\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 187, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/187\hfil Besov-Morrey spaces]
{Besov-Morrey spaces associated with Hermite operators and applications
to fractional Hermite equations}

\author[N. A. Dao, N. N. Trong, L. X. Truong \hfil EJDE-2018/187\hfilneg]
{Nguyen Anh Dao, Nguyen Ngoc Trong, Le Xuan Truong}

\address{Nguyen Anh Dao \newline
Applied Analysis Research Group,
Faculty of Mathematics and Statistics,
Ton Duc Thang University,
HoChiMinh City, Vietnam}
\email{daonguyenanh@tdtu.edu.vn}

\address{Nguyen Ngoc Trong (corresponding author)\newline
Faculty of Mathematics and Computer Science,
VUNHCM - University of Science,
HoChiMinh city, Vietnam. \newline
Department of Primary Education,
HoChiMinh City University of Education, Vietnam}
\email{trongnn37@gmail.com}

\address{Le Xuan Truong \newline
Department of Mathematics and Statistics,
University of Economics HoChiMinh City, Vietnam}
\email{lxuantruong@gmail.com}

\dedicatory{Communicated by Jesus Ildefonso Diaz}

\thanks{Submitted September 26, 2018. Published November 20, 2018.}
\subjclass[2010]{42B35, 42B20}
\keywords{Fractional Hermite equations; Hermite-Besov-Morrey space; 
\hfill\break\indent molecular decomposition}

\begin{abstract}
 The purpose of this article is to establish the molecular decomposition
 of the homogeneous Besov-Morrey spaces associated with the Hermite operator
 $\mathbb{H} = -\Delta+|x|^2$ on the Euclidean space $\mathbb{R}^n$.
 Particularly, we obtain some estimates for the operator $\mathbb{H}$ on the
 Hermite-Besov-Morrey spaces and the regularity results to the fractional
 Hermite equations
 \[
 (-\Delta +|x|^2 )^su=f,
 \]
 and
 \[
 (-\Delta +|x|^2 +I)^su=f.
 \]
 Our results generalize some results by Anh and Thinh \cite{Anh1}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

In this article, we study the Besov-Morrey spaces associated with the
Hermite operator $\mathbb{H}=-\Delta + |x|^2$ on
$\mathbb{R}^n$, $n \geq 1$. It is known that the classical theory of the
Besov and Triebel-Lizorkin spaces plays a crucial role not only in the
theory of function spaces, but also in the theory of partial differential
equations and harmonic analysis, see e.g.\
\cite{f, Kozono, Cheng, Mazzucato1, Mazzucato2, Sawano, Sawano3},
and the references therein.

Recently, the theory of the Besov and Triebel-Lizorkin spaces associated with
the operators has been developed by many authors when one observed that the
classical Besov and Triebel-Lizorkin spaces are not always the most suitable
to investigate a number of operators, see
\cite{Anh1,Anh2,Anh3,Anh4, Qui2, Cheng, Mazzucato1, wang}, and their references.
For example, Petrusev and Xu \cite{P1} studied the characterization of the
inhomogeneous Besov and Triebel-Lizorkin spaces in terms of Littlewood-Paley
decomposition in the context of Hermite expansions that the frame elements
have almost exponential localization. Note that these frame elements
can be viewed as an analogue of the $\varphi$-transform of Frazier and Jawerth
 \cite{f}.
Another approach introduced by Anh and Thinh \cite{Anh1}
is of defining the Besov and Triebel-Lizorkin spaces in terms of the heat
kernels via square functions. Their approach adapted to the study of the
theory of both homogeneous and inhomogeneous Besov and
Triebel-Lizorkin spaces. This allows them to extend the range of indices
 $1\leq p, q\leq \infty$ of the homogeneous Besov space
$\mathrm{BM}_{p,q}^{\alpha,\mathbb{H}}$ (resp. Triebel-Lizorkin spaces
$\mathrm{FM}_{p,q}^{\alpha,\mathbb{H}}$) to $0< p, q\leq \infty$,
compare to the results in \cite{k1}.

One of the most interesting studies of the theory of Besov spaces is the
Besov-Morrey spaces, introduced first by Kozono and Yamazaki \cite{Kozono}
to investigate time-local solutions of the Navier-Stokes equations with the
initial data in the spaces of this type.
As a matter of fact, the Besov-Morrey spaces share several features of
Besov and Morrey spaces. They represent the local oscillations and singularities
of functions more precisely than the classical Besov spaces.
Thus, they behav better in many aspects, particularly under the action of
singular integrals and pseudo-differential operators.
In addition, Mazzucato \cite{Mazzucato1, Mazzucato2} established the wavelet
decompositions to characterize the homogeneous and inhomogeneous
Besov-Morrey spaces. For more results on the Besov-Morrey spaces, we refer
the reader to \cite{Kozono,Cheng,Mazzucato1,Mazzucato2,Sawano,Sawano3, Tang,wang}
and the references therein.

Inspired by the above results, we would like to generalize the theory of
the homogeneous Besov spaces associated with the Hermite operator
$\mathrm{BM}_{p,q}^{\alpha,\mathbb{H}}$ to the one of the homogeneous Besov-Morrey
spaces associated with the Hermite operator $\mathrm{BM}_{p,q,r}^{\alpha,\mathbb{H}}$
in this paper. To study $\mathrm{BM}_{p,q,r}^{\alpha,\mathbb{H}}$, we use the results
in \cite{Anh1}, specifically, the estimates on the heat kernels via the square
functions. Beside, we also establish the molecular decompositions for
$\mathrm{BM}_{p,q,r}^{\alpha,\mathbb{H}}$. As applications, we obtain the regularity
of solutions to the fractional Hermite equations:
\[
\mathbb{H}^s u=f,
\]
and
\[
(\mathbb{H}+I)^su=f.
\]

We organize this paper as follows: Section 2 contains some preliminary results
and definitions of functional spaces. Section 3 is devoted to the study
of the molecular decomposition for the Hermite-Besov-Morrey space.
Finally, we investigate the regularity of solutions
on Hermite-Besov-Morrey spaces to the fractional Hermite equations in Section 4.

Throughout this paper, we always use $C$ and $c$ to denote positive constants
that are independent of the main parameters involved but whose values may
differ from line to line. We write $A \lesssim B$ if there is a universal
constant $C$ such that $A \leq C B$; and $A \sim B$ if $A \lesssim B$ and
$B \lesssim A$. We use the following notation:
$\mathbb{N}=\{0,1,2,\dots\}$, $\mathbb{N}_+=\{1,2,3,\dots\}$,
$\mathbb{Z}^-=\{-1,-2,\dots\}$, $\mathbb{Z}^-_0=\{0,-1,-2,\dots\}$
$a \wedge b =\min\{a,b\}$, $a \vee b =\max \{a,b\}$, and
$\operatorname{int}[a]$ is the integer part of $a$.

\section{Preliminaries}\label{sec:2}
\subsection{Dyadic cube}
The set of all dyadic cubes $\mathcal{D}$ in $\mathbb{R}^n$ is defined by
$$
\mathcal{D}=\big\{ \prod_{j=1}^n [ m_j 2^k, (m_j+1)2^k ):
m_1,m_2,\dots,m_n, k \in \mathbb{Z} \big\}.
$$
For a dyadic cube $Q:=\prod_{j=1}^n \Big[ m_j 2^k, (m_j+1)2^k \Big)$,
for some $m_1, m_2,\dots, m_n, k \in \mathbb{Z}$ we denote by $\ell(Q)$
and $x_Q$ the length and the center of the dyadic cube $Q$.
In this case, $\ell(Q)=2^k$ and $x_Q=\left((m_j+1/2)2^k\right)_{j=1}^n$.
Moreover, for every $\nu \in \mathbb{Z}$, we set
\[
\mathcal{D}_\nu=\{Q \in \mathcal{D}: \ell(Q) = 2^\nu\}.
\]


\subsection{Morrey space}

Let us first recall the definition of the Morrey spaces.

\begin{definition} \rm
 For every $0< p \leq r < \infty$, the Morrey space $\mathrm{M}^r_p$ is defined by
 \[
 \mathrm{M}^r_p \equiv \big\{ f \in L_{\rm loc}^p(\mathbb{R}^n):
\| f \|_{\mathrm{M}^r_p}
= \sup_{x_0 \in \mathbb{R}^n} \sup_{R>0}
R^{n (\frac{1}{r} - \frac{1}{p}) }
\| f \|_{{L^p}( B( x_0,R))} < \infty \big\}.
 \]
\end{definition}

Next, we point out some known results about the Morrey norms.

\begin{proposition}\label{Prop1}
Let $0< p \leq r < \infty$. Then
 \begin{gather}\label{tr1}
 \quad \|f\|_{\mathrm{M}^r_p} \sim \sup_{Q \in \mathcal{D} }
 |Q|^{ \frac{1}{r} -\frac{1}{p} } \|f\|_{L^p(Q)}, \\
\label{bdt}
 \|f^\theta\|_{\mathrm{M}^r_p} = \|f\|^\theta_{\mathrm{M}^{r\theta}_{p\theta}},
\quad \forall \theta>0, \\
\label{bdt1}
 \big\|\Big(\int_a^b|F(\cdot,t)|^q \frac{dt}{t}\Big)^{1/q} \big\|_{\mathrm{M}^r_p}
 \leq \Big(\int_a^b
 \|F(\cdot,t)\|_{\mathrm{M}^r_p}^q \frac{dt}{t}\Big)^{1/q}, \quad \text{for }
 0<q\leq p.
 \end{gather}
\end{proposition}

\begin{proof}
Note that \eqref{tr1} and \eqref{bdt} follow from the definition of the Morrey
spaces. While, \eqref{bdt1} can be obtained by using Minkowski integral
inequality, see also \cite[(2.20)]{k1}.
\end{proof}

For $\theta > 0$, we denote by $\mathbb{M}_\theta$ the Hardy-Littlewood
maximal function
\[
\mathbb{M}_\theta f(x)= \sup_{x\in B}
\Big(\frac{1}{|B|} \int_B |f(y)|^\theta dy \Big)^{1/\theta},\quad x\in \mathbb{R}^n,
\]
where the supremum is taken over all balls $B \subset \mathbb{R}^n$ containing $x$.

Then, we have a version of the Fefferman-Stein vector-valued maximal inequality
for the Morrey spaces, see \cite[Proposition 2.1]{Sawano2}.

\begin{proposition}\label{nnt}
 Let $0< q\leq\infty$, $0<p\leq r<\infty$, and $0<\theta<\min\{p,q\}$.
Then
 \[
 \big\| \Big(\sum_{k\in \mathbb{Z}}|\mathbb{M}_\theta f_k |^q\Big)^{1/q}
\big\|_{\mathrm{M}^r_p} \lesssim \big\| \Big(\sum_{k\in \mathbb{Z}}| f_k |^q\Big)^{1/q}
\big\|_{\mathrm{M}^r_p}.
 \]
\end{proposition}

\begin{remark}\label{rem1} \rm
 As a consequence of Proposition \ref{nnt}, the Hardy-Littlewood maximal operator
 $\mathbb{M}_\theta$ is bounded on $\mathrm{M}^r_p$.
\end{remark}

Next, we put
\[
A_v= \Big(\sup_{J \in \mathcal{D},\ell(J)\geqslant 2^v}
 \Big(\frac{1}{|J|}\Big)^{1-p/r}\sum_{Q \in \mathcal{D}_v,Q \subset J
}|Q|^{1-p/r}|s_Q|^p\Big)^{1/p}.
\]
We borrow a result of Wang \cite[p.779]{wang} involving the
characterization of $A_v$ in the Morrey norms.

\begin{lemma}\label{tr} Let $0< p \leq r < \infty$, and $\nu\in \mathbb{Z}$.
Assume that the sequence $\left\{s_Q : Q \in \mathcal{D}_\nu\right\}$ satisfies
 $$
 \| \sum_{Q\in \mathcal{D}_\nu}|Q|^{-1/r}|s_Q|\chi_Q\|_{\mathrm{M}^r_p}<\infty.
 $$
 Then
 $$
 \| \sum_{Q\in \mathcal{D}_v}|Q|^{-1/r}|s_Q|\chi_Q\|_{\mathrm{M}^r_p} \sim A_\nu.
 $$
\end{lemma}

\subsection{Kernel estimates on Hermite operators}

For any $k \geq 0$ and for $t > 0$, we denote the kernel associated with
$(t\sqrt{\mathbb{H}})^k e^{-t\sqrt{\mathbb{H}}}$ by $p_{t,k}(x, y)$.
We recall here the results of \cite[Lemma 2.1 and Propisition 2.2]{Anh1}.

\begin{proposition}\label{prop2.1}
 For $k \in \mathbb{N}$, there exist $C > 0$ and $\delta > 0$ so that
 \begin{enumerate}
 \item $|p_{t,k}(x,y)| \leq C \frac{t^k}{(t+|x-y|)^{n+k}}$, for $x, y\in \mathbb{R}^n$.

 \item for any $|h|<t$, we have
 \[
 |p_{t,k}(x+h,y)-p_{t,k}(x,y)|
\leq C \Big(\frac{|h|}{t}\Big)^\delta
\frac{t^k}{(t+|x-y|)^{n+k}}, \quad \text{for } x, y\in \mathbb{R}^n.
 \]
 \end{enumerate}
\end{proposition}

\begin{proposition}\label{proposition2.2}
 For every $y \in \mathbb{R}^n$, we have $p_{t,k}(\cdot, y) \in \mathcal{S}$.
\end{proposition}

\subsection{Calder\'on reproducing formulas}
In this part we recall Calder\'on's formula from \cite{Anh1}, that  is useful
for studying the homogeneous Besov-Morrey spaces.

\begin{proposition}\label{proposition2.10}
 Let $m_1, m_2 \in \mathbb{N}^+$ and $f \in \mathcal{S}'$. Then
 \[
 f =-\frac{1}{2^{m-1}(m-1)!}\int_0^{\infty}(t\sqrt{\mathbb{H}})^{m_1}
e^{-t\sqrt{\mathbb{H}}}(t\sqrt{\mathbb{H}})^{m_2}
e^{-t\sqrt{\mathbb{H}}}f \frac{dt}{t}\quad\text{in }\mathcal{S}',
 \]
 where $m=m_1+m_2$, and $\mathcal{S}'$ is the dual space of the
Schwartz functions $\mathcal{S}$ as usual.
\end{proposition}


\section{Besov-Morrey Spaces associated with the Hermite operators}

It is convenient for us to introduce first the homogeneous Besov-Morrey
spaces corresponding to the Hermite operator $\mathbb{H}$.

\begin{definition}\label{def1} \rm
 Let $\alpha \in \mathbb{R}$, $0 < p, q \leq \infty$, $p\leq r \leq \infty$, and
for every positive integer $m>n+\max\{\alpha,0\}+ \operatorname{int}
[n(\frac{1}{\theta_0}-1)] +1$, with $\theta_0=\min\{1,p,q\}$.
Then, we define the homogeneous Hermite-Besov-Morrey space
$\mathrm{BM}^{\alpha, \mathbb{H},m}_{p,q,r}$ as follows
\begin{align*}
 \mathrm{BM}^{\alpha, \mathbb{H},m}_{p,q,r}
:=\Big\{ &f \in \mathcal{S}':\|f\|_{\mathrm{BM}^{\alpha, \mathbb{H},m}_{p,q,r}}
=\Big(\int_0^\infty \Big(t^{-\alpha} \|(t\sqrt{\mathbb{H}})^m
 e^{-t\sqrt{\mathbb{H}}}f\|_{\mathrm{M}^r_p}\Big)^q\frac{dt}{t}
\Big)^{1/q} \\
&< \infty\Big\}.
 \end{align*}
\end{definition}

\begin{remark} \rm
If $r=p$, then the space $\mathrm{BM}^{\alpha,\mathbb{H},m}_{p,q,r}$ is exactly the space
$\mathrm{BM}^{\alpha, \mathbb{H},m}_{p,q}$ in \cite{Anh1}.
\end{remark}

We will show that $\mathrm{BM}^{\alpha,\mathbb{H},m}_{p,q,r}$ is independent of the choice of $m$ when $m$
is large enough. Precisely, we have the following result.

\begin{theorem}\label{t0}
 Let $\alpha \in \mathbb{R}$, $0 < p, q \leq \infty$, and $p \leq r \leq \infty$.
Let $m_1, m_2$ be the positive integers such that
 $$
m_1, m_2>n+\max\{\alpha,0\}+\operatorname{int} [n(\frac{1}{\theta_0}-1)] +1,
$$
 with $\theta_0=\min\{1,p,q\}$. Then, the spaces
$\mathrm{BM}^{\alpha, \mathbb{H},m_1}_{p,q,r}$ and
$\mathrm{BM}^{\alpha, \mathbb{H},m_2}_{p,q,r}$ coincide with equivalent norms.
\end{theorem}

As a consequence of Theorem \ref{t0}, we can define the Besov space
$\mathrm{BM}^{\alpha, \mathbb{H}}_{p,q,r}$ as any space $\mathrm{BM}^{\alpha,\mathbb{H},m}_{p,q,r}$, for any
positive integer $m>n+\max\{\alpha,0\}+\operatorname{int}
[n(\frac{1}{\theta_0}-1)] +1$.

We now recall the definition of the molecules associated with the Hermite
operator in \cite{Anh1}.

\begin{definition}\label{def3.3} \rm
 Let $0<r\leq \infty,\alpha \in \mathbb{R}$, and $N,M \in \mathbb{N}_+$.
A function $u$ is said to be an $(\mathbb{H},M,N,\alpha,r)$ molecule if there exist
a function $b$ from the domain $(\sqrt{\mathbb{H}})^M$ and a dyadic cube $Q \in \mathcal{D}$
so that
(i) $u=(\sqrt{\mathbb{H}})^Mb$, and
(ii)
\[
|(\sqrt{\mathbb{H}})^k b(x)| \leq \ell(Q)^{M-k}|Q|^{\alpha/n-1/r}
\Big(1+\frac{|x-x_Q|}{\ell(Q)}\Big)^{-n-N},\quad\text{for } k= 0,\dots, 2M.
\]
Briefly, we denote $u=m_Q$, for every dyadic cube $Q \in \mathcal{D}$.
\end{definition}

Next, we have some elementary estimates.

\begin{lemma}\label{nt}
Let $N \in \mathbb{N}_+$ and $a>t>0$. For any $x,z\in \mathbb{R}^n$,
 $$
\int_{\mathbb{R}^n} \Big(1+\frac{|x-y|}{t}\Big)^{-n-N}
\Big(1+\frac{|z-y|}{a}\Big)^{-n-N}dy
\lesssim t^n \Big(1+\frac{|x-z|}{t}\Big)^{-n-N}.
$$
\end{lemma}

For a proof of the above lemma, we refer to \cite[Lemma 3.6]{Anh1}.
Next, we have a result of the molecular decomposition for $\mathrm{BM}^{\alpha,\mathbb{H},m}_{p,q,r}$.

\begin{theorem}\label{theo3.4}
 Let $\alpha \in \mathbb{R}$, $0 < p, q \leq \infty$, $p \leq r \leq \infty$, and
$\theta_0=\min\{1,p,q\}$.


(i) For every $M,N \in \mathbb{N}_+$ and
$m>n+\max\{\alpha,0\}+\operatorname{int} [n(\frac{1}{\theta_0}-1) ] +1$,
if $f \in \mathrm{BM}^{\alpha,\mathbb{H},m}_{p,q,r}$, then there exist a sequence of
$(\mathbb{H}, M, N, \alpha, r)$ molecules
$\{m_Q\}_{Q\in\mathcal{D}_v, v \in \mathbb{Z}}$ and a sequence of coefficients
$\{s_Q\}_{Q \in \mathcal{D}_v, v \in \mathbb{Z}}$
 so that
 \[
 f =\sum_{v \in \mathbb{Z}}\sum_{Q \in \mathcal{D}_v} s_Q m_Q, \quad \text{in } \mathcal{S}'.
 \]
 Moreover,
 \begin{equation}\label{20a}
 \Big(\sum_{v\in \mathbb{Z}} A^q_v \Big)^{1/q}
 \lesssim \|f\|_{\mathrm{BM}^{\alpha, \mathbb{H},m}_{p,q,r}}.
 \end{equation}

(ii) Conversely, if
 \[
 f =\sum_{v \in \mathbb{Z}}\sum_{Q \in \mathcal{D}_v}s_Q m_Q, \quad \text{in } \mathcal{S}',
 \]
where $\{m_Q\}_{Q\in\mathcal{D}_v, v \in \mathbb{Z}}$ is a sequence of
$(\mathbb{H}, M, N, \alpha, r)$ molecules
and $\{s_Q\}_{Q \in \mathcal{D}_v, v \in \mathbb{Z}}$ is a sequence of
coefficients satisfying
 $\big(\sum_{v\in \mathbb{Z}} A^q_v \big)^{1/q} <\infty$, then $f \in \mathrm{BM}^{\alpha,\mathbb{H},m}_{p,q,r}$, and
 \begin{equation}\label{21}
 \|f\|_{\mathrm{BM}^{\alpha,\mathbb{H},m}_{p,q,r}} \lesssim \Big(\sum_{v\in \mathbb{Z}} A^q_v \Big)^{1/q},
 \end{equation}
provided that $N, M\in \mathbb{N}_+$ such that $\frac{n}{n+N}<\theta_0$,
$M > \max\{\frac{n}{\theta_0}-\alpha, m\}$, with
$m >\max\{\alpha, 0\}+N+n$.
\end{theorem}

\begin{proof}[Proof of part (i)]
 For every $f\in\mathrm{BM}^{\alpha,\mathbb{H},m}_{p,q,r}$, it follows from Proposition
\ref{proposition2.10} that
\[
 f =c_{m,M,N}\int_0^{\infty}(t\sqrt{\mathbb{H}})^{M+N}
e^{-t\sqrt{\mathbb{H}}}(t\sqrt{\mathbb{H}})^{m}
e^{-t\sqrt{\mathbb{H}}}f \frac{dt}{t},\quad \text{in } \mathcal{S}',
\]
with $c_{m,M,N}=-\frac{1}{2^{m+M+N-1}(m+M+N-1)!}$.
 Thus,
 \begin{align*}
f& =c_{m,M,N}\sum_{v\in \mathbb{Z}}\int_{2^v}^{2^{v+1}}
(t\sqrt{\mathbb{H}})^{M+N} e^{-t\sqrt{\mathbb{H}}}(t\sqrt{\mathbb{H}})^{m}
 e^{-t\sqrt{\mathbb{H}}}f \frac{dt}{t} \\
&=c_{m,M,N}\sum_{v\in \mathbb{Z}}\sum_{Q \in \mathcal{D}_v}\int_{2^v}^{2^{v+1}}
(t\sqrt{\mathbb{H}})^{M+N} e^{-t\sqrt{\mathbb{H}}}
[(t\sqrt{\mathbb{H}})^{m} e^{-t\sqrt{\mathbb{H}}}f.\chi_Q] \frac{dt}{t}.
 \end{align*}
For any $v \in \mathbb{Z}$ and $Q \in \mathcal{D}_v$, we set
 \begin{equation}\label{b1}
 s_Q=2^{-v(\alpha-n/r)} \sup_{(y,t)\in Q \times [2^v,2^{v+1})}
|(t\sqrt{\mathbb{H}})^me^{-t\sqrt{\mathbb{H}}}f(y)|,
 \end{equation}
and $m_Q=\mathbb{H}^{M/2}b_Q$, with
 \[
 b_Q=\frac{1}{s_Q}\int_{2^v}^{2^{v+1}}t^M(t\sqrt{\mathbb{H}})^{N}
e^{-t\sqrt{\mathbb{H}}}[(t\sqrt{\mathbb{H}})^{m} e^{-t\sqrt{\mathbb{H}}}f.\chi_Q]
\frac{dt}{t}.
 \]
Obviously, we have
 \[
 f =\sum_{v \in \mathbb{Z}}\sum_{Q \in \mathcal{D}_v}s_Q m_Q, \quad \text{in } \mathcal{S}'.
 \]
Thus, it remains to show that $m_Q$ is an $(\mathbb{H},M,N,\alpha,r)$ molecule.
Indeed, for $k = 0,\dots, 2M$, and for any $x \in \mathbb{R}^n$,
 from Proposition \ref{prop2.1} we have
\begin{equation} \label{22}
 \begin{aligned}
|\mathbb{H}^{k/2}b_Q(x)|
&=\big|\frac{1}{s_Q}\int_{2^v}^{2^{v+1}}t^{M-k}(t\sqrt{\mathbb{H}})^{N+k}
e^{-t\sqrt{\mathbb{H}}}[(t\sqrt{\mathbb{H}})^{m}
e^{-t\sqrt{\mathbb{H}}}f.\chi_Q] \frac{dt}{t}\big| \\
&\leq \frac{1}{s_Q}\int_{2^v}^{2^{v+1}}t^{M-k}\int_Q|p_{t,N+k}(x,y)|
\big|(t\sqrt{\mathbb{H}})^{m} e^{-t\sqrt{\mathbb{H}}}f(y)\big|dy \frac{dt}{t} \\
& \lesssim \frac{1}{s_Q} \sup_{(z,t)\in Q \times [2^v,2^{v+1})}
\big|(t\sqrt{\mathbb{H}})^{m} e^{-t\sqrt{\mathbb{H}}}f(z)\big| \\
&\quad\times \int_{2^v}^{2^{v+1}} t^{M-k}\int_Q \frac{t^N}{(t+|x-y|)^{n+N}}dy
\frac{dt}{t}.
 \end{aligned}
\end{equation}
 On the other hand, it is not difficult to verify that
 \begin{equation}\label{23}
 \int_Q \frac{t^N}{(t+|x-y|)^{n+N}}dy
\leq C(n,N) \Big(1+\frac{|x-x_Q|}{2^v}\Big)^{-n-N}, \quad
\forall t\in [2^v, 2^{v+1}).
 \end{equation}
Combination \eqref{b1}, \eqref{22} and \eqref{23} yields
 \[
|\mathbb{H}^{k/2}b_Q(x) | \lesssim 2^{v(\alpha +M-k-n/r)}
\Big(1+\frac{|x-x_Q|}{2^v}\Big)^{-n-N}.
\]
This implies that $m_Q$ is an $(\mathbb{H},M,N,\alpha,r)$ molecule.

 Next, we prove \eqref{20a}.
 We observe that $w(x,t) \equiv \mathbb{H}^{m/2}e^{-t\sqrt{\mathbb{H}}}f(x)$ is a solution of
the equation
 $$
-(\Delta_{x,t}+|x|^2) w = 0, \quad\text{with } \Delta_{x,t}w=w_{tt}+ \Delta w.
$$
So, $w$ is a subharmonic function. Thanks to \cite[Lemma 5.2]{ali},
for every $\theta\in(0,\infty)$ we obtain
 $$
 \sup_{(y,t)\in \widetilde{Q}}|\mathbb{H}^{m/2} e^{-t\sqrt{\mathbb{H}}}f(y)|
\lesssim \Big(\frac{1}{|\widetilde{Q}|} \int_{\frac{3}{2}\widetilde{Q}}
 |\mathbb{H}^{m/2} e^{-t\sqrt{\mathbb{H}}}f(y)|^\theta dydt \Big)^{1/\theta},
 $$
 where $\widetilde{Q}=Q \times [2^v,2^{v+1})$ is a cube in $\mathbb{R}^{n+1}$.

 Note that $|\widetilde{Q}| \sim 2^v |Q|$ and $t \sim 2^v$, for any
$(y,t) \in \widetilde{Q}$. Hence, it follows from the last inequality that
\begin{equation} \label{11}
 \begin{aligned}
 \sup_{(y,t)\in \widetilde{Q}}| (t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}}f(y)|
& \lesssim \Big(\frac{1}{|Q|} \int_{\frac{3}{4}2^v}^{\frac{9}{8}2^{v+1}}
\int_{\frac{3}{2}Q}|(t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}}f(y)|^\theta dy
\frac{dt}{t} \Big)^{1/\theta} \\
& \lesssim \Big(\int_{\frac{3}{4}2^v}^{\frac{9}{8}2^{v+1}}
[\mathbb{M}_\theta (|(t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}}f|)(x) ]^\theta
\frac{dt}{t} \Big)^{1/\theta},
 \end{aligned}
\end{equation}
 for any $x \in Q$. From \eqref{b1} and \eqref{11}, we obtain
 $$
 |s_Q|\chi_Q(x) \lesssim 2^{-v(\alpha-n/r)}
\Big(\int_{\frac{3}{4}2^v}^{\frac{9}{8} 2^{v+1}}
[\mathbb{M}_\theta (|(t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}}f|)(x)]^\theta \frac{dt}{t}
\Big)^{1/\theta}\chi_Q(x),
 $$
or
 $$
 \sum_{Q\in \mathcal{D}_v}|Q|^{-1/r}|s_Q|\chi_Q(x)
\lesssim 2^{-v\alpha}\Big(\int_{\frac{3}{4}2^v}^{\frac{9}{8} 2^{v+1}}
[\mathbb{M}_\theta (|(t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}}f|)(x)]^\theta \frac{dt}{t}\Big)^{1/\theta}.
 $$
Thanks to Lemma \ref{tr}, we have
\[
 A_v
\lesssim 2^{-v\alpha}
\big\| \Big(\int_{\frac{3}{4}2^v}^{\frac{9}{8} 2^{v+1}}
[\mathbb{M}_\theta (|(t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}}f|)]^\theta \frac{dt}{t}\Big)^{1/\theta}
\big\|_{\mathrm{M}^r_p}.
\]
 Next, Minkowski integral inequality (see $\eqref{bdt1}$) yields
\[
 A_v
\lesssim 2^{-v\alpha}\Big[ \int_{\frac{3}{4}2^v}^{\frac{9}{8}
2^{v+1}}\|\mathbb{M}_\theta \big(|(t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}}f|\big)
\|^\theta_{\mathrm{M}^r_p} \frac{dt}{t} \Big]^{1/\theta}.
\]
At the moment, for a fixed $\theta\in (0, \theta_0)$, then
 $\mathbb{M}_\theta$ is a bounded operator on
 $\mathrm{M}^{r}_{p}$, likewise
\begin{align*}
 A_v
 & \lesssim 2^{-v\alpha}\Big[\int_{\frac{3}{4}2^v}^{\frac{9}{8} 2^{v+1}}\|
(t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}}f \|^\theta_{\mathrm{M}_p^r} \frac{dt}{t} \Big]^{1/\theta}\\
&\lesssim \Big[\int_{\frac{3}{4}2^v}^{\frac{9}{8} 2^{v+1}}\Big(t^{-\alpha}
 \| (t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}}f \|_{\mathrm{M}_p^r}
 \Big)^\theta \frac{dt}{t} \Big]^{1/\theta}\\
& \lesssim \Big[\int_{\frac{3}{4}2^v}^{\frac{9}{8} 2^{v+1}}\Big(t^{-\alpha}
 \| (t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}}f \|_{\mathrm{M}_p^r}
 \Big)^q \frac{dt}{t} \Big]^{1/q},
 \end{align*}
 where the last inequality is obtained by using H\"older's inequality.
Therefore,
 \[
 \Big(\sum_{v\in \mathbb{Z}} A_v^q\Big)^{1/q}
\lesssim \Big[\sum_{v\in \mathbb{Z}} \int_{\frac{3}{4}2^v}^{\frac{9}{8} 2^{v+1}}
\Big(t^{-\alpha}\| (t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}}f \|_{\mathrm{M}_p^r}\Big)^q
\frac{dt}{t} \Big]^{1/q}.
 \]
By noting that
$\sum_{v\in \mathbb{Z}} \chi_{(\frac{3}{4}2^v, \frac{9}{8} 2^{v+1})}\leq 2$,
we obtain
 \[
 \sum_{v\in \mathbb{Z}} \int_{\frac{3}{4}2^v}^{\frac{9}{8} 2^{v+1}}
\Big(t^{-\alpha}\| (t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}}f \|_{\mathrm{M}_p^r}\Big)^q \frac{dt}{t}
\leq 2 \int_{0}^{\infty}\Big(t^{-\alpha}
 \| (t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}}f \|_{\mathrm{M}_p^r}
 \Big)^q \frac{dt}{t},
 \]
which implies
 \begin{equation*}
\Big(\sum_{v\in \mathbb{Z}} A_v^q\Big)^{1/q}
\lesssim \Big[ \int_{0}^{\infty}\Big(t^{-\alpha}
 \| (t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}}f \|_{\mathrm{M}_p^r}
 \Big)^q \frac{dt}{t} \Big]^{1/q} = \|f\|_{\mathrm{BM}^{\alpha,\mathbb{H},m}_{p,q,r}}.
 \end{equation*}
 This completes the proof of part (i).
\end{proof}

 To prove (ii) of Theorem \ref{theo3.4}, we need the following auxiliary lemmas.

 \begin{lemma}\label{lem3.5}
 Let $N > 0$, and let $\eta, v \in \mathbb{Z}$ be such that $v \leq \eta$.
Let $\{f_Q\}_{Q \in \mathcal{D}_v}$ be a sequence of functions satisfying
 \[
 |f_Q(x)| \lesssim \left(1+2^{-\eta}|x-x_Q|\right)^{-n-N}.
 \]
Then, for any $\theta\in (\frac{n}{n+N}, \infty)$ and for a sequence of numbers
$\{s_Q\}_{Q \in \mathcal{D}_v}$, we have
 $$
 \sum_{Q \in \mathcal{D}_v}|s_Q||f_Q(x)|
\lesssim 2^{\frac{(\eta-v)n}{\theta}} \mathbb{M}_\theta
\Big(\sum_{Q \in \mathcal{D}_v} |s_Q|\chi_Q\Big)(x).
 $$
 \end{lemma}

The proof of the above lemma can be found in \cite[p.147]{f}.
 Next, we recall \cite[Lemma 3.6]{Anh1}.

\begin{lemma}\label{lem3.6}
 Under the assumptions as in (ii) of Theorem \ref{theo3.4}, we have
\begin{gather*}
|(t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}}m_Q(x)|\lesssim |Q|^{ \frac{\alpha}{n}-\frac{1}{r} }
\Big(\frac{t}{2^v}\Big)^{m-N-n} \Big(1+\frac{|x-x_Q|}{2^{v}}\Big)^{-n-N}, \quad
 \forall t < 2^v, \\
|(t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}}m_Q(x)|\lesssim |Q|^{\frac{\alpha}{n}-\frac{1}{r}}
\Big(\frac{2^v}{t}\Big)^M \Big(1+\frac{|x-x_Q|}{t}\Big)^{-n-N},
 \quad \forall t\geq 2^v.
\end{gather*}
 \end{lemma}

\begin{proof}[Proof of part (ii) of Theorem \ref{theo3.4}]
We begin by writing
\begin{align*}
 \|f\|^q_{\mathrm{BM}^{\alpha,\mathbb{H},m}_{p,q,r}}
& = \sum_{k\in \mathbb{Z}} \int_{2^k}^{2^{k+1}}
 \Big( t^{-\alpha}  \| \sum_{v \in \mathbb{Z}} \sum_{Q \in \mathcal{D}_v} s_Q (t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}} m_Q
 \|_{\mathrm{M}^r_p}
 \Big)^q\frac{dt}{t}\\
 & \lesssim \sum_{k\in \mathbb{Z}}
 \Big( 2^{-k\alpha}  \| \sum_{v>k} \sum_{Q \in \mathcal{D}_v} |s_Q| 
 \sup_{t\in [2^k,2^{k+1})}  |(t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}} m_Q | \|_{\mathrm{M}^r_p}
 \Big)^q\\
 &\quad + \sum_{k\in \mathbb{Z}}
 \Big( 2^{-k\alpha} \| \sum_{v \leq k} \sum_{Q \in \mathcal{D}_v} |s_Q| 
\sup_{t\in [2^k,2^{k+1})}|(t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}} m_Q |  \|_{\mathrm{M}^r_p}
 \Big)^q\\
 & :=I_1+I_2.
 \end{align*}
Thus, the proof is complete if we can demonstrate that
 \begin{equation}\label{12}
 I_1, \, I_2\lesssim \sum_{v\in\mathbb{Z}} A^q_v.
 \end{equation}
We first prove \eqref{12} for $I_1$. Keep in mind that $v\geq k+1$ in this case.
 Since $\theta_0 > \frac{n}{n+N}$ and $M>\max \{\frac{n}{\theta_0}-\alpha, m\}$, 
we can choose a real number $\theta\in(\frac{n}{n+N},\theta_0)$ such that 
$M>\frac{n}{\theta} -\alpha$.
 By noting that $2^v\geq 2^{k+1}>t$, Lemma \ref{lem3.6} implies
 \[
 \sup_{t\in [2^k,2^{k+1})}  |(t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}} m_Q (x)|
\lesssim |Q|^{\frac{\alpha}{n}-\frac{1}{r}} 2^{(k-v)(m-N-n)} 
\left(1+ 2^{-v}|x-x_Q|\right)^{-n-N}.
\]
 Thus,
 \begin{equation} \label{14}
\begin{aligned}
&\sum_{Q \in \mathcal{D}_v} |s_Q| \sup_{t\in [2^k,2^{k+1})}|(t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}} m_Q(x) | \\
&\lesssim \sum_{Q \in \mathcal{D}_v} |Q|^{\frac{\alpha}{n}-\frac{1}{r}} 2^{(k-v)(m-N-n)} |s_Q|
 \left(1+ 2^{-v}|x-x_Q|\right)^{-n-N} \\
& \lesssim 2^{v \alpha} 2^{(k-v)(m-N-n)} \sum_{Q \in \mathcal{D}_v} |Q|^{-1/r}
|s_Q| \left(1+ 2^{-v}|x-x_Q|\right)^{-n-N}.
 \end{aligned}
\end{equation}
 Now, we apply Lemma \ref{lem3.5} with $\eta=v$ and
$f_Q(x)=\left(1+ 2^{-v}|x-x_Q|\right)^{-n-N}$ to obtain
 \begin{equation}\label{15}
 \sum_{Q \in \mathcal{D}_v}|Q|^{-1/r} |s_Q|
\left(1+ 2^{-v}|x-x_Q|\right)^{-n-N}
\lesssim \mathbb{M}_\theta \Big(\sum_{Q \in \mathcal{D}_v} |Q|^{-1/r} |s_Q|\chi_Q\Big)(x),
\end{equation}
for $\theta\in(\frac{n}{n+N}, \theta_0)$.
Inserting \eqref{15} into \eqref{14} yields
 \begin{align*}
&\sum_{Q \in \mathcal{D}_v} |s_Q|  \sup_{t\in [2^k,2^{k+1})}|(t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}} m_Q(x)| \\
&\lesssim 2^{v\alpha} 2^{(k-v)(m-N-n)}\mathbb{M}_\theta
\Big(\sum_{Q \in \mathcal{D}_v} |Q|^{-1/r} |s_Q|\chi_Q\Big)(x).
\end{align*}
 Then
\begin{equation} \label{30}
 \begin{aligned}
 I_1 & \lesssim \sum_{k \in \mathbb{Z}} \Big[ 2^{-k \alpha} \big\|
 \sum_{v > k} 2^{\alpha v}2^{(k-v)(m-N-n)}\mathbb{M}_\theta
 \Big(\sum_{Q \in \mathcal{D}_v}|Q|^{-1/r}|s_Q|\chi_Q\big)
 \big\|_{\mathrm{M}^r_p} \Big]^q \\
&= \sum_{k \in \mathbb{Z}}
 \big\| \sum_{v > k} 2^{(k-v)(m-N-n-\alpha)}\mathbb{M}_\theta
\Big(\sum_{Q \in \mathcal{D}_v}|Q|^{-1/r}|s_Q|\chi_Q\Big) \big\|_{\mathrm{M}^r_p}^q
 \\
& \lesssim  \sum_{k \in \mathbb{Z}}
 \Big[ \sum_{v > k} 2^{(k-v)(m-N-n-\alpha)}
\big\|\mathbb{M}_\theta \Big(\sum_{Q \in \mathcal{D}_v}|Q|^{-1/r}|s_Q|\chi_Q\Big)
\big\|_{\mathrm{M}^r_p} \Big]^q.
 \end{aligned}
\end{equation}
 Again the fact that $\mathbb{M}_\theta$ is bounded on $\mathrm{M}_p^r$ implies
 \begin{equation}\label{31}
\big\| \mathbb{M}_\theta
 \Big( \sum_{Q \in \mathcal{D}_v}|Q|^{-1/r}|s_Q|\chi_Q \Big)
\big\|_{\mathrm{M}_p^r}
\lesssim  \| \sum_{Q \in \mathcal{D}_v}|Q|^{-1/r}|s_Q|\chi_Q
 \|_{\mathrm{M}_p^r} \sim A_v.
 \end{equation}
Combination  \eqref{30} and \eqref{31} yields
 \begin{equation*}
 I_1 \lesssim \sum_{k \in \mathbb{Z}} \Big[\sum_{v>k}
 2^{(k-v)(m-N-n-\alpha)} A_v\Big]^{q}.
 \end{equation*}
Applying Young's inequality yields
 \begin{align*}
&\sum_{v>k} 2^{(k-v)(m-N-n-\alpha)} A_v \\
&\leq  \Big( \sum_{v>k}
 2^{ \frac{(k-v)(m-N-n-\alpha)q} {2(q-1)} } \Big)^{\frac{q-1}{q}}
\Big( \sum_{v>k} 2^{ \frac{(k-v)(m-N-n-\alpha)q} {2} } A^q_v \Big)^{1/q}.
\end{align*}
Since $m>N+n+\alpha$, $\sum_{v>k}
 2^{ \frac{(k-v)(m-N-n-\alpha)q} {2(q-1)} }$ is then bounded by a constant
independent of $k, v$. Thus,
 \begin{align*}
 I_1 & \lesssim \sum_{k \in \mathbb{Z}} \sum_{v>k}
 2^{\frac{(k-v)(m-N-n-\alpha)q}{2}} A_v^q\\
 &= \sum_{v \in \mathbb{Z}} \left(\sum_{k<v}
 2^{\frac{(k-v)(m-N-n-\alpha)q}{2}} \right) A_v^q
 \lesssim
 \sum_{v \in \mathbb{Z}} A_v^q.
 \end{align*}

It remains to show that estimate \eqref{12} holds for $I_2$. Actually, 
the proof for $I_2$ is most likely to the one for $I_1$, with only one 
different point that we use Lemma \ref{lem3.6} for $v\leq k$, i.e.
 \[
 \sup_{t\in [2^k,2^{k+1})}|(t\sqrt{\mathbb{H}})^m e^{-t\sqrt{\mathbb{H}}} m_Q(x) \big|
 \lesssim |Q|^{\frac{\alpha}{n}-\frac{1}{r}} 2^{(v-k)M} 
\Big(1+\frac{|x-x_Q|}{2^v}\Big)^{-n-N}.
 \]
Proceed similarly to the proof (from \eqref{14} to \eqref{31}) above, we obtain
 \[
I_2 \lesssim \sum_{k \in \mathbb{Z}} \Big[\sum_{v\leq k}
 2^{(v-k)(M+\alpha)} A_v\Big]^{q}.
\]
By noting that $M+\alpha>0$, apply Young's inequality yields the result.
This completes the proof of Theorem \ref{theo3.4}.
\end{proof}


\begin{proof}[Proof of Theorem \ref{t0}] 
Let  $N = \operatorname{int}[n(\frac{1}{\theta_0}-1)] +1$, and
 $M>\max\{m_1, m_2, \frac{n}{\theta_0} -\alpha \}$.
 Because $m_1$ and $m_2$ play the same role, it then suffices to prove that 
$\mathrm{BM}^{\alpha, \mathbb{H},m_1}_{p,q} \hookrightarrow \mathrm{BM}^{\alpha,
\mathbb{H},m_2}_{p,q}$.

 In fact, for $f \in \mathrm{BM}^{\alpha, \mathbb{H},m_1}_{p,q,r}$, thanks to (i)
of Theorem \ref{theo3.4}, there exist a sequence of $(\mathbb{H}, M, N,\alpha, r)$ 
molecules $\Big\{m_Q:Q \in \mathcal{D}_v,v \in \mathbb{Z}\Big\}$, and a sequence 
of coefficients $\Big\{s_Q:Q \in \mathcal{D}_v,v \in \mathbb{Z}\Big\}$ so that
 \[
 f =\sum_{v \in \mathbb{Z}}\sum_{Q \in \mathcal{D}_v}s_Q m_Q, \quad \text{in } \mathcal{S}',
 \]
 and
 $$
\Big(\sum_{v\in \mathbb{Z}} A^q_v \Big)^{1/q} 
\lesssim \|f\|_{\mathrm{BM}^{\alpha, \mathbb{H},m_1}_{p,q,r}}.
$$
In other words, $(\sum_{v\in \mathbb{Z}} A^q_v )^{1/q} $ is finite.


By (ii) of Theorem \ref{theo3.4}, we obtain 
$f \in \mathrm{BM}^{\alpha, \mathbb{H},m_2}_{p,q,r}$. Furthermore, $f$ satisfies
 $$
 \|f\|_{\mathrm{BM}^{\alpha, \mathbb{H},m_2}_{p,q,r}}
\lesssim \Big(\sum_{v\in \mathbb{Z}} A^q_v \Big)^{1/q}.
 $$
 Or, we obtain the result.
\end{proof}


\section{Regularity on Besov-Morrey spaces for fractional Hermite equations}

In this part, we study the regularity results of solutions of the two 
fractional Hermite equations:
\[
\mathbb{H}^s u=f,\quad\text{and}\quad (I+\mathbb{H})^{s}=f, \quad \text{on } \mathbb{R}^n,
\]
for any $s>0$, and for $f\in \mathrm{BM}^{\alpha, \mathbb{H}}_{p,q,r}$.
To solve the indicated equations, it is necessary to investigate the operators
$\mathbb{H}^{-s}$ and $(I+\mathbb{H})^{-s}$, named by the Riesz potential of Hermite 
operator and the Bessel potential of Hermite operator respectively.

In fact, by following \cite[Proposition 2.5]{Anh1}, we can define the operators
 $\mathbb{H}^{-s}:\mathcal{S}'\to \mathcal{S}'$ and 
$(I+\mathbb{H})^{-s}:\mathcal{S}'\to \mathcal{S}'$ by setting
\[
 \langle \mathbb{H}^{-s}f,\phi\rangle=\langle f,\mathbb{H}^{-s}\phi\rangle,\quad
 \text{and} \quad
 \langle (I+\mathbb{H})^{-s}f,\phi\rangle=\langle f,(I+\mathbb{H})^{-s}\phi\rangle,
\]
for any $f\in \mathcal{S}'$, and for $\phi\in \mathcal{S}$.
 Note that $\langle \cdot,\cdot \rangle$ is the pair between a linear function 
in $\mathcal{S'}$ and a function in $\mathcal{S}$.
Moreover,  for any $\phi\in\mathcal{S}$ we have
\begin{gather*}
 \mathbb{H}^{-s}\phi=\frac{1}{\Gamma(s)}\int_0^\infty t^s e^{-t\mathbb{H}}\phi\frac{dt}{t}
 \in \mathcal{S}, \\
 (I+\mathbb{H})^{-s}\phi=\frac{1}{\Gamma(s)}\int_0^\infty t^s 
e^{-t}e^{-t\mathbb{H}}\phi\frac{dt}{t}\in \mathcal{S}.
\end{gather*}
Let $K_t(x,y)$ (resp. $K_{t,k}(x,y)$) be the kernel of $e^{-t\mathbb{H}}$ 
(resp. $(t \mathbb{H})^k e^{-t \mathbb{H}}$).
 Thanks to \cite[Lemma 2.5]{Cou}, and \cite[Lemma 2.4]{Anh1}, 
we have the following results.

\begin{lemma}\label{lem2.4}
 For $k\in\mathbb{N}$, there exists $c, C > 0$ so that for all $y \in \mathbb{R}$
 \begin{gather}\label{20aa}
 | \partial^k_x K_t(x,y)| \leq
\begin{cases}
 Ct^{-\frac{k+1}{2}}\exp\big(-c\frac{|x-y|^2}{t}\big), & 0<t \leq 1;\\
 e^{-t}e^{-|x-y|^2},& t>1.
 \end{cases} \\
\label{20}
 K_{t,k}(x,y)\leq \frac{C}{t^{n/2}}\exp\Big(-c\frac{|x-y|^2}{t}\Big),
\end{gather}
\end{lemma}

Our regularity results are as follows.

\begin{theorem}\label{theo5.1}
Let $\alpha\in \mathbb{R}$, $0<q\leq \infty$, $0<p \leq r \leq \infty$, and 
$f\in\mathrm{BM}_{p,q,r}^{\alpha,\mathbb{H}}$. Assume that $u$ is a solution of equation
 $\mathbb{H}^s u=f$, Then, there exists a constant $C>0$ such that
 \[
 \| u \|_{\mathrm{BM}_{p,q,r}^{\alpha+2s,\mathbb{H}}}
\leq C \| f \|_{\mathrm{BM}_{p,q,r}^{\alpha,\mathbb{H}}}.
\]
\end{theorem}

\begin{theorem}\label{theo4}
 Let $\alpha\in \mathbb{R}$, $0<q\leq \infty$, $0<p \leq r \leq \infty$, 
and $f\in\mathrm{BM}_{p,q,r}^{\alpha,\mathbb{H}}$. Assume that $u$ is a solution of equation
 $(\mathbb{H}+I)^s u=f$. Then, there exists a constant $C>0$ such that
 \[
 \| u \|_{\mathrm{BM}_{p,q,r}^{\alpha+2s,\mathbb{H}}}
\leq C \| f \|_{\mathrm{BM}_{p,q,r}^{\alpha,\mathbb{H}}}.
\]
\end{theorem}

Theorems \ref{theo5.1} and \ref{theo4} are just a consequence of the theorem below.

\begin{theorem}\label{theo5.1bis}
 Let $\alpha\in \mathbb{R}$, $0<p \leq r < \infty$, and
 $0<q \leq \infty$. For any $s>0$, the operator $\mathbb{H}^{-s}$ 
(resp. $(I+\mathbb{H})^{-s}$) is bounded from $\mathrm{BM}_{p,q,r}^{\alpha,\mathbb{H}}$
to $\mathrm{BM}_{p,q,r}^{\alpha+2s,\mathbb{H}}$.
\end{theorem}

\begin{proof}[Proof of Theorem \ref{theo5.1bis}]
 Let $\{m_Q:Q \in \mathcal{D}_v,v \in \mathbb{Z}\}$ be a sequence of 
$(\mathbb{H},4M,N,\alpha,r)$ molecules, with $M,N\in \mathbb{N}$, and
$M>s+n/2+N/2$.

We first prove that $\mathbb{H}^{-s}(m_Q)$ is an $(H,2M,N,\alpha+2s,r)$ molecule 
associated with the cube $Q$.
 Indeed, let $m_Q=\mathbb{H}^{2M}b_Q$ as in Definition \ref{def3.3}, and put 
$y_Q=\mathbb{H}^{-s}\mathbb{H}^Mb_Q$. Then
 $$
\mathbb{H}^{-s}m_Q=\mathbb{H}^M y_Q= (\sqrt{\mathbb{H}})^{2M} y_Q.
$$
Thus, it suffices to show that
 \begin{equation}\label{4.1}
 | (\sqrt{\mathbb{H}})^k y_Q(x)|
\lesssim \ell(Q)^{2M-k}| Q |^{\frac{\alpha+2s}{n}-\frac{1}{r}}
 \Big(1+\frac{|x-x_Q|}{\ell(Q)}\Big)^{-n-N},
 \end{equation}
for $k=0,\dots,4M$.
In fact, we have
\[
 y_Q(x) =\mathbb{H}^{-s}\mathbb{H}^Mb_Q=\frac{1}{\Gamma(s)}\int_0^\infty t^s 
e^{-t\mathbb{H}}\mathbb{H}^Mb_Q(x)\frac{dt}{t}.
\]
 Therefore,
 \begin{align*}
 | (\sqrt{\mathbb{H}})^k y_Q(x) | 
&\leq \frac{1}{\Gamma(s)}\int_0^{4^v} | t^s e^{-t\mathbb{H}}(\sqrt{\mathbb{H}})^{2M+k}b_Q(x) | 
\frac{dt}{t} \\
&+\frac{1}{\Gamma(s)}\int_{4^v}^\infty | t^s e^{-t\mathbb{H}}(\sqrt{\mathbb{H}})^{2M+k}b_Q(x) | 
 \frac{dt}{t}
 :=I_1+I_2.
 \end{align*}

 First, we estimate $I_1$. Thanks to Lemma \ref{lem2.4}, we have
 \begin{align*}
|e^{-t\mathbb{H}}(\sqrt{\mathbb{H}})^{2M+k}b_Q(x)|
 & = \int_{\mathbb{R}^n}|K_t(x,y)(\sqrt{\mathbb{H}})^{2M+k}b_Q(y)|dy \\
& \lesssim \int_{\mathbb{R}^n}\frac{1}{t^{n/2}}
 \exp\Big(-2c\frac{|x-y|^2}{t}\Big)|(\sqrt{\mathbb{H}})^{2M+k}b_Q(y)|dy.
\end{align*}
 Taking Definition \ref{def3.3} into account, we obtain
 \begin{align*}
|e^{-t\mathbb{H}}(\sqrt{\mathbb{H}})^{2M+k}b_Q(x)|
&\lesssim \int_{\mathbb{R}^n}\frac{1}{t^{n/2}}\exp\Big(-c\frac{|x-y|^2}{t}\Big)
\Big(1+\frac{|x-y|}{\sqrt{t}}\Big)^{-n-N} \\
&\quad\times |Q|^{\frac{\alpha}{n}-\frac{1}{r}}
2^{v(2M-k)}\Big(1+\frac{|y-x_Q|}{2^v}\Big)^{-n-N}dy.
 \end{align*}
Next, we apply the inequality $(1+a+b) \leq (1+a)(1+b)$,  for all $a,b\geq 0$
 and the fact $t<4^v$ to the right hand side of the above inequality  
to obtain
\begin{align*}
&|e^{-t\mathbb{H}}(\sqrt{\mathbb{H}})^{2M+k}b_Q(x)| \\
&\lesssim |Q|^{\frac{\alpha}{n}-\frac{1}{r}}2^{v(2M-k)}
 \Big(1+\frac{|x-x_Q|}{2^v}\Big)^{-n-N} \int_{\mathbb{R}^n}\frac{1}{t^{n/2}}
\exp\Big(-c\frac{|x-y|^2}{t}\Big)dy;
\end{align*}
thus
\[
|e^{-t\mathbb{H}}(\sqrt{\mathbb{H}})^{2M+k}b_Q(x)|
 \lesssim |Q|^{\frac{\alpha}{n}-\frac{1}{r}}2^{v(2M-k)} 
\Big(1+\frac{|x-x_Q|}{2^v}\Big)^{-n-N}.
\]
 This implies
 \begin{equation}\label{4.4}
\begin{aligned}
 I_1 &\lesssim |Q|^{\frac{\alpha}{n}-\frac{1}{r}}2^{v(2M-k)} 
\Big(1+\frac{|x-x_Q|}{2^v}\Big)^{-n-N}\int_0^{4^v} t^{s} \frac{dt}{t} \\
&\lesssim|Q|^{\frac{\alpha+2s}{n}-\frac{1}{r}}2^{v(2M-k)} 
\Big(1+\frac{|x-x_Q|}{2^v}\Big)^{-n-N}.
\end{aligned}
\end{equation}

 It remains to consider $I_2$. By \eqref{20}, we have
 \begin{align*}
|\mathbb{H}^M e^{-t\mathbb{H}}(\sqrt{\mathbb{H}})^{k}b_Q(x)|
& =t^{-M}|(t\mathbb{H})^M e^{-t\mathbb{H}}(\sqrt{\mathbb{H}})^{k}b_Q(x)|\\
& = t^{-M} \int_{\mathbb{R}^n}|K_{t,M}(x,y)(\sqrt{\mathbb{H}})^k b_Q(y)|dy\\
& \lesssim t^{-M} \int_{\mathbb{R}^n}\frac{1}{t^{n/2}}\exp\left(-c\frac{|x-y|^2}{t}\right)|(\sqrt{\mathbb{H}})^k b_Q(y)|dy.
 \end{align*}
In similar to the above proof, we also have
 \begin{align*}
&|\mathbb{H}^M e^{-t\mathbb{H}}(\sqrt{\mathbb{H}})^{k}b_Q(x)|\\
&\lesssim t^{-M}\int_{\mathbb{R}^n}\frac{1}{t^{n/2}}
 \Big(1+\frac{|x-y|}{\sqrt{t}}\Big)^{-n-N}|Q|^{\frac{\alpha}{n}
 -\frac{1}{r}}2^{v(4M-k)}\Big(1+\frac{|y-x_Q|}{2^v}\Big)^{-n-N}dy.
 \end{align*}
By Lemma \ref{nt}, and noting that $t \geq 4^v$, we obtain
 \begin{align*}
&|\mathbb{H}^M e^{-t\mathbb{H}}(\sqrt{\mathbb{H}})^{k}b_Q(x)| \\
& \lesssim t^{-M}|Q|^{\frac{\alpha}{n}-\frac{1}{r}}2^{v(4M-k)} 
\Big(1+\frac{|x-x_Q|}{\sqrt{t}}\Big)^{-n-N}\\
& \lesssim \big(\frac{t}{4^v}\big)^{(n+N)/2}t^{-M}
 |Q|^{\frac{\alpha}{n}-\frac{1}{r}}2^{v(4M-k)} 
\Big(1+\frac{|x-x_Q|}{2^v}\Big)^{-n-N}.
 \end{align*}
Thus
\begin{equation} \label{4.5}
 \begin{aligned}
 I_2
& \lesssim |Q|^{\frac{\alpha}{n}-\frac{1}{r}}2^{v(4M-k-N-n)}
\Big(1+\frac{|x-x_Q|}{2^v}\Big)^{-n-N}
 \int_{4^v}^ \infty t^{s+\frac{n+N}{2}-M} \frac{dt}{t} \\
& \lesssim|Q|^{\frac{\alpha+2s}{n}-\frac{1}{r}}2^{v(2M-k)}
 \Big(1+\frac{|x-x_Q|}{2^v}\Big)^{-n-N}.
 \end{aligned}
\end{equation}
 Hence, \eqref{4.1} follows from \eqref{4.4} and \eqref{4.5}.
Thus $\mathbb{H}^{-s}(m_Q)$ is an $(H,2M,N,\alpha+2s,r)$ molecule associated with
 the cube $Q$. By Theorem \ref{theo3.4} and a suitable choice of $M, N$,
 we obtain the boundedness of $\mathbb{H}^{-s}$ from $\mathrm{BM}_{p,q,r}^{\alpha,\mathbb{H}}$
to $\mathrm{BM}_{p,q,r}^{\alpha+2s,\mathbb{H}}$.

 Similarly, we can also establish the boundedness of the Bessel potential 
$(I+H)^{-s}$ from $\mathrm{BM}_{p,q,r}^{\alpha,\mathbb{H}}$ to
 $\mathrm{BM}_{p,q,r}^{\alpha+2s,\mathbb{H}}$.
We leave the proof to the reader.
\end{proof}

\subsection*{Acknowledgments} 
We would like to thank the anonymous referees for their valuable comments.

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\end{document}