\documentclass[twoside]{article}
\usepackage{amsfonts, amsmath} % used for R in Real numbers
\pagestyle{myheadings}

\markboth{Function spaces of  $BMO$ and Campanato type }
{ Azzeddine El Baraka }

\begin{document}
\setcounter{page}{109}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
2002-Fez conference on Partial Differential Equations,\newline
Electronic Journal of Differential Equations,
Conference 09, 2002, pp 109--115. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
%
  Function spaces of  $BMO$ and Campanato type
%
\thanks{ {\em Mathematics Subject Classifications:} 46E35, 46B70.
\hfil\break\indent
{\em Key words:} $BMO$-space, Campanato spaces, Real interpolation, Sobolev embeddings.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Published December 28, 2002.} }

\date{}
\author{Azzeddine El Baraka}
\maketitle

\begin{abstract}
  To obtain the Littlewood-Paley characterization for Campanato spaces
  $\mathcal{L}^{2,\lambda}$ modulo polynomials (which contain as special case the John and Nirenberg space $BMO$), we define and study a
  scale of function spaces on $\mathbb{R}^{n}$. We discuss the real
  interpolation of these spaces and some embeddings between these spaces
  and the classical spaces. These embeddings cover some classical results
  obtained by  Campanato, Strichartz, Stein and Zygmund.
\end{abstract}


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

\section{Introduction}

In this work, we introduce and study a scale of function spaces on
$\mathbb{R}^{n}$. The homogeneous version of these spaces contains Campanato
spaces $\mathcal{L}^{2,\lambda}$ and John and Nirenberg space
$BMO=\mathcal{L}^{p,n}$. It is classical that the homogeneous space of
Triebel-Lizorkin $\dot{F}_{p,q}^{s}(\mathbb{R}^{n})$ coincides with $BMO$ modulo
polynomials for some values of $p,q$ and $s$. Namely, $BMO=\dot %
{F}_{\infty,2}^{0}$ \cite[chapter 5]{tr} and
$I^{s}(BMO)=\dot {F}_{\infty,2}^{s}$, where
$I^{s}=\mathcal{F}^{-1}(|.|^{-s}\mathcal{F}) $ is the Riesz potential operator.
The spaces $I^{s}(BMO)$ were studied by Strichartz \cite{str}. We use a
Littlewood-Paley partition to define these spaces denoted by
$\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$ and their homogeneous version
$\dot {\mathcal{L}}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$. These spaces allow us
to give the Littlewood-Paley characterization of Campanato spaces
$\mathcal{L}^{2,\lambda}$ and more generally of
$I^{s}(\mathcal{L}^{2,\lambda})$ modulo
polynomials (cf. Theorem \ref{223a}). If we denote $L_{p}^{s}$ the local
approximation Campanato spaces defined for instance in the book
\cite[Definition 1.7.2. (5)]{tr2}
 for $s\geq-n/p$ and $1\leq p<+\infty$,
then we recall that $L_{p}^{s}=C^{s}$ for any $s>0$, $L_{p}^{-n/p}=L^{p}$ and
$L_{p}^{0}=bmo$ the local version of $BMO$, cf. \cite{c2}, \cite{jn},
\cite{wa} and \cite{tr2} for the proof and more references. The spaces of
Campanato $\mathcal{L}^{p,\lambda}$\ considered here (Definition \ref{camp})
coincide with the local approximation Campanato spaces $L_{p}^{s}$
with$\;s=(\lambda-n)/p$ for $-\frac{n}{p}<s<0$ (ie. $0<\lambda<n$)
which are themselves equal to Morrey spaces. The characterization given here
is of interest for $L_{2}^{s}$ spaces in the case $-n/2<s<0$.

Next we give a result concerning the real interpolation of these spaces, and
we extend some injections due to  Strichartz \cite{str} and  Stein
and  Zygmund \cite{sz} by showing some embeddings between the spaces
$\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$\ and Triebel-Lizorkin ones
$F_{p,q}^{s}(\mathbb{R}^{n})$ and Besov-Peetre ones
$B_{p,q}^{s}(\mathbb{R}^{n})$, and on the other hand 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}).
To define the spaces we will need the following partition of unity: we denote
$x\in\mathbb{R}^{n}$ and $\xi$ its dual variable.\ Let $\varphi\in
C_{0}^{\infty}( \mathbb{R}^{n})$, $\varphi\geq0$, $\varphi$
equal to $1$ on $|\xi| \leq1$, and equal to $0$ on $|\xi | \geq2$.
Let $\theta(\xi)=\varphi(\xi)-\varphi(2\xi)$,
$\mathop{\rm \mathop{\rm supp}}\theta\subset\{\frac{1}{2}\leq| \xi| \leq2\}$. For
$j\in\mathbf{Z}$ we set $\dot {\Delta}_{j}u=\theta(2^{-j}D_{x})u$,
$\Delta_{0}u=\varphi(D_{x})u$ and if $j\geq1$ we set also
$\Delta_{j}u=\dot {\Delta}_{j}u$.

\begin{remark} \label{ca} \rm
We recall that if $u\in\mathcal{S}'(\mathbb{R}^{n})$ then
$u=\sum_{k\geq0}\Delta_{k}u$ and $u=\sum_{k\in\mathbf{Z}}\dot {\Delta}_{k}u$ modulo polynomials.
\end{remark}

Now we give the definition of the nonhomogeneous spaces.

\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}'(\mathbb{R}^{n}) $ 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{11z}%
\end{equation}
where $J^{+}=\max( J,0) $, $| B| $ is the measure of
$B$ and the supremum is taken over all $J\in\mathbf{Z}$ and all balls $B$ of
$\mathbb{R}^{n}$ of radius $2^{-J}$.
\end{definition}

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

To define the homogeneous spaces we recall the notation of \cite{tr}:
$Z'( \mathbb{R}^{n}) :=\mathcal{S}'(
\mathbb{R}^{n}) /\mathcal{P}$ is the space of all tempered
distributions modulo the set $\mathcal{P}$ of polynomials of $\mathbb{R}^{n}$
with complex coefficients.

\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'(
\mathbb{R}^{n}) $ such that
\begin{equation}
\| u\| _{\dot {\mathcal{L}}_{p,q}^{\lambda,s}
(\mathbb{R}^{n})}=\Big( \sup_B \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 $J\in\mathbf{Z}$ and all balls $B$ of
$\mathbb{R}^{n}$ of radius $2^{-J}$\textbf{.}
\end{definition}

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 $\mathcal{P(}\mathbb{R}^{n}\mathcal{)}$ and
$u\in\mathcal{S}'\mathcal{(}\mathbb{R}^{n}\mathcal{)}$, it follows
immediately 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.

Now we recall the definition of Campanato spaces and $BMO$.

\begin{definition} \label{camp}\rm
Let $\lambda\geq0$ and $1\leq p<+\infty$.
(i) We say $u\in\mathcal{L}^{p,\lambda}(\mathbb{R}^{n})$ if $u\in L_{\rm loc}%
^{p}( \mathbb{R}^{n}) $ and
\[
\| u\| _{\mathcal{L}^{p,\lambda}(\mathbb{R}^{n})}=\Big(
\sup_{B}\frac{1}{| B| ^{\lambda/n}}\int_{B}|
u-m_{B}u| ^{p}dx\Big) ^{1/p}<+\infty
\]
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 the balls $B$ of $\mathbb{R}^{n}$.
The space $\mathcal{L}^{p,\lambda}(\mathbb{R}^{n})$ is a Banach space modulo
constants and is equal to $\{0\}$ for $\lambda>n+p$.\\ 
Let us denote $BMO$ the space $\mathcal{L}^{2,n}(\mathbb{R}^{n})$.  Note that $BMO$ is equal to $\mathcal{L}^{p,n}(\mathbb{R}^{n})$ for any $1\leq p<+\infty$, cf. \cite{jn}.

\noindent (ii) For $0\leq\lambda<n+p$, we define the space
$\dot {\mathcal{L}}^{p,\lambda}(\mathbb{R}^{n})$ as the set of all
equivalence classes modulo
$\mathcal{P}$ of elements of $\mathcal{L}^{p,\lambda}(\mathbb{R}^{n})$,
equipped with the norm
$\| U\| _{\dot {\mathcal{L}}^{p,\lambda}(\mathbb{R}^{n})}
=\| u\| _{\mathcal{L}^{p,\lambda}(\mathbb{R}^{n})}$ where $u$ is the
unique (modulo constants) element of
$\mathcal{L}^{p,\lambda}(\mathbb{R}^{n})$ belonging to the class $U$.
\end{definition}

\section{Results}

The following proposition yields the dyadic characterization of
$\dot{\mathcal{L}}^{2,\lambda}(\mathbb{R}^{n})$.

\begin{proposition} \label{222}
Let $0\leq\lambda<n+2$. The space
$\dot {\mathcal{L}}^{2,\lambda,0}(\mathbb{R}^{n})$ coincides
algebraically and topologically with
Campanato space $\dot {\mathcal{L}}^{2,\lambda}(\mathbb{R}^{n})$.
\end{proposition}

This proposition allows us to deduce the link between the discrete scale built
on $\dot {\mathcal{L}}^{2,\lambda}(\mathbb{R}^{n})$ and the continuous scale.

\begin{corollary} \label{disc}
Let $0\leq\lambda<n+2$ and $m\in\mathbb{N}$.
\begin{itemize}
\item[(i)] $\dot {\mathcal{L}}^{2,\lambda,m}(\mathbb{R}^{n})\hookrightarrow
\dot {\mathcal{H}}^{2,\lambda,m}:=\{u\in\dot {\mathcal{L}
}^{2,\lambda}(\mathbb{R}^{n});\,D^{\alpha}u\in\dot {\mathcal{L}
}^{2,\lambda}(\mathbb{R}^{n}),| \alpha| \leq m\}$.
\item[(ii)] $\dot {\mathcal{H}}^{2,\lambda,m}\bigcap\dot {H}
^{m+\lambda/2}(\mathbb{R}^{n})\hookrightarrow\dot {\mathcal{L}
}^{2,\lambda,m}(\mathbb{R}^{n})$, here
$\dot {H}^{m+\lambda/2}(\mathbb{R}^{n})$ is the classical homogeneous
Sobolev space.
\end{itemize}
Therefore, $\dot {\mathcal{H}}^{2,\lambda,m}\bigcap
\dot {H}^{m+\lambda/2}(\mathbb{R}^{n})\equiv\dot {\mathcal{L}}^{2,\lambda
,m}(\mathbb{R}^{n})\bigcap\dot {H}^{m+\lambda/2}(\mathbb{R}^{n})$.
\end{corollary}

To state a more general result than the proposition \ref{222}, we recall the
definition of the Riesz potential operator
\[
I^{s}f=\mathcal{F}^{-1}\{| .| ^{-s}\mathcal{F}f\},\quad f\in
Z'(\mathbb{R}^{n})\text{ and } s\in\mathbb{R}
\]

\begin{theorem} \label{223a}
Let $s\in\mathbb{R}$ and $0\leq\lambda<n+2$. The space
$\dot {\mathcal{L}}^{2,\lambda,s}(\mathbb{R}^{n})$ coincides
algebraically and topologically with the space
$I^{s}( \dot{\mathcal{L}}^{2,\lambda}(\mathbb{R}^{n})) $ image of
$\dot{\mathcal{L}}^{2,\lambda}(\mathbb{R}^{n})$ under $I^{s}$.
\end{theorem}

\begin{remark} \label{rmk8} \rm
These results are not true in general for the spaces
$\dot {\mathcal{L}}^{p,\lambda}$, $p\neq 2$. For this, G. Bourdaud notes
that $\dot {\mathcal{L}}^{p,0,0}=\dot {B}_{p,p}^{0}$ for any
$1\leq p<+\infty$, and it is classical that $\dot {\mathcal{L}}^{p,0}=L^{p}$.
\end{remark}

The following lemma shows that these spaces are independent of the partition
$(\Delta_{j})_{j}$.

\begin{lemma} \label{w0}
Let $R>1$. Let $(u_{j})_{j\geq0}$ be a sequence of
$L_{\rm loc}^{p}(\mathbb{R}^{n})$ satisfying the following assumptions:
\begin{itemize}
\item[(i)] $\mathop{\rm supp}\mathcal{F}u_{0}\subset\{| \xi| \leq R\}$ and
$\mathop{\rm supp}\mathcal{F}u_{j}\subset\{\frac{1}{R}2^{j}\leq| \xi| \leq
R2^{j}\}$ for $j\geq1$.
\item[(ii)]
\[
M:=( \sup_{B}\frac{1}{| B| ^{\lambda/n}}
\sum_{j\geq J^{+}} 2^{jsq}\| u_{j}\| _{L^{p}( B) }^{q})
^{1/q}<+\infty
\]
where the supremum is taken over all $J\in\mathbf{Z}$ and all balls $B$ of
$\mathbb{R}^{n}$ of radius $2^{-J}$.
\end{itemize}
Then the series $\sum_{j}u_{j}$ converges in $\mathcal{S}'(\mathbb{R}^{n})$,
and its sum $u$ belongs to $\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$
with $\|u\| _{\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})}\leq CM$,
where the constant $C$ depends only from $s,p,n,R$ and the partition
$(\Delta_{j})_{j\geq0}$.
We have an analogous result for the dotted spaces.
\end{lemma}

\begin{corollary} \label{coro10}
The derivation $D_{x}^{\alpha}$ is a bounded operator from the sapce
$\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$ to the space
$\mathcal{L}_{p,q}^{\lambda,s-| \alpha| }(\mathbb{R}%
^{n})$ and from $\dot {\mathcal{L}}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$
to $\dot {\mathcal{L}}_{p,q}^{\lambda,s-| \alpha|
}(\mathbb{R}^{n})$. \end{corollary}

For this it suffices to note that $D_{x}^{\alpha}u=\sum_{j\geq0}\Delta
_{j}D_{x}^{\alpha}u=\sum_{j\geq0}2^{j|\alpha|}L_{j}u$, where $\mathcal{F}%
L_{j}u(\xi)=\theta_{\alpha}(2^{-j}\xi)\mathcal{F}u(\xi)$, with $\theta
_{\alpha}(\xi)=\xi^{\alpha}\theta(\xi)$. We apply lemma \ref{w0} then.

We can remove the spectral assumption (i) of lemma \ref{w0} by giving a result
dealing with the real interpolation of these spaces:

\begin{theorem}[Interpolation]\label{inter}
Let $N$, be an integer $\geq1$, $0<s<N$,
$\lambda\geq0$ and $p,q\in\lbrack1,+\infty\lbrack$. Let $(u_{j})_{j}$ be a
sequence of functions belonging to
$C^{\infty}( \mathbb{R}^{n}) \cap L_{\rm loc}^{p}(\mathbb{R}^{n}) $.
We assume that there is a sequence $(\varepsilon_{j}) _{j}$ $\in l^{q}$
such that for any ball $B$ of $\mathbb{R}^{n}$ of radius
$2^{-J},J\in\mathbf{Z,}$,
\begin{equation}
\| D_{x}^{\alpha}u_{j}\| _{L^{p}( B) }
\leq\varepsilon_{j}2^{j( | \alpha| -s) }
|B|^{\lambda/(qn)}\inf\{1,2^{-JN}\}\;\text{for any }j\geq0\text{ and
}|\alpha|\leq N \label{in1}
\end{equation}
Then, the series$\underset{j\geq0}{\sum}$ $u_{j}$ converges in
$L_{\rm loc}^{p}( \mathbb{R}^{n}) $ and its sum $u$ belongs to
$\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$ with
$$\| u\| _{\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})}\leq
C\| ( \varepsilon_{j}) _{j}\| _{l^{q}},
$$
where $C$ depends only on $N,s,p,n,\lambda$ and the partition defining the
norm of $\mathcal{L}_{p,q}^{\lambda,s}(\mathbb{R}^{n})$.
\end{theorem}

The following lemma gives the inclusion property among these spaces
in dependance of their parameters:

\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 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 any $\mu
\leq\lambda$. 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 any $\alpha\geq0$ and $\lambda\geq0$.
Now we give the connection 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{itemize}
\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}(
\mathbb{R}^{n}) $ 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{itemize}\end{lemma}

\begin{remark} \label{rmk14} \rm
It follows that if $s>0$ then
\[
L^{p}(\mathbb{R}^{n})\cap\dot {\mathcal{L}}_{p,q}^{\lambda,s}(
\mathbb{R}^{n}) =L^{p}(\mathbb{R}^{n})\cap\mathcal{L}_{p,q}^{\lambda
,s}( \mathbb{R}^{n}) .
\]
\end{remark}

Finally we give the connection between these spaces and the classical spaces.
For the definitions of the spaces $B_{p,q}^{s},C^{s},F_{p,q}^{s}$ and the
dotted ones we refer to \cite{tr}.

\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
\begin{gather*}
\mathcal{L}_{p,q}^{\lambda,s+\frac{n}{p}-\frac{\lambda}{q}}(
\mathbb{R}^{n}) \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}}(
\mathbb{R}^{n}) \quad \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})
\quad\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})
\quad \text{ provided }\lambda\geq n\frac{q}{p}
\end{gather*}
and finally
\[
\sum_{j\geq0}2^{jq(s+\frac{\lambda-n}{q})}| \Delta_{j}u| ^{q}\in L^{\infty
}(\mathbb{R}^{n})\text{ implies }u\in\mathcal{L}^{q,\lambda,s}(
\mathbb{R}^{n}) \text{ provided }\lambda\geq n
\]
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 \begin{itemize}
\item[(i)] These embeddings cover theorem 2.1 of \cite{ab} and theorem
3.4 of \cite{str} which asserts that $\overset{\cdot}{B}_{\infty,2}^{s}(
\mathbb{R}^{n}) \hookrightarrow I^{s}(BMO)\hookrightarrow\overset
{\cdot}{C}^{s}( \mathbb{R}^{n}) $, where $I^{s}$ is the Riesz
potential operator and $I^{s}(BMO)=\dot {\mathcal{L}}^{2,n,s}
(\mathbb{R}^{n})$, $BMO$ is defined modulo polynomials.

\item[(ii)] In the case $s=0$, S. Campanato \cite{c1} and \cite{c2}
showed that if
$n<\lambda<n+p$ we have
$\mathcal{L}^{p,\lambda}\cong C^{\frac{\lambda-n}{p}}$ and
$\mathcal{L}^{p,n+p}=Lip$ (we refer also to \cite{g}).

\item[(iii)] If we do $s=0,\;p=q=2$ and $\lambda=n$ in the third embedding,
then we find again a result due to  Stein and 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{itemize}\end{remark}

From this theorem we deduce 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 $s\in\mathbb{R},\,\lambda\geq0,\,1\leq p<+\infty,\,1\leq
q<+\infty,\,1<p'\leq+\infty$ 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}))'
\hookrightarrow F_{p',q'}^{-s-\frac{\lambda}{p}}%
(\mathbb{R}^{n})\text{ provided }q\leq p \label{d2}%
\end{equation}
in particular
\[
F_{1,1}^{\frac{n}{2}-\frac{\lambda}{2}}(\mathbb{R}^{n})\hookrightarrow(
\overset{\circ}{\mathcal{L}}^{2,\lambda}(\mathbb{R}^{n}))'
\hookrightarrow F_{2,q'}^{-\frac{\lambda}{2}}(\mathbb{R}^{n})
\quad\text{for any }q'\geq2
\]
We have the same injections for the dotted spaces.
\end{corollary}

All the previous results are proved in \cite{e4} and \cite{e2}.

\paragraph{Acknowledgement.}
I am grateful to Professors Pascal Auscher, Gerard
Bourdaud and Jacques Camus. They read the proofs of these results and made
many remarks and clarifications.

\begin{thebibliography}{99} \frenchspacing

\bibitem{b}G. Bourdaud, \textit{Analyse fonctionnelle dans l'espace
euclidien}, Publ. Math. Univ. Paris VII, 23 (1987), r\'{e}\'{e}dit\'{e} 1995.

\bibitem{bl}J. Bergh - J. L\"{o}fstr\"{o}m, \textit{Interpolation spaces},
Springer Verlag (1976).

\bibitem{c1}S. Campanato, \textit{Propriet\`{a} di Holderianit\`{a} di alcune
classi di funzioni,} Ann. Scuola Norm. Sup. Pisa 17 (1963), 175-188.

\bibitem{c2}S. Campanato,\textit{\ Propriet\`{a} di una famiglia di spazi
funzionali}, Ann. Scuola Norm. Sup. Pisa 18 (1964), 137-160.

\bibitem{ab}A. EL Baraka, \textit{Optimal }$BMO$\textit{\ and }$\mathcal{L}%
^{p,\lambda}$\textit{\ estimates near the boundary for solutions of a class of
degenerate elliptic problems}. In: ''\textit{Partial differential
equations}'', Proc. Conf. Fez (Morocco, 1999). Lecture Notes Pure Appl. Math.
229, New York, Marcel Dekker 2002, 183-193.

\bibitem{e4}A. EL\ Baraka, \textit{An embedding theorem for Campanato spaces},
Electron. J. Diff. Eqns., Vol. 2002 (2002), N. 66, pp. 1-17. 

\bibitem{e2}A. EL\ Baraka, \textit{Dyadic characterization of Campanato
spaces}, Preprint, Fez (2001).

\bibitem{fs}C. Feffermann - E. M. Stein, $H^{p}$\textit{\ spaces of several
variables}, Acta math.129 (1972), 167-193.

\bibitem{g}M. Giaquinta, \textit{Introduction to regularity theory for
nonlinear elliptic systems} Birkhauser (1993).

\bibitem{jn}F. John - L. Nirenberg, \textit{On functions of bounded mean
oscillations}, Comm. Pure Appl. Math. 14 (1961), 415-426.

\bibitem{sz}E. S. Stein - A. Zygmund, \textit{Boundedness of translation
invariant operators on H\"{o}lder and }$L^{p}$\textit{\ spaces} Ann. of Math.
85 (1967), 337-349.

\bibitem{str}R. S. Strichartz, \textit{Bounded mean oscillation and Sobolev
spaces}, Indiana Univ. Math. J. 29 (1980), 539-558.

\bibitem{tr}H. Triebel, \textit{Theory of function spaces,} Birkh\"{a}user (1983).

\bibitem{tr2}H. Triebel, \textit{Theory of function spaces}, Birkh\"{a}user (1992).

\bibitem{wa}H. Wallin, \textit{New and old function spaces}. In:
\textit{``Function spaces and applications''}, Proc. Conf. Lund 1986, Lect.
Notes Math. 1302, Berlin, Springer (1988), 97-114.
\end{thebibliography}

\noindent\textsc{Azzeddine El Baraka}\\
University M. Ben Abdellah, FST Fez-Saiss\\
Department of mathematics\\
BP 2202, Route Immouzer\\
Fez 30000 Morocco\\
e-mail: aelbaraka@yahoo.com
\end{document}