Singular measures and convolution operators
J. M. Aldaz
We show that in the study of certain convolution operators, functions can be replaced by measures without changing the size of the constants appearing in weak type $(1,1)$ inequalities. As an application, we prove that the best constants for the centered Hardy-Littlewood maximal operator associated to parallelotopes do not decrease with the dimension.
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\journal{J.\ Math.\ Anal.\ Appl.}

\date{30 July 2002}

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\begin{document}

\begin{frontmatter}


\title{Singular measures and convolution operators}
\thanks{Research supported by grant BFM2000-0206-C04-03}


%%%%%%%%%%%%
%% Authors:
%
\author{J. M. Aldaz},
\ead{aldaz@dmc.unirioja.es}
%
\author{Juan L. Varona}
\ead{jvarona@dmc.unirioja.es}

\address{Departamento de Matem\'aticas y Computaci\'on,
Universidad de La Rioja,\\
26004 Logro\~no, Spain}

%%%%%%%%%%%%

\begin{abstract}
We show that in the study of certain convolution operators,
functions can be replaced by measures without changing the size of the
constants appearing in weak type $(1,1)$ inequalities. As an application,
we prove that the best constants for the centered
Hardy-Littlewood maximal operator associated to parallelotopes do not
decrease with the dimension.
\end{abstract}


\begin{keyword}
Hardy-Littlewood maximal function
\MSC 42A85
\end{keyword}

\end{frontmatter}


%---------------------------------------------------
\section{Introduction}
%---------------------------------------------------

The method of discretization for convolution
operators, due to M.~de Guzm\'an
(cf.~\cite{Guz}, Theorem~4.1.1), and further developed by
M.~T.~Men\'ar\-guez and F.~Soria (cf.~Theorem~1 of~\cite{MeSo})
consists in replacing functions by finite sums of Dirac
deltas in the study of the operator.
So far, the main applications of these theorems have been
related to the Hardy-Littlewood maximal function, and
more precisely, to the determination of bounds for the best
constants $c_d$ appearing in the weak type $(1,1)$ inequalities
(cf.~\cite{MeSo},~\cite{A1},~\cite{Mel1}, and \cite{Mel2} for the
one dimensional case, and for higher dimensions,
\cite{MeSo} and~\cite{A2}).
In this paper we complement de Guzm\'an's Theorem by proving
that one can
consider arbitrary measures instead of finite discrete measures, and the same
conclusions still hold (Theorem~\ref{th:equiv}). A special case of our
theorem (where the space is the real line and the convolution operator
is precisely the Hardy-Littlewood maximal function) appears
in~\cite{Mel2} (see Theorem~2).

Regarding upper bounds
for $c_d$, E.~M.~Stein and J.~Str\"omberg (see~\cite{StSt}) showed
that the constants grow at most like $O(d \log d)$ for arbitrary balls, and
like $O(d)$ in the case of euclidean balls. With respect to lower
bounds for the maximal function associated to cubes, it is shown
in~\cite{MeSo}, Theorem~6,
that $c_d \ge \big( \frac{1+2^{1/d}}{2} \big)^d$. These
bounds, which decrease with the dimension to $\sqrt{2}$,
where conjectured to be optimal in~\cite{Me}.
The ``optimality  part" of the conjecture was refuted
in~\cite{A2}, where
it was proved that $\liminf_d c_d \ge \frac{47 \sqrt{2}}{36}$.
It is an easy consequence of Theorem~\ref{th:equiv}
that the ``decreasing part'' of the conjecture
is also false:
For cubes the inequality $c_d\le c_{d+1}$ holds in every dimension $d$
(Theorem~\ref{th:ineq}). In dimensions $1$ and $2$ the stronger result
$c_1 < c_2$ is known, thanks to the recent determination by Antonios
D.~Melas of the exact value of $c_1$ as $\frac{11
+ \sqrt{61}}{12}$ (Corollary~1 of \cite{Mel2}). Since  $c_2 \ge
\sqrt\frac32 + \frac{3-\sqrt2}4$, by Proposition~1.4 of \cite{A2},
Melas's result entails that the first inequality is strict.


Finally, we note that the original question of Stein and Str\"omberg (see
also~\cite{BH}, Problem~7.74~c, proposed by A.~Carbery) as to whether
$\lim_d c_d < \infty$ or $\lim_d c_d = \infty$, remains open.

%---------------------------------------------------
\section{Convolution operators and measures}
%---------------------------------------------------

We shall state the main theorem of this note in terms of a
locally compact group~$X$.
Denote by $C(X)$ the family of all continuous
functions $g \colon X\to \mathbb{R}$, by $C_c(X)$ the continuous
functions with compact support, and by $\lambda$ the left Haar
measure on~$X$. If $X = \mathbb{R}^d$,
$\lambda^d$ will stand for the $d$-dimensional Lebesgue
measure. As usual, we shall write $dx$ instead of $d\lambda (x)$.
A finite real valued Borel measure $\mu$ on $X$ is \emph{Radon} if
$|\mu|$ is inner
regular with respect to the compact sets. It is well
known that if $X$ is a locally compact separable metric space, then
every finite Borel measure is automatically Radon.
Let $\mathcal{N}$ be a neighborhood base at $0$ such that each element
of $\mathcal{N}$ has compact closure, and let
$\{h_U : U \in \mathcal{N}\}$ be an approximate identity, i.e.,
a family of nonnegative Borel functions such that for every
$U \in \mathcal{N}$, $\supp h_U \subset U$ and $\|h_U\|_1 =1$.
Furthermore, since for every neighborhood $U$ of $0$ there
is a continuous function $g_U$ with values in $[0,1]$,
$g_U(0) = 1$, and $\supp g_U \subset U$, we may assume that each
function in the approximate identity is continuous (obtain $h_U$
by normalizing~$g_U$).
Let $\mu$ be a finite, nonnegative Radon measure on~$X$.
Recall that
\[
    h*f(x) = \int f(y^{-1}x) h(y) \, dy
    \text{\qquad and \qquad}
    \mu*f(x) = \int f(y^{-1}x) \, d\mu(y).
\]
Let $g\in C_c(X)$;
we shall utilize the following well known results:
$\mu*(h_U*g) = (\mu*h_U)*g$, and
$h_U*g \to g$ uniformly as $U \downarrow 0$.
The idea of the proof below consists simply in replacing the measure
$\mu$ with the continuous function $\mu*h_U$, using the fact that
$\| \mu*h_U \|_1 = \mu(X)$.

The $L_1$ norm refers always in this paper to Haar measure.

\begin{lem}
\label{lem:family}
Let $\{k_{\beta}\}$ be a family of
nonnegative lower semicontinuous real valued functions, defined
on~$X$. Set $k^*v := \sup_{\beta}|v*k_{\beta}|$, where $v$ is either a
function or a measure. Then, for every
finite real valued Radon measure $\mu$ on $X$,
and every $\alpha > 0$,
\[
    \lambda^d \{k^* \mu > \alpha\} \le
    \sup \big\{ \lambda^d \{k^* f > \alpha\} : \|f\|_1 = |\mu|(X) \big\}.
\]
The same result holds if $\{k_n\}$ is a sequence of
nonnegative real valued Borel functions.
\end{lem}

\begin{pf}
Consider first the case where $\{k_{\beta}\}$ is a family of
lower semicontinuous functions. We shall assume that
functions and measures are nonnegative.
There is no loss of generality in doing so since
$k^* f \le k^*|f|$ and
$k^*\mu \le k^*|\mu|$
always. Also,
by lower semicontinuity,
$\int k_{\beta} \, d\mu = \sup \{\int g_{\gamma,\beta} \, d\mu :
0\le g_{\gamma ,\beta} \le k_\beta$,
$g_{\gamma,\beta} \in C_c(X)\}$ (Corollary~7.13 of~\cite{Fo}).
It follows that for every $x$,
$\sup_\beta \mu *k_{\beta} (x) = \sup_{\gamma, \beta}
\{\mu *g_{\gamma,\beta} (x): 0\le g_{\gamma ,\beta} \le k_\beta$,
$g_{\gamma ,\beta} \in C_c(X)\}$. Therefore
we may assume that the family $\{k_{\beta}\}$
consists of nonnegative continuous functions with compact support.

Next, let $\{h_U: U \in \mathcal{N}\}$ be an approximate identity as above,
with each $h_U$ continuous, and
let $C \subset \{k^* \mu > \alpha\}$
be a compact set. It suffices to show that there exists
a function $f$ with $\| f \|_1 = \mu(X)$ and
$C \subset \{k^* f > \alpha\}$. We shall take $f$ to be $\mu*h_{U_0}$,
for a suitably chosen neighborhood $U_0$. Since
$\{k^* \mu > \alpha\} = \cup_\beta \{\mu *k_\beta > \alpha\}$ and each
$\mu *k_\beta $ is continuous, there exists a finite subcollection of
indices $\{\beta_1, \dots , \beta_\ell\}$ with
$C \subset \cup_1^\ell \{\mu *k_{\beta_i} > \alpha\}$, so the continuous
function $\max_{1\le i\le \ell} \mu *k_{\beta_i} $ attains a minimum
value $\alpha + a$ on $C$, with $a$ strictly positive.
Because $\mu$ is a finite measure and $h_U * k_{\beta_i}$
converges uniformly to $k_{\beta_i}$ as $U \to 0$,
$\mu * h_U * k_{\beta_i}$ also
converges uniformly to $\mu * k_{\beta_i}$. Hence, there exists an $U_0 \in
\mathcal{N}$ such that for every $V \subset U_0$, $V \in \mathcal{N}$, and
every $i\in \{1, \dots, \ell\}$,
$\| \mu * k_{\beta_i} - \mu * h_V * k_{\beta_i} \|_\infty < a /2$.
In particular, it follows that
\[
    C \subset \big\{ \max_{1 \le i \le \ell} \mu * h_{U_0} * k_{\beta_i}
      > \alpha \big\}
    \subset \big\{ k^*(\mu*h_{U_0})> \alpha \big\}.
\]

The case where $\{k_n\}$ is a sequence of
nonnegative bounded Borel functions, can be proven by reduction
to the previous one. Choose a finite Radon measure $\mu$
and fix $\alpha > 0$.
Given $\ep \in (0, 1)$, for every $n$ let
$g_n \ge k_n$ be a bounded, lower semicontinuous function with
\[
    \| g_n - k_n \|_1 < \frac{\ep^2}{2^{n+1}\mu(X)}
\]
(cf.~Proposition~7.14 of~\cite{Fo}).
Then, for any $ f \in L_1(\lambda)$, using the Fubini-Tonelli Theorem and
left invariance we have
\begin{gather*}
    \| g^*f - k^*f \|_1
\\
    = \Big\| \sup_n \int g_n (y^{-1}x) f(y) \,dy
       - \sup_n \int k_n (y^{-1}x) f(y) \,dy \Big\|_1
\\
    \le \sum_n \iint (g_n (y^{-1}x) - k_n (y^{-1}x)) |f(y)| \,dy\,dx
\\
    = \sum_n \int |f(y)| \int (g_n (y^{-1}x) - k_n (y^{-1}x)) \,dx\,dy
\\
    = \sum_n \|f\|_1 \|g_n - k_n\|_1
    < \|f\|_1 \ep^2 (\mu(X))^{-1}.
\end{gather*}
In particular, if $\| f \|_1 = \mu(X)$,
we have that
\[
    \| g^*f - k^*f \|_1 < \ep^2,
\]
from which
\[
    \lambda \{g^*f - k^*f \ge \ep\}
    \le \frac{\| g^* f - k^* f\|_1}{\ep} < \ep
\]
follows.
Now
$\{g^* f > \alpha + \ep \} \subset \{k^* f > \alpha\} \cup
\{g^* f - k^* f > \ep\}$, so
\begin{gather*}
    (\alpha + \ep) \lambda \{ k^*\mu > \alpha + \ep \}
    \le (\alpha + \ep ) \lambda \{ g^*\mu > \alpha + \ep \}
\\
    \le (\alpha + \ep) \sup \{ \lambda \{g^*f > \alpha + \ep\}
      : \|f\|_1 = \mu(X) \}
\\
    \le (\alpha + \ep) ( \sup \{\lambda \{k^*f > \alpha\}
      : \|f\|_1 = \mu(X)\} + \ep ),
\end{gather*}
and the result is obtained by letting $\ep \downarrow 0$.
\end{pf}

\begin{thm}
\label{th:equiv}
Let $\{k_{\beta}\}$ be a family of
nonnegative lower semicontinuous real valued functions, defined on
$X$, and let $c>0$ be a fixed constant. Then the
following are equivalent:

(i) For every function $f\in L_1 (\lambda )$,
and every $\alpha > 0$,
\[
    \alpha \lambda \{k^*f > \alpha\} \le c \|f\|_1.
\]

(ii) For every finite real valued Radon measure $\mu$ on $X$,
and every $\alpha > 0$,
\[
    \alpha \lambda \{k^*\mu > \alpha\} \le c |\mu|(X).
\]
The same result holds if $\{k_{n}\}$ is a sequence of
nonnegative real valued Borel functions.
\end{thm}

\begin{pf}
(i) is the special case of~(ii) where $d\mu(y) = f(y)\,dy$.
For the other direction, by Lemma~\ref{lem:family} and part~(i) we have
\[
    \alpha \lambda \{k^* \mu > \alpha\}
    \le \alpha \sup \{ \lambda \{k^*f > \alpha\} : \|f\|_1 = |\mu|(X) \}
    \le c |\mu|(X).
\]
\end{pf}

\begin{rem}
By the discretization theorem of M.~de Guzm\'an
(see~\cite{Guz}, Theorem~4.1.1), further refined by
M.~T.~Men\'arguez and F.~Soria (Theorem~1 of~\cite{MeSo}), in
$\mathbb{R}^d$ conditions~(i) and~(ii) of Theorem~\ref{th:equiv} are
both equivalent to

\textit{
(iii) For every finite collection $\{\delta_{x_1}, \dots , \delta_{x_N}\}$
of Dirac deltas on $X$, and every $\alpha > 0$,
\[
    \alpha \lambda \big\{ k^* \sum_1^N \delta_{x_i} > \alpha \big\} \le c N.
\]
}

From the viewpoint of obtaining lower
bounds, the usefulness of~(ii) is due to the fact that it allows to choose
among a wider class of potential examples than just finite sums of Dirac deltas.
Both~(ii) and~(iii) will be utilized in the sext section.
\end{rem}


%---------------------------------------------------
\section{Behavior of constants for the Hardy-Littlewood maximal operator}
\label{sec:constants}
%---------------------------------------------------

Let $B \subset \mathbb{R}^d$ be an open, bounded, convex set,
symmetric about zero.
We shall call $B$ a ball, since each norm on $\mathbb{R}^d$ yields sets of this
type, and each bounded $B$, convex and symmetric about zero, defines a norm.
The (centered) Hardy-Littlewood maximal operator associated to $B$
is defined for locally integrable
functions $f \colon \mathbb{R}^d \to \mathbb{R}$ as
\[
    M_{d,B} f(x) := \sup_{r > 0}
      \frac{\chi_{rB}}{r^d \lambda^d (B)}*|f| (x).
\]
We denote by $c_{d,B}$ the best constant in the weak type $(1,1)$
inequality
$\alpha \lambda^d \{ M_{d,B} f > \alpha \} \le c \|f\|_1$,
where $c$ is independent of
$f \in L^1(\mathbb{R}^n)$ and $\alpha > 0$. Let $s :=\{r_n\}_{-\infty}^\infty$
be a lacunary (bi)sequence (i.e., a sequence
that satisfies $r_{n+1}/r_n \ge c$ for some fixed constant
$c > 1$ and every $n \in \mathbb{Z}$). Then the associated maximal operator
is defined via
\[
    M_{s,d,B} f(x) := \sup_{n \in \mathbb{Z}}
    \frac{\chi_{r_n B}}{r_n^d \lambda^d(B)}*|f| (x).
\]
The arguments given below are applicable to both the maximal function
and to lacunary versions of it, so we shall not introduce
a different notation for the best constants in the lacunary case.
In particular, Lemma~\ref{lem:linear} and Theorem~\ref{th:ineq}
refer to all of these maximal operators, but only the usual maximal operator
shall be mentioned in the proofs.

Given a finite sum
$\mu = \sum_1^k \delta_{x_i}$ of Dirac deltas, where
the $x_i$'s need not be all different, let $\sharp (x + B)$ be the number
of point masses from $\mu$ contained in $x + B$.

\begin{lem}
\label{lem:linear}
Let $B$ be a ball in $\mathbb{R}^d$. Then for
every linear transformation $T \colon \mathbb{R}^d \to \mathbb{R}^d$ with
$\det T \ne 0$, $c_{d,B} = c_{d,T(B)}$.
\end{lem}

\begin{pf}
Given $\mu := \sum_1^k \delta_{x_i}$ and
$T \mu := \sum_1^k \delta_{T(x_i)}$, we have that
\[
    M_{d,B} \mu (x) := \sup_{r > 0} \frac{\sharp (x + rB)}{r^d\lambda^d (B)}
\]
and
\[
    M_{d,T(B)} T\mu (x) := \sup_{r > 0}
      \frac{\sharp (x + rT(B))}{r^d\lambda^d (T(B))}.
\]
Then
$x \in \{ M_{d,B} \mu > \alpha \}$
iff
$T(x)\in \{ M_{d,T(B)} T\mu > (\alpha / |\det T|)\}$.
Since
\[
    |\det T| \lambda^d \{ M_{d,B} \mu > \alpha \}
    = \lambda^d \{ M_{d,T(B)} T\mu > (\alpha / |\det T|) \},
\]
we have
\[
    \alpha \lambda^d \{ M_{d,B} \mu > \alpha \}
    = (\alpha / |\det T|) \lambda^d \{ M_{d,T(B)} T\mu
      > (\alpha / |\det T|) \},
\]
and the result follows.
\end{pf}


\begin{thm}
\label{th:ineq}
For each $d \in \mathbb{N} \setminus \{0\}$ let $B_d$ be
a $d$-dimensional parallelotope centered at zero. Then
$c_{d,B_d} \le c_{d+1,B_{d+1}}$ for both the maximal operator
and for lacunary operators.
\end{thm}

\begin{pf}
Since every such $B_d$ is the image under a nonsingular
linear transformation of the $d$-dimensional cube $Q_d$ centered at zero
with sides parallel to the axes and
volume $1$, we may assume that in fact $B_d = Q_d$.
With the convex bodies fixed, we will write
$c_d$ and $M_d$ rather than $c_{d,B_d}$ and $M_{d,B_d}$. Given $\alpha > 0$,
$\mu_d = \sum_1^k \delta_{x_i}$ on $\mathbb{R}^d$
and a constant $c > 0$ such that
$\alpha \lambda^d\{ M_d \mu_d > \alpha \} > c \mu_d (\mathbb{R}^d)$, we want to
find a measure $\mu_{d+1}$ on $\mathbb{R}^{d+1}$ such that
$\alpha \lambda ^{d+1}\{ M_{d+1} \mu_{d+1} > \alpha \} >
c \mu_{d+1} (\mathbb{R}^{d+1})$. This will imply that $c_d\le c_{d+1}$.
Let $L := (k/\alpha)^{1/d}$. Note that if $r \ge L$, then for every
$x\in \mathbb{R}^d$,
$\frac{\sharp (x + rQ_d) }{r^d} \le \alpha$. Choose $N \gg L$
such that $\alpha \frac{N-L}{N} \lambda^d \{ M_d \mu_d > \alpha \} > c k$,
and let $\mu_{d+1} := \mu_d \times \lambda_{[-N,N]}$, where
$\lambda_{[-N,N]}$ stands for the restriction of linear Lebesgue measure
to the interval $[-N, N]$. We claim that
$\{ M_d \mu_d > \alpha \} \times [-N+L, N-L] \subset
\{ M_{d+1} \mu_{d+1} > \alpha \}$. In order to establish the
claim, the following notation shall be used:
If $x = (x_1, \dots , x_d) \in \mathbb{R}^d$,
by $(x, x_{d+1})$ we denote the point
$(x_1, \dots , x_d, x_{d+1}) \in \mathbb{R}^{d+1}$. Now if
$x\in \{ M_d \mu_d > \alpha \}$, then there exists an $r(x)\in (0, L)$
such that $r(x)^{-d} \mu_d (x + r(x) Q_d) > \alpha$, so for every
$y\in [-N + L, N -L]$,
\begin{gather*}
    r(x)^{-d-1} \mu_{d+1} ((x, y) + r(x) Q_{d+1})
\\
    = r(x)^{-d-1} (\mu_d (x + r(x) Q_d)
      \times \lambda_{[-N,N]}( [y - \tfrac{r(x)}2, y + \tfrac{r(x)}2] )
\\
    = r(x)^{-d} \mu_d (x + r(x) Q_d)> \alpha,
\end{gather*}
as desired. But now
\begin{gather*}
    \alpha \lambda^{d+1} \{ M_{d+1} \mu_{d+1} > \alpha \}
    \ge 2 \alpha (N-L) \lambda^d \{ M_d \mu_d > \alpha \}
\\
    = 2 \alpha N \frac{N-L}{N} \lambda^d \{ M_d \mu_d > \alpha \}
       > 2 N c k = c \mu_{d+1} (\mathbb{R}^{d+1}).
\end{gather*}
\end{pf}

\begin{rem}
Recall from the Introduction that for the
$\ell_\infty$ balls (i.e., cubes with sides parallel to the axes)
$
c_1 < c_2.
$
Since the $\ell_1$ unit ball in dimension 2 is a square, it
follows from Lemma~\ref{lem:linear} that the best constant in dimension 2
is equal for the $\ell_1$ and the $\ell_\infty$ norms. It follows that
$c_1 < c_2$ in the $\ell_1$ case also. It would
be interesting to know whether or not the best constants associated
to the $\ell_p$ balls are all the same.
Note that  establishing bounds of the type
$a^{-1} c_{d,2} \le c_{d,p} \le a c_{d,2}$ (where the constant
$a \ge 1$ is independent of the dimension $d$ and $c_{d,p}$
denotes the best constant associated to the $\ell_p$ ball),
would show that the bounds
$O(d)$ (which hold for euclidean balls by
\cite{StSt}) extend to $\ell_p$ balls.
\end{rem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{thebibliography}{00}

\bibitem{A1}
J. M. Aldaz,
Remarks on the Hardy-Littlewood maximal function,
\textit{Proc. Roy. Soc. Edinburgh Sect. A}
\textbf{128}
(1998),
% no. 1,
1--9.

\bibitem{A2}
J. M. Aldaz,
A remark on the centered $n$-dimensional Hardy-Littlewood maximal
function,
\textit{Czech. Math. J.}
\textbf{50(125)} (2000),
% no. 1,
103--112.

\bibitem{BH}
D. A. Brannan and W. K. Hayman,
Research problems in complex analysis,
\textit{Bull. London Math. Soc.}
\textbf{21}
(1989),
1--35.

\bibitem{Fo}
Gerald B. Folland,
``Real Analysis. Modern Techniques and Their Applications,''
Pure and Applied Mathematics,
John Wiley and Sons, New York,
1984.

\bibitem{Guz}
Miguel de Guzm\'an,
``Real variable methods in Fourier analysis,''
North-Holland Mathematics
Studies, 46. Notas de Matem\'atica [Mathematical Notes], 75.
North-Holland Publishing Co., Amsterdam-New York, 1981.

\bibitem{Mel1}
Antonios D. Melas,
On the centered Hardy-Littlewood
maximal operator,
\textit{Trans. Amer. Math. Soc.}
\textbf{354} (2002),
% no. 8,
3263--3273.

\bibitem{Mel2}
Antonios D. Melas,
The best constant for the centered Hardy-Littlewood
maximal inequality,
\textit{Ann. of Math.}, to appear.

\bibitem{Me}
M. Trinidad Men\'arguez,
``T\'ecnicas de discretizaci\'on en an\'alisis
arm\'onico para el estudio de acotaciones debiles
de operadores maximales e integrales singulares,''
Ph. D. Thesis, Universidad Complutense de Madrid,
1990.

\bibitem{MeSo}
M. Trinidad Men\'arguez and Fernando Soria,
Weak type $(1,1)$ inequalities for maximal convolution operators,
\textit{Rend. Circ. Mat. Palermo (2)}
\textbf{41}
(1992),
% no. 3,
342--352.


\bibitem{StSt}
E. M. Stein and J. O. Str\"omberg,
Behaviour of maximal functions in $\mathbb{R}^n$ for large~$n$,
\textit{Ark. Mat.}
\textbf{21}
(1983),
259--269.

\end{thebibliography}

\end{document}






math0207285.extracted_bib
\begin{thebibliography}{00}

\bibitem{A1}
J. M. Aldaz,
Remarks on the Hardy-Littlewood maximal function,
\textit{Proc. Roy. Soc. Edinburgh Sect. A}
\textbf{128}
(1998),
% no. 1,
1--9.

\bibitem{A2}
J. M. Aldaz,
A remark on the centered $n$-dimensional Hardy-Littlewood maximal
function,
\textit{Czech. Math. J.}
\textbf{50(125)} (2000),
% no. 1,
103--112.

\bibitem{BH}
D. A. Brannan and W. K. Hayman,
Research problems in complex analysis,
\textit{Bull. London Math. Soc.}
\textbf{21}
(1989),
1--35.

\bibitem{Fo}
Gerald B. Folland,
``Real Analysis. Modern Techniques and Their Applications,''
Pure and Applied Mathematics,
John Wiley and Sons, New York,
1984.

\bibitem{Guz}
Miguel de Guzm\'an,
``Real variable methods in Fourier analysis,''
North-Holland Mathematics
Studies, 46. Notas de Matem\'atica [Mathematical Notes], 75.
North-Holland Publishing Co., Amsterdam-New York, 1981.

\bibitem{Mel1}
Antonios D. Melas,
On the centered Hardy-Littlewood
maximal operator,
\textit{Trans. Amer. Math. Soc.}
\textbf{354} (2002),
% no. 8,
3263--3273.

\bibitem{Mel2}
Antonios D. Melas,
The best constant for the centered Hardy-Littlewood
maximal inequality,
\textit{Ann. of Math.}, to appear.

\bibitem{Me}
M. Trinidad Men\'arguez,
``T\'ecnicas de discretizaci\'on en an\'alisis
arm\'onico para el estudio de acotaciones debiles
de operadores maximales e integrales singulares,''
Ph. D. Thesis, Universidad Complutense de Madrid,
1990.

\bibitem{MeSo}
M. Trinidad Men\'arguez and Fernando Soria,
Weak type $(1,1)$ inequalities for maximal convolution operators,
\textit{Rend. Circ. Mat. Palermo (2)}
\textbf{41}
(1992),
% no. 3,
342--352.


\bibitem{StSt}
E. M. Stein and J. O. Str\"omberg,
Behaviour of maximal functions in $\mathbb{R}^n$ for large~$n$,
\textit{Ark. Mat.}
\textbf{21}
(1983),
259--269.

\end{thebibliography}