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\markboth{\hfil An embedding theorem for Campanato spaces \hfil 
EJDE--2002/66}
{EJDE--2002/66\hfil Azzeddine El Baraka \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 66, pp. 1--17. \newline
ISSN: 1072-6691. URL: <http://ejde.math.swt.edu> or 
<http://ejde.math.unt.edu>
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  An embedding theorem for Campanato spaces
 %
\thanks{ {\em Mathematics Subject Classifications:} 46E35.
\hfil\break\indent
{\em Key words:} Sobolev embeddings, BMO, Campanato spaces.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted April 15, 2002. Published July 11, 2002.} }
\date{}
%
\author{Azzeddine El Baraka}
\maketitle

\begin{abstract} 
  The purpose of this paper is to give a Sobolev type embedding theorem
  for the spaces $\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$.
  The homogeneous versions of these spaces contain well known spaces
  such as the Bounded Mean Oscillation spaces (BMO) and  the 
  Campanato spaces  $\mathcal{L}^{2,\lambda}$. Our result extends some 
  injections obtained by Campanato \cite{c1,c2}, Strichartz \cite{str}, 
  and Stein and Zygmund \cite{sz}.
\end{abstract}


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


\section{Introduction and statement of results}

The main goal of this work is to give a Sobolev type embedding theorem 
for the
appropriate scaled functions in $\mathcal{L}_{p,q}^{\lambda,s}
(\mathbb{R}^{n})$ whose homogeneous version 
$\dot{\mathcal{L}}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$ 
contains some well known spaces as special
cases: John and Nirenberg space BMO (Bounded Mean Oscillation)
and more generally Campanato spaces
$\mathcal{L}^{2,\lambda}$ modulo polynomials \cite{e2}.

It is well known that the homogeneous  Triebel-Lizorkin 
$\dot{F}_{p,q}^{s}(\mathbb{R}^{n})$ spaces coincide with 
BMO modulo polynomials for
some values of $p,q$ and $s$. Namely, $BMO=\dot{F}_{\infty,2}^{0}$ 
\cite[chap.~5]{tr}  and thus $I^{s}(BMO)=\dot{F}_{\infty,2}^{s}$,
where $I^{s}=\mathcal{F}^{-1}( |\cdot |^{-s}\mathcal{F})  $ 
is the Riesz potential operator.  Strichartz \cite{str}
discussed the connexion between $I^{s}(BMO)$ and the homogeneous Besov 
space
$\dot{\Lambda}_{\infty,q}^{s}$ and proved the following injections
(Theorem 3.4.)
\[
\dot{\Lambda}_{\infty,2}^{s}\subseteq I^{s}(BMO)\subseteq\overset
{.}{\Lambda}_{\infty,\infty}^{s}
\]
Let us mention that others classical embeddings have been obtained
respectively by Stein and Zygmund \cite{sz} and Campanato
\cite{c1,c2}; they proved that
\[
H_{p}^{n/p}\hookrightarrow BMO\quad \text{and}\quad
\mathcal{L}^{p,\lambda}\cong C^{\frac{\lambda-n}{p}}\quad
\text{if }n<\lambda<n+p
\]
here $H_{p}^{n/p}$ is the Bessel potential space and 
$\mathcal{L}^{p,\lambda}$
is the Campanato space (cf. Definition \ref{camp}). In this paper we 
extend
these injections to a more general frame (Theorem \ref{221}), by giving 
some
embedding results between the spaces $\mathcal{L}_{p,q}^{\lambda,s}
(\mathbb{R}^{n})$\ (which include $I^{s}(BMO)$ and $I^{s}(\mathcal{L}
^{2,\lambda})$ as special cases), Triebel-Lizorkin spaces 
$F_{p,q}^{s}(\mathbb{R}^{n})$ (containing $H_{p}^{n/p}$ as particular 
case) 
and Besov-Peetre spaces $B_{p,q}^{s}(\mathbb{R}^{n})$, 
and on the other side between
the same spaces $\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$ and
H\"{o}lder-Zygmund ones $C^{s}(\mathbb{R}^{n})$. Such embeddings shed 
some
light on duals of the closure of Schwartz space 
$\mathcal{S}(\mathbb{R}^{n})$
in BMO and in Campanato spaces $\mathcal{L}^{2,\lambda}$ (Corollary 
\ref{cor}).

The present article is organised as follows: first, we give the 
definition of
the $\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$ spaces and their
homogeneous counterparts 
$\dot{\mathcal{L}}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$ 
which invoke a Littlewood-Paley partition of unity and
some expressions with balls (or cubes) of $\mathbb{R}^{n}$, and we 
state the
main result (Theorem \ref{221}). Next, we proceed to a discussion of 
some
properties of these spaces: we give the connexion between the 
nonhomogeneous
spaces $\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$ and the 
homogeneous
ones, and we specify the differential dimension of the dotted spaces 
which
depends on $p,q,s$, and $\lambda$. Finally, we prove Theorem \ref{221} 
and
Corollary \ref{cor}.


To define these spaces, we will need a Littlewood-Paley partition of 
unity.
Denote $x\in\mathbb{R}^{n}$ and $\xi$ its dual variable. Consider the 
cutoff
function $\varphi\in C_{0}^{\infty}(\mathbb{R}^n)$,
$\varphi\geq0$ and $\varphi$ equal to $1$ for $|  \xi|  \leq1$, $0$
for $|  \xi|  \geq2$. Let $\theta(\xi)=\varphi(\xi)-\varphi(2\xi)$
which satisfies $\mathop{\rm supp}\theta\subset\{\frac{1}{2}\leq|  \xi|
\leq2\}$. For $j\in\mathbb{Z}$ we set
\[
\dot{\Delta}_{j}u=\theta(2^{-j}D_{x})u
\]
and
\[
\Delta_{0}u=\varphi(D_{x})u\text{, }\Delta_{j}u=\dot{\Delta}_{j}u\text{
if }j\geq1
\]

\begin{remark} \label{ca} \rm
We have $\varphi(\xi)+\sum_{k\geq1}\theta(2^{-k}\xi)=1$ for
all $\xi\in\mathbb{R}^{n}$, thus the nonhomogeneous partition of
$u\in\mathcal{S}'(\mathbb{R}^{n})$ is given by the formula
\[
u=\sum_{k\geq0}\Delta_{k}u
\]
Set $f(\xi)=\sum_{k\in\mathbb{Z}}\theta(2^{-k}\xi)$; for each fixed
$\xi\neq0$, $f(\xi)$ contains at most two non-vanishing terms, we 
deduce
$f(\xi)=1$ for any $|  \xi|  \geq2$. Note that $f(2^{-j}\xi
)=f(\xi)$ for all $j\in\mathbf{Z,}$ and choose $j$ more and more large, 
to
obtain $f(\xi)=1$ for any $\xi\neq0$. Then, for 
$u\in\mathcal{S}'(\mathbb{R}^{n})$, with 
$0\notin \mathop{\rm supp}\mathcal{F}u$, the homogeneous partition
of $u$ is given by the formula
\[
u=\sum_{k\in\mathbb{Z}}\dot{\Delta}_{k}u
\]
\end{remark}

Now, let us define the nonhomogeneous space 
$\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$.

\begin{definition} \label{def1} \rm
Let $s\in\mathbb{R},\lambda\geq0$, $1\leq p<+\infty$ and $1\leq
q<+\infty$. The space $\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$ 
denotes
the set of all tempered distributions $u\in\mathcal{S}'\left(
\mathbb{R}^{n}\right)  $ such that
\begin{equation}
\|  u\|  _{\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})}
=\Big\{
\sup_B \frac{1}{|  B|^{\lambda/n}}\sum_{j\geq
J^{+}}2^{jqs}\|  \Delta_{j}u\|  _{L^{p}(B)  }
^q \Big\}^{1/q}<+\infty\label{11}
\end{equation}
where $J^{+}=\max\left(  J,0\right)  $, $|  B|  $ is the measure of
$B$ and the supremum is taken over all $J\in\mathbb{Z}$ and all balls 
$B$ of
$\mathbb{R}^{n}$ with radius $2^{-J}$. 
When  $p=q$, the space $\mathcal{L}_{p,p}^{\lambda,s}(\mathbb{R}^{n})$ 
will be
denoted $\mathcal{L}^{p,\lambda,s}(\mathbb{R}^{n})$.
\end{definition}


Note that the space $\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$ 
equipped with the
norm (\ref{11}) is a Banach space.

To give the homogeneous counterpart of the space $\mathcal{L}_{p,q}
^{\lambda,s}(\mathbb{R}^{n})$, we recall the notations of 
\cite[chap.~5]{tr}. Let
\[
Z(\mathbb{R}^{n})=\big\{\varphi\in\mathcal{S}(\mathbb{R}^{n});(D^{\alpha
}\mathcal{F}\varphi)(0)=0\text{ for every multi-index }\alpha\big\}.
\]
The space $Z(\mathbb{R}^{n})$ is considered as a subspace of 
$\mathcal{S}(\mathbb{R}^{n})$ with the induced topology, and 
$Z'(\mathbb{R}^{n})$ its topological dual. 
We may identify $Z'(\mathbb{R}^{n})$ with
$\mathcal{S}'(\mathbb{R}^{n})/\mathcal{P}(\mathbb{R}^{n})$, where 
$\mathcal{P}(\mathbb{R}^{n})$ is
the set of all polynomials of $\mathbb{R}^{n}$ with complex 
coefficients i.e.
$Z'(\mathbb{R}^{n})$ is interpreted as $\mathcal{S}'(\mathbb{R}^{n})$ 
modulo polynomials.

\begin{definition}\label{def2} \rm
Let $s\in\mathbb{R},\lambda\geq0$, $1\leq p<+\infty$ and $1\leq
q<+\infty$. The dotted space $\dot{\mathcal{L}}_{p,q}^{\lambda
,s}(\mathbb{R}^{n})$ denotes the set of all $u\in Z'\left(
\mathbb{R}^{n}\right)  $ such that
\begin{equation}
\|  u\|  _{\dot{\mathcal{L}}_{p,q}^{\lambda,s}
(\mathbb{R}^{n})}=\Big\{  \underset{B}{\sup}\frac{1}{|  B|
^{\lambda/n}}\sum_{j\geq J}2^{jqs}\|  \dot{\Delta}
_{j}u\|  _{L^{p}(B)  }^q \Big\}^{1/q}
<+\infty\label{21}
\end{equation}
where the supremum is taken over all $J\in\mathbb{Z}$ and all balls $B$ 
of
$\mathbb{R}^{n}$ with radius $2^{-J}$.
\end{definition}

If $p=q$, the space $\dot{\mathcal{L}}_{p,p}^{\lambda,s}(\mathbb{R}
^{n})$ will be denoted 
$\dot{\mathcal{L}}^{p,\lambda,s}(\mathbb{R}^{n})$.
If $P$ is a polynomial of $\mathcal{P}(\mathbb{R}^{n})$ and
$u\in\mathcal{S}'(\mathbb{R}^{n})$, it follows
immediatly that
\[
\|  u+P\|  _{\dot{\mathcal{L}}_{p,q}^{\lambda,s}
(\mathbb{R}^{n})}=\|  u\|  _{\dot{\mathcal{L}}
_{p,q}^{\lambda,s}(\mathbb{R}^{n})}
\]
This shows that the norm (\ref{21}) is well defined. Further, the space
$\dot{\mathcal{L}}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$ equipped with
this norm is a Banach space.

\begin{remark} \rm
The supremum given in expressions (\ref{11}) and (\ref{21}) may be 
taken over
all $J\in\mathbb{Z}$ and all cubes $B$ of $\mathbb{R}^{n}$ of length 
side
$2^{-J}.$
\end{remark}

Now we define Campanato spaces modulo polynomials $\dot{\mathcal{L}
}^{p,\lambda}(\mathbb{R}^{n})$.

\begin{definition} \rm
\label{camp}Let $\lambda\geq0$ and $1\leq p<+\infty$.
\begin{enumerate}
\item[(i)] The space $\mathcal{L}^{p,\lambda}(\mathbb{R}^{n})$ 
denotes the set of (classes of) functions
$u\in L_{loc}^{p}(\mathbb{R}^n)$ such that
\[
\|  u\|  _{\mathcal{L}^{p,\lambda}(\mathbb{R}^{n})}=\left\{
\sup_B \frac{1}{|  B|^{\lambda/n}}\int_B |
u-m_B u|^{p}dx\right\}^{1/p}<+\infty
\]
where $m_B u=\frac{1}{|  B|  }\int_B u(y)dy$ is the mean value of
$u$ and the supremum is taken over all balls $B$ of $\mathbb{R}^{n}$.
\end{enumerate}
Note that $\mathcal{L}^{p,\lambda}(\mathbb{R}^{n})$ is a Banach space 
modulo
constants equal to $\{0\}$ if $\lambda>n+p$.
\begin{enumerate}

\item[(ii)] The space $\dot{\mathcal{L}}^{p,\lambda}(\mathbb{R}^{n})$ 
denotes
the set of functions  $u\in Z'(\mathbb{R}^n)$ such that
$\|  u\|  _{\dot{\mathcal{L}}^{p,\lambda}(\mathbb{R}^{n})}<+\infty$, 
where
\[
\|  u\|  _{\dot{\mathcal{L}}^{p,\lambda}(\mathbb{R}^{n}
)}=\left\{
\begin{array}
{ll}
\inf_{P\in\mathcal{P}}\|  u+P\|  _{\mathcal{L}^{p,\lambda
}(\mathbb{R}^{n})}&\text{if }u\in L_{loc}^{1}(\mathbb{R}^{n})\\[3pt]
+\infty&\text{if }u\in Z'(\mathbb{R}^n)  \text{ but}\\
&\text{not locally integrable}
\end{array}
\right.
\]
the infimum is taken over the set $\mathcal{P}$ of all polynomials
of $\mathbb{R}^{n}$ with complex coefficients.
\end{enumerate}
\end{definition}

\begin{remark}\label{primo} \rm
(i) Modulo polynomials, the space 
$\dot{\mathcal{L}}^{p,\lambda}(\mathbb{R}^{n})$
is the Campanato space $\mathcal{L}^{p,\lambda}(\mathbb{R}^{n})$. 
For $0\leq \lambda<n+p$, 
$\dot{\mathcal{L}}^{p,\lambda}(\mathbb{R}^{n})$ 
can be defined as the set of all equivalence classes modulo
 $\mathcal{P}$ of elements
of $\mathcal{L}^{p,\lambda}(\mathbb{R}^{n})$, equipped with the 
following norm
$\|  U\|  _{\dot{\mathcal{L}}^{p,\lambda}(\mathbb{R}^{n}
)}=\|  u\|  _{\mathcal{L}^{p,\lambda}(\mathbb{R}^{n})}$ where $u$
is the unique (modulo a constant) element of $\mathcal{L}^{p,\lambda
}(\mathbb{R}^{n})$ belonging to the class $U$.

(ii) Using a result of John and Nirenberg \cite{jn} it is classical
that if 
$\lambda=n$, then for every $1\leq p<+\infty$, $\mathcal{L}
^{p,n}(\mathbb{R}^{n})$ coincides with the space $BMO\left(  \mathbb{R}
^{n}\right)  $ of all functions $u\in L_{loc}^{1}\left(  \mathbb{R}
^{n}\right)  $ such that there exists a constant $C>0$ satisfying the
following inequality
\[
\frac{1}{|  B|  }\int_B |  u-m_B u|  dx\leq C
\]
for all balls $B$ of $\mathbb{R}^{n}$. We show in \cite{e2} that in 
general
$\dot{\mathcal{L}}^{2,\lambda,0}(\mathbb{R}^{n})
\equiv\overset{.}{\mathcal{L}}^{2,\lambda}(\mathbb{R}^{n})$ provided 
$0\leq\lambda<n+2$;
therefore the characterization of Littlewood-Paley type of Campanato 
spaces
$\dot{\mathcal{L}}^{2,\lambda}(\mathbb{R}^{n})$ is given. This result
is not true in general for $p\neq2$, nevertheless if $p\geq2$ and
$0\leq\lambda<n+p$ we show that $\dot{\mathcal{L}}^{p,\lambda
}(\mathbb{R}^{n})$ is continuously embedded in $\dot{\mathcal{L}
}^{p,\lambda,0}(\mathbb{R}^{n}).$
\end{remark}

Now, recall the definition of some function spaces.

\begin{definition} \rm
Let $s\in\mathbb{R}$, $1\leq p<+\infty$ and $1\leq q<+\infty$.
\begin{enumerate}
\item[(i)] We denote $F_{p,q}^{s}(\mathbb{R}^n)  $ the set of all
tempered distributions $u\in\mathcal{S}'(\mathbb{R}^{n})$ such that
\[
\|  u\|  _{F_{p,q}^{s}(\mathbb{R}^n)  }=\Big(
{\int_{\mathbb{R}^{n}}}\Big({\sum_{j\geq0}}
2^{jqs}|  \Delta_{j}u|^q \Big)^{p/q}dx\Big)^{1/p}<+\infty
\]

\item[(ii)] The space $B_{\infty,q}^{s}(\mathbb{R}^{n})$ denotes the 
set of all
tempered distributions
$u\in\mathcal{S}'(\mathbb{R}^{n})$ such that
\[
\|  u\|  _{B_{\infty,q}^{s}(\mathbb{R}^{n})}=\Big\{
{\sum_{k\geq0}}
2^{kqs}\|  \Delta_{k}u\|  _{L^{\infty}(\mathbb{R}^{n})}
^q \Big\}^{1/q}<+\infty
\]

\item[(iii)] We denote $C^{s}(\mathbb{R}^n)  $ the set of all tempered
distributions $u\in\mathcal{S}'(\mathbb{R}^{n})$ such that
\[
\|  u\|  _{C^{s}(\mathbb{R}^n)  }=\sup_{j\geq
0}2^{js}\|  \Delta_{j}u\|  _{L^{\infty}(\mathbb{R}^n)  }<+\infty
\]

\item[(iv)] If we replace in (i), (ii) and (iii) 
$\mathcal{S}'(\mathbb{R}
^{n})$ by $\mathcal{Z}'(\mathbb{R}^{n})$, $\Delta_{j}$ by 
$\overset{.}{\Delta}_{j}$ and $j\geq0$ by $j\in\mathbb{Z}$, 
we obtain respectively the
definition of the homogeneous spaces $\dot{F}_{p,q}^{s}\left(
\mathbb{R}^{n}\right)$, $\dot{B}_{\infty,q}^{s}\left(  \mathbb{R}
^{n}\right)  $\ and $\dot{C}^{s}(\mathbb{R}^n)$.
\end{enumerate}
\end{definition}

Here is the main result of this paper.

\begin{theorem}
\label{221}Let $s\in\mathbb{R},\,1\leq p<+\infty,\,1\leq q<+\infty$ and
$\lambda\geq0$. We have the following continuous embeddings
\[
\mathcal{L}_{p,q}^{\lambda,s+\frac{n}{p}-\frac{\lambda}{q}}\left(
\mathbb{R}^{n}\right)  \hookrightarrow C^{s}(\mathbb{R}^n)
\]
\[
F_{p,q}^{s+\frac{n}{p}}(\mathbb{R}^n)  \hookrightarrow
\mathcal{L}_{p,q}^{\lambda,s+\frac{n}{p}-\frac{\lambda}{q}}\left(
\mathbb{R}^{n}\right)  \text{ provided }q\geq p
\]
\[
F_{p,q}^{s+\frac{n}{p}}(\mathbb{R}^n)  \hookrightarrow
\mathcal{L}^{p,\lambda,s-\frac{\lambda-n}{p}}(\mathbb{R}^n)
\text{ provided }p\geq q.
\]
\[
B_{\infty,q}^{s-\frac{n}{p}+\frac{\lambda}{q}}(\mathbb{R}^n)
\hookrightarrow\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^n)
\text{ provided }\lambda\geq n\frac{q}{p}
\]
and finally if $\lambda\geq n$ then
\[
{\sum_{j\geq0}}
2^{jq(s+\frac{\lambda-n}{q})}|  \Delta_{j}u|^q \in L^{\infty
}(\mathbb{R}^{n})\text{ implies }u\in\mathcal{L}^{q,\lambda,s}\left(
\mathbb{R}^{n}\right)
\]
We have also the same continuous embeddings if we replace $B,C,F$ and
$\mathcal{L}$ respectively by the dotted spaces 
$\dot{B},\dot{C},\dot{F}$ and $\dot{\mathcal{L}}$.
\end{theorem}

\begin{remark}\label{strich} \rm 
(i) In the case $s=0$, S. Campanato \cite{c1,c2} 
showed that if $n<\lambda<n+p$ we have
 $\mathcal{L}^{p,\lambda}\cong C^{\frac{\lambda-n}{p}}$ 
and $\mathcal{L}^{p,n+p}=Lip$ (cf. \cite{g} too).

(ii) If $\lambda=n$ and $p=q=2$, we obtain the theorem 3.4. of R. S.
Strichartz \cite{str}, namely $\overset{\cdot}{B}_{\infty,2}^{s}\left(
\mathbb{R}^{n}\right)  \hookrightarrow 
I^{s}(BMO)\hookrightarrow\overset
{\cdot}{C}^{s}(\mathbb{R}^n)  $, where $I^{s}$ is the Riesz
potential operator and $I^{s}(BMO)=\dot{\mathcal{L}}^{2,n,s}
(\mathbb{R}^{n})$, cf. \cite{e2}, BMO is defined modulo polynomials. We note that Theorem \ref{221} generalise Theorem 2.1 of \cite{e4}.

(iii) Choosing $s=0,\;p=q=2$ and $\lambda=n$ in the third embedding, we 
find a
result due to E. M. Stein and A. Zygmund \cite{sz}
\[
\dot{H}^{\frac{n}{2}}=\dot{F}_{2,2}^{\frac{n}{2}}\hookrightarrow
\dot{\mathcal{L}}^{2,n,0}(\mathbb{R}^{n})=BMO\text{ modulo polynomials}
\]
\end{remark}

\section{Properties of the spaces}

We start with some helpful lemmas needed in the further considerations.

\begin{lemma}\label{sup}
Let $1\leq p<+\infty$, and $A$ a real $<0$. If $(a_{j\nu})_{j,\nu}$
is a sequence of positive real numbers satisfying $(a_{j\nu})_{j}\in 
l^{p}$
for all $\nu\geq1$, then there exists a constant $C>0$ such that
\[
{\sum_{j\geq1}}\Big({\sum_{\nu\geq1}}
2^{\nu A}a_{j\nu}\Big)^{p}\leq C\sup_{\nu\geq1}
{\sum_{j\geq1}}a_{j\nu}^{p}.
\]
\end{lemma}

\paragraph{Proof}
Let $\frac{1}{p}+\frac{1}{q}=1$. Using H\"{o}lder's inequality we get
\begin{eqnarray*}
{\sum_{j\geq1}}
\Big({\sum_{\nu\geq1}}2^{\nu A}a_{j\nu}\Big)^{p}
&\leq&{\sum_{j\geq1}}\Big({\sum_{\nu\geq1}}
  2^{\frac\nu2A}(2^{\frac\nu2A}a_{j\nu})\Big)^{p}\\
&\leq&{\sum_{j\geq1}}\Big(  \Big({\sum_{\nu\geq1}}
2^{\frac\nu2Aq}\Big)^{p/q}
{\sum_{\nu\geq1}}2^{\frac\nu2Ap}a_{j\nu}^{p}\Big)  \\
&\leq& C{\sum_{j\geq1}}{\sum_{\nu\geq1}}
2^{\frac\nu2Ap}a_{j\nu}^{p}\\
&\leq& C{\sum_{\nu\geq1}}2^{\frac\nu2Ap}{\sum_{j\geq1}}
a_{j\nu}^{p}\leq C\sup_{\nu\geq1}{\sum_{j\geq1}}a_{j\nu}^{p}
\end{eqnarray*}
It is known that $\dot{\Delta}_{l}$ is uniformly bounded on
$L^{p}(\mathbb{R}^n)  $. On the other hand, on 
$L^{p}(B)  ,B\subset\mathbb{R}^{n}$, we have the following result. 
We denote $\alpha B$, $\alpha>0$, the ball with the same center 
$x_{0}$ as $B$ and of radius $\alpha r$, $r$ is the radius of $B$.

\begin{lemma} \label{pb}
For each integer $M>0$, there exists a constant $C_{M}>0$, such that
for any $J\in\mathbb{Z}$, $x_{0}\in \mathbb{R}^{n}$, 
for any ball $B$ centered at $x_{0}$ and of radius
$2^{-J}$, for any $l\in\mathbb{Z}$ and $u\in L^{p}\left(
\mathbb{R}^{n}\right)  ,1\leq p\leq+\infty$,
\[
\|  A_{l}u\|  _{L^{p}(B)}  \leq  C_{M}\Big\{  \|
u\|  _{L^{p}(2B)}+\sum_{\nu\geq-J+1}2^{-(l+\nu)M}\|
u\|  _{L^{p}(F_{\nu})}\Big\}
\]
 holds with $F_{\nu}=\left\{  x\in\mathbb{R}^{n};2^{\nu}\leq|
x-x_{0}|  \leq2^{\nu+1}\right\}  $ and $A_{l}$ denotes any product of
 $\Delta_{l}$, $S_{l}$ and the dotted operators.
\end{lemma}

\paragraph{Proof}
We give the proof for $A_l=\dot{\Delta}_{l}$. Remark that
\[
u=\chi_{2B}u+\sum_{\nu\geq-J+1}\chi_{F_{\nu}}u
\]
where $\chi_{\Omega}$ is the characteristic function of the set 
$\Omega$. Thus
\begin{eqnarray*}
\|  \dot{\Delta}_{l}u\|  _{L^{p}(B)} & \leq&\|
\dot{\Delta}_{l}\chi_{2B}u\|  _{L^{p}(B)}+\sum_{\nu
\geq-J+1}\|  \dot{\Delta}_{l}\chi_{F_{\nu}}u\|  _{L^{p}
(B)}\\
& \leq &C\|  u\|  _{L^{p}(2B)}+\sum_{\nu\geq-J+1}\|
\dot{\Delta}_{l}\chi_{F_{\nu}}u\|  _{L^{p}(B)},
\end{eqnarray*}
where $C$ is a constant independent of $l$ and $B$.
Now
\[
\dot{\Delta}_{l}\chi_{F_{\nu}}u(x)=2^{nl}\int_{F_{\nu}}u(y)\mathcal{F}
^{-1}\theta(2^{l}(x-y))dy
\]
For $x\in B$ and $y\in F_{\nu}$, we have $|  x-y|  \sim|
x_{0}-y|  \sim2^{\nu}$. Since $\,\theta\in\mathcal{S}\left(
\mathbb{R}^{n}\right)  $, for every integer $N$, there exists $C_{N}>0$ 
such that
\[
|  \mathcal{F}^{-1}\theta\left(  2^{l}\left(  x-y\right)  \right)
|  \leq C_{N}2^{-\left(  l+\nu\right)  N}
\]
We deduce
\begin{eqnarray*}
|  \dot{\Delta}_{l}\chi_{F_{\nu}}u(x)|   
&\leq& C_{N}2^{nl}2^{-\left(  l+\nu\right)  N}\int_{F_{\nu}}|  u(y)|  
dy\\
&  \leq& C_{N}2^{nl}2^{-\left(  l+\nu\right)  N}\|  u\|
_{L^{p}(F_{\nu})}|  F_{\nu}|^{1-\frac{1}{p}}\\
&  \leq& C_{N}2^{-\left(  l+\nu\right)  (N-n)}2^{-\nu\frac{n}{p}}\|
u\|  _{L^{p}(F_{\nu})}
\end{eqnarray*}
It follows
\begin{equation}
\|  \dot{\Delta}_{l}\chi_{F_{\nu}}u(x)\|  _{L^{p}\left(
B\right)  }\leq C_{N}2^{-\left(  l+\nu\right)  
(N-n)}2^{-\frac{n}{p}(\nu
+J)}\|  u\|  _{L^{p}(F_{\nu})}\nonumber
\end{equation}
Therefore
\[
\|  \dot{\Delta}_{l}u\|  _{L^{p}(B)}\leq C\|
u\|  _{L^{p}(2B)}+C_{N}\sum_{\nu\geq-J+1}2^{-\left(
l+\nu\right)  (N-n)}\|  u\|  _{L^{p}(F_{\nu})}
\]
Put $M=N-n>0$, and the proof is complete. \hfill$\diamondsuit$

\begin{remark} \rm
Note that
\[
|  \mathcal{F}^{-1}\theta\left(  2^{l}\left(  x-y\right)  \right)
|  \leq\|  \mathcal{F}^{-1}\theta\|  _{L^{\infty}
(\mathbb{R}^{n})}
\]
Thus we may improve the statement of the last lemma by replacing the 
term
$2^{-\left(  l+\nu\right)  M}$ by $inf(1,\,2^{-\left(  l+\nu\right)  
M}).$
\end{remark}

Here is a classical lemma \cite{b}.

\begin{lemma}\label{2100}
Let $\lambda>0$, $1\leq p\leq q\leq+\infty$. For any 
$u\in L^{p}(\mathbb{R}^n) $ with
$\mathop{\rm supp}\mathcal{F}u\subset\{  |  \xi|  \leq
\lambda\}  $ and for any multi-index $\alpha$, there exists $C_{\alpha
}>0$ such that
\[
\|  D_{x}^{\alpha}u\|  _{L^q (\mathbb{R}^{n})}\leq C_{\alpha
}\lambda^{|  \alpha|  +n(\frac{1}{p}-\frac{1}{q})}\|
u\|  _{L^{p}(\mathbb{R}^{n})}
\]
\end{lemma}

\begin{lemma}\label{210}
Let $R$ be a real $>1$; there exists $C>0$ such that for any
$\lambda>0$ and $1\leq p\leq+\infty$, for any 
$u\in L^{p}\left(\mathbb{R}^{n}\right)  $ with 
$\mathop{\rm supp}\mathcal{F}u\subset$ 
$\{\lambda\leq|\xi|\leq R\lambda\}$ and for any $k\in\mathbb{N}$
we have
\[
\|  u\|  _{L^{p}(\mathbb{R}^{n})}\leq C^{k}\lambda^{-k}
\sup_{|  \alpha|  =k}\|  D_{x}^{\alpha}u\|
_{L^{p}(\mathbb{R}^{n})}
\]
\end{lemma}

\paragraph{Proof}
The above lemma gives that $D_{x}^{\alpha}u\in L^{p}(\mathbb{R}^{n})$ 
for all multi-index $\alpha$. Let $\psi\in 
C_{0}^{\infty}(\mathbb{R}^{n})$, 
$\psi(\xi)=1$ on a neighbourhood of the annulus
$\{1\leq|  \xi|  \leq R\}$ and $\psi(\xi)=0$ on a
neighbourhood of $0$. Setting 
$\psi_{\lambda}(\xi)=\psi(\frac{\xi}{\lambda})$
and $\psi_{\lambda,j}=\frac{\xi_{j}}{|  \xi|^{2}}\psi_{\lambda}$
we obtain $\sum_{j=1}^{n}\xi_{j}\psi_{\lambda,j}(\xi)=1$ on a
neighbourhood of $\mathop{\rm supp}\mathcal{F}u$. We deduce
\begin{equation}
u=\sum_{j=1}^{n}F_{\lambda,j}\ast D_{x_{j}}u\quad \text{where }
F_{\lambda,j}=\mathcal{F}^{-1}\psi_{\lambda,j}\in\mathcal{S}(\mathbb{R}
^{n}) \label{8000}
\end{equation}
We have $F_{\lambda,j}(x)=\lambda^{n-1}F_{1,j}(\lambda x)$, $\|
F_{\lambda,j}\|  _{L^{1}}=\lambda^{-1}\|  F_{1,j}\|  _{L^{1}
}$ and using (\ref{8000}) we deduce lemma \ref{210} for $k=1$. For the 
general
case we do an induction on $k>1$. \hfill$\diamondsuit$

The following lemma yields the homogeneous decomposition modulo 
polynomials of
a distribution $u\in\mathcal{S}'(\mathbb{R}^{n})$ without taking in
account the condition $0\notin \mathop{\rm supp}\mathcal{F}u$ that we 
have 
met in remark \ref{ca}.

\begin{lemma}\label{2101}
For any $u\in\mathcal{S}'(\mathbb{R}^{n})$ we have the decomposition
\[
u={\sum_{j\in\mathbb{Z}}} \dot{\Delta}_{j}u
\quad \text{in }Z'(\mathbb{R}^{n}).
\]
\end{lemma}

\paragraph{Proof}
The series ${\sum_{j=k}^{+\infty}}
\dot{\Delta}_{j}u$ converges in $\mathcal{S}'(\mathbb{R}^{n})$
for all $k\in\mathbb{Z.}$ It suffices to show that for any $\psi\in
Z(\mathbb{R}^{n})$,
\[
\lim_{k\to-\infty}\Big\langle
{\sum_{j=k}^{+\infty}}
\dot{\Delta}_{j}u,\check{\psi}\Big\rangle _{\mathcal{S}
'\times\mathcal{S}}
=\Big\langle u,\check{\psi}\Big\rangle_{\mathcal{S}'\times\mathcal{S}}
\]
where $\check{\psi}(x)=\psi(-x)$ for $x\in\mathbb{R}^{n}$. Now
\[
{\sum_{j=k}^{+\infty}}
\dot{\Delta}_{j}u\,-u=
{\sum_{j=k}^{0}}
\dot{\Delta}_{j}u-\Delta_{0}u=\varphi(2^{-k}D_{x})u
\]
which is a smooth function. Thus
\[
(2\pi)^{n}\Big\langle{\sum_{j=k}^{+\infty}}
\dot{\Delta}_{j}u\,-u,\check{\psi}\Big\rangle _{\mathcal{S}
'\times\mathcal{S}}=\left\langle \mathcal{F}u,\varphi(2^{-k}
.)\mathcal{F}\psi\right\rangle _{\mathcal{S}'\times\mathcal{S}}
\]
Since $\mathcal{F}u\in\mathcal{S}'(\mathbb{R}^{n})$, there exists a
constant $C>0$ and an integer $N$ such that
\begin{eqnarray*}
\lefteqn{(2\pi)^{n}\Big|  \Big\langle{\sum_{j=k}^{+\infty}}
\dot{\Delta}_{j}u\,-u,\check{\psi}\Big\rangle _{\mathcal{S}
'\times\mathcal{S}}\Big|  }\\
&\leq& C\sup_{|  \alpha|+|  \beta|  \leq 
N}\,\sup_{\xi\in\mathbb{R}^{n}}|
\xi^{\alpha}D_{\xi}^{\beta}(\varphi(2^{-k}\xi)\mathcal{F}\psi(\xi))|\\
&\leq& C'\sup_{|  \alpha|  +|  \beta|  \leq N}\,\sup_{|  \xi|  \leq 
2^{k+1}}
{\sum_{\gamma\leq\beta}}
2^{k(|  \alpha|  -|  \beta-\gamma|  )}|  (D_{\xi
}^{\beta-\gamma}\varphi)(2^{-k}\xi)|  |  D_{\xi}^{\gamma
}\mathcal{F}\psi(\xi)|
\end{eqnarray*}
By the assumption $\psi\in\mathcal{Z(}\mathbb{R}^{n})$ and Taylor's
formula, for all nonnegative large integer $M$
\[
(2\pi)^{n}\Big|  \Big\langle{\sum_{j=k}^{+\infty}}
\dot{\Delta}_{j}u\,-u,\check{\psi}\Big\rangle _{\mathcal{S}
'\times\mathcal{S}}\Big|  \leq C2^{k(M-N)}\underset{k\to
-\infty}{\longrightarrow}0
\]
The proof is complete. \hfill$\diamondsuit$

Now we state the connexion between 
$\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})  $ and 
$\dot{\mathcal{L}}_{p,q}^{\lambda,s}(\mathbb{R}^n)  $.

\begin{lemma} \label{r}
Let $1\leq p,q<+\infty$, $\lambda\geq0$ and $s\in\mathbb{R}$.
\begin{enumerate}
\item[(i)] If the class of $u$ modulo $\mathcal{P}$ belongs to 
$\dot{\mathcal{L}}_{p,q}^{\lambda,s}(\mathbb{R}^n)  $ and if
$\Delta_{0}u\in L^{p}(\mathbb{R}^n)  $, then $u\in
\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^n)$.

\item[(ii)] $L^{p}(\mathbb{R}^{n})\cap\dot{\mathcal{L}}_{p,q}^{\lambda
,s}(\mathbb{R}^n)  \subset\mathcal{L}_{p,q}^{\lambda,s}\left(
\mathbb{R}^{n}\right)  $ with the same meaning as (i).

\item[(iii)] $\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^n)
\subset\dot{\mathcal{L}}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$ 
provided $s>0$.
\end{enumerate}
\end{lemma}

\begin{remark} \rm
It follows that if $s>0$ then
\[
L^{p}(\mathbb{R}^{n})\cap\dot{\mathcal{L}}_{p,q}^{\lambda,s}\left(
\mathbb{R}^{n}\right)  
=L^{p}(\mathbb{R}^{n})\cap\mathcal{L}_{p,q}^{\lambda
,s}(\mathbb{R}^n)  .
\]
\end{remark}

\paragraph{Proof of Lemma \ref{r}} 
(i) If $u\in\dot{\mathcal{L}}_{p,q}^{\lambda,s}\left(  \mathbb{R}
^{n}\right)  $, then $u\in\mathcal{S}'(\mathbb{R}^{n})$ and there
exists a constant $M>0$ such that for any ball $B$ of $\mathbb{R}^{n}$ 
with
radius $2^{-J}$ with  $J\in\mathbb{Z}$,
\begin{equation}
\frac{1}{|  B|^{\lambda/n}}
{\sum_{l\geq J}}
2^{lsq}\|  \overset{\cdot}{\Delta}_{l}u\|  _{L^{p}(B)}^q \leq
M<+\infty\,.\label{cj}
\end{equation}
Let $J\in\mathbb{Z}$ and $B$ be a ball of $\mathbb{R}^{n}$ with radius
$2^{-J}$. If $J\geq1$, then inequality (\ref{cj}) gives
\[
\dfrac1{|  B|^{\frac\lambda n}}
{\sum_{l\geq J^{+}}}
2^{lsq}\|  \Delta_{l}u\|  _{L^{p}(B)}^q \leq M
\]
If $J\leq0$, we have
\begin{align*}
\frac{1}{|  B|^{\lambda/n}}
{\sum_{l\geq J^{+}}}
2^{lsq}\|  \Delta_{l}u\|  _{L^{p}(B)}^q   &  =\frac{1}{|
B|^{\lambda/n}}\|  \Delta_{0}u\|  _{L^{p}(B)}^q +\\
\frac{1}{|  B|^{\lambda/n}}
{\sum_{l\geq1}}
2^{lsq}\|  \Delta_{l}u\|  _{L^{p}(B)}^q   &  \leq C2^{J\lambda
}\|  \Delta_{0}u\|  _{L^{p}(B)}^q +M\\
&  \leq C\|  \Delta_{0}u\|  _{L^{p}(\mathbb{R}^{n})}^q +M
\end{align*}
Since $\Delta_{0}u\in L^{p}(\mathbb{R}^n)  $, we obtain
\[
\frac1{|  B|^{\frac\lambda n}}{\sum_{l\geq J^{+}}}
2^{lsq}\|  \Delta_{l}u\|  _{L^{p}(B)}^q \leq C'M
\]
Hence $u\in\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^n) $.

\noindent (ii) follows from (i).

\noindent(iii) Let $u\in\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^n)$.
There exists a constant $M>0$ such that for any ball $B$ of 
$\mathbb{R}^{n}$
with radius $2^{-J}$ and $J\in\mathbb{Z}$,
\[
\frac{1}{|  B|^{\lambda/n}}
{\sum_{l\geq J^{+}}}
2^{lsq}\|  \Delta_{l}u\|  _{L^{p}(B)}^q \leq M<+\infty\,.
\]
We have to show (\ref{cj}). Let $J\in\mathbb{Z}$ and $B$ a ball of
$\mathbb{R}^{n}$ of radius $2^{-J}$.
If $J\geq1$, then $J^{+}=J$ and (\ref{cj}) is valid.
If $J\leq0\;($i.e. $J^{+}=0)$. We have
\begin{equation}
\begin{gathered}
\frac{1}{|  B|^{\lambda/n}}
{\sum_{l\geq J}}
2^{lsq}\|  \overset{\cdot}{\Delta}_{l}u\|  _{L^{p}(B)}^q 
=\frac{1}{|  B|^{\lambda/n}}
{\sum_{l=J}^{0}}
2^{lsq}\|  \overset{\cdot}{\Delta}_{l}u\|  _{L^{p}(B)}^q +\\
\frac{1}{|  B|^{\lambda/n}}
{\sum_{l\geq1}}
2^{lsq}\|  \overset{\cdot}{\Delta}_{l}u\|  _{L^{p}(B)}^q 
\leq\frac{1}{|  B|^{\lambda/n}}
{\sum_{l=J}^{0}}
2^{lsq}\|  \overset{\cdot}{\Delta}_{l}u\|  _{L^{p}(B)}^q +M
\end{gathered} \label{n1}
\end{equation}
Using the nonhomogeneous decomposition of $u$ we have for $l\leq0$
\[
\dot{\Delta}_{l}u=\sum_{k\geq0}\Delta_{k}\dot{\Delta}
_{l}u=\Delta_{0}\dot{\Delta}_{l}u+\Delta_{1}\dot{\Delta}_{l}u
\]
and then
\begin{equation}
\|  \dot{\Delta}_{l}u\|  _{L^{p}(B)  }
\leq\|  \Delta_{0}\dot{\Delta}_{l}u\|  _{L^{p}\left(
B\right)  }+\|  \Delta_{1}\dot{\Delta}_{l}u\|
_{L^{p}(B)  }. \label{n2}
\end{equation}
Lemma \ref{pb} gives for $M$ large,
\[
\|  \dot{\Delta}_{l}\Delta_{0}u\|  _{L^{p}(B)}^q  
\leq C_{M}\{\|  \Delta_{0}u\|  _{L^{p}(2B)}^q +
  2^{-(l-J)Mq}[\sum_{\nu\geq1}2^{-\nu M}\|  \Delta_{0}u\|
_{L^{p}(B_{\nu-J})}]^q \}
\]
where $B_{\nu-J}=2^{\nu+1}B$. Thus
\begin{eqnarray*}
\lefteqn{\frac{1}{|  B|^{\lambda/n}}{\sum_{l=J}^{0}}
2^{lsq}\|  \overset{\cdot}{\Delta}_{l}\Delta_{0}u\|  _{L^{p}
(B)}^q  }\\
&\leq&\frac{C}{|  B|^{\lambda/n}}\| \Delta_{0}u\|  _{L^{p}(2B)}^q 
{\sum_{l=J}^{0}}2^{lsq}+ \frac{1}{|  B|^{\lambda/n}}
{\sum_{l=J}^{0}}2^{lsq}\Big(  \sum_{\nu\geq1}2^{-\nu M}\|  
\Delta_{0}u\|_{L^{p}(B_{\nu-J})}\Big)^q 
\end{eqnarray*}
Since $s>0$, ${\sum_{l=J}^{0}}2^{lsq}\leq C$
\begin{equation}
\begin{aligned}
\frac{1}{|  B|^{\lambda/n}}
{\sum_{l=J}^{0}}
2^{lsq}\|  \overset{\cdot}{\Delta}_{l}\Delta_{0}u\|  _{L^{p}
(B)}^q \leq\frac{C}{|  B|^{\lambda/n}}\|
\Delta_{0}u\|  _{L^{p}(2B)}^q &\\
+C\Big(  \sum_{\nu\geq1}2^{\nu(-M+\frac{\lambda}{q})}\frac{1}{|
B_{\nu-J}|^{\frac{\lambda}{nq}}}\|  \Delta_{0}u\|
_{L^{p}(B_{\nu-J})}\Big)^q \\
\leq
CM+C\sup_{\nu}\frac{1}{|  B_{\nu-J}|^{\lambda/n}
}\|  \Delta_{0}u\|  _{L^{p}(B_{\nu-J})}^q &\leq CM
\end{aligned}
\label{n3}
\end{equation}
In the same way we have
\begin{equation}
\frac{1}{|  B|^{\lambda/n}}
{\sum_{l=J}^{0}}
2^{lsq}\|  \overset{\cdot}{\Delta}_{l}\Delta_{1}u\|  _{L^{p}
(B)}^q \leq CM \label{n4}
\end{equation}
 From (\ref{n1}), (\ref{n2}), (\ref{n3}) and (\ref{n4}) it follows
\[
\frac{1}{|  B|^{\lambda/n}}
{\sum_{l\geq J}}
2^{lsq}\|  \overset{\cdot}{\Delta}_{l}u\|  _{L^{p}(B)}^q \leq CM
\]
and then $u\in\dot{\mathcal{L}}_{p,q}^{\lambda,s}\left(  \mathbb{R}
^{n}\right)  $. \hfill$\diamondsuit$

Now we give the differential dimension of 
$\dot{\mathcal{L}}_{p,q}^{\lambda,s}(\mathbb{R}^n)$.

\begin{proposition}
For any $u\in\dot{\mathcal{L}}_{p,q}^{\lambda,s}\left(  \mathbb{R}
^{n}\right)  $ and any $t>0$, we have
\[
\|  u(t^{-1}\cdot)\|  _{\dot{\mathcal{L}}_{p,q}^{\lambda
,s}(\mathbb{R}^n)  }\approx t^{d}\|  u\|
_{\dot{\mathcal{L}}_{p,q}^{\lambda,s}(\mathbb{R}^n)  }
\]
where $d=\frac{n}{p}-\frac{\lambda}{q}-s$.
\end{proposition}

\paragraph{Proof}
Let $t=2^{N}$, $N\in\mathbb{Z}$ and set $v(x)=u(\frac{x}{2^{N}})$. From
\[
\overset{\cdot}{\Delta}_{l}v(x)=(\overset{\cdot}{\Delta}_{l+N}u)(\dfrac
{x}{2^{N}})
\]
we get
\[
\|  \overset{\cdot}{\Delta}_{l}v\|  _{L^{p}(B)}=2^{N\frac{n}{p}
}\|  \overset{\cdot}{\Delta}_{l+N}u\|  _{L^{p}(2^{-N}B)}
\]
where $B$ is a ball of $\mathbb{R}^{n}$ with radius $2^{-J}$, 
$J\in\mathbb{Z}$.
Then we have
\begin{align*}
\|  v\|  _{\dot{\mathcal{L}}_{p,q}^{\lambda,s}\left(
\mathbb{R}^{n}\right)  }^q   &  \approx\sup_B 2^{J\lambda}
{\sum_{l\geq J}}
2^{lsq}2^{N\frac{n}{p}q}\|  \overset{\cdot}{\Delta}_{l+N}u\|
_{L^{p}(2^{-N}B)}^q \\
&  \approx2^{N\frac{n}{p}q}2^{-N\lambda}2^{-Nsq}\sup_B 2^{(J+N)\lambda}
{\sum_{l+N\geq J+N}}
2^{(l+N)sq}\|  \overset{\cdot}{\Delta}_{l+N}u\|  _{L^{p}(2^{-N}
B)}^q \\
&  \approx2^{(\frac{n}{p}-\frac{\lambda}{q}-s)qN}\sup_B 2^{J\lambda}
{\sum_{l\geq J}}
2^{lsq}\|  \overset{\cdot}{\Delta}_{l}u\|  _{L^{p}(B)}^q 
\end{align*}

\begin{lemma}\label{2102}
Let $1\leq p\leq p'<+\infty$, $1\leq q'\leq
q<+\infty$ and $s\in\mathbb{R}$. Further, let $\lambda$ and $\mu\geq0$ 
such
that $\frac{n}{p'}-\frac{\mu}{q'}\geq\frac{n}{p}-\frac
{\lambda}{q}$. Then we have the continuous embedding
\[
\mathcal{L}_{p',q'}^{\mu,s+\frac{n}{p'}-\frac{\mu
}{q'}}(\mathbb{R}^{n})\hookrightarrow\mathcal{L}_{p,q}^{\lambda
,s+\frac{n}{p}-\frac{\lambda}{q}}(\mathbb{R}^{n})\,.
\]
We also have the same result for the dotted spaces $\dot{\mathcal{L}}$.
\end{lemma}

In particular, if $p=p'$ and $q=q'$ then $\mathcal{L}
_{p,q}^{\mu,s-\frac{\mu}{q}}(\mathbb{R}^{n})\hookrightarrow\mathcal{L}
_{p,q}^{\lambda,s-\frac{\lambda}{q}}(\mathbb{R}^{n})$ holds for all 
$\mu
\leq\lambda$. Furthermore if $p=p'=q=q'$ we get $\mathcal{L}
^{p,\lambda,s+\frac{\alpha}{p}}(\mathbb{R}^{n})\hookrightarrow\mathcal{L}
^{p,\lambda+\alpha,s}(\mathbb{R}^{n})$ for all $\alpha\geq0$ and 
$\lambda\geq0$.

\paragraph{Proof}
Let $u\in\mathcal{L}_{p',q'}^{\mu,s+\frac{n}{p'}
-\frac{\mu}{q'}}(\mathbb{R}^{n})$. Let $B$ be a ball of $\mathbb{R}
^{n}$ with radius $2^{-J}$, $J\in\mathbb{Z}$. Since $p\leq p'$ and
$\frac{q}{q'}\geq1$ we obtain
\begin{eqnarray*}
\lefteqn{\frac1{|  B|^{\frac\lambda n}}
{\sum_{l\geq J^{+}}}
2^{l(s+\frac np-\frac\lambda q)q}\|  \Delta_{l}u\|  _{L^{p}(B)} ^q }\\
&\leq&\dfrac1{|  B|^{\frac\lambda n}}
{\sum_{l\geq J^{+}}}
2^{l(s+\frac np-\frac\lambda q)q}\left(  \|  \Delta_{l}u\|
_{L^{p'}(B)}|  B|^{\frac1p-\frac1{p'}}\right) ^q \\
&\leq&\frac1{|  B|^{\frac\lambda n}}
{\sum_{l\geq J^{+}}}
\left(  2^{l(s+\frac np-\frac\lambda q)q'}\|  \Delta
_{l}u\|  _{L^{p'}(B)}^{q'}\right)^{q/q'}|  B|^{q\left(  
\frac1p-\frac1{p'}\right)  }\\
&\leq&\frac1{|  B|^{\frac\lambda n}}\Big({\sum_{l\geq J^{+}}}
2^{l(s+\frac np-\frac\lambda q)q'}\|  \Delta_{l}u\|
_{L^{p'}(B)}^{q'}\Big)^{q/q'}|
B|^{q\left(  \frac1p-\frac1{p'}\right)  }
\end{eqnarray*}
Now $|  B|  \sim2^{-nJ}$, $l\geq J^{+}\geq J$ and the assumption
$\frac n{p'}-\frac\mu{q'}\geq\frac np-\frac\lambda q$ yield
\begin{eqnarray*}
\lefteqn{\Big\{  \dfrac1{|  B|^{\frac\lambda n}}
{\sum_{l\geq J^{+}}}
2^{l(s+\frac np-\frac\lambda q)q}\|  \Delta_{l}u\|  _{L^{p}(B)}
^q \Big\}^{1/q}}\\
&\leq&\Big(  \dfrac1{|  B|^{\frac\mu n}}
{\sum_{l\geq J^{+}}}
2^{l(s+\frac np-\frac\lambda q)q'}\|  \Delta_{l}u\|
_{L^{p'}(B)}^{q'}\Big)^{1/q'}|
B|^{\frac1p-\frac1{p'}}|  B|^{\frac\mu
{nq'}-\frac\lambda{nq}}\\
&\leq& C\Big(  \frac1{|  B|^{\frac\mu n}}
{\sum_{l\geq J^{+}}}
2^{J\left(  (\frac n{p'}-\frac\mu{q'})-(\frac np-\frac\lambda
q)\right)  q'}2^{l(s+\frac np-\frac\lambda q)q'}\|
\Delta_{l}u\|  _{L^{p'}(B)}^{q'}\Big)^{1/q'}\\
&\leq& C\Big(  \dfrac1{|  B|^{\frac\mu n}}
{\sum_{l\geq J^{+}}}2^{l(s+\frac n{p'}-\frac\mu{q'})q'}\|  \Delta
_{l}u\|  _{L^{p'}(B)}^{q'}\Big)^{1/q'}
\end{eqnarray*}
The proof of lemma \ref{2102} is complete. \hfill$\diamondsuit$

\begin{lemma} \label{derivation}
The derivation $D_{x}^{\alpha}$ is a bounded operator from
$\mathcal{L}_{p,q}^{\lambda,s+|  \alpha|  }\left(  \mathbb{R}
^{n}\right)  $ to $\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^n)
$ and from $\dot{\mathcal{L}}_{p,q}^{\lambda,s+|  \alpha|
}(\mathbb{R}^n)  $ to $\dot{\mathcal{L}}_{p,q}
^{\lambda,s}(\mathbb{R}^n)$.
\end{lemma}

\paragraph{Proof}
Let $|\alpha|=1$. We have
\begin{align*}
\|  D_{x}u\|  _{\mathcal{L}_{p,q}^{\lambda,s}\left(  \mathbb{R}
^{n}\right)  }^q   &  =\|  D_{x}
{\sum_{l\geq0}}
\Delta_{l}u\|  _{\mathcal{L}_{p,q}^{\lambda,s}\left(  \mathbb{R}
^{n}\right)  }^q \\
&  =\sup_B \frac{1}{|  B|^{\lambda/n}}
{\sum_{j\geq J^{+}}}2^{jsq}\|
{\sum_{l=j-1}^{j+1}}D_{x}\Delta_{j}\Delta_{l}u\|  _{L^{p}(B)}^q \\
&  \leq C\sup_B \frac{1}{|  B|^{\lambda/n}}
{\sum_{j\geq J^{+}}}2^{jsq}
{\sum_{l\sim j}}\|  D_{x}\Delta_{j}\Delta_{l}u\|  _{L^{p}(B)}^q 
\end{align*}
(in the case of the dotted spaces we use lemma \ref{2101}).
Let $B$ a ball of $\mathbb{R}^{n}$ of radius $2^{-J},J\in\mathbf{Z.}$ 
Set
$f_{l}=\Delta_{l}u$. Note that
\[
f_{l}=\chi_{2B}f_{l}+\sum_{\nu\geq-J+1}\chi_{F_{\nu}}f_{l}
\]
and use lemma \ref{2100} to get
\begin{equation}
\begin{aligned}
\|  D_{x}\Delta_{j}f_{l}\|  _{L^{p}(B)  }  \leq &
\|  D_{x}\Delta_{j}\chi_{2B}f_{l}\|  _{L^{p}(B)
}+\sum_{\nu\geq-J+1}\|  D_{x}\Delta_{j}\chi_{F_{\nu}}
f_{l}\|  _{L^{p}(B)  }\\
 \leq &  C2^{j}\|  \Delta_{j}\chi_{2B}f_{l}\|  _{L^{p}\left(
\mathbb{R}^{n}\right)  }+\sum_{\nu\geq-J+1}\|  D_{x}\Delta_{j}
\chi_{F_{\nu}}f_{l}\|  _{L^{p}(B)  }\\
\leq &  C2^{j}\|  f_{l}\|  _{L^{p}\left(  2B\right)  }
+\sum_{\nu\geq-J+1}\|  D_{x}\Delta_{j}\chi_{F_{\nu}}f_{l}\|
_{L^{p}(B)  }
\end{aligned}
\label{2251a}
\end{equation}
On the other hand,
\[
D_{x}\Delta_{j}\chi_{F_{\nu}}f_{l}\left(  x\right)
=2^{(n+1)j}\int_{F_{\nu}}f_{l}\left(  y\right)
\psi\left(  2^{j}(x-y)\right)  dy
\]
where\ $\psi=D_{x}\mathcal{F}^{-1}\theta$.
Therefore, if $x\in B$ and $y\in F_{\nu},\nu\geq-J+1$, then $|x-y|\sim
|x_{0}-y|\sim2^{\nu}$. Since $\psi\in\mathcal{S}(\mathbb{R}^{n})$, for 
each
integer $N$ there exists a constant $C_{N}$ such that
\begin{align*}
|  D_{x}\Delta_{j}\chi_{F_{\nu}}f_{l}\left(  x\right)  |   &  \leq
C_{N}2^{j\left(  n+1-N\right)  }2^{-\nu N}
{\int_{F_{\nu}}}
|f_{l}(y)|\,dy\\
&  \leq C_{N}2^{j\left(  n+1-N\right)  }2^{-\nu N}\|  f_{l}\|
_{L^{p}\left(  F_{\nu}\right)  }|F_{\nu}|^{1-\frac{1}{p}}
\end{align*}
We deduce
\[
\|  D_{x}\Delta_{j}\chi_{F_{\nu}}f_{l}\|  _{L^{p}(B)
}\leq C_{N}2^{j\left(  n+1-N\right)  }|  B|^{1/p}
2^{\nu\left(  -N+\frac{\lambda}{q}-\frac{n}{p}+n\right)  }\frac{1}{|
F_{\nu}|^{\frac{\lambda}{nq}}}\|  f_{l}\|  _{L^{p}\left(
F_{\nu}\right)  }
\]
Thus
\begin{eqnarray}
\lefteqn{\sum_{\nu\geq-J+1}\|  D_{x}\Delta_{j}\chi_{F_{\nu}}f_{l}\|
_{L^{p}(B)  }  }\nonumber\\
&\leq& C_{N}2^{j\left(  n+1-N\right)  }
2^{-J\frac{n}{p}}\sum_{\nu\geq-J+1}2^{\nu\left(  -N+\frac{\lambda}
{q}-\frac{n}{p}+n\right)  }\frac{\|  f_{l}\|  _{L^{p}\left(
F_{\nu}\right)  }}{|  F_{\nu}|^{\frac{\lambda}{nq}}}\label{vb}\\
&  \leq & C2^{j}2^{(j-J)(n-N)}2^{-J\frac{\lambda}{q}}\sum_{\nu\geq
1}2^{\nu\left(  -N+\frac{\lambda}{q}-\frac{n}{p}+n\right)  }\frac{\|
f_{l}\|  _{L^{p}\left(  F_{\nu-J}\right)  }}{|  F_{\nu-J}|
^{\frac{\lambda}{nq}}}\nonumber
\end{eqnarray}
Choose $N$ large and use inequalities (\ref{2251a}), (\ref{vb}) and 
lemma
\ref{sup} to deduce
\begin{eqnarray*}
\lefteqn{\frac{1}{|  B|^{\lambda/n}}{\sum_{j\geq J^{+}}}
2^{jsq}{\sum_{l\sim j}}
\|  D_{x}\Delta_{j}\Delta_{l}u\|  _{L^{p}(B)}^q }\\
&\leq& C\frac{1}{|  B|^{\lambda/n}}\sum_{j\geq J^{+}
}2^{j(s+1)q}\|  f_{j}\|  _{L^{p}\left(  2B\right)  }^q \\
&&+ C_{N}\sum_{j\geq J^{+}
}\Big\{  \sum_{\nu\geq1}2^{\nu\left(  -N+\frac{\lambda}{q}-\frac{n}
{p}+n\right)  }\frac{2^{j(s+1)}}{|  F_{\nu-J}|^{\frac{\lambda
}{nq}}}\|  f_{j}\|  _{L^{p}\left(  F_{\nu-J}\right)  }\Big\}^q \\
&\leq&
C\frac{1}{|  3B|^{\lambda/n}}\sum_{j\geq
(J-2)^{+}}2^{j(s+1)q}\|  \Delta_{j}u\|  _{L^{p}\left(  2B\right)}^q \\
&&+C_{N}'\sup_{\nu\geq1} \frac{1}{|  F_{\nu-J}|^{\lambda/n}}
\sum_{j\geq(J-\nu-1)^{+}}2^{j(s+1)q}\|  \Delta_{j}u\|
_{L^{p}\left(  F_{\nu-J}\right)}^q 
\end{eqnarray*}
Finally,
\[
\|  D_{x}u\|  _{\mathcal{L}_{p,q}^{\lambda,s}\left(  \mathbb{R}
^{n}\right)  }^q \leq C\|  u\|  _{\mathcal{L}_{p,q}^{\lambda
,s+1}(\mathbb{R}^n)  }^q 
\]
which completes the proof of lemma \ref{derivation}. 

\section{Proof of theorem \ref{221}}

For the first embedding, let $u\in\mathcal{L}_{p,q}^{\lambda,s+\frac{n}
{p}-\frac{\lambda}{q}}(\mathbb{R}^n)  $. Let $j\geq1$ and
$x\in\mathbb{R}^{n}$ be fixed. Let $\phi\in\mathcal{S}\left(  
\mathbb{R}
^{n}\right)  $ with $\mathcal{F}\phi$ $=1$ on 
$\frac{1}{2}\leq|\xi|  \leq2$.

 From 
$\mathcal{F}\Delta_{j}u(\xi)=\mathcal{F}(\Delta_{j}u)(\xi)\mathcal{F}
\phi(2^{-j}\xi)$ we deduce
\begin{equation}
\Delta_{j}u\left(  x\right)  =2^{nj}\int_{\mathbb{R}^{n}}\Delta_{j}
u(y)\phi(2^{j}(x-y))dy \label{conv}
\end{equation}
(in the case $j=0$, we assume $\mathcal{F}\phi$ $=1$ on 
$|\xi|\leq2$).
We decompose $\mathbb{R}^{n}$ into disjunctive cubes with the 
side-length
$2^{-j}$,
\[
Q_{\nu}=\left\{  y\in\mathbb{R}^{n} : \|  y-x-2^{-j}\nu\|
_{\infty}\leq2^{-j-1}\right\},
\]
 $\nu\in\mathbb{Z}^{n}$, and we set
$a_{\nu}=\sup_{\|  y-\nu\|  _{\infty}\leq\frac{1}{2}}|  \phi(y)|$
 which is a rapidly decreasing sequence. Then,
we have
\begin{eqnarray*}
|  \Delta_{j}u\left(  x\right)  |  & \leq & 2^{nj}\sum_{\nu}
\int_{Q_{\nu}}|  \Delta_{j}u(y)\phi(2^{j}(x-y))|  dy\\
& \leq & 2^{nj}\sum_{\nu}a_{\nu}\int_{Q_{\nu}}|  \Delta_{j}u(y)|dy\\
& \leq & 2^{nj}\sum_{\nu}a_{\nu}\|  \Delta_{j}u\|  _{L^{p}\left(
Q_{\nu}\right)  }|  Q_{\nu}|^{1-\frac1p}\\
& \leq &  C2^{nj}\sum_{\nu}a_{\nu}\Big(  2^{-j(s+\frac np-\frac\lambda
q)}\|  u\|  _{\mathcal{L}_{p,q}^{\lambda,s+\frac np-\frac\lambda
q}(\mathbb{R}^n)  }|  Q_{\nu}|^{\frac\lambda{nq}
}\Big)  |  Q_{\nu}|^{1-\frac1p}\\
& \leq &  C2^{nj}\sum_{\nu}a_{\nu}2^{-j(s+\frac np-\frac\lambda q)}\|
u\|  _{\mathcal{L}_{p,q}^{\lambda,s+\frac np-\frac\lambda q}\left(
\mathbb{R}^{n}\right)  }2^{-jn(\frac\lambda{nq}+1-\frac1p)}\\
& \leq &  C2^{-js}\|  u\|  _{\mathcal{L}_{p,q}^{\lambda,s+\frac
np-\frac\lambda q}(\mathbb{R}^n)  }.
\end{eqnarray*}
The first embedding is proved.

Now let $u\in F_{p,q}^{s+\frac{n}{p}}(\mathbb{R}^n)  $ and $B$
a ball of $\mathbb{R}^{n}$ centered at $x_{0}$ with radius $2^{-J}$,
$J\in\mathbf{Z}$. The assumption $\frac{q}{p}\geq1$ gives
\begin{eqnarray*}
\lefteqn{\frac{1}{|  B|^{\frac{\lambda}{nq}}}\Big\{
{\sum_{j\geq J^{+}}}
2^{jq(s+\frac{n}{p}-\frac{\lambda}{q})}\left(
{\int_B }|  \Delta_{j}u|^{p}dx\right)^{q/p}\Big\}^{1/q}}\\
&=&
\frac{1}{|  B|^{\frac{\lambda}{nq}}}\Big\|  \Big(
{\int_B }2^{jp(s+\frac{n}{p}-\frac{\lambda}{q})}|  \Delta_{j}u|
^{p}dx\Big)  _{j\geq J^{+}}\Big\|  _{l^{\frac{q}{p}}}^{1/p}\\
&\leq&
\frac{1}{|  B|^{\frac{\lambda}{nq}}}C\Big\{
{\int_{\mathbb{R}^{n}}}
\|  \left(  2^{jp(s+\frac{n}{p}-\frac{\lambda}{q})}|  \Delta
_{j}u|^{p}\right)  _{j\geq J^{+}}\|  _{l^{\frac{q}{p}}}dx\Big\}^{1/p}\\
&\leq&
C2^{J\frac{\lambda}{q}}\Big\{
{\int_{\mathbb{R}^{n}}}\Big({\sum_{j\geq J^{+}}}
2^{jq(s+\frac{n}{p}-\frac{\lambda}{q})}|  \Delta_{j}u|
^q \Big)^{p/q}\Big\}^{1/p}\\
&\leq&
C\Big\{
\int_{\mathbb{R}^{n}}\Big({\sum_{j\geq J^{+}}}
2^{J\lambda}2^{jq(s+\frac{n}{p}-\frac{\lambda}{q})}|  \Delta_{j}u|
^q \Big)^{p/q}\Big\}^{1/p}
\leq
C\|u\|  _{F_{p,q}^{s+\frac{n}{p}}(\mathbb{R}^n)  }.
\end{eqnarray*}
Therefore the second embedding yields true.

For the third embedding, use $\frac pq\geq1$ to obtain
\begin{eqnarray*}
\dfrac1{|  B|^{\frac\lambda n}}{\int_B }{\sum_{j\geq J^{+}}}
2^{jp(s- \frac{\lambda-n}p)}|  \Delta_{j}u|^{p}dx 
&=& \dfrac1{|  B|^{\frac\lambda n}} {\int_B }{\sum_{j\geq J^{+}}}
\Big(  2^{jq(s- \frac{\lambda-n}p)}|  \Delta_{j}u|^q \Big)^{p/q}dx\\
&\leq& 
C2^{J\lambda}{\int_B }\Big({\sum_{j\geq J^{+}}}
2^{jq(s- \frac{\lambda-n}p)}|  \Delta_{j}u|^q \Big)^{p/q}dx\\
&\leq&
{\int_B }\Big({\sum_{j\geq J^{+}}}2^{J\lambda\frac qp}
2^{jq(s- \frac{\lambda-n}p)}|  \Delta_{j}u|^q \Big)^{p/q}dx\\
&\leq&
C{\int_{\mathbb{R}^{n}}}\Big({\sum_{j\geq J^{+}}}
2^{jq(s+\frac np)}|  \Delta_{j}u|^q \Big)^{p/q}dx\\
&\leq& C\|  u\|  _{F_{p,q}^{s+\frac np}(\mathbb{R}^n)}^{p}
\end{eqnarray*}
Let $B$ be a ball of $\mathbb{R}^{n}$ centered at $x_{0}$ with radius 
$2^{-J}$
for $J\in\mathbb{Z}$.
Let $u\in B_{\infty,q}^{s-\frac np+\frac\lambda q}(\mathbb{R}^{n})$. 
Then
\begin{eqnarray*}
\dfrac1{|  B|^{\frac\lambda n}}{\sum_{j\geq J^{+}}}
2^{jqs}\|  \Delta_{j}u\|  _{L^{p}(B)}^q 
&\leq& C2^{J\lambda} {\sum_{j\geq J^{+}}}
 2^{jqs}|  B|^{\frac qp}\|  \Delta_{j}u\|  _{L^{\infty}(B)}^q \\
 &\leq&
C{\sum_{j\geq J^{+}}}2^{jqs}2^{J(\lambda-n\frac qp)}
\|  \Delta_{j}u\|  _{L^{\infty}(B)}^q \\
&\leq& 
C{\sum_{j\geq0}}2^{jq(s-\frac np+\frac\lambda q)}
\|  \Delta_{j}u\|  _{L^{\infty}(B)}^q 
\end{eqnarray*}
Therefore the fourth injection is proved. 
For the last assertion, 
\begin{eqnarray*}
\frac{1}{|  B|^{\lambda/n}}{\sum_{j\geq J^{+}}}
2^{jqs}\|  \Delta_{j}u\|  _{L^q (B)}^q 
&\leq& C\frac{1}{|B|^{\lambda/n}}{\int_B }{\sum_{j\geq J^{+}}}
2^{jqs}|  \Delta_{j}u|^q dx\\
&\leq& 
C\sup_{x\in\mathbb{R}^{n}}{\sum_{j\geq J^{+}}}
2^{jqs}|  \Delta_{j}u(x)|^q |  B|^{-\frac{\lambda}{n}+1}\\
&\leq& C\sup_{x\in\mathbb{R}^{n}}{\sum_{j\geq0}}
2^{jq(s+\frac{\lambda-n}{q})}|  \Delta_{j}u|^q 
\end{eqnarray*}
and the proof of theorem \ref{221} is complete. \hfill$\diamondsuit$

Now we can state a partial result on the topological dual of 
$\overset{\circ}{\mathcal{L}}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$, 
the closure of Schwartz space $\mathcal{S}(\mathbb{R}^{n})$ in 
$\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$.

\begin{corollary}
\label{cor}Let $s\in\mathbb{R}$, $\lambda\geq 0$, $1\leq p<+\infty$,
$1\leq q<+\infty$, $1<p'\leq+\infty$ and $1<q'\leq+\infty$ with
$\frac{1}{p}+\frac{1}{p'}=1$, $\frac{1}{q}+\frac{1}{q'}=1$.
We have
\begin{equation}
F_{1,1}^{-s-\frac{\lambda}{q}+\frac{n}{p}}(\mathbb{R}^{n})\hookrightarrow
\overset{\circ}{(\mathcal{L}}_{p,q}^{\lambda,s}(\mathbb{R}^{n}))^{\prime
}\hookrightarrow F_{p',q'}^{-s-\frac{\lambda}{q}}
(\mathbb{R}^{n})\text{ provided }p\leq q. \label{d1}
\end{equation}
\begin{equation}
F_{1,1}^{-s-\frac{\lambda}{p}+\frac{n}{p}}(\mathbb{R}^{n})\hookrightarrow
(\overset{\circ}{\mathcal{L}}^{p,\lambda,s}(\mathbb{R}^{n}))^{\prime
}\hookrightarrow F_{p',q'}^{-s-\frac{\lambda}{p}}
(\mathbb{R}^{n})\text{ provided }q\leq p \label{d2}
\end{equation}
in particular
\[
F_{1,1}^{\frac{n}{2}-\frac{\lambda}{2}}(\mathbb{R}^{n})\hookrightarrow\left(
\overset{\circ}{\mathcal{L}}^{2,\lambda}(\mathbb{R}^{n})\right)^{\prime
}\hookrightarrow F_{2,q'}^{-\frac{\lambda}{2}}(\mathbb{R}^{n})\text{
for all }q'\geq2.
\]
We have also the same injections for the dotted spaces.
\end{corollary}

\paragraph{Proof}
We prove only (\ref{d2}).
Applying the first and the third embedding of theorem \ref{221} with
$s+\frac{\lambda-n}p$ instead of $s$ we obtain
\[
F_{p,q}^{s+\frac\lambda p}(\mathbb{R}^{n})\hookrightarrow\mathcal{L}
^{p,\lambda,s}(\mathbb{R}^{n})\hookrightarrow C^{s+\frac{\lambda-n}
p}(\mathbb{R}^{n})\equiv B_{\infty,\infty}^{s+\frac{\lambda-n}p}
(\mathbb{R}^{n})
\]
hence
\[
F_{p,q}^{s+\frac\lambda p}(\mathbb{R}^{n})\hookrightarrow\overset{\circ
}{\mathcal{L}}^{p,\lambda,s}(\mathbb{R}^{n})\hookrightarrow\overset{\circ}
{B}_{\infty,\infty}^{s+\frac{\lambda-n}p}(\mathbb{R}^{n})
\]
where 
$\overset{\circ}{B}_{\infty,\infty}^{s+\frac{\lambda-n}p}(\mathbb{R}
^{n})$ denotes the closure of $\mathcal{S}(\mathbb{R}^{n})$ in 
$B_{\infty,\infty}^{s+\frac{\lambda-n}p}(\mathbb{R}^{n})$.
Now 
\[
\Big(\overset{\circ}{B}_{\infty,\infty}^{s+\frac{\lambda-n}{p}
}(\mathbb{R}^{n})\Big)'
=F_{1,1}^{-s-\frac{\lambda}{p}+\frac{n}{p}
}(\mathbb{R}^{n})
\]
yields the result. \hfill$\diamondsuit$

\textbf{Acknowledgement}
The author wants to express his gratitude to Professors
Pascal Auscher, Gerard Bourdaud and Jacques Camus 
who read this manuscript and made many valuable corrections and
comments.

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\noindent\textsc{Azzeddine El Baraka} \\
Universit\'{e} Mohamed Ben Abdellah, FST Fes-Saiss\\
D\'{e}partement de Math\'{e}matiques\\
BP2202, Route Immouzer\\
Fes 30000 Morocco\\
e-mail: aelbaraka@yahoo.com
\end{document}