(r)|+|C(u)|)$ $\displaystyle\leq
W(6\epsilon^{-3}d(r)+12\epsilon^{-4}d(u))$ $\displaystyle\leq
W(6\epsilon^{-3}d(r)+24\epsilon^{-5}d(r))$ $\displaystyle\leq
30W\epsilon^{-5}d(r)$

According to Hoeffding’s inequality,

$\displaystyle\mathbf{Pr}\left[\left|\frac{|S^{\prime}\oplus
C(r)|}{\eta^{\prime}}\sum_{j=1}^{\eta^{\prime}}X_{u_{j}}-(\textsc{Cost}_{w}(\mathcal{C})-\textsc{Cost}_{w}(\mathcal{C}+S^{\prime}))\right|\geq\frac{\epsilon^{13}}{4}\cdot\frac{\epsilon^{8}}{576}|D(r)|\cdot|N(r)|\right]$
$\displaystyle\qquad\leq
2\exp\left(-\frac{2\left(\frac{\epsilon^{13}}{4}\cdot\frac{\epsilon^{8}}{576}|D(r)|\cdot|N(r)|\right)^{2}}{\eta^{\prime}\left(\frac{|S^{\prime}\oplus
C(r)|}{\eta^{\prime}}60W\epsilon^{-5}d(r)\right)^{2}}\right)$
$\displaystyle\qquad\leq
2\exp\left(-\frac{2\eta^{\prime}\epsilon^{42}(|D(r)|\cdot|N(r)|)^{2}}{19110297600W^{2}\epsilon^{-10}|D(r)|^{2}d(r)^{2}}\right)$
$\displaystyle\qquad\leq
2\exp\left(-\frac{\eta^{\prime}\epsilon^{44}}{38220595200W^{2}\epsilon^{-10}}\right)$
$\displaystyle\qquad\leq 2\exp\left(-\zeta\right)$

The last two inequality hold because $|N(r)|\geq\frac{1}{2}\epsilon d(r)$ and
$\eta^{\prime}>38220595200\epsilon^{-54}W^{2}\zeta$. This finishes the proof
of this lemma. ∎

##  8 Acknowledgements

David Rasmussen Lolck, Mikkel Thorup, Shuyi Yan and Hanwen Zhang are part of
Basic Algorithms Research Copenhagen (BARC), supported by the VILLUM
Foundation grant 16582. Marcin Pilipczuk is partially supported by the VILLUM
Foundation grant 16582 while visiting Basic Algorithms Research Copenhagen.
Hanwen Zhang is partially supported by Starting Grant 1054-00032B from the
Independent Research Fund Denmark under the Sapere Aude research career
programme.

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en-US

# A Mixed Graph Achieving a Moore-like Bound

Geoffrey Exoo
Department of Mathematics and Computer Science
Indiana State University
Terre Haute, IN 47802
<EMAIL_ADDRESS>
###### Abstract

Mixed graphs have both directed and undirected edges. A mixed cage is a
regular mixed graph of given girth with minimum possible order. In this paper
we construct a mixed cage of order $30$ that achieves the mixed graph analogue
of the Moore bound for degree 3, out-degree 1, and girth 6.

Mathematics Subject Classifications: 05C35, 05C20, 05C38

##  1 Notation and Terminology

A mixed graph is a graph with both directed and undirected edges. We refer to
directed edges as arcs and undirected edges as edges. The degree of a vertex
$v$ in a mixed graph $G$ is the number of edges incident with $v$, whereas the
in-degree and out-degree are the numbers of arcs incident to and from $v$. $G$
is regular if all three degrees are constant as $v$ ranges over $V(G)$.

A cycle in a mixed graph is a sequence of vertices $v_{0},v_{1},\cdots,v_{k}$
such that there are no repeated vertices except that $v_{0}=v_{k}$, each pair
of consecutive vertices $(v_{i},v_{i+1})$ is either an edge or an arc, and
there are no repeated edges or arcs. The girth of a mixed graph is the length
of a shortest cycle. Note that this definition considers the possibility of
$1$-cycles (loops) and $2$-cycles.

Mixed graphs have been studied in the context of the degree/diameter problem.
We follow the notation used in that literature and denote the degree and
outdegree of a regular mixed graph by $r$ and $z$, respectively.

An $(r,z,g)$-graph is a regular mixed graph with degree $r$, out-degree $z$,
and girth $g$. An $(r,z,g)$-cage is an $(r,z,g)$-graph of minimum possible
order. We denote this minimum order by $f(r,z,g)$.

Recall [2] that the Moore bound for an $r$-regular graph of diameter $d$ is
given by:

$n(r,d)\geq\frac{r(r-1)^{d}-2}{r-2}$ (1)

In [1] Araujo-Pardo, Hernández-Cruz, and Montellano-Ballesteros consider the
problem of finding mixed cages. They focus on the case $z=1$ and determine a
lower bound for $f(r,1,g)$ based on the Moore bound. Their idea is to attach
undirected Moore trees to each vertex of a directed path of length $g{-}1$,
choosing trees whose depth is as large as possible while still guaranteeing
that all tree vertices are distinct.

So let $v_{0},v_{1},\cdots,v_{g-1}$ be the vertices of a directed path of
length $g{-}1$. Using edges, attach a Moore tree of depth $i$ to both $v_{i}$
and $v_{g-1-i}$ for $0\leq i\leq\lfloor g/2\rfloor$. Note: if $i=g{-}i{-}1$ we
attach only one tree. The base path contains arcs and the Moore trees contain
edges. This gives the following bound.

######  Theorem 1 (The AHM Bound [1])

$f(r,1,g)\geq{{\sum}}_{i=0}^{g-1}n(r,\,{\tt min}(i,g{-}i{-}1))$

###### Theorem 2

$f(3,1,6)=30$

Proof. The AHM bound for $r=3$, $z=1$, and $g=6$ is $30$. The graph $G$ shown
in the figure has order $30$ and has the require parameters.

The graph can be describe algebraically as follows. Let

$V(G)=\\{v(i,j)\,|\,0{\leq}i{<}3,\,0{\leq}j{<}10\\}$

be the vertex set. For $0\leq i<3$, let

$V_{i}=\\{v(i,j)\,|\,0{\leq}j{<}10\\}.$

Each $V_{i}$ induces a directed $10$-cycle. These three cycles are shown in
figure. Let $V_{0}$ be the set of vertices in the lower partion of the figure
and let $V_{1}$ and $V_{2}$ be the sets of vertices on the outer and inner
(resp.) cycles in the upper part of the figure. Imagine the vertices labeled
such that vertex $v_{i,0}$ is the rightmost vertex on each of the three
directed $10$-cycles and that the other vertices are labeled in counter-
clockwise order. The arcs and edges are given as follows, where second indices
are computed module $10$:

  1. (a)

arc($v(0,j),\,v(0,j{+}1)$)

  2. (b)

edge($v(0,j),\,v(1,j)$)

  3. (c)

edge($v(0,j),\,v(2,j{+}5)$)

  4. (d)

edge($v(1,j),\,v(2,j{+}2)$)

  5. (e)

edge($v(1,j),\,v(2,j{-}2)$)

In the figure, edges of type (b) and (c) are not drawn but indicated by color:
vertices in $V_{0}$ are adjacent to those vertices in $V_{1}$ and $V_{2}$ that
have the matching color.

Note: Recently another graph achieving the AHM bound was found by the author.
This graph is for the case $(6,1,6)$. The graph has order $90$, which is the
AHM bound, and can be constructed as a lift of the complement of the line
graph of $K_{6}$. The adjacency matrix for the graph can be found at the
following location.

`https://cs.indstate.edu/ge/MixedCages/g90.txt`

Figure 1: The unique smallest $(3,1,6)$-graph of order $30$. Vertices in the
lower figure are adjacent to vertices in the upper figure that have the
matching color. The automorphism group has two generators: a rotation of
$\pi/5$ (in both figures) and an involution that transposes the inner and
outer cycles in the upper figure. These generators commute, so the group is
$Z_{2}\times Z_{10}$.

## References

  * [1] G. Araujo-Pardo, C. Hernández-Cruz, and J. J. Montellano-Ballesteros, Mixed Cages. Graphs and Combinatorics, 35 (2019), 989–999.
  * [2] M. Miller and J. Sirán, Moore Graphs and Beyond: A Survey of the Degree/Diameter Problem, Electronic Journal of Combinatorics DS14, May 16, 2013.


Different approach to the decomposition theory of $HM^{p}_{q,{\Delta_{\nu}}}$
Hardy-Morrey spaces

Cansu Keskina, 111 Corresponding author.
Key words and phrases: Atomic decomposition, B-maximal function, Hardy-Morrey
spaces, Laplace-Bessel equation.
2010 Mathematics Subject Classification : 42B30, 42B20, 42B10, 42B25. ,

a Dumlupinar University, Department of Mathematics, Kutahya, Turkey

e-mail<EMAIL_ADDRESS>
###### Abstract

The Hardy-Morrey spaces related to Laplace-Bessel differential equations are
introduced in terms of maximal functions. The atomic decomposition theory
which has the same cancellation properties of the
$H^{p}_{\Delta_{\nu}}(\mathbb{R}^{n}_{+})$ Hardy spaces is established.

##  1 Introduction

Lebesgue spaces $L^{p}$ and Hardy spaces $H^{p}$ play an important role in
function spaces theory and in harmonic analysis as well. So many people have
shown that Hardy spaces $H^{p}(\mathbb{R}^{n})$ $(0<p\leq\infty)$ can be more
informative than Lebesgue spaces $L^{p}(\mathbb{R}^{n})$ when we investigate
the boundedness of some singular integral operators. For example, the
classical Riesz transforms are bounded from $H^{1}(\mathbb{R}^{n})$ to
$L^{1}(\mathbb{R}^{n})$, although they are not bounded on
$L^{1}(\mathbb{R}^{n})$. We can also give an equivalent definition for Hardy
spaces by using the decomposition theory. This definition states that any
elements of Hardy spaces can be represented as the series of atoms. One of the
advantages of the decompositions in Hardy spaces is that we can prove the
boundedness of some operators can be verified only for the collection of
atoms.

It is worthwhile to mention that the development of a theory of Hardy spaces
(or other generalizations as Hardy-Morrey) related to some classes of
differential operators has attracted a great interest. It can be developed to
other function spaces. Some results in this direction can be encountered in
[2], the decomposition of Hardy-Morrey spaces, the decomposition of Hardy
spaces with variable exponent in [4], and the atomic decomposition of Morrey
spaces [1]. Motivated by these advantages that Hardy spaces enjoy, in the
present paper, we investigate the atomic decomposition for
$HM^{p}_{q,{\Delta_{\nu}}}$ Hardy-Morrey spaces.

Hardy-Morrey spaces are important due to the connections with $L^{p}$, Hardy
space, $BMO^{-1}$ and etc. The maximal characterization, the atomic
decomposition of this space, which have the same cancellation properties as
the classical Hardy space, were studied by Jia and Wang [2, 3]. These results
extend those of E. M. Stein for the Hardy space $H^{p}(\mathbb{R}^{n})(p\leq
1)$, of A. L. Mazzucato for the Besov–Morrey space [10]
$\mathcal{N}_{p,q,r}^{s}$ and of M. E. Taylor [9] for the Morrey space
$\mathcal{M}^{p}_{q}(q\geq 1)$.

The atomic decomposition is one of the remarkable features for Hardy type
spaces. There are many attempts of obtaining these decompositions [2, 3, 5]. A
study of the classical Hardy-Morrey spaces by using the maximal function
approach is given in [2] and some applications of these spaces are given in
[3]. Also weighted version is investigated by Ho in [5]. An interesting
problem is whether we can give an atomic characterization of Hardy-Morrey
spaces associated with Laplace-Bessel operator $\Delta_{\nu}$ defined as
follows

$\Delta_{\nu}=\sum\limits_{i=1}^{n}\frac{\partial^{2}}{\partial
x_{i}^{2}}+\frac{\nu}{x_{n}}\frac{\partial}{\partial x_{n}},~{}~{}~{}\nu>0.$

Before finishing we would like to point out that in this paper, maximal
function characterization and atomic decomposition are studied in the Laplace-
Bessel setting of Hardy-Morrey spaces. We introduce some new spaces called
$HM^{p}_{q,{\Delta_{\nu}}}$ Hardy-Morrey spaces which generalize the Morrey
spaces $M^{p}_{q}(q>1)$ and Hardy spaces $H^{p}_{\Delta_{\nu}}(p<1)$ [7].

The structure of the remaining part of the present paper is as follows: First,
we recall some informations for both Hardy and Morrey spaces in Section 2. In
Section 3, the maximal function characterization is given. Finally, as
applications of the atomic characterization we will obtain some estimates of
the singular integral operators associated with the generalized shift
operator. We apply all our results to the Laplace-Bessel setting, obtaining
boundedness properties for $0<q\leq p<\infty$ of the singular integral
operators appearing. We also derive the characterization of the Laplace
Bessel-Hardy spaces in terms of the Riesz-Bessel transforms, extending the
results in [6] to a wider class of $HM^{p}_{q,{\Delta_{\nu}}}$ Hardy-Morrey
spaces.

##  2 Hardy-Morrey spaces for $\triangle_{\nu}$

The classical Hardy spaces $H^{p}$ are defined by [7],

$H^{p}=\bigg{\\{}f\in\mathcal{S}^{{}^{\prime}}\setminus\mathcal{P}:||f||_{H^{p}}=\bigg{|}\bigg{|}\sup\limits_{t>0}|\phi_{t}\ast
f|\bigg{|}\bigg{|}_{L^{p}}<\infty\bigg{\\}}.$

Here, $\mathcal{S}^{{}^{\prime}}$ is the topological dual of $\mathcal{S}$
Schwartz space is the collection of all tempered distributions on
$\mathbb{R}^{n}$ equipped with the strong topology, and
$\phi_{t}(x)=t^{-n}\phi(t^{-1}x)$ for $\phi\in\mathcal{S}$ with
$\int\phi(x)dx=1$.

The classical Morrey space was first introduced by Morrey in [8], since then a
large number of investigations have been given to them by mathematicians.
Recently, many authors established the boundedness of some type operators
associated with Morrey spaces.

######  Definition 2.1.

For $p$ and $q$ satisfying $0<q\leq p<\infty$, the homogeneous Morrey spaces
$M^{p}_{q}$, are defined as

$M^{p}_{q}=\\{f\in
L^{q}_{\text{loc}}:||f||_{M^{p}_{q}}=\sup\limits_{x\in\mathbb{R}^{n},R>0}|B(x,R)|^{1/p-1/q}||f||_{L^{q}(B(x,R))}<\infty\\}$

where $B(x,R)$ is the closed ball of $\mathbb{R}^{n}$ with center $x$ and
radius $R$.

Let $j\in\mathbb{Z}$, $k\in\mathbb{Z}^{n}$. The set

$Q_{jk}=\\{x\in\mathbb{R}^{n}:2^{-j}k_{i}\leq x_{i}\leq
2^{-j}(k_{i}+1),i=1,\ldots,n\\}$

is a called a dyadic cube.

######  Remark 2.1.

Note that

$||f||_{M^{p}_{q}}\approx\sup\limits_{J:\text{dyadic}}|J|^{1/p-1/q}||f||_{L^{q}(J)}.$

Based on the above notions and notations, we can state the Hardy-Morrey spaces
in Laplace-Bessel setting.

######  Definition 2.2.

We say that a function $f\in\mathcal{S}_{+}\setminus\mathcal{P}$ belongs to
the Hardy-Morrey space $HM^{p}_{q,{\Delta_{\nu}}}$ for $0<q\leq p<\infty$, if

$||f||_{HM^{p}_{q,{\Delta_{\nu}}}}=||\sup\limits_{t>0}|\phi(t)\otimes
f|||_{M^{p}_{q,\nu}}<\infty.$

Here $\phi\in\mathcal{S}_{+}$($\mathcal{S}_{+}(\mathbb{R}^{n}_{+})$ be the
space of functions which are the restrictions to $\mathbb{R}^{n}_{+}$ of the
test functions of the Schwartz that are even with respect to $x_{n}$,
decreasing sufficiently rapidly at infinity) satisfies
$\int\phi(x)x_{n}^{\nu}dx=1$ and $\mathcal{P}$ is the set of polynomials.

The $HM^{p}_{q,{\Delta_{\nu}}}$ Hardy-Morrey spaces cover Hardy spaces. From
the definition, $H^{p}_{\Delta_{\nu}}=HM^{p}_{p,{\Delta_{\nu}}}\subset
HM^{p}_{q,{\Delta_{\nu}}}$ for $q\leq p<\infty$. Thus, the Hardy space
$H^{p}_{\Delta_{\nu}}(\mathbb{R}^{n}_{+})$ is the space of those
$f\in\mathcal{S}^{\prime}_{+}(\mathbb{R}^{n}_{+})$ (a function called bounded
tempered distribution on $\mathbb{R}^{n}_{+})$ for which maximal function
belongs to $L^{p}_{\nu}(\mathbb{R}^{n}_{+})$, and defined by [6]

$H^{p}_{\Delta_{\nu}}=\\{f\in\mathcal{S}^{\prime}_{+}\setminus\mathcal{P}:||f||_{H^{p}_{\Delta_{\nu}}}=||\sup\limits_{t>0}|\phi(t)\otimes
f|||_{L^{p}_{\nu}}<\infty\\}.$

Here, the generalized convolution($\Delta_{\nu}$-convolution) is defined by

$(f\otimes g)(x)=\int_{\mathbb{R}^{n}_{+}}f(y)\;T^{y}g(x)\,y_{n}^{\nu}dy,$

and $T^{y}$ denotes the generalized shift operator given by [6].

##  3 Maximal Characterization of $HM^{p}_{q,{\Delta_{\nu}}}$ spaces

The Hardy-Morrey spaces are defined in this section by using the maximal
function characterizations. One of the remarkable and fundamental properties
of Hardy type spaces are the equivalence of definitions of Hardy type spaces
by different maximal functions such as the nontangential maximal function
characterizations and the grand maximal function characterizations. The
equivalence of these maximal function characterizations are established in
this section.

We review some needed background on Hardy-Morrey spaces. For any
$N\in\mathbb{Z}^{+}$, define

$p_{N,\nu}(\varphi)=\sup\limits_{x\in\mathbb{R}^{n}_{+}}\big{(}1+|x|\big{)}^{N}\sum\limits_{|\alpha|\leq
N}\big{|}D_{x^{\prime}}^{\alpha^{\prime}}B_{n}^{\alpha_{n}}\varphi(x)\big{|}~{}x_{n}^{\nu}dx$

and

$\mathfrak{F}_{N,\nu}=\Big{\\{}\varphi\in\mathcal{S}_{+}(\mathbb{R}^{n}_{+}):p_{N,\nu}(\varphi)\leq
1\Big{\\}}.$

For any positive $t$ and $\varphi\in\mathcal{S}_{+}(\mathbb{R}^{n}_{+})$, we
write $\varphi_{t}(x)=t^{-n-\nu}\varphi(x/t)$.

Let $\varphi\in\mathcal{S}_{+}(\mathbb{R}^{n}_{+})$ with
$\int_{\mathbb{R}^{n}_{+}}\varphi(x)~{}x_{n}^{\nu}dx=1$ and any
$f\in\mathcal{S}^{\prime}_{+}(\mathbb{R}^{n}_{+})$, the maximal function
associated with the Laplace-Bessel differential operator is defined by V.S.
Guliyev in [11, 12, 13]

$\displaystyle\mathcal{M}_{\nu}f(x)=\sup\limits_{r>0}\frac{1}{|E(0,r)|_{\nu}}\int_{E(0,r)}T^{y}|f(x)|y_{n}^{\nu}dy.$

Note that $\mathcal{M}_{\nu}$ is the Hardy-Littlewood maximal operator
associated with the Laplace-Bessel differential operator.

The grand maximal function is defined by setting

$\displaystyle\mathcal{M}_{N,\nu}f(x)=\sup\\{|(f\otimes\varphi_{t})(x)|:t>0,~{}~{}~{}\varphi\in\Im_{N,\nu}\\}$

where we choose and fix a large integer $N$.

We define the ”nontangential” version of $\mathcal{M}_{\nu}$, given by

$\mathcal{M}_{\varphi,\nu}^{*}f(x)=\sup\limits_{y\in\Gamma_{t}(x)}\left|(f\otimes\varphi_{t})(y)\right|,$

and also the ”radial(vertical)” maximal functions defined by

$\displaystyle\mathcal{M}^{0}_{\varphi,\nu}f(x)=\sup\limits_{0<t<\infty}\left|(f\otimes\varphi_{t})(x)\right|,$

Here $\Gamma_{t}(x)=\\{y\in\mathbb{R}^{n}_{+}:|x-y|<t\\}$ and
$\varphi_{t}(y)=t^{-n-\nu}\varphi\big{(}\frac{y}{t}\big{)}$.

Beside this, the $\varphi$-tangential maximal function

$\displaystyle\mathcal{M}_{\varphi,\nu}^{\lambda}f(x)=\sup\limits_{y\in\mathbb{R}^{n}_{+},t>0}|(f\otimes{\varphi_{t}})(y)|\bigg{(}\frac{t}{|x-y|+t}\bigg{)}^{\lambda}.$

Let

$\begin{array}[]{ll}P^{\nu}(x,t)=P_{t}^{\nu}(x)=\dfrac{C_{k,\nu}t}{(t^{2}+|x|^{2})^{\frac{n+k+\nu}{2}}},\quad
C_{k,\nu}=\dfrac{2^{\frac{n+\nu}{2}}\Gamma\big{(}\frac{n+k+\nu}{2}\big{)}}{\Gamma\big{(}{\frac{k}{2}}\big{)}}.\end{array}$

be the $\Delta_{\nu}$-Poisson type kernel.

######  Definition 3.3.

The Hardy-Morrey space $HM^{p}_{q,{\Delta_{\nu}}}(\mathbb{R}^{n}_{+})$ is the
space of those $f\in\mathcal{S}^{\prime}_{+}(\mathbb{R}^{n}_{+})$ for which
$\mathcal{M}_{P^{\nu}}f$, the $\Delta_{\nu}$-Poisson type maximal function and
$\mathcal{M}_{N,\nu}f$, the grand maximal function of $f$ all belongs to
$M^{p}_{q,\nu}$, and define

$\|f\|_{HM^{p}_{q,{\Delta_{\nu}}}}=\|\mathcal{M}_{P^{\nu}}f\|_{M^{p}_{q,\nu}}=\|\mathcal{M}_{N,\nu}f\|_{M^{p}_{q,\nu}},$

where

$\mathcal{M}_{P^{\nu}}f(x)=\sup\limits_{t>0}\Big{|}\big{(}P^{\nu}_{t}\otimes
f\big{)}(x)\Big{|}.$

We prove one of the fundamental theorem for Hardy–Morrey spaces. It is an
extension for the corresponding theorem for Hardy spaces [ See [6], Theorem
5].

######  Theorem 3.1.

Let $0<q\leq 1$ and $q\leq p<\infty$. Then the following statements are valid:

  1. (i)

There exists a Schwartz function $\varphi$ with
$\int\limits_{\mathbb{R}^{n}_{+}}\varphi(x)x_{n}^{\nu}dx=1$ and a constant
$C_{1}$ such that

$\displaystyle\|\mathcal{M}_{\varphi,\nu}^{0}f\|_{M^{p}_{q,\nu}}\leq
C_{1}\|f\|_{HM^{p}_{q,{\Delta_{\nu}}}}$ (3.1)

for all $f\in\mathcal{S}^{\prime}_{+}(\mathbb{R}^{n}_{+})$.

  2. (ii)

For every $\varphi\in\mathcal{S}_{+}(\mathbb{R}^{n}_{+})$ there exists a
constant $C_{2}$ such that

$\displaystyle\|\mathcal{M}_{\varphi,\nu}^{*}f\|_{M^{p}_{q,\nu}}\leq
C_{2}\|\mathcal{M}_{\varphi,\nu}^{0}f\|_{M^{p}_{q,\nu}}$ (3.2)

for all $f\in\mathcal{S}^{\prime}_{+}(\mathbb{R}^{n}_{+})$.

  3. (iii)

For every $\varphi\in\mathcal{S}_{+}(\mathbb{R}^{n}_{+})$ and
$\lambda>\frac{n+\nu}{p}$ there exists a constant $C_{3}$ such that

$\displaystyle\|\mathcal{M}_{\varphi,\nu}^{\lambda}f\|_{M^{p}_{q,\nu}}\leq
C_{3}\|\mathcal{M}_{\varphi,\nu}^{*}f\|_{M^{p}_{q,\nu}}$ (3.3)

for all $f\in\mathcal{S}^{\prime}_{+}(\mathbb{R}^{n}_{+})$.

  4. (iv)

For every $\lambda\in\mathbb{Z}^{+}$ and
$\varphi\in\mathcal{S}_{+}(\mathbb{R}^{n}_{+})$ with
$\int\limits_{\mathbb{R}^{n}_{+}}\varphi(x)x_{n}^{\nu}dx=1$ there exists a
constant $C_{4}$ such that if $N\geq\lambda+1$ we have

$\displaystyle\|\mathcal{M}_{{N,\nu}}f\|_{M^{p}_{q,\nu}}\leq
C_{4}\|\mathcal{M}_{\varphi,\nu}^{\lambda}f\|_{M^{p}_{q,\nu}}$ (3.4)

for all $f\in\mathcal{S}^{\prime}_{+}(\mathbb{R}^{n}_{+})$.

  5. (v)

For every positive integer $N$ there exists a constant $C_{5}$ such that every
tempered distribution $f$ with
$\|\mathcal{M}_{{N,\nu}}f\|_{M^{p}_{q,\nu}}<\infty$ is a bounded distribution
and satisfies

$\displaystyle\|f\|_{HM^{p}_{q,{\Delta_{\nu}}}}\leq
C_{5}\|\mathcal{M}_{{N,\nu}}f\|_{M^{p}_{q,\nu}}$ (3.5)

that is, lies in the Hardy-Morrey space
$HM^{p}_{q,{\Delta_{\nu}}}(\mathbb{R}^{n}_{+})$.

We deduce that for $f\in HM^{p}_{q,{\Delta_{\nu}}}(\mathbb{R}^{n}_{+})$, the
inequality (3.5) can be reversed, and therefore for any Schwartz function
$\varphi$ with $\int\limits_{\mathbb{R}^{n}_{+}}\varphi(x)x_{n}^{\nu}dx=1$, we
have

$\displaystyle\|\mathcal{M}_{\varphi,\nu}^{*}f\|_{M^{p}_{q,\nu}}\leq
C_{6}\|f\|_{HM^{p}_{q,{\Delta_{\nu}}}}.$

We remark the pointwise following inequalities for all
$x\in\mathbb{R}^{n}_{+}$

$\displaystyle\mathcal{M}_{{N,\nu}}f(x)\leq\mathcal{M}_{\nu}f(x)\leq\mathcal{M}_{\varphi,\nu}^{0}f\leq
C\mathcal{M}_{\varphi,\nu}^{*}f(x)\leq
C\mathcal{M}_{{\varphi,\nu}}^{\lambda}f(x).$

It is a fact that all the maximal functions have comparable quasinorms for all
$0<q\leq 1$ and $q\leq p<\infty$. This presents a variety of characterizations
for Hardy-Morrey spaces.

###### Proof.

The proof of the theorem is similar to [6, Theorem 5]. For a complete and
detail account of the proof, the reader is referred to [6, Theorem 5].

Part (i). By the proof of [6, Theorem 5(i)], we have the pointwise
inequalities

$\displaystyle\mathcal{M}_{\varphi,\nu}^{0}f(x)\leq
C_{1}\mathcal{M}_{P^{\nu}}f(x),$ (3.6)

$\forall x\in\mathbb{R}^{n}_{+}$. Thus, (3.1) follows from (3.6).

Part (ii). Similar to the proof [6, Theorem 5(ii)], we present the proof for
$\mathcal{M}_{\varphi,\nu}^{*}f$ only as the general case follows similarly.
To this end, let us now set

$\displaystyle\mathcal{M}_{\varphi,\nu,\varepsilon,N}^{*}f=\sup_{\\{y\in\mathbb{R}^{n}_{+}:|x-y|<t<1/\varepsilon\\}}\left|\left(f\otimes\varphi_{t}\right)(y)\right|\left(\dfrac{t}{t+\varepsilon}\right)^{N}\left(1+\varepsilon\left|y\right|\right)^{-N}.$

where $0<\varepsilon<1$. Firstly, we deduce that if
$\mathcal{M}_{\varphi,\nu}^{0}f\in M^{p}_{q,\nu}$, then
$\mathcal{M}_{\varphi,\nu}^{*}f\in M^{p}_{q,\nu}\cap
L^{\infty}_{\nu}(\mathbb{R}^{n}_{+})$. From [6, p. 14], we obtain

$\displaystyle\mathcal{M}_{\varphi,\nu,\varepsilon,N}^{*}$
$\displaystyle\leq\frac{C}{\left(1+\varepsilon\left|x\right|\right)^{N-\ell}}$
$\displaystyle\leq
C_{\epsilon}\bigg{(}\chi_{E(0,1)}(x)+\sum\limits_{k=1}^{\infty}2^{-k(N-\ell)}\chi_{E(0,2^{k})\setminus
E(0,2^{k-1})}(x)\bigg{)}.$ (3.7)

Since the characteristic function is in $M^{p}_{q,\nu}$, we may write

$\displaystyle||\chi_{E(0,2^{k})\setminus E(0,2^{k-1})}||_{M^{p}_{q,\nu}}\leq
C\max(||\chi_{E(0,2^{k+1})}||_{L^{p}_{\nu}(\mathbb{R}^{n}_{+})},||\chi_{E(0,2^{k})}||_{L^{p}_{\nu}(\mathbb{R}^{n}_{+})}).$
(3.8)

By the inequalities (3) and (3.8), we further conclude that, for any $N$ is
large enough, $\mathcal{M}_{\varphi,\nu}^{*}(f)^{\varepsilon,N}\in
M^{p}_{q,\nu}\cap L^{\infty}_{\nu}(\mathbb{R}^{n}_{+})$.

Next, we recall two functions used in [6, p. 14-15]. For any
$M>\frac{n+\nu}{p}$, we introduce functions

$\displaystyle
V_{\varepsilon,N,M,\nu}(x)=\sup_{y\in\mathbb{R}^{n}_{+},t<1/\varepsilon}\left|\left(f\otimes\varphi_{t}\right)(y)\right|\left(\dfrac{t}{t+\varepsilon}\right)^{N}\left(1+\varepsilon\left|y\right|\right)^{-N}\left(\dfrac{t}{\left|x-y\right|+t}\right)^{M}$

and

$\displaystyle
U_{\varepsilon,N,\nu}(x)=\sup_{\\{y\in\mathbb{R}^{n}_{+}:|x-y|<t<1/\varepsilon\\}}t\left|\nabla_{\nu}\left(f\otimes\varphi_{t}\right)(y)\right|\left(\dfrac{t}{t+\varepsilon}\right)^{N}\left(1+\varepsilon\left|y\right|\right)^{-N}$

where $\nabla_{\nu}=\big{(}\partial_{1},\ldots,\partial_{n-1},B_{n}\big{)}$.
From [6, p. 16], it is easy to see that

$\displaystyle\left|\left(f\otimes\varphi_{t}\right)(y)\right|\left(\dfrac{t}{t+\varepsilon}\right)^{N}\left(1+\varepsilon\left|y\right|\right)^{-N}\leq\left(\dfrac{t}{\left|x-y\right|+t}\right)^{M}\\{\mathcal{M}_{\nu}[\mathcal{M}_{\varphi,\nu,\varepsilon,N}^{*}f]^{s}\\}^{1/s}(x)$

where $0<s<\infty$. By the properties of the $\triangle_{\nu}$-maximal
function (see [11, 12]), we also have

$\displaystyle||V_{\varepsilon,N,M,\nu}||_{M^{p}_{q,\nu}}$ $\displaystyle\leq
C||\mathcal{M}_{\nu}[\mathcal{M}_{\varphi,\nu,\varepsilon,N}^{*}f]^{s}||^{1/s}_{M^{p}_{q/s,\nu}}$
$\displaystyle\leq
C||[\mathcal{M}_{\varphi,\nu,\varepsilon,N}^{*}f]^{s}||^{1/s}_{M^{p}_{q/s,\nu}}\leq
C||\mathcal{M}_{\varphi,\nu,\varepsilon,N}^{*}f||_{M^{p}_{q,\nu}}$ (3.9)

for some $C>0$ independent of $f$. Moreover, according to [6, (15)], we obtain
the pointwise inequality

$\displaystyle U_{\varepsilon,N,\nu}(x)\leq CV_{\varepsilon,N,M,\nu}(x).$
(3.10)

Hence (3) and (3.10) yield

$\displaystyle||U_{\varepsilon,N,\nu}||_{M^{p}_{q,\nu}}\leq
C_{0}||\mathcal{M}_{\varphi,\nu,\varepsilon,N}^{*}f||_{M^{p}_{q,\nu}}$ (3.11)

where $C_{0}$ is a constant to be determined in [6, p. 16]. Now, if we set
$E_{\varepsilon}=\\{x:U_{\varepsilon,N,\nu}(x)\leq
2C_{0}\mathcal{M}_{\varphi,\nu,\varepsilon,N}^{*}f(x)\\}$, where $C_{0}$ is a
constant to be determined in (3.11), we may write

$\displaystyle||\chi_{(E_{\varepsilon})^{c}}\mathcal{M}_{\varphi,\nu,\varepsilon,N}^{*}f||_{M^{p}_{q,\nu}}\leq\frac{1}{2C_{0}}||\chi_{(E_{\varepsilon})^{c}}U_{\varepsilon,N,\nu}||_{M^{p}_{q,\nu}}\leq\frac{1}{2C_{0}}||U_{\varepsilon,N,\nu}||_{M^{p}_{q,\nu}}.$

It follows from (3.11),

$\displaystyle||\chi_{(E_{\varepsilon})^{c}}\mathcal{M}_{\varphi,\nu,\varepsilon,N}^{*}f||_{M^{p}_{q,\nu}}\leq\frac{1}{2}||\mathcal{M}_{\varphi,\nu,\varepsilon,N}^{*}f||_{M^{p}_{q,\nu}}.$
(3.12)

Moreover, by [6, p. 16], we obtain

$\displaystyle\mathcal{M}_{\varphi,\nu,\varepsilon,N}^{*}f\leq
C\\{\mathcal{M}_{\nu}[\mathcal{M}_{\varphi,\nu}^{0}f]^{s}(x)\\}^{1/s},~{}\text{if}~{}x\in
E_{\varepsilon}$

where $0<s<\infty$.

Similar to the proof of (3), we have

$\displaystyle||\chi_{(E_{\varepsilon})}\mathcal{M}_{\varphi,\nu,\varepsilon,N}^{*}f||_{M^{p}_{q,\nu}}\leq
C||\mathcal{M}_{\varphi,\nu}^{0}f||_{M^{p}_{q,\nu}}.$ (3.13)

The inequalities (3.12) and (3.13) yield the following estimate

$\displaystyle||\mathcal{M}_{\varphi,\nu,\varepsilon,N}^{*}f||_{M^{p}_{q,\nu}}\leq
C||\mathcal{M}_{\varphi,\nu}^{0}f||_{M^{p}_{q,\nu}}+\frac{1}{2}||\mathcal{M}_{\varphi,\nu,\varepsilon,N}^{*}f||_{M^{p}_{q,\nu}}.$

Since $\mathcal{M}_{\varphi,\nu,\varepsilon,N}^{*}f\in M^{p}_{q,\nu}$, by
applying Lebesgue monotone theorem, we obtain

$\displaystyle||\mathcal{M}_{\varphi,\nu}^{*}f||_{M^{p}_{q,\nu}}\leq
C||\mathcal{M}_{\varphi,\nu}^{0}f||_{M^{p}_{q,\nu}}$

for some $C>0$ depending on $N$. This assure that

$\displaystyle||\mathcal{M}_{\varphi,\nu}^{0}f||_{M^{p}_{q,\nu}}<\infty\Rightarrow||\mathcal{M}_{\varphi,\nu}^{*}f||_{M^{p}_{q,\nu}}<\infty.$

With this assertion, (3.2) can be established by repeating the above arguments
with $U_{\varepsilon,N,\nu}$ and $V_{\varepsilon,N,M,\nu}$ replaced by two new
auxiliary functions

$\displaystyle
U_{\varepsilon,N,\nu}(x)=\sup_{\\{y\in\mathbb{R}^{n}_{+}:|x-y|<t<1/\varepsilon\\}}t\left|\nabla_{\nu}\left(f\otimes\varphi_{t}\right)(y)\right|\left(\dfrac{t}{t+\varepsilon}\right)^{N}\left(1+\varepsilon\left|y\right|\right)^{-N}$

and

$\displaystyle
V_{\varepsilon,N,M,\nu}(x)=\sup_{y\in\mathbb{R}^{n}_{+},t<1/\varepsilon}\left|\left(f\otimes\varphi_{t}\right)(y)\right|\left(\dfrac{t}{t+\varepsilon}\right)^{N}\left(1+\varepsilon\left|y\right|\right)^{-N}\left(\dfrac{t}{\left|x-y\right|+t}\right)^{M}.$

For a detail account the reader is referred to [6, pp. 14-17].

Part (iii). Again return to the proof [6, Theorem 6(iii)], we have the
following inequality

$\displaystyle\\{\mathcal{M}_{\varphi,\nu}^{\lambda}f(x)\\}^{q}\leq
C_{3}\mathcal{M}_{\nu}[\mathcal{M}_{\varphi,\nu}^{*}f]^{q}(x).$

Then, by the above inequality with $q<p$ and the boundedness properties of
$\triangle_{\nu}$-maximal functions in $M^{p}_{q,\nu}$ Morrey spaces give
(3.3) whenever $\lambda>\frac{n+\nu}{p}$.

Part (iv). The proof of [6, Theorem 6(iv)] asserts that

$\displaystyle\mathcal{M}_{{N,\nu}}f\leq
C_{4}\mathcal{M}_{\varphi,\nu}^{\lambda}f.$

Clearly, (3.4) follows.

Part (v). Let $f\in\mathcal{S}^{\prime}(\mathbb{R}^{n}_{+})$ satisfy
$||\mathcal{M}_{{N,\nu}}f||_{M^{p}_{q,\nu}}<\infty$ for some $N\in\mathbb{N}$.
For any $\theta\in\mathcal{S}(\mathbb{R}^{n}_{+})$, we obtain a fixed constant
$c_{0}>0$ such that $c_{0}\theta\in\Im_{N,\nu}$. Hence,
$\mathcal{M}_{c_{0}\theta,\nu}^{*}f\leq\mathcal{M}_{{N,\nu}}f$.

By the proof of [6, Theorem 6(v)], we obtain the following stronger estimate

$\displaystyle\mathcal{M}_{P^{\nu}}f(x)=\sup\limits_{t>0}|(P^{\nu}_{t}\otimes
f)(x)|\leq C_{5}\mathcal{M}_{{N,\nu}}f(x),$

hence, (3.5) follows.

Finally, it follows that the quasinorm
$\|\mathcal{M}_{P^{\nu}}f(x)\|_{M^{p}_{q,\nu}}$ is also equivalent to
$\|f\|_{HM^{p}_{q,{\Delta_{\nu}}}}$. ∎

######  Corollary 3.1.

The Hardy-Morrey space $HM^{p}_{q,{\Delta_{\nu}}}$ is complete in the metric
$d(f,g)=||f-g||_{HM^{p}_{q,{\Delta_{\nu}}}}^{q}$ for $q\leq 1$.

We note that if $f\in HM^{p}_{q,{\Delta_{\nu}}}$ and
$\varphi\in\mathcal{S}(\mathbb{R}^{n}_{+})$, then

$\displaystyle|f\otimes\varphi(x)|^{q}\leq\frac{C}{|B(x,1)|_{\nu}}\int_{B(x,1)}\mathcal{M}_{{N,\nu}}^{*}f(y)y_{n}^{\nu}dy.$
(3.14)

Hence if $f_{n}\to f$ in $f\in HM^{p}_{q,{\Delta_{\nu}}}$, then $f_{n}\to f$
in $\varphi\in\mathcal{S}^{\prime}$. So, the embedding is valid.

##  4 Atomic decomposition

The main result of this paper is presented in this section. The atomic
decomposition for the Hardy-Morrey spaces is established. We next define a
$(p,q,s)$-atom associated to the operator $\triangle_{\nu}$.

######  Definition 4.4.

Let $0<q\leq 1$, $q\leq p<\infty$ and $s\in\mathbb{N}\cup\\{0\\}$. A
$(p,q,s)$-atom $a(x)$ is a function which satisfies the following properties:

  1. (i)

$a$ be supported on a cube $3Q$, namely, $\text{supp}~{}~{}a\subset 3Q$,

  2. (ii)

$\|a(x)\|_{L^{\infty}_{\nu}}\leq|Q|_{\nu}^{-\frac{1}{p}}$,

  3. (iii)

$\int_{\mathbb{R}^{n}_{+}}a(x)x^{\alpha}x_{n}^{\nu}dx=0$ for all
$s\geq[(n+k+\nu)\big{(}\frac{1}{q}-1\big{)}]$ with $|\alpha|\leq s,$.

Here $3Q$ is the cube concentric with $Q$ of side-length $3\ell(Q)$.

The so called atomic decomposition theorem in $HM^{p}_{q,{\Delta_{\nu}}}$
space is as follows:

######  Theorem 4.2.

Let $0<q\leq 1$, $q\leq p<\infty$. Then for any $f\in
HM^{p}_{q,{\Delta_{\nu}}}$, there exist a sequence of scalars $\lambda_{j}$, a
sequence of $(p,q,s)$-atoms $a_{j}=\\{a_{j}\\}$ for
$HM^{p}_{q,{\Delta_{\nu}}}$ such that

$\displaystyle f=\sum\limits_{j=1}^{\infty}\lambda_{j}a_{j}$ (4.1)

with

$\displaystyle||\lambda||_{p,q,\nu}=\bigg{\\{}\sup\limits_{Q}\bigg{(}\frac{1}{|Q|_{\nu}}\bigg{)}^{1-q/p}\sum\limits_{j=1}^{\infty}|Q|^{1-q/p}_{\nu}|\lambda_{j}|^{q}\bigg{\\}}^{1/q}<\infty.$

The sum converges in $\mathcal{S}^{{}^{\prime}}_{+}\setminus\mathcal{P}$ and
$f\in HM^{p}_{q,{\Delta_{\nu}}}$ with $||f||_{HM^{p}_{q,{\Delta_{\nu}}}}\leq
C||\lambda||_{p,q,\nu}$ for some $C=C(n,p,q,\nu)$. Conversely, every function
$f\in HM^{p}_{q,{\Delta_{\nu}}}$ has the atomic decomposition (4.1) in
$\mathcal{S}^{{}^{\prime}}_{+}\setminus\mathcal{P}$, here $a_{j}$’s are
$(p,q,s)$-atoms and $\lambda$ satisfies $||\lambda||_{p,q,\nu}\leq
C||f||_{HM^{p}_{q,{\Delta_{\nu}}}}$ for some $C=C(n,p,q,\nu)$.

Before proving the decomposition theorem, we state a well known preliminary
supporting result for the atomic decomposition.

######  Proposition 1.

Suppose that $f\in HM^{p}_{q,{\Delta_{\nu}}}$ with $0<q\leq 1$, $q\leq
p<\infty$ and $\delta>0$. Then for any
$f\in\mathcal{S}_{+}(\mathbb{R}^{n}_{+})$, there exist
$g\in\mathcal{S}_{+}^{\prime}(\mathbb{R}^{n}_{+})$, a collection of cubes
$\\{Q\\}_{k\in\mathbb{N}}$ and a family of smooth functions with compact
supports $\\{\xi_{k}\\}$ such that

  * (i)

$f=g+b$ where $b=\sum\limits_{k\in\mathbb{N}}b_{k}$,

  * (ii)

The cubes $\\{Q_{k}\\}_{k\in\mathbb{N}}$ which are mutually disjoint, satisfy

$\displaystyle\bigcup\limits_{k\in\mathbb{N}}Q_{k}=\\{x\in\mathbb{R}^{n}_{+}:(\mathcal{M}_{N,\nu}f)(x)>\delta\\},$

  * (iii)

$\text{supp}~{}\xi_{k}\subset Q_{k}$, $0\leq\xi_{k}\leq 1$ and
$\sum\limits_{k\in\mathbb{N}}\xi_{k}=\chi_{\\{x\in\mathbb{R}^{n}_{+}:(\mathcal{M}_{N,\nu}f)(x)>\delta\\}}$

  * (iv)

The function $g$ satisfies

$\displaystyle(\mathcal{M}_{N,\nu}g)(x)$
$\displaystyle\leq(\mathcal{M}_{N,\nu}f)(x)\chi_{\\{x\in\mathbb{R}^{n}_{+}:(\mathcal{M}_{N,\nu}f)(x)\leq\delta\\}}(x)$
$\displaystyle+\delta\sum\limits_{k\in\mathbb{N}}\dfrac{\ell(Q_{k})^{n+\nu+s+1}}{(\ell(Q_{k})+|x-x_{k}|)^{n+\nu+s+1}},$

  * (v)

The function $b_{k}$ is given by $b_{k}=(f-c_{k})\xi_{k}$ where
$c_{k}\in\mathcal{P}_{s}$ satisfying

$\displaystyle\int_{\mathbb{R}^{n}_{+}}b_{k}(x)m(x)x_{n}^{\nu}dx=0,~{}~{}~{}\forall
m\in\mathcal{P}_{s},$

and

$\displaystyle(\mathcal{M}_{N,\nu}b_{k})(x)\leq
C(\mathcal{M}_{N,\nu}f)(x)\chi_{Q_{k}}(x)+\delta\dfrac{\ell(Q_{k})^{n+\nu+s+1}}{|x-x_{k}|^{n+\nu+s+1}}\chi_{\mathbb{R}^{n}_{+}\setminus
Q_{k}}(x)$

for some $C>0$.

We are now ready to prove the main theorem.

Proof of Our Main Theorem. In the rest of this section, for simplicity, we use
the notation $\mathcal{M}f=\mathcal{M}_{N,\nu}f$.

It suffices to show the atomic decomposition for $(p,\infty,s)$ atoms since
$(p,\infty,s)$ atoms are $(p,q,s)$ atoms for any $1\leq q<\infty$. According
to the Proposition 1, for any $\delta=2^{j},j\in\mathbb{Z}$, we have
distributions $g^{j},b^{j}$ satisfying the properties (i)-(v) in the
Proposition 1 and $f=g^{j}+b^{j}$. We can write

$\displaystyle
O^{j}=\\{x\in\mathbb{R}^{n}_{+}:(\mathcal{M}f)(x)>2^{j}\\}=\bigcup\limits_{k\in\mathbb{N}}Q^{j}_{k}$
(4.2)

where $\bigcup\limits_{k\in\mathbb{N}}Q^{j}_{k}$ is the decomposition of
$O^{j}$ given in the Proposition below.

Let $\\{\xi_{k}^{j}\\}$ be the family of smooth functions with respect to
decomposition $O^{j}$ given by Proposition 1, property (iii). In addition, as
$\mathcal{M}f\in M^{p}_{q}$, we obtain $O^{j+1}\subset O^{j}$ and for all
$j\in\mathbb{N}$, $O^{j}$ are mutually disjoint. For any
$\varphi\in\mathcal{S}(\mathbb{R}^{n})$, we have a constant $c>0$ such that
$c\varphi\in\mathfrak{F}_{N,\nu}$. Proposition 1 gives

$\displaystyle c|\varphi\ast g^{j}(x)|$
$\displaystyle\leq(\mathcal{M}g^{j})(x)$
$\displaystyle\leq(\mathcal{M}f)(x)\chi_{x\in\mathbb{R}^{n}_{+}:(\mathcal{M}f)(x)\leq
2^{j}}(x)+2^{j}\sum\limits_{k\in\mathbb{N}}\dfrac{\ell(Q_{k})^{n+\nu+s+1}}{(\ell(Q_{k})+|x-x_{k}^{j}|)^{n+\nu+s+1}}\leq
C2^{j}$

for some $C>0$ where $x_{k}^{j}$ is the center of $Q_{k}^{j}$. That is
$g^{j}\to 0$ in $\mathcal{S}^{\prime}(\mathbb{R}^{n}_{+})$ as $j\to-\infty$.

Observe next that $b^{j}\to 0$ in $\mathcal{S}^{\prime}(\mathbb{R}^{n}_{+})$
as $j\to\infty$. By the item (v) of Proposition 1, similar to the above proof
works and (4.2), for any $n/(n+\nu+s+1)<q$, we obtain

$\displaystyle\int_{Q}|(\mathcal{M}b^{j})(x)|^{q}x_{n}^{\nu}dx\leq\int_{O^{j}}|(\mathcal{M}f)(x)|^{q}(\mathcal{M}_{\nu}\chi_{Q})(x)x_{n}^{\nu}dx$
(4.3)

for some $C>0$.

In fact, for any $\varphi\in\mathcal{S}(\mathbb{R}^{n}_{+})$, using the
inequalities (3.14) and (4.3), we find that

$\displaystyle|b^{j}\otimes\varphi(x)|^{q}$ $\displaystyle\leq
C\frac{1}{|Q(x,1)|_{\nu}}\int_{Q(x,1)}|\mathcal{M}_{\varphi,\nu}^{*}(b^{j})(y)|^{q}y_{n}^{\nu}dy\leq
C\int_{Q(x,1)}|(\mathcal{M}b^{j})(y)|^{q}y_{n}^{\nu}dy$ $\displaystyle\leq
C\int_{O^{j}}|(\mathcal{M}f)(y)|^{q}(\mathcal{M}_{\nu}\chi_{Q(x,1)})(y)y_{n}^{\nu}dy$
$\displaystyle
C\int_{O^{j}}|(\mathcal{M}f)(y)|^{q}(1+|x-y|)^{-n-\nu}y_{n}^{\nu}dy.$

By using the Hölder inequality for the pair $L_{p/q,\nu}(\mathbb{R}^{n}_{+})$
and $L_{(p/q)^{\prime},\nu}(\mathbb{R}^{n}_{+})$, we obtain

$\displaystyle\int_{\mathbb{R}^{n}_{+}}|(\mathcal{M}f)(y)|^{q}(1+|x-y|)^{-n-\nu}y_{n}^{\nu}dy$
$\displaystyle\leq
C\sum\limits_{k=0}^{\infty}2^{-k(n+\nu)}\int_{\mathbb{R}^{n}_{+}}|(\mathcal{M}f)(y)|^{q}\chi_{B_{k}}(y)y_{n}^{\nu}dy$
$\displaystyle\leq
C\sum\limits_{k=0}^{\infty}\frac{1}{|B(x,2^{k})|_{\nu}}||(\mathcal{M}f)(y)^{q}||_{L_{\nu}^{p/q}(\mathbb{R}^{n}_{+})}||\chi_{B(x,2^{k})}||_{L_{\nu}^{(p/q)^{\prime}}(\mathbb{R}^{n}_{+})}$
$\displaystyle\leq
C\sum\limits_{k=0}^{\infty}\dfrac{||\mathcal{M}f||^{q}_{M_{q,\nu}^{p}}}{||\chi_{B(x,2^{k})}||_{L_{\nu}^{p/q}(\mathbb{R}^{n}_{+})}}\leq
C||\mathcal{M}f||^{q}_{M_{q,\nu}^{p}},$

where $B_{k}=B(x,2^{k})\setminus B(x,2^{k-1})$ when $k\geq 1$ and
$B_{0}=B(x,1)$.

In view of the fact that $O^{j}\downarrow\emptyset$, the dominated convergence
theorem gives that

$\displaystyle\lim\limits_{j\to\infty}|b^{j}\otimes\varphi(x)|^{q}\leq
C\lim\limits_{j\to\infty}\int_{O^{j}}|(\mathcal{M}f)(y)|^{q}(1+|x-y|)^{-n-\nu}y_{n}^{\nu}dy=0.$

Thus, $b^{j}\otimes\varphi\to 0$ pointwisely. That is, $b^{j}\to 0$ in
$\mathcal{S}^{\prime}(\mathbb{R}^{n}_{+})$ when $j\to\infty$.

Now we return to the atomic decomposition. The telescoping sum
$f=\sum\limits_{k\in\mathbb{Z}}(g^{j+1}-g^{j})$ converges in the sense of
distributions. By using the property $(v)$ of Proposition 1, we obtain

$\displaystyle
g^{j+1}-g^{j}=b^{j+1}-b^{j}=\sum\limits_{k\in\mathbb{N}}((f-c_{k}^{j+1})\xi_{k}^{j+1}-(f-c_{k}^{j})\xi_{k}^{j})$

where $c_{k}^{j}\in\mathcal{P}_{s}$. Also, $\forall m\in\mathcal{P}_{s}$

$\displaystyle\int_{\mathbb{R}^{n}_{+}}(f(x)-c_{k}^{j}(x))m(x)\xi_{k}^{j}(x)x_{n}^{\nu}dx=0$

holds. By the fact $\sum_{k\in\mathbb{N}}\xi_{k}^{j}=1$ on the support
$\xi_{k}^{j+1}$, we obtain

$\displaystyle f=\sum_{j,k}A^{j}_{k}=\sum_{j}(g^{j+1}-g^{j}),$ (4.4)

where

$\displaystyle
A^{j}_{k}=(f-c_{k}^{j})\xi_{k}^{j}-\sum\limits_{l\in\mathbb{N}}(f-c_{l}^{j+1})\xi_{l}^{j+1}\xi_{k}^{j}+\sum\limits_{l\in\mathbb{N}}c_{k,l}\xi_{l}^{j+1},~{}~{}~{}~{}c_{k,l}\in\mathcal{P}_{s}$

and

$\displaystyle
a_{k}^{j}=\lambda^{-1}_{j,k}A^{j}_{k}~{}~{}~{}~{}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}\lambda_{j,k}=C2^{j}||\chi_{Q^{j}_{k}}||_{L^{p}_{\nu}(\mathbb{R}^{n}_{+})}.$

The function $a_{k}^{j}$’s are obviously $(p,\infty,s)$ atoms.

According to the definition of $Q_{k}^{j}$ and following from the fact that
the family $\\{Q^{j}_{k}\\}_{k\in\mathbb{N}}$ has the finite intersection
property, we compute that for any $0<q<\infty$

$\displaystyle\sum\limits_{k\in\mathbb{N}}\bigg{(}\frac{|\lambda_{j,k}|}{||\chi_{Q^{j}_{k}}||_{L^{p}_{\nu}(\mathbb{R}^{n}_{+})}}\bigg{)}^{q}\chi_{Q^{j}_{k}}(x)\leq
C2^{qj}\chi_{O^{j}}(x)$

for some $C>0$. Consequently,

$\displaystyle\sum\limits_{j,k}\bigg{(}\frac{|\lambda_{j,k}|}{||\chi_{Q^{j}_{k}}||_{L^{p}_{\nu}(\mathbb{R}^{n}_{+})}}\bigg{)}^{s}\chi_{Q^{j}_{k}}(x)\leq
C\sum\limits_{j\in\mathbb{Z}}2^{qj}\chi_{O^{j}}(x)\leq
C(\mathcal{M}f)(x)^{q}.$

By applying the quasi-norm $||.||^{1/q}_{M_{p/q,\nu}}$ on both sides of the
above inequality,

$\displaystyle\bigg{|}\bigg{|}\sum\limits_{j,k}\bigg{(}\frac{|\lambda_{j,k}|}{||\chi_{Q^{j}_{k}}||_{L^{p}_{\nu}(\mathbb{R}^{n}_{+})}}\bigg{)}^{q}\chi_{Q^{j}_{k}}\bigg{|}\bigg{|}^{1/q}_{M_{p/q,\nu}}\leq
C||f||_{HM^{p}_{q,{\Delta_{\nu}}}},~{}~{}~{}0<q<\infty$

for some $C>0$ independent of $f$.

It follows from that

$\displaystyle||\lambda||_{p,q,\nu}\leq C||f||_{HM^{p}_{q,{\Delta_{\nu}}}}.$

We can replace (4.4) by $f=\sum\limits_{j=1}^{\infty}\lambda_{j}a_{j}.$ Thus,
the proof of our main theorem is completed.

### Acknowledgement

The author would like to thank the reviewers for valuable suggestions and
corrections.

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