b$.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Archimedean orders have a simple characterization
in terms of real-valued homomorphisms
due to O.\ H\"{o}lder
(\cite[Theorem 1, p. 45]{fu}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{archimedean}
An order $P$ on $\Gamma$ is Archimedean
if and only if $\Gamma$ is isomorphic to a subgroup of $\R$.
\label{archimedean}
\end{archimedean}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
There is another useful characterization of
Archimedean orders in terms of convex subgroups
(\cite[Corollary 5, p.\ 47]{fu}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem2}
Suppose that $\Gamma$ is an ordered group,
then $\Gamma$ is Archimedean ordered
if and only if the only convex subgroups
of $\Gamma$ are $\{0\}$ and $\Gamma$.
\label{theorem2}
\end{theorem2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Following \cite[Chapter IV, Section 3]{fu},
we let $\Sigma$ denote the system of all
convex subgroups in $\Gamma$. This system
is in fact a chain containing $\{0\}$
and $\Gamma$. Hence if $C$ and $D$ are in $\Sigma$, then
either $C\subset D$ or $D\subset C$. Also,
whenever
$\{C_\lambda\}_{\lambda\in\Lambda}$
is a collection from $\Sigma$, then
$\bigcap_{\lambda\in\Lambda}C_\lambda$
and
$\bigcup_{\lambda\in\Lambda}C_\lambda$
are again in $\Sigma$. By a jump in
$\Sigma$ we mean a pair of subgroups
$C$ and $D$ such that $D\subset C, D\neq C$,
and $\Sigma$ contains no subgroups
between $C$ and $D$. A jump will be denoted by
$D\prec C$. \\
It is a fact that a
subgroup $C\in\Sigma$ is the greater
member of a jump (i.\ e.\ $D\prec C$) if and only
if $C$ is a principal convex subgroup. That is,
$$C= \{a\}_\Box$$
for some $a\in \Gamma$ (see p. 54 of \cite{fu}).\\
Let
$\Sigma_0$
denote the system of principal convex
subgroups, and let $\Pi$ be an indexing set for
$\Sigma_0$. Order $\Pi$ as follows:
for $\rho, \pi\in \Pi$,
set
$\pi\leq\rho$ if and only if
$C_\rho\subset C_\pi$.
With this order, $\Pi$ has a
maximal element
$\alpha_0$
corresponding to $\{0\}\in \Sigma_0$.
Thus $C_{\alpha_0}=\{0\}$.
For notational convenience, let
$D_{\alpha_0}=\emptyset$.
Let
$D_\pi\prec C_\pi$
denote a jump in $\Sigma$, with $\pi<\alpha_0$.
The quotient group $C_\pi/ D_\pi$ has
no nontrivial convex subgroups.
By Theorem (\ref{theorem2}), $C_\pi/ D_\pi$ is
Archimedean ordered. Let
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\psi_\pi :\ \ C_\pi/ D_\pi \rightarrow \R
\label{psi-sub-pi}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
denote the order isomorphism mapping $C_\pi/D_\pi$
into a subgroup of $\R$, and let
%
%
%
\begin{equation}
L_\pi :\ \ C_\pi \rightarrow \R
\label{L-sub-pi}
\end{equation}
%
%
%
denote the composition of $\psi_\pi$
with the natural homomorphism of
$C_\pi$ onto $C_\pi/ D_\pi$. Then $L_\pi$
is an order homomorphism of $C_\pi$ into
$\R$ with $\ker L_\pi=D_\pi$.
Since $\R$ is a divisible group,
the homomorphism $L_\pi$
can be extended to a homomorphism of the
entire group $\Gamma$ into $\R$.
(See \cite[Theorem A.7, p. 441]{hr1}.)
We keep the same notation for this extension.
The next
theorem summarizes this discussion.
It is a basic result of this section
and will be used in defining our construction
of the conjugate function.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%THEOREM3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem3}
Let $\Gamma$ be an infinite discrete
(torsion-free) ordered group with order
$P$. Let $\Sigma$ denote the chain of
convex subgroups of $\Gamma$ and $\Sigma_0$
the subcollection of principal convex
subgroups indexed by the ordered set $\Pi$.
There is a collection of real-valued
homomorphisms
$\{L_\pi, \pi\in \Pi \}$
of $\Gamma$ into $\R$
such that, for every jump
$D_\pi\prec C_\pi$,
we have \\
(i) \ \ $ L_\pi\left( D_\pi\right)=\{0\}$; and\\
(ii) $\sgn (L_\pi(\chi))=\sgn_P(\chi)$\\
for all $\chi\in C_\pi\setminus D_\pi$.
\label{theorem3}
\end{theorem3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
Observe that, in the notation of the previous theorem, we have
%
%
%
\begin{equation}
\Gamma =\bigcup_{\alpha\in\Pi} C_\alpha\backslash D_\alpha.
\label{slices}
\end{equation}
%
%
%
In fact, given $x\in\Gamma$, we have
$\{x\}_\Box=C_\alpha$ for some $\alpha\in \Pi$.
Since $D_\alpha$ is strictly contained
in $C_\alpha$, it follows that $x$
belongs to $C_\alpha\backslash
D_\alpha$ which proves (\ref{slices}).\\
The following example will illustrate many of the results of this section.\\
{\bf Example} Suppose that $( T, <)$ is an
ordered set and that $\gamma$ is a limit
ordinal. Let $\Z^{(T,\gamma)}$ be the group
of sequences $(z_t)_{t\in T}$ such that the set
$\{ t:\ \ z_t\neq 0\}$ is reverse well ordered (i.\ e.,
every subset has a largest element) with order type less than
$\gamma$. Define an order as follows: If
$z=(z_t)_{t\in T}\in \Z^{(T,\gamma)}$ and
$t_0=\max \{t:\ \ z_t\neq 0\}$, then $z\in P$
if and only if $z_{t_0} \ge 0$.\\
In this example we see that $\Sigma_0$
is order isomorphic $-T$, that
$C_\alpha=\{(z_t):\ z_t=0,\ {\rm for}\ t\geq\alpha\}$,
that
$D_\alpha=\{(z_t):\ z_t=0,\ {\rm for}\ t> \alpha\}$,
and that $L_\alpha ((z_t))=z_\alpha$.\\
%
We can also construct $\R^{(T,\gamma)}$ similarly.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Orders on locally compact abelian groups}
%
\newtheorem{theorem3.1}{Theorem}[section]
\newtheorem{remark3.2}[theorem3.1]{Remark}
\newtheorem{addedlemma}[theorem3.1]{Lemma}
\newtheorem{theorem3.3}[theorem3.1]{Theorem}
\newtheorem{remarks3.4}[theorem3.1]{Remarks}
\newtheorem{proposition31}[theorem3.1]{Proposition}
\newtheorem{lemma31}[theorem3.1]{Lemma}
\newtheorem{theorem32}[theorem3.1]{Theorem}
\newtheorem{theorem33}[theorem3.1]{Theorem}
\newtheorem{theorem34}[theorem3.1]{Theorem}
\newtheorem{theorem35}[theorem3.1]{Theorem}
\newtheorem{theorem36}[theorem3.1]{Theorem}
%
%
In this section we prove a general version
of Theorem (\ref{theorem3}) for measurable orders.
%
The difficulty here is due to
the fact that, in general, the chain
of convex subgroups in Theorem (\ref{theorem3})
may contain nonmeasurable
subgroups, or jumps of the form $D_\alpha\prec C_\alpha$
with $C_\alpha\setminus D_\alpha$ having measure zero.
To overcome these measure theoretic problems, we will
find a smallest open principal convex subgroup
of $\Gamma$ which will determine when to stop the chain
while still being able to separate with
continuous real-valued homomorphisms as in Theorem
(\ref{theorem3}).
This section is based on the
study of orders of Hewitt and Koshi \cite{hk}.
Indeed, Theorem \ref{theorem35}
below, is a combination of results from \cite{hk} and the
material from the previous section.
%
%
%
Throughout this section $\Gamma$ will
denote an infinite locally compact torsion-free
abelian group.
The following
basic properties of measurable orders will be needed.
%%
%%
%%
\begin{theorem3.1}
(a) If $P$ is a measurable order, then $P$
has a nonvoid interior (\cite[Theorem (3.1)]{hk}.
Consequently, if
$P$ is a measurable order, then $(-P)$ has nonvoid interior.\\
(b) If $\Gamma$ is an infinite compact
torsion-free group, then every
order on $\Gamma$ is dense and has
void interior (\cite[Theorem (3.2)]{hk}.
Consequently, every order on a compact
infinite group is nonmeasurable.\\
\label{theorem3.1}
\end{theorem3.1}
%
%
%
\begin{remark3.2}
{\rm Suppose that $\Gamma$ is a locally compact
abelian group, and that $P$ is a
measurable order on $\Gamma$.
%
Use the structure theorem for locally
compact abelian groups to write $\Gamma$
as $\Gamma=\R^a\times\O$ where $a$ is a nonnegative
integer and $\O$
contains a compact open subgroup $\O_0$
(\cite[Theorem (24.30)]{hr1}).
The fact that $P$ is measurable automatically
implies that either $\Gamma$ is
discrete or $a>0$.
%
In fact, if $a=0$, then $\Gamma=\O$ and so $\O_0$
is a compact open
subgroup of $\Gamma$.
%
The restriction of $P$ to $\O_0$ is
a measurable order in $\O_0$.
But since $\O_0$ is compact, it
follows from Theorem (\ref{theorem3.1})(b) that $\O_0=\{0\}$,
and so $\Gamma$ is discrete. }
\label{remark3.2}
\end{remark3.2}
%
%
%
%
Henceforth to avoid the cases
treated in the previous section,
we will assume that $\Gamma=\R^a\times\O$ where $a>0$.
For use in the sequel, we need the following result due to
Hewitt and Koshi \cite[Theorem (3.12)]{hk}. %
%
%
%
%
\begin{theorem3.3}
Let $P$ be an order on $\R^a\times B$\ where
$a>0$ and $B$ is an infinite torsion-free
locally compact abelian group that is
the union of its compact open subgroups.
Suppose that $P$ has nonempty interior.
Then there is a continuous real-valued
homomorphism $L:\ \R^a\longrightarrow \R$\ such that
\begin{equation}
L^{-1}(] 0,\infty [ )\times B\subset {\rm int} (P) \subset P\subset
L^{-1}([ 0,\infty [ )\times B.
\label{orders-on-R-a}
\end{equation}
The order $P\cap (L^{-1} (\{0\})\times B)$
is arbitrary.
\label{theorem3.3}
\end{theorem3.3}
%
%
%
%
%
%
%
%
\begin{remarks3.4}
{\rm (a) It follows from Theorems (\ref{theorem3.1}) and
(\ref{theorem3.3}) that an order
$P$ on $\R^a$ is measurable if and only if
it is not dense in $\R^a$.\\
(b) Let $P$ be a measurable order on $\R^a \times \O$,
and let $B$ denotes the union of all the compact open
subgroups of $\O$.
The restriction of $P$ to $\R^a \times B$
is a nondense order, and so by
Theorem (\ref{theorem3.3}) there is a
continuous real-valued homomorphism
$L:\ \R^a \longrightarrow \R$ such that
$$ L^{-1} (]0,\infty]) \times B \subset P .$$}
\label{remarks3.4}
\end{remarks3.4}
%
%
%
We can now describe our candidate
for a smallest open convex
principal subgroup of $\Gamma$.
Let
%
%
%
\begin{equation}
H=\left\{
y\in\O:\ \ \R^a\times\{y\} \
{\rm \ has\ nonvoid\ intersections\ with}\ P\
{\rm \ and\ } \ (-P)
\right\}.
\label{candidate}
\end{equation}
%
%
%
%
%
%
\begin{proposition31}
Let $P$ be a measurable order
on $\R^a\times \O$, and let $H$ be as in (\ref{candidate}).
%
Then $H$ is a subgroup of $\O$ that contains all the compact
open subgroups of $\O$;
and $R^a\times H$ is an open convex
subgroup of $\Gamma$.
\label{proposition31}
\end{proposition31}
%
%
%
{\bf Proof.} The fact that $H$ contains all the
compact open subgroups of
$\O$ follows from Theorem (\ref{theorem3.3}).
Also, the fact that $H$ is a subgroup is easily verified.
Since $\R^a\times H$ is a subgroup with nonvoid interior
it follows immediately that the subgroup is open.
%
%
%
To establish the convexity of $\R^a\times H$,
suppose that
%
%
%
$$0< (t,y) < (t^\prime, y^\prime)$$
%
%
%
with $y^\prime\in H,\ t,\ t^\prime \in \R^a$.
To show that
$(t,y)\in \R^a\times H$, it is enough to
find $x\in \R^a$ with $(x,y)\in -P$.
Since $y^\prime \in H$, we can find
$x\in \R^a$ such that $(x,y^\prime)\in -P$.
Hence
$(t-t^\prime ,y-y^\prime) +(x,y^\prime)\in (-P)$,
or,
$(t-t^\prime +x , y)\in (-P)$. \\
%
%
%
\begin{lemma31}
Let $P$ be a measurable order in $\Gamma$
and let $H$ be as in (\ref{candidate}).
%
For $y\in H$, let $A_y=(\R^a\times\{y\})\cap P$ and
$B_y=(\R^a\times\{y\})\cap (-P)$.
%
Then $A_y$ and $B_y$ are nondense in $\R^a\times\{y\}$.
\label{lemma31}
\end{lemma31}
%
%
%
{\bf Proof.}
It is enough to deal with the set $A_y$
with $y\neq 0$.
%
Assume that
$(\R^a\times\{y\})\cap P$ is dense
in $\R^a\times\{y\}$.
%
Let $(s,y)$ be any element of $\R^a\times\{y\}$, and let
$(s_0,y)\in (\R^a\times\{y\}) \cap P$
be such that $L(s_0)> L(s)$.
%
We have $L(s_0-s)>0$, and so from Remark (\ref{remarks3.4}) (b)
we see that $(s_0-s,0)\in P$.
%
Hence $(s_0,y)+(s-s_0,0)=(s,y)\in P$
implying that $\R^a\times\{y\}\subset P$
which contradicts the fact that $y\in H$.
%
Thus $P\cap(\R^a\times\{y\})$ is
nondense in $\R^a\times\{y\}$.\\
%
%
%
%
We can now prove a special separation theorem for orders.
(Compare with \cite[Theorem 3.7]{hk}.)
%
%
%
\begin{theorem32}
Let $P$ be a measurable
order on $\Gamma$, let
$H$ be as in (\ref{candidate}),
and let $L$ be as in Remarks (\ref{remarks3.4}) (b).
For every $y\in H$, there is a real number
$\alpha ( y )$ such that \\
(i) $L^{-1}\left(]-\infty,\alpha(y)
[\right) \times \{y\}\subset -P,$\\
%
%
(ii) $L^{-1}\left(]\alpha(y),\infty[\right) \times \{y\}\subset P.$\\
Moreover, the mapping $y\mapsto \alpha(y)$ is a continuous real-valued homomorphism from $H$.
\label{theorem32}
\end{theorem32}
%
{\bf Proof.} We will write the elements of
$\Gamma=\R^a\times \O$ as $(x,y)$ where $x \in \R^a$,
and $y\in \O$.
%
Suppose that $(x_1,y) \in P$ and $L(x_2)>L(x_1)$.
%
Then, $L(x_2-x_1)>0$, and so from Remarks (\ref{remarks3.4}) (b)
we have that $(x_2-x_1,0)\in P\backslash \{0\}$,
and consequently, $(x_2,y)\in P\backslash \{0\}$.
%
Similarly, if $(x_1,y)\in (-P)$, and $L(x_2) y\}\right)\leq
\frac{A}{y} \|f\|_1$$
for all $y>0$, where $A$ is the weak type $(1,1)$
norm of the Hilbert transform on $L^1(\R)$.\\
(iv) For $f\in L^2(G)$, we have
$$\widehat{H_L f}(\chi)=- i \sgn (L (\chi))\widehat{f}(\chi)$$
for almost all $\chi\in \Gamma$.
\label{ergodic-hilbert-transform}
\end{ergodic-hilbert-transform}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%ERGODIC_HILBERT_TRANSFORM%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The usefulness of this theorem
is due in great part to the fact that all the
estimates are independent of $L$ or $G$.
Property (iv) justifies using the terminology
"the Hilbert transform in the direction of
$L$" and shows a clear
connection between the ergodic Hilbert transform
and the conjugate function on groups.
The proof of (iv) is straightforward,
using (ii) and (\ref{adjoint-map}). (See
\cite[Theorem (6.7)]{ah}.) For use in the sequel,
we recall the generalizations of M. Riesz's
Theorem and Kolmogorov's Theorem from \cite{ah}.
(These results are due to Helson \cite{hel1} and \cite{hel2},
when $G$ is compact.)
Also, having all the necessary ingredients to prove
these results, we will sketch short proofs
to make the paper more self contained
and to illustrate the use of the separation theorems.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%generalized-m-riesz%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{generalized-m-riesz}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $G$ be a locally compact abelian group with
dual group $\Gamma$, and let $P$ denote an arbitrary
measurable order on $\Gamma$. For all
$f\in L^p(G)$, $1 0$, we have
$$\mu\left(\{x\in G :\ \ | \H f(x)|>y\}\right)\leq
\frac{A}{y} \|f\|_1$$
where $A$ is the weak type $(1,1)$ norm of the
Hilbert transform on $L^1(\R)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{generalized-kolmogorov}
\end{generalized-kolmogorov}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Both theorems are proved in a similar way.
It is enough to consider $f\in L^2(G)$
with compactly supported Fourier transform.
Let $K\subset\Gamma$ denote the compact support of
$\widehat{f}$. Apply Theorem (\ref{theorem36})
and obtain a real-valued homomorphism $L$
of $\Gamma$ such that
$$\sgn_P(\chi)=\sgn(L(\chi))$$
for almost all $\chi \in K$. Thus,
from Theorem (\ref{ergodic-hilbert-transform})(iv)
and the fact that $\widehat{f}$ is supported in
$K$, it follows from the uniqueness
of the Fourier transform that
$$\H f=H_L f$$
a.e. on $G$. The inequalities in
Theorems (\ref{generalized-m-riesz})
and (\ref{generalized-kolmogorov})
follow now from the corresponding ones
for $H_L$ in
Theorem (\ref{ergodic-hilbert-transform}).\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Because of Theorem (\ref{generalized-kolmogorov}),
the operator $\H$ extends from $L^2 \cap L^1 (G)$
to an operator on $L^1(G)$ satisfying the
same weak type
$(1,1)$ estimate. We keep the same notation for the
extended operator.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The next theorem is our first step toward
building the conjugate function. We continue
with the notation leading to (\ref{difference-series}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%conjugate-of-slice%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{conjugate-of-slice}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $f\in L^p(G)$ where $1\leq p<\infty$,
and let $\alpha\in \Pi$. Then\\
(i)\ \ $\H (d_\alpha f)= H_{L_\alpha} (d_\alpha f)$
$\mu-$a.e.\\
If $f\in L^2\cap L^p(G)$, then we also have\\
(ii)\ \ $\H (d_\alpha f)= d_\alpha (\H f)$ and
$H_{L_\alpha} (d_\alpha f)= d_\alpha (H_{L_\alpha} f)$\ $\mu-$a.e.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%conjugate-of-slice%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{conjugate-of-slice}
\end{conjugate-of-slice}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf Proof.} The equalities in (ii)
are clear since all operators in question are
multiplier operators and so they commute.
To prove (i) we note that
since $d_\alpha$ is a bounded operator from
$L^1(G)$ into $L^1(G)$, and since
$\H$ and $H_{L_\alpha}$ are bounded from
$L^1(G)$ into $L^{1,\infty}(G)$, it is enough
to consider $f\in L^2(G)$. Since $\sgn_P$ and
$\sgn (L_\alpha(\cdot))$ agree a.e. on
$C_\alpha\setminus D_\alpha$, and since
$d_\alpha$ projects the Fourier transform
on $C_\alpha\setminus D_\alpha$, it
is easy to see that the Fourier
transforms of $\H (d_\alpha f)$ and
$ H_{L_\alpha} (d_\alpha f)$ agree almost everywhere
on $\Gamma$,
and so (i) follows.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As we argued for (\ref{difference-series}),
we will agree that, for $f\in L^p(G)$
$(1\leq p<\infty)$, the formal
series
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%conjugate-series%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\sum_{\alpha\in \Pi} H_{L_\alpha}(d_\alpha f)
\label{conjugate-series}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
has only countably many terms.
We will refer to (\ref{conjugate-series}) as
the conjugate (difference) series of $f$.\\
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\section{Unconditional convergence of conjugate difference series}
\newtheorem{conjugate-projection}{Theorem}[section]
\newtheorem{finite-conjugate-projection}[conjugate-projection]{Corollary}
\newtheorem{unconditional-convergence}[conjugate-projection]{Theorem}
\newtheorem{partial-sum}[conjugate-projection]{Corollary}
\newtheorem{theorem-square-function}[conjugate-projection]{Theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We will show that the conjugate series
(\ref{conjugate-series})
converges unconditionally in $L^p(G)$
when $1 0}
y\left( \lambda_f (y)\right)^\frac{1}{p}=
\sup_{y>0}y^{1/p} f^*(y),
\label{lorentz-pseudo-norm}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
and
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\| f\|_{p,\infty}=\sup_{y>0}
y^\frac{1}{p} m_f (y)
\label{lorentz-norm}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$ m_f(y)=\frac{1}{y} \int_0^y f^*(u)du.$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $L^{p,\infty}(G)$ consist of
all measurable functions on $G$ such that
$\| f\|^*_{p,\infty}<\infty$.
It is well-known that, when $1 0$, we can make the left side $<\e$
by first choosing $n$ so that
$\| f-g_n\|^*_{1,\infty}<\frac{\e}{12 A}$ and
then choosing $N=N(n)$ so that
$\| \sum_{j=1}^N H_{L_{\alpha_j}} d_{\alpha_j} g_n -
\H g_n\|^*_{1,\infty}=0$.
This completes the proof.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf The conjugate square function} We end this section
with a study of the square function
associated with the conjugate series (\ref{conjugate-series}).
We start with a definition. For
$f\in L^p(G), \ 1\leq p<\infty$, let
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\widetilde{S} f =\left( \sum_{\alpha\in \Pi}
| H_{L_\alpha} ( d_\alpha f )|^2
\right)^\frac{1}{2},
\label{square-function}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where the index of
summation runs over those
$\alpha$'s for which $d_\alpha f\not\equiv 0$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem-square-function}
(i) Let $1 0$,
we have
$$ \mu \left( \left\{ x\in G: \ \ | \widetilde{S} f (x) |
> y \right\} \right) \leq \frac{B}{y} \| f \|_1 .
$$
\label{theorem-square-function}
\end{theorem-square-function}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf Proof.} Part (i) is a well-known consequence
of Theorem (\ref{unconditional-convergence}) (ii).
We will omit the proof. (In fact one can
prove it by reproducing the argument that
we present for part (ii)).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To prove (ii), let $p$ be an arbitrary but
fixed number in $]0,1[$. We will need Kintchine's
Inequality (\cite[Theorem V. 8.4, p.213]{zyg},
which we will cite here in a notation
convenient for our proof. Let
$a_1 , a_2,\ldots , a_N$ be arbitrary
complex numbers, and write $\E$ for the
expected value over the probability space
$\{-1,1\}^N$. Then,
Kintchine's Inequality asserts that there are
constants $\alpha_p$ and $\beta_p$,
depending only on $p$, such that
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\alpha_p \left\{ \sum_{j=1}^N |a_j |^2
\right\}^\frac{1}{2}
\leq
\left\{ \E \left| \sum_{j=1}^N a_j \e_j \right|^p
\right\}^\frac{1}{p}
\leq
\beta_p \left\{ \sum_{j=1}^N |a_j |^2
\right\}^\frac{1}{2}.
\label{kintchine-inequality}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Returning to the proof of (ii), we note
by monotone convergence that it is enough
to consider a finite sum
$$ \left( \sum_{j=1}^N
| H_{L_{\alpha_j}} d_{\alpha_j} f|^2
\right)^\frac{1}{2}. $$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Applying Kintchine's Inequality, we see that,
pointwise on $G$, we have
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$$
\left( \sum_{j=1}^N
| H_{L_{\alpha_j}} d_{\alpha_j} f|^2
\right)^\frac{1}{2}
\leq C_p
\left( \E | \sum_{j=1}^N
\e_j H_{L_{\alpha_j}} d_{\alpha_j} f|^p
\right)^\frac{1}{p}.
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We think of each $\e\in \{-1,1\}^N$
as an element of $\{-1,1 \}^\Pi$ by setting
$\e(\alpha_j)=\e(j)$ for $j=1,2,\ldots,N$, and
$\e(\pi)=1$ for $\pi \not\in
\{\alpha_1, \alpha_2,\ldots , \alpha_N\}$.
Let $\eta=\eta(\e) $
be defined as in the proof of
Corollary (\ref{finite-conjugate-projection}),
(see (\ref{conjugate-eta-projection})) so that
$$\sum_{j=1}^N
\e_j H_{L_{\alpha_j}} d_{\alpha_j} f=
\widetilde{{\cal P}}_{\eta (\epsilon), \e(P)}f.$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Then
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\| \left( \sum_{j=1}^N
| H_{L_{\alpha_j}} d_{\alpha_j} f|^2
\right)^\frac{1}{2} \|^*_{1,\infty}
\leq C_p
\| \left( \E | \widetilde{{\cal P}}_{\eta(\e), \e(P)}f|^p
\right)^\frac{1}{p}
\|^*_{1,\infty}.
\label{kintchine}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
It is easy to prove from definitions that,
for any $s>0$ and for any
measurable function $f$ on $G$,
$\| | f |^s \|^*_{p,\infty}=
\| f \|^{*\ s}_{s p,\infty}.$
%
%
%
The
fact that $\| \cdot\|^*_{\frac{1}{p},\infty}$
is equivalent to a norm (see (\ref{equivalent-lorentz})),
implies that
$$\| \E f \|^*_{\frac{1}{p},\infty}\leq
\frac{1}{1-p} \E\| f\|^*_{\frac{1}{p},\infty}.$$
%
We can now estimate the right side of (\ref{kintchine})
as follows
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray*}
C_p
\| \left( \E | \widetilde{{\cal P}}_{\eta(\e), \e(P)}f|^p
\right)^\frac{1}{p}
\|^*_{1,\infty}
&=&
C_p \left( \| \E | \widetilde{{\cal P}}_{\eta (\e), \e (P)}f|^p
\|^*_{\frac{1}{p} ,\infty} \right)^\frac{1}{p}\\
&\leq&
C_p \left(
\frac{1}{1-p}
\E \| | \widetilde{{\cal P}}_{\eta(\e), \e(P)}f |^p
\|^{*}_{\frac{1}{p} ,\infty} \right)^\frac{1}{p}\\
&=&
C_p \left(
\frac{1}{1-p}
\E \| \widetilde{{\cal P}}_{\eta(\e), \e(P)}f
\|^{*\ p}_{1 ,\infty} \right)^\frac{1}{p}\\
&\leq&
C_p \left(
\frac{2^p A^p}{1-p}
\E \| f
\|^p_1 \right)^\frac{1}{p}\\
&=&
2 A C_p \left( \frac{1}{1-p}\right)^\frac{1}{p}
\| f\|_1 .
\end{eqnarray*}
%
%
%
The penultimate inequality follows from
Corollary (\ref{finite-conjugate-projection}).
This completes the proof of the theorem.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf Acknowledgements} The research of the authors was supported by grants from the National Science Foundation (U.\ S.\ A.) and the Research Board of the University of Missouri. Both authors are grateful for conversations with Professors Nigel Kalton, and Saleem Watson.
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\end{document}