Extensions of Weak-Type Multipliers
P. Mohanty
In this paper we prove that if $\Lambda\in M_p(\mathbb R^N)$ and has compact support then $\Lambda$ is a weak summability kernel for $1<p<\infty$, where $ M_p(\mathbb R^N)$ is the space of multipliers of $L^p(\mathbb R^N)$.
math0201084.tex
\documentclass[12pt]{amsart}

%\usepackage{amsmath,  amsfonts,  amssymb}
\renewcommand{\baselinestretch}{1.5}
               %ENVIRONMENTS%
\newtheorem{theorem}{Theorem}[section]
\newtheorem{defn}{Definition}[section]
\newtheorem{lem}{Lemma}[section]
\newtheorem{cor}{Corollary}[section]
\newtheorem{rem}{Remark}[section]
\newtheorem{prop}{Proposition}[section]
\renewcommand{\thesection}{\thechapter.\arabic{section}}
\renewcommand{\thesection}{\arabic{section}}
\renewcommand{\thesubsection}{\thesection. \arabic{subsection} }
\renewcommand{\thelem}{\thesection.\arabic{lem}}
\renewcommand{\thecor}{\thesection.\arabic{cor}}
\renewcommand{\thedefn}{\thesection.\arabic{defn}}
\renewcommand{\therem}{\thesection.\arabic{rem}}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\renewcommand{\thetheorem}{\thesection.\arabic{theorem}}
%\newcommand{\abs}[2]{\lvert#2\rvert}
%\newcommand{\blankbox}[2]{%
 % \parbox{\columnwidth}{\centering
 %   \setlength{\fboxsep}{0pt}%
  %  \fbox{\raisebox{0pt}[#1]{\hspace{#1}}}%
%}  
%}
\allowdisplaybreaks[4]
\begin{document}

\title{Extensions of Weak-Type Multipliers}



%    Information for first author

\author{P. Mohanty}

%    Address of record for the research reported here

\address{Department of Mathematics, Indian Institute of Technology, 
Kanpur-208016, India}

%    Current address

%\curraddr{Department of Mathematics and Statistics,

%Case Western Reserve University, Cleveland, Ohio 43403}

\email{parasar@iitk.ac.in}

%    \thanks will become a 1st page footnote.

\thanks{The first author was supported by CSIR}



%    Information for second author

\author{S. Madan}

\address{Department of Mathematics, Indian Institute of Technology, 
Kanpur-208016, India}

\email{madan@iitk.ac.in}

%\thanks{Support information for the second author.}



%    General info

\subjclass{46E30, 42B15}



%\date{January 1, 1994 and, in revised form, June 22, 1994.}



%\dedicatory{This paper is dedicated to our authors.}



\keywords{Weak-type multipliers, transference}

\begin{abstract}
 In this paper we prove that if $\Lambda\in  M_p(\mathbb R^N)$ and has 
 compact support then $\Lambda$ is a weak summability kernel for 
 $1<p<\infty$, where $  M_p(\mathbb R^N)$ is the space of multipliers 
 of $L^p(\mathbb R^N)$.
\end{abstract}

%\footnote{{\bf Key Words and Phrases} Weak-type multipliers, 
%transference\\{\bf Math Subject %Classification} 46E30, 42B15, } 

\maketitle

\clearpage

\section{Introduction}

 Let $G$ be a locally compact abelian group, with Haar measure 
 $\mu$ and
 let $\hat G$ be its dual. We call an operator 
$T: L^p(G)\longrightarrow L^{p,\infty}(G),\;1\leq p<\infty$, a multiplier of weak type 
$(p,p)$, 
 if it is bounded and translation invariant i.e.
$\tau_x T=T\tau_x \;\;\;\forall x\in G$, and there exists a constant 
$C>0$ such that 
\begin{eqnarray}
\mu\{x\in G : |Tf(x)|>t\}\leq \frac{C^p}{t^p}\|f\|_p^p 
\label{1.1}
\end{eqnarray}
for all 
$f\in L^p(G)$ and $t>0$. (Here $L^{p,\infty}$ denotes the standard weak $L^p$ spaces.) 
 Asmar, Berkson and Gillespie 
in \cite{ABG3} proved that for 
all such operators $T$ there exists a $\phi\in L^\infty(\hat G)$ 
such that 
$(Tf)^\wedge =\phi \hat f$ for all $f\in L^2\cap L^p(G)$. We will also 
call such $\phi$'s to be multipliers of weak type $(p,p)$. Let  
$M_p^{(w)}(\hat G)$ denote the space of multipliers of 
weak type $(p,p)$ for $1\leq p<\infty$, and let $  N_p^{(w)}(\phi)$ be 
the  smallest constant $C$ such that inequality (\ref{1.1}) holds. 


In this paper we are concerned with extensions of weak type multipliers 
from $\mathbb Z^N$ to $\mathbb R^N$ through summability kernels. For 
similar results on strong type 
multipliers , see \cite{J}, \cite{BPW}. 
Here we identify $\mathbb T^N$ with  $[0, 1)^N$ 
and for $f\in {L^1(\mathbb R^N)}$ we define its Fourier transform as
$\hat{f}(\xi)=\int\limits_{\mathbb R^N}f(x)\;e^{-2\pi i\xi\cdot x}dx$ 
for $\xi\in\mathbb R^N$. Let us 
define summability kernels for weak type multipliers as follows
\begin{defn}
 A bounded measurable function 
 $\Lambda :\;\mathbb R^N\longrightarrow \;\mathbb C$ is 
 called a weak summability kernel for $  M_p^{(w)}(\mathbb R^N)$ if 
 for $\phi\in  M_p^{(w)}(\mathbb Z^N)$ the function 
$W_{\phi, \Lambda}(\xi)= \sum\limits_{n\in\mathbb Z^N}\phi(n)
\Lambda(\xi-n)$  is defined and belongs
to $  M_p^{(w)}(\mathbb R^N)$. 
\end{defn}
This definition is just the weak type analouge of summability kernel 
for 
strong type multipliers~\cite{BPW}. We first cite two important results regarding
 the summability kernels of strong type multipliers from the work of 
Jodeit \cite{J} and of Berkson, Paluszynski and Weiss \cite{BPW}:
\begin{theorem}{\rm\cite{J}}
Let $S\in L^1(\mathbb R^N)$ and 
$supp\;S\subseteq[\frac{1}{4},\frac{3}{4}]^N$ with
$\tau= \sum\limits_{n\in\mathbb Z^N} |\hat s(n)|<\infty$ , where 
$s$ is the 1-periodic extension of 
 $S$, then the function defined by 
$W_{\phi,\hat S}(\xi)= \sum\limits_{n\in\mathbb Z^N}\phi(n)\hat 
S(\xi-n)$
 belongs to $M_p(\mathbb R^N)$, for $1\leq p<\infty$ with
 $\|W_{\phi,\hat S}\|_{M_p(\mathbb R^N)}\leq 
C_p\tau\|\phi\|_{M_p(\mathbb Z^N)}$
\end{theorem}
\begin{theorem}{\rm \cite{BPW}}
For $1\leq p<\infty$, let $\Lambda\in M_p(\mathbb R^N)$ and 
$supp \Lambda\subseteq [\frac{1}{4},\frac{3}{4}]^N$. For 
$\phi\in M_p(\mathbb Z^N)$ define 
$W_{\phi, \Lambda}(\xi)= \sum\limits_{n\in\mathbb Z^N}\phi(n)
\Lambda(\xi-n)$ on $\mathbb R^N$. Then 
$W_{\phi, \Lambda}\in  M_p^(\mathbb R^N)$ and
 $\|W_{\phi, \Lambda}\|_{M_p(\mathbb R^N)}\leq C_p\|
 \Lambda\|_{M_p(\mathbb R^N)}\|\phi\|_{M_p(\mathbb Z^N)}$
 where  $C_p$ is a constant. (Further, if $\Lambda$ has arbitary compact 

support the same result holds except that the constant $C_p$ necessarily depends 

on the support of $\Lambda$, as shown in ~\cite{BPW}
\label{thm:9.4}
\end{theorem}
Asmar, Berkson and Gillespie proved a weak type analogue of 
Theorem 1.1 in 
\cite{ABG6}.  In this same paper they also proved that 
$\Lambda$ defined by  
$\Lambda(\xi)=\prod\limits_{j=1}^N \max(1-|\xi_j|,0)$ for 
$\xi=(\xi_1,...,\xi_N)$ is a weak type summability kernel. 
 In this paper, we prove the weak type analouge of Theorem 1.2 in 
 $\S 2$, for $1<p<\infty$. In $\S 3$ we relax the hypothesis that 
 $supp~\Lambda\subseteq [\frac{1}{4},\frac{3}{4}]^N$. For the proof of 
our main result , as in \cite{BPW}, we will obtain the weak type 
inequalities 
by applying the technique of transference couples due to Berkson, 
Paluszy${\rm{\check n}}$ski,  and Weiss~\cite{BPW}. 
\begin{defn}
 For a locally compact group $G$, a transference couple is a pair 
$(S,T)=(\{S_u\},\{T_u\})$ , $u\in G$, of 
strongly continuous mappings defined on $G$ with values in 
${\mathcal B}(X)$, 
where $X$ is a Banach space, satisfying\\
(i) $C_S=\sup\{\|S_u\|:u\in G\}<\infty$\\
(ii) $C_T=\sup\{\|T_u\|:u\in G\}<\infty$\\
(iii) $S_v T_u=T_{vu}\;\;\;\forall u,v\in G$
\end{defn}
  In $\S 4$, as an application of our result, we prove a weak type 
analogue of an extension theorem by 
de Leeuw.
\section{Weak-Type Inequality for Transference Couples and The 
Main Theorem}
Let $\Lambda\in L^\infty(\mathbb R^N)$ and $supp\;\Lambda\subseteq 
[\frac{1}{4},\frac{3}{4}]^N$. 
Consider the following  transference couple $(S,T)$ used by 
Berkson, Paluszy$\rm{\acute n}$ski, and Weiss in~\cite{BPW}. For 
$u\in\mathbb T^N$ the family 
$T=\{T_u\}$ is given by 
\begin{equation}
(T_u f)^\wedge(\xi)= \sum\limits_{n\in\mathbb Z^N}\Lambda(\xi-n)e^{2\pi 
iu.n} \hat f(\xi),\; \;{\rm for}\;f\in L^p(\mathbb R^N)
\label{eqn:9.4}
\end{equation}
and the family $S=\{S_u\}$ is defined by
\begin{equation}
(S_u f)^\wedge(\xi)= \sum\limits_{n\in\mathbb Z^N} b(\xi-n)e^{2\pi 
iu.n} \hat f(\xi).\; \;{\rm for}\;f\in L^p(\mathbb R^N),
\label{eqn:10.4}
\end{equation}

 where $b(\xi)=\prod\limits _{i=1}^Nb_i(\xi_i)$ for 
 $\xi=(\xi_1,......,\xi_N)$ and
for each $i$, $b_i$ is the continuous function defined on 
$\mathbb R$ as 
$b_i(x)=1$ if $x\in[\frac{1}{4},\frac{3}{4}]$, linear in 
$[0,\frac{1}{4})\cup(\frac{3}{4},1]$ and $0$ otherwise.
 It is easy to see that
\begin{eqnarray}
S_u f(x)= \sum\limits_{l\in\mathbb Z^N}\check \beta 
_u(l)f(x+u-l)\;\;\;a.e.,
\label{eqn:11.4}
\end{eqnarray}
where $\check \beta_u$ is the inverse Fourier transform of the function
 $\beta_u(\xi)=b(\xi)e^{2\pi i\xi.u}$, given explicitily by
 $$\check\beta_u(\xi)=\prod_{i=1}^N\check\beta_{u_i}(\xi_i),$$
where 
\begin{equation}
\check\beta_{u_i}(\xi_i)=
\begin{cases}
  \frac{2e^{2\pi i\frac{(\xi_i+u_i)}{2}}}{\pi^2 
(\xi_i-u_i)^2}(cos\frac{\pi}{2}(\xi_i-u_i)-cos\pi(\xi_i-u_i))
\; & {\mbox{if}}\;\; \xi_i\not= u_i\\
\frac{3e^{2\pi i\frac{(\xi_i+u_i)}{2}}}{4}\; \; & {\mbox{if}} 
\;\xi_i=u_i.
\end{cases}
\label{betau}
\end{equation}
Then by a straightforward calculation using Eqn.(\ref{betau}) we have
\begin{equation}
  \sum\limits_{l\in\mathbb Z^N}|\check\beta_u(l)|\leq  
\sum\limits_{l\in\mathbb Z^N}\beta(l)=C<\infty,
\label{eqn:12.4}
\end{equation}
where $\beta(l)=\prod_{i=1}^N\beta_i(l_i)~{\rm and}$
\begin{eqnarray*}
\beta_i(l_i) = 
\begin{cases}
\frac{1}{(l_i-1)^2} &if\;l_i>1\\
\frac{1}{(l_i+1)^2} &if\;l_i<1 \\ 
\|b_i\|_1 &{\mbox{otherwise.}}
\end{cases}
\end{eqnarray*}
 In the following theorem we shall show that the  operator transferred 
by $T$ (of the transfernce couple $(S,T)$ defined in 
Eqn.~(\ref{eqn:9.4}) and Eqn.~(\ref{eqn:10.4})) given  by
$$H_k f(.)=\int_{\mathbb T^N}k(u)T_{u^{-1}}f(.)du,$$
where  $k\in L^1(\mathbb T^N)$ and $f\in L^p(\mathbb R^N)$, satisfies a
 weak  $(p,p)$ inequality.
\begin{theorem}
 Let $(S,T)$ be the transference couple as defined in 
 Eqn.~(\ref{eqn:9.4}) and Eqn.~(\ref{eqn:10.4}). Then
 for $1<p<\infty$ and $t>0$\\
\[\lambda\{x\in\mathbb R^N\;:\;|H_k f(x)|>t\}\leq 
(\frac{C\;C_p}{t}C_T  N_p^{(w)}(k)\|f\|_p)^p,\]
where $\lambda$ denotes the Lebesgue measure of $\mathbb R^N$, 
$C= \sum\limits_{l\in\mathbb Z^N} \beta(l)$ as in 
Eqn.~(\ref{eqn:12.4}), $C_T$ is the uniform bound for the 
family $T=\{T_u\}$, and $C_p=\frac{p}{p-1}$.
\label{thm:10.4}
\end{theorem}
\noindent{\bf Proof:}
Assume $f\in {\mathcal S}(\mathbb R^N)$. For $t>0$ define 
$E_t=\{x\;:\;|H_k f(x)|>t\}$.



 \noindent Notice that 

$H_k f(x)= S_{v^{-1}}S_v H_k f(x)=\sum\limits_{l\in\mathbb Z^N}\check\beta_{v^{-1}}(l)\int_
{\mathbb T^N}k(u)T_{u^{-1}v}f(x-v-l) du|>t\}.$
Let ${\mathcal F}_t=\{(v,x)\in{\mathbb T^N\times\mathbb R^N}\;:\;| 
\sum\limits_{l\in\mathbb Z^N}\check\beta_{v^{-1}}(l)\int_
{\mathbb T^N}k(u)T_{u^{-1}v}f(x-l) du|>t\}.$


Then, using translation invariant of Lebesgue measure
\begin{eqnarray*}
\lambda(E_t)& = &\lambda\{x\in\mathbb R^N\;:\;|S_{v^{-1}}\int_{\mathbb 
T^N}k(u)T_{u^{-1}v} f(x)du|>t\}\\
& = &\lambda\{x\in\mathbb R^N:| \sum\limits_{l\in\mathbb 
Z^N}\check\beta_{v^{-1}}(l)\int_{\mathbb T^N}k(u)T_{u^{-1}v} f(x-l) du|>t\}\\
& = &\int_{\mathbb T^N}\int_{\mathbb R^N}\chi_{{\mathcal F}_t}(v,x) dx 
dv\\
& = &\int_{\mathbb R^N}|\{v:| \sum\limits_{l\in\mathbb 
Z^N}\check\beta_{v^{-1}}(l)\int_{\mathbb T^N}k(u)T_{u^{-1}v} f(x-l) du|>t\}|dx,
\end{eqnarray*}
 where $|E|$ denotes the measure of the subset $E\subseteq\mathbb T^N$. 
 Thus
\begin{eqnarray*}
\lambda(E_t)& \leq & \int_{\mathbb R^N}|\{v: \sum\limits_
{l\in\mathbb Z^N}\beta(l)|\int_{\mathbb  T^N}k(u)T_{u^{-1}v} 
f(x-l)| du>t\}|dx\\
& = &\int_{\mathbb R^N}|\{v: \sum\limits_{l\in\mathbb 
Z^N}\beta(l)|k*F(.,x-l)(v)|>t\}|dx ,\;\;\; {\rm where}\;F(v,x)=T_v f(x)~~~{\rm a.e.}.
\end{eqnarray*}
 We know that $\sup\limits_{t>0}~ 
t{\lambda_f(t)}^{\frac{1}{p}}=\|f\|_{L^{p,\infty}}$
 for $f\in L^{p,\infty}$. Also, since $p>1$, $\|\;\|_{p,\infty}$ is 
equivalent to a norm $\|\;\|_{p,\infty}^*$ (\cite{SW}), using traingle inequality for norms 

we have 
\begin{eqnarray*}
\lambda(E_t)& \leq &\int_{\mathbb R^N}\frac{1}{t^p}\| 
\sum\limits_{l\in\mathbb Z^N} \beta(l)|k*F(.,x-l)\|_{L^{p,\infty}(\mathbb T^N)}^p dx\\
& \leq &C_p^p\int_{\mathbb R^N}\frac{1}{t^p}( \sum\limits_{l\in\mathbb 
Z^N} \beta(l)\|k*F(.,x-l)\|_{L^{p,\infty}(\mathbb T^N)}^*)^p dx, 
\;\;\;{\mbox{where}}\;\;C_p=\frac{p}{p-1}\\
& \leq &C_p\int_{\mathbb R^N}\frac{1}{t^p}( \sum\limits_{l\in\mathbb 
Z^N} \beta(l)  N_p^{(w)}(k)\|F(.,x-l)\|_{L^p(\mathbb T^N)})^p dx,
\end{eqnarray*}
where $  N_p^{(w)}(k)$ is the weak-type norm of the convolution 
operator 
 $f\longmapsto k*f$ for $f\in L^p(\mathbb T^N)$. Thus,
\begin{eqnarray*}
\lambda(E_t)& \leq &C_p^p\frac{1}{t^p} \sum\limits_{l\in\mathbb 
Z^N}\beta(l)  N_p^{(w)}(k)(\int_{\mathbb R^N}\int_{\mathbb T^N}|T_v f(x-l)|^p 
dx dv)^{\frac{1}{p}})^p\\
& = &C_p^p\frac{1}{t^p}( \sum\limits_{l\in\mathbb Z^N}\beta(l)  
N_p^{(w)}(k)(\int_{\mathbb T^N}\int_{\mathbb R^N}|T_v f(x-l)|^p dx 
dv)^{\frac{1}{p}})^p\\
& \leq &(\frac{C C_pC_T}{t^p}  N_p^{(w)}(k)\|f\|_p)^p.
\end{eqnarray*}
Hence, $H_k f$ satisfies a weak $(p,p)$ inequality.


 In order to prove the weak-type analogue of Theorem~\ref{thm:9.4} we 
 need the 
following Lemma proved by Asmar, Berkson, and Gillespie in \cite{ABG5}. 
\begin{lem}{\rm{\cite{ABG5}}}
 Suppose that $1\leq p<\infty$ , $\{\phi_j\}\subseteq M_p^{(w)}(\hat 
G)$; 
$\sup\{|\phi_j(\gamma)|:j\in \mathbb N,\gamma\in\hat G\}~~<\infty$ and 
 suppose $\phi_j$ converges pointwise a.e. on $\hat G$ to a function 
 $\phi$ . 
If $\liminf\limits_j  N_p^{(w)}(\phi_j)<\infty$ then $\phi\in 
M_p^{(w)}(\hat G)$ 
and $  N_p^{(w)}(\phi)\leq \liminf\limits_j  N_p^{(w)}(\phi_j)$. 
\label{lem:2.4}
\end{lem}

In the following theorem, we use the family of operators $\{T_u\}$ 

defined in (~\ref{eqn:9.4}) with $\Lambda\in M_p(\mathbb R^N$) and 

$supp\Lambda\subseteq [\frac{1}{4},\frac{3}{4}]^N$. In ths case, 

by ~\cite{BPW} we have $C_T\leq c_p\|\Lambda\|_{M_p(\mathbb R^N})$, 

where $c_p$ is a constant.
\begin{theorem}
 Suppose $1<p<\infty$ and $\Lambda\in   M_p(\mathbb R^N)$  is 
supported
  in the set
 $[\frac{1}{4},\frac{3}{4}]^N$. For $\phi\in  M_p^{(w)}(\mathbb Z^N)$ 
 define
$$W_{\phi, \Lambda}(\xi)= \sum\limits_{n\in\mathbb 
Z^N}\phi(n)\Lambda(\xi-n) \;\;\;\;{\mbox{on}} \;\mathbb R^N.$$
Then $W_{\phi, \Lambda}\in  M_p^{(w)}(\mathbb R^N)$ and $  
N_p^{(w)}(W_{\phi, \Lambda})\leq C  N_p^{(w)}(\phi)\|\Lambda\|_{M_p(\mathbb R^N)}$.
\label{eqn:11a.4}
\end{theorem}
\noindent{\bf Proof:}
Using Lemma~\ref{lem:2.4} we first show that it is enough to prove  the
  theorem for  $\phi\in  M_p^{(w)}(\mathbb Z^N)$ 
having finite support. Suppose the theorem is true for finitely 
supported $\phi$. 
Then for  arbitrary $\phi\in  M_p^{(w)}(\mathbb Z^N)$, 
define $\phi_j=\hat k_j\phi$, where $k_j$ is the j-th
F$\acute{\mbox{e}}$jer  kernel. Then for each $j$ , $\phi_j$'s have 
finite support and 
$(T_{\phi_j}f)^\wedge(n)=\phi_j(n)\hat f(n)=(T_\phi(k_j*f))^\wedge(n)$. 
So $\phi_j\in  M_p^{(w)}(\mathbb Z^N)\;\;$ for each $j$ and 
$  N_p^{(w)}(\phi_j)\leq   N_p^{(w)}(\phi)$. Define 
$W_{\phi_j, \Lambda}(\xi)= \sum\limits_{n\in\mathbb 
Z^N}\phi_j(n)\Lambda(\xi-n)$. Now $\liminf\limits_jW_{\phi_j, \Lambda}(\xi)=W_{\phi, 
\Lambda}(\xi)$.
 Also, by our assumption 
\begin{eqnarray*}
  N_p^{(w)}(W_{\phi_j, \Lambda})& \leq & C  
N_p^{(w)}(\phi_j)\|\Lambda\|_{M_p(\mathbb R^N)}\\
& \leq &C  N_p^{(w)}(\phi)\|\Lambda\|_{M_p(\mathbb R^N)}
\end{eqnarray*}
and $|W_{\phi_j, \Lambda}|\leq 2\|\Lambda\|_\infty\|\phi_j\|_\infty\leq 
2\|\Lambda\|_\infty\|\phi\|_\infty$.
 Thus by Lemma~\ref{lem:2.4}, applied to $W_{\phi_j,\Lambda}$'s ,we 
 conclude that
 $W_{\phi, \Lambda}\in  M_p^{(w)}(\mathbb R^N)$. Hence it is enough 
 to assume that $\phi\in   M_p^{(w)}(\mathbb Z^N)$ 
has finite support.


Now let $\phi\in M_p^{(w)}(\mathbb Z^N)$ be finitely supported. Define
 $k(u)= \sum\limits_{n\in\mathbb Z^N}\phi(n)e^{-2\pi iu.n}$ then 
 $k\in L^1(\mathbb T^N)$ and 
$\hat k(n)=\phi(n)$. For this particular $k$ and the transference 
couple 
$(S,T)$ defined above. We have
$$(H_k f)^\wedge(\xi)=(T_{W_{\phi,\Lambda}}f)^\wedge(\xi).$$
 Thus $T_{W_{\phi,\Lambda}}f=H_kf$. Hence from Theorem~\ref{thm:10.4} 
 and since $C_T\leq c_p\|\Lambda\|_{M_p(\mathbb R^N}$, we have
$$\lambda\{x\in\mathbb R^N:|T_{W_{\phi,\Lambda}}f(x)|>t\}\leq 
(\frac{C}{t}  N_p^{(w)}(\phi)\|\Lambda\|_{M_p(\mathbb 
R^N)}\|f\|_p)^p.$$
\section{Lattice Preserving Linear Transformations and Multipliers}
We shall now relax the hypothesis that  
$supp\;\Lambda\subseteq[\frac{1}{4},\frac{3}{4}]^N$ to allow 
$\Lambda$ to have
 arbitrary compact support. In fact this can be done by a 
partition of identity argument as in \cite{BPW}. Here we give  a 
different method by proving Lemma~\ref{lem:5.4} below.  Particular 
cases of this lemma occur in \cite{J} and in  \cite{ABG6}. 
Suppose $supp\;\Lambda\subseteq [-M,M]^N$; define 
$\Lambda_M(\xi)=\Lambda_1(4M\xi)$, 
where $\Lambda_1(\xi)=\Lambda(\xi-\frac{1}{2})$.  
 So $supp\;\Lambda_M\subseteq [\frac{1}{4},\frac{3}{4}]^N$. Thus if we 
 define a non-singular 
transformation $A:\mathbb R^N\longrightarrow \mathbb R^N$ such that 
$Ax=4Mx$ then 
$\Lambda_M=\Lambda_1\circ A$. In order to replace the support condition 
we need
 to prove $\Lambda_M\circ A^{-1}$ is a summability kernel. In the work 
 of Jodeit and of Asmar, Berkson and  Gillespie they assume $A$ 
in Lemma~\ref{lem:5.4} to be multiplication by 2. We have combined some 
of the results proved by 
  Gr$\ddot{\mbox{o}}$chenig and Madych ~\cite{GM} in the following 
  lemma which will help us to prove 
Lemma~\ref{lem:5.4}. In the proof of Theorem~\ref{thm:12.4}, we only 
use the case of a diagonal linear transform, but the more general 
results proved below are of some interest in their own right.
\begin{lem}{\rm{\cite{GM}}}
 Let $A:\mathbb R^N\longrightarrow \mathbb R^N$ be a non-singular 
linear 
 transformation   which preserves the lattice $\mathbb Z^N$ (i.e.  
$A(\mathbb Z^N)\subseteq \mathbb Z^N$). Then the following are true.\\
(i) The number of distinct coset representatives of 
$\mathbb Z^N/{A\mathbb Z^N}$ is equal to $q=|\det A|$.\\
(ii) If $Q_0=[0,1)^N$ and $k_1,.....,k_q$ are the distinct  coset 
representatives  of $\mathbb Z^N/{A\mathbb Z^N}$ then the sets 
$A^{-1}(Q_0+k_i)$ are mutually disjoint.\\
(iii) Let $Q=\cup _{i=1}^q A^{-1}(Q_0+k_i)$, then $\lambda(Q)=1$ and 
$\cup_{k\in \mathbb Z^N}(Q+k)\simeq \mathbb R^N.$\\
(iv) $AQ\simeq\cup_{i=1}^q (Q_0+k_i)$.\\

Where $E\simeq F$ if $\lambda (F\bigtriangleup E)=0$.
\label{lem:3.4}
\end{lem}
The above result is essentially contained  in \cite{GM}. 
\begin{lem}
 Let $A$ be as in Lemma~\ref{lem:3.4}. Denote $A^t=B$, where $A^t$ is 
 the transpose of $A$. For $\phi\in l_\infty(\mathbb Z^N)$ define
$$\psi(n)=\phi(Bn)$$
and
\begin{center}
$\eta(n)=
\begin{cases}
\phi(B^{-1}n) &n\in B\mathbb Z^N\\
 0 &otherwise.
\end{cases}$
\end{center}
(i) If $\phi\in  M_p(\mathbb Z^N)$ then $\psi, \eta\in  M_p(\mathbb 
Z^N)$
 with multiplier norms 
not exceeding the multiplier norm of $\phi$.\\
(ii) If $\phi\in  M_p^{(w)}(\mathbb Z^N)$ then 
$\psi, \eta\in  M_p^{(w)}(\mathbb Z^N)$ with weak 
multiplier norms not exceeding the weak multiplier norm of $\phi$.
\label{lem:5.4}
\end{lem}
\noindent{\bf Proof:}
(i) For $f\in L^p(Q_0)$, we let $f$ again denote the periodic extension 
to $\mathbb R^N$. Define $Sf(x)=f(Ax)$ , then $Sf$ is also periodic and 
\begin{eqnarray*}
\int_{Q_0}|Sf(x)|^pdx & = 
&\int_{Q_0}|Sf(x)|^p\sum\limits_j\chi_Q(x-j)dx\\
& = &\sum\limits_j\int_{Q_0+j}|Sf(x)|^p\chi_Q(x)dx\\
& = &\int_Q|Sf(x)|^p dx\\
& = &\frac{1}{|\det A|}\int_{AQ}|f(x)|^p dx\\
& = &\frac{1}{q}\sum\limits_{i=1}^q\int_{Q_0+k_i}|f(x)|^p 
dx\;\;\;\;\;\;\;((iv) \;{\mbox{of Lemma~\ref{lem:3.4}}})\\
& = &\int_{Q_0}|f(x)|^p dx.
\end{eqnarray*}
Thus $S$ is an isometry, i.e., 
$\parallel Sf\parallel _{L^p(Q_0)}=\parallel f\parallel _{L^p(Q_0)}$. 
\noindent
Further,  from the orthogonality relations of the characters (Lemma 1, 
\cite{M}) we have 
\begin{eqnarray*}
(Sf)^\wedge(n)=
\begin{cases}
\hat f(B^{-1}n) &if \;n\in B\mathbb Z^N\\
 0 &otherwise.
\end{cases}
\end{eqnarray*}
For $f\in L^p(Q_0)$ we define an  operator $W$ on $L^p(Q_0)$ given by
 $Wf(x)=\frac{1}{q}\sum\limits_{i=1}^q f(A^{-1}(x+~~k_i)),$ $\;\;$ 
where $k_1,\dots,k_q$ are distinct  cosets representations of 
$\mathbb Z^N/ {A\mathbb Z^N}$. 
Then for a trigonometric polynomial $f$,
$$(Wf)^\wedge(n)  =  \hat f(Bn),$$
and so 
\begin{eqnarray*}
(\int_{Q_0}|Wf(x)|^pdx)^{\frac{1}{p}} & = 
&(\int_{Q_0}|\frac{1}{q}\sum\limits_{i=1}^q f(A^{-1}(x+k_i))|^p dx)^{\frac{1}{p}}\\
& \leq 
&\frac{1}{q}\sum\limits_{i=1}^q(\int_{Q_0}|f(A^{-1}(x+k_i))|^pdx)^{\frac{1}{p}}\\
& = 
&\frac{q^{\frac{1}{p}}}{q}\sum\limits_{i=1}^q(\int_{A^{-1}(Q_0+k_i)}|f(x)|^p dx)^{\frac{1}{p}}.
\end{eqnarray*}
Therefore $\parallel Wf\parallel_{L^p(Q_0)}\leq q^
{\frac{1-p}{p}}\parallel f\parallel _{L^p(Q_0)}$, since 

$\int_{Q_0}|f(x)|^p dx=\int_Q|f(x)|^p dx$ as above.  
It is easy to see that
\begin{equation}
ST_\phi W=T_\eta
\label{eqn:13.4}
\end{equation}
and
\begin{equation}
WT_\phi S=T_\psi
\label{eqn:14.4}
\end{equation}
It follows that, if  $\phi\in  M_p(\mathbb Z^N)$  then 
$\|T_\psi f\|\leq C_p\|\phi\|_{M_p(\mathbb Z^N)}\|f\|_{L^p(Q_0)}.$  
 Also $\|T_\eta f\|_{L^P(Q_0)}\leq C_p\|\phi\|_
 {M_p(\mathbb Z^N)}\|f\|_{L^p(Q_0)}$. Hence 
$\psi,\eta\in M_p(\mathbb Z^N)$. 


(ii) For $\phi\in  M_p^{(w)}(\mathbb Z^N)$, we need to calculate the 
distribution 
function of $Sf$ and $Wf$. Denote $E_t=\{x\in Q_0 :|Sf(x)|>t>0\}$. Then
\begin{eqnarray*}
|E_t| & = &\int_{Q_0}\chi_{E_t}(x)dx\\
& = &\int_{Q_0}\chi_{\mathbb R_+}(|f(Ax)|-t)dx\\
& = &\frac{1}{q}\int_{AQ}\chi_{\mathbb R_+}(|f(x)|-t)dx\\
& = &\frac{1}{q}\sum\limits_{i=1}^q\int_{Q_0+k_i}\chi_{\mathbb 
R_+}(|f(x)|-t)dx\\
& = &|\{x:|f(x)>t\}|.
\end{eqnarray*}
\noindent
Therefore, 
\begin{equation}
|\{x\in Q_0 :|Sf(x)>t\}|=|\{x\in Q_0 :|f(x)>t\}|
\label{eqn:15.4}
\end{equation}
Also
\begin{eqnarray*}
|\{x\in Q_0:|Wf(x)|>t\}| & = & |\{x\in Q_0 :| \sum\limits_{i=1}^q 
f(A^{-1}(x+k_i))|>tq\}|\\
& \leq &|\{x\in Q_0 : \sum\limits_{i=1}^q|f(A^{-1}(x+k_i))|>tq\}|\\
& = & \sum\limits_{i=1}^q\int_{Q_0}\chi_{\mathbb 
R_+}(|f(A^{-1}(x+k_i))|-t)dx\\
& = & q \sum\limits_{i=1}^q\int_{A^{-1}(Q_0+k_i)}\chi_{\mathbb 
R_+}(|f(x)|-t)dx.
\end{eqnarray*}
Thus
\begin{equation}
|\{x\in Q_0 :|Wf(x)|>t\}\leq q|\{x\in Q_0 :|f(x)|>t\}|.
\label{eqn:16.4}
\end{equation}
From the relations (\ref{eqn:13.4}) - (\ref{eqn:16.4}), we conclude 
that 
$\psi , \eta\in  M_p^{(w)}(\mathbb Z^N)$ whenever 
$\phi\in  M_p^{(w)}(\mathbb Z^N)$ . Also 
$  N_p^{(w)}(\psi)\leq C  N_p^{(w)}(\phi)$ and
 $  N_p^{(w)}(\eta)\leq C  N_p^{(w)}(\phi)$.


 As an application of this Lemma we get the following result regarding
  weak summability kernels.
\begin{lem}
 Let $A$ be as in Lemma~\ref{lem:3.4}. Suppose $\Lambda$ is  a 
 weak (strong) summabiliy kernel then
 $\Lambda\circ B$ and $\Lambda\circ B^{-1}$ are also weak (strong) 
 summability kernels.
\label{lem:6.4}
\end{lem}
\noindent{\bf Proof:}
Define $W_{\phi,\Lambda\circ B}$ on $\mathbb R^N$ for 
$\phi\in  M_p^{(w)}(\mathbb Z^N)$.
\begin{eqnarray*}
W_{\phi,\Lambda\circ B}(x)& = & \sum\limits_{n\in\mathbb Z^N}
\phi(n)\Lambda\circ B(x-n)\\
& = & \sum\limits_{n\in\mathbb Z^N}\eta(n)\Lambda(Bx-n)\\
& = &W_{\eta,\Lambda}(Bx).
\end{eqnarray*}
As $\eta\in  M_p^{(w)}(\mathbb Z^N)$ (by Lemma~\ref{lem:5.4}) and 
since $\Lambda$ is a  summability kernel
 we have $W_{\eta,\Lambda}\in  M_p^{(w)}(\mathbb R^N)$. Hence 
$W_{\phi,\Lambda\circ B}\in  M_p^{(w)}(\mathbb R^N)$.
\noindent
 Similarly 
\begin{eqnarray*}
W_{\phi,\Lambda\circ B^{-1}}(x)& = &   \sum\limits_{n\in\mathbb 
Z^N}\phi(n)\Lambda(B^{-1}x-B^{-1}n)\\
& = & \sum\limits_{j=1}^q\sum\limits_{n\in B\mathbb Z^N +
p_{j}}\phi(n)\Lambda(B^{-1}x-B^{-1}n)\\
\end{eqnarray*}
where $p_1....p_{q}$ are distinct  coset representatives of 
$B\mathbb Z^N/ \mathbb Z^N\;(p_1=0).$
\begin{eqnarray*}
W_{\phi,\Lambda\circ B^{-1}}(x)& = & \sum\limits_{j=1}^q\sum\limits_{n\in \mathbb 
Z^N}\phi(Bn+p_{j})\Lambda(B^{-1}x+B^{-1}p_{j}-n)\\
& = &W_{\psi,\Lambda}(B^{-1}x)+...+W_{\psi_{p_{q-1}},\Lambda}(B^{-1}x-B^{-1}p_{q})\;\;
\end{eqnarray*}
 where $\psi_{p_i}(l)=\phi(Bl+p_j),\;i=1,2,..,q$.
 As $\psi\in  M_p^{(w)}(\mathbb Z^N)$ and $\Lambda$ is a summability 
kernel we conclude that
$W_{\phi,\Lambda\circ B^{-1}}\in  M_p^{(w)}(\mathbb R^N)$.



 Hence from Lemma~\ref{lem:6.4} and the discusssion preceeding 
 Lemma~\ref{lem:3.4} we conclude the following theorem.
\begin{theorem}
 Suppose $\Lambda\in  M_p(\mathbb R^N)$ and 
 $supp\;\Lambda\subseteq [-M,M]$; for $\phi\in  M_p^{(w)}(\mathbb Z^N)$
  define $W_{\phi, \Lambda}(\xi)= \sum
 \limits_{n\in\mathbb Z^N}\phi(n)\Lambda(\xi-n)$ on $\mathbb R^N$, 
 then $W_{\phi, \Lambda}\in  M_p^{(w)}(\mathbb R^N)$ and
$  N_p^{(w)}(W_{\phi, \Lambda})\leq C_\Lambda   N_p^{(w)}(\phi)
\|\Lambda\|_{M_p(\mathbb R^N)}$, where $C_\Lambda$ is 
a constant depending  on $\Lambda$.
\label{thm:12.4}
\end{theorem}
\section{An Application}
 As an application of Theorem~\ref{thm:12.4}, we prove a weak-type 
 version of a result proved by de Leeuw~\cite{SW}.
\begin{theorem}
 For $1<p<\infty$ , and $\epsilon>0$; let 
 $\{\phi_\epsilon\}\subseteq M_p^{(w)}(\mathbb Z)$ satisfy \\
(i) $\lim\limits_{\epsilon\rightarrow 0}\phi_\epsilon
([\frac{x}{\epsilon}])=\phi(x)$ a.e.\\
(ii) $\sup\limits_\epsilon  N_p^{(w)}(\phi_\epsilon)=K<\infty.$\\
Then $\phi\in M_p^{(w)}(\mathbb R)$ and 
$  N_p^{(w)}(\phi)\leq\sup\limits_\epsilon  N_p^{(w)}(\phi_\epsilon).$
\label{thm:13.4}
\end{theorem}
\noindent{\bf Proof:}
 For each $\epsilon>0$, define $W_{\phi_\epsilon}$ on $\mathbb R$ by
\begin{equation}
W_{\phi_\epsilon}(x)=\sum\limits_{n\in\mathbb 
Z}\phi_\epsilon(n)\chi_{[0,1)}(x-n).
\label{eqn:17.4}
\end{equation}
 As $\chi_{[0,1)}\in  M_p(\mathbb R)$ for $1<p<\infty$, from 
 Theorem~\ref{thm:12.4} we have 
 $W_{\phi_\epsilon}\in M_p^{(w)}(\mathbb R)$ and 
$  N_p^{(w)}(W_{\phi_\epsilon})\leq C  N_p^{(w)}(\phi_\epsilon)\leq CK$ 
.
 We define another function $\psi_\epsilon,$ 
for each $\epsilon>0,$  by 
$\psi_\epsilon(x)=W_{\phi_\epsilon}(\frac{x}{\epsilon})$. Then 
$\psi_\epsilon \in M_p^{(w)}(\mathbb R)$ and 
\begin{equation}
  N_p^{(w)}(\psi_\epsilon)\leq  N_p^{(w)}(W_{\phi_\epsilon})\leq CK.
\label{eqn:18.4}
\end{equation}
From (\ref{eqn:17.4}) we have
\begin{eqnarray*}
\psi_\epsilon(x)\; = \;W_{\phi_\epsilon}(\frac{x}
{\epsilon})& = & \sum\limits_{n\in\mathbb Z}\phi_\epsilon(n)
\chi_{[0,1)}(\frac{x}{\epsilon}-n)\\
& = &\phi_\epsilon([\frac{x}{\epsilon}]).
\end{eqnarray*}
So from our hypothesis  
\begin{equation}
\lim\limits_{\epsilon\rightarrow 0}\psi_\epsilon(x)=\phi(x)\;\;\; a.e.
\label{eqn:19.4}
\end{equation}
Also we have $|\psi_\epsilon(x)|<\infty\;\;\;\;$ (as 
$\sup\limits_{\epsilon,n}|\phi_\epsilon(n)|<\infty$).


 Hence from (\ref{eqn:17.4}), (\ref{eqn:18.4}) and (\ref{eqn:19.4}) 
 along with Lemma~\ref{lem:2.4} we have $\phi \in M_p^{(w)}(\mathbb R)$ 
 and $  N_p^{(w)}(\phi)\leq\lim\limits_\epsilon  
 N_p^{(w)}(\phi_\epsilon)\leq CK.$
\begin{thebibliography}{99}
\bibitem{ABG5} Asmar,  N.,    Berkson,  E.  and Gillespie, 
 T.  A.    
\newblock Maximal estimates on Measure 
spaces by weak type multipliers. 
\newblock The J. of Geom. Analysis, 5, No.2, 1995, 167-179.
\bibitem{ABG6} Asmar,  N.,    Berkson,  E.  and Gillespie, 
 T.  A.
\newblock Generalized de Leeuw Theorems and Extension Theorems for Weak 
Type Multipliers.
\newblock  Interaction between Functional Analysis, Harmonic Analysis 
and Probability. N. Kalton   et al. (ed.), Marcel Dekker Lect. Notes 
Pure and Appl. Math. , 175, 1996, 41-67.
\bibitem{ABG3} Asmar,  N.,    Berkson,  E.  and Gillespie, 
 T.  A.  
\newblock  Note on norm convergence in the space 
of a weak type multipliers.
\newblock  J. of Op.Theory, 39, 1998, 139-149.
\bibitem{BPW} Berkson, E., Paluszy${\rm{\check n}}$ski, M., and Weiss, 
G.
\newblock Transference Couples and Their Applications to convolution 
Operators and Maximal operators.
\newblock Interaction between Functional Analysis, Harmonic Analysis, 
and Probabilty,  N. Kalton et al. (ed.), Lect. Notes in Pure and Applied 
Maths., Marcel Dekker, 175, 1996, 69-84.
\bibitem{GM} Gr${\rm\ddot o}$chenig, M. and Madych, W.R.
\newblock Multiresolution Analysis, Haar Basis, and Self-Similar 
Tilings of $\mathbb R^N$.
\newblock IEEE Transactions of Information Theory, Vol. 38, No.2, 1992, 
556-568.
\bibitem{J} Jodeit, M.
\newblock Restrictions and Extensions of Fourier Multipliers.
\newblock Studia Math., 34, 1970, 215-226.
\bibitem{M} Madych, W. R.
\newblock Some elementary properties of Multiresolution Analysis of 
$L^2(\mathbb R^N)$.
\newblock Wavelets a Tutorial in Theory and Applications, C.K. Chui 
(ed.), Academic Press Inc., 1992, 259-294.
\bibitem{SW} Stein, E. M. and Weiss, G.
\newblock Introduction to Fourier
 Analysis on Euclidean Spaces.
\newblock Princeton Math.Ser., Vol. 32, Princeton 
University Press, Princeton, New Jersey, 1971.
\end{thebibliography}
\end{document}
math0201084.extracted_bib
\begin{thebibliography}{99}
\bibitem{ABG5} Asmar,  N.,    Berkson,  E.  and Gillespie, 
 T.  A.    
\newblock Maximal estimates on Measure 
spaces by weak type multipliers. 
\newblock The J. of Geom. Analysis, 5, No.2, 1995, 167-179.
\bibitem{ABG6} Asmar,  N.,    Berkson,  E.  and Gillespie, 
 T.  A.
\newblock Generalized de Leeuw Theorems and Extension Theorems for Weak 
Type Multipliers.
\newblock  Interaction between Functional Analysis, Harmonic Analysis 
and Probability. N. Kalton   et al. (ed.), Marcel Dekker Lect. Notes 
Pure and Appl. Math. , 175, 1996, 41-67.
\bibitem{ABG3} Asmar,  N.,    Berkson,  E.  and Gillespie, 
 T.  A.  
\newblock  Note on norm convergence in the space 
of a weak type multipliers.
\newblock  J. of Op.Theory, 39, 1998, 139-149.
\bibitem{BPW} Berkson, E., Paluszy${\rm{\check n}}$ski, M., and Weiss, 
G.
\newblock Transference Couples and Their Applications to convolution 
Operators and Maximal operators.
\newblock Interaction between Functional Analysis, Harmonic Analysis, 
and Probabilty,  N. Kalton et al. (ed.), Lect. Notes in Pure and Applied 
Maths., Marcel Dekker, 175, 1996, 69-84.
\bibitem{GM} Gr${\rm\ddot o}$chenig, M. and Madych, W.R.
\newblock Multiresolution Analysis, Haar Basis, and Self-Similar 
Tilings of $\mathbb R^N$.
\newblock IEEE Transactions of Information Theory, Vol. 38, No.2, 1992, 
556-568.
\bibitem{J} Jodeit, M.
\newblock Restrictions and Extensions of Fourier Multipliers.
\newblock Studia Math., 34, 1970, 215-226.
\bibitem{M} Madych, W. R.
\newblock Some elementary properties of Multiresolution Analysis of 
$L^2(\mathbb R^N)$.
\newblock Wavelets a Tutorial in Theory and Applications, C.K. Chui 
(ed.), Academic Press Inc., 1992, 259-294.
\bibitem{SW} Stein, E. M. and Weiss, G.
\newblock Introduction to Fourier
 Analysis on Euclidean Spaces.
\newblock Princeton Math.Ser., Vol. 32, Princeton 
University Press, Princeton, New Jersey, 1971.
\end{thebibliography}