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In the remainder of this paper,
we transfer a result from Littlewood-Paley theory in
to spaces of measures on which
is acting.
We then show
how this transferred result implies
with ease several of the main results of Bochner [3],
de Leeuw and Glicksberg [6], and Forelli [9].
We start by setting our notation. Let
Let
denote a space of measures and
let
denote a sup path attaining representation of
by isomorphisms of
.
According to Forelli [9],
a measure
is called
-analytic if
.
We now introduce another equivalent definition.
Definition 4.1
Suppose that
is a sup path attaining representation of
by isomorphisms of
. A measure
is
called weakly analytic if the mapping
is in
for every
.
It is easy to see that if
is
-analytic then it is
weakly analytic. The converse is also true. The proof is based on the
fact that
is a set of spectral synthesis (see [1, Proposition 1.7]).
For
,
let
be the function whose Fourier
transform is piecewise linear and satisfies
![$\displaystyle \widehat{m_n}(s)=\left\{ \begin{array}{ll} 0 & \mbox{if $s\not\in [2^{n-1},2^{n+1}]$;}\\ 1 & \mbox{if $s=2^n$.} \end{array} \right.$](img265.png) |
(22) |
Let
be the function whose Fourier transform is piecewise
linear and satisfies
![$\displaystyle \widehat{h}(s)=\left\{ \begin{array}{ll} 0 & \mbox{if $s\not\in [-1,1]$,}\\ 1 & \mbox{if $\vert s\vert\leq \frac{1}{2}$.} \end{array} \right.$](img267.png) |
(23) |
It is easy to check that
![$\displaystyle \widehat{h}(s)+\sum^\infty_{n=0} \widehat{m_n}(s) =\left\{ \begin...
...mbox{if $s\geq -\frac{1}{2}$,}\\ 0 & \mbox{if $s\leq -1 $,} \end{array} \right.$](img268.png) |
(24) |
and that the left side of
(24) is continuous and piecewise linear.
The following theorem is a consequence of
standard facts from Littlewood-Paley theory.
We postpone its proof to the end of this section.
Our main theorem is the following.
Proof. To prove (27),
combine Theorems 1.6 and 4.2.
Inequality (27) states that the partial sums of the series
are unconditionally bounded.
To prove that they converge unconditionally, we recall
the Bessaga-Pe
czynski Theorem from [2]. This theorem
tells us that for any Banach
space, every unconditionally bounded series is
unconditionally convergent if and
only if the Banach space does not contain an isomorphic copy of
.
Now since
is weakly complete and
is not
(see [7, Chap IV.9, Theorem 3, and IV.13.9]), we conclude that
does not contain
.
Applying the Bessaga-Pe
czynski Theorem, we infer that there is a measure
such that
![$\displaystyle \eta=h*_T\mu+\sum_{n=0}^\infty m_n*_T \mu$](img287.png) |
(29) |
unconditionally in
. Moreover,
is weakly measurable,
because of (29).
It remains to show that
.
By Proposition 2.4,
it is enough to show that
for every
, we have
![$\displaystyle T_t\mu(A)=T_t\eta(A)$](img290.png) |
(30) |
for almost every
.
Since
is weakly analytic, the function
is in
.
By Theorem 4.2 (iii),
we have
![$\displaystyle T_t\mu(A)=h*T_t\mu(A)+\sum_{n=0}^\infty m_n* T_t \mu(A)$](img291.png) |
(31) |
for almost every
. On the other hand, from
the unconditional convergence of the series in
(29), (1), and (3), it
follows that
![$\displaystyle T_t\eta(A)=h*T_t\mu(A)+\sum_{n=0}^\infty m_n* T_t \mu(A)$](img292.png) |
(32) |
for all
. Comparing (31) and
(32), we see that (30)
holds, completing the proof of the theorem.
Remarks 4.4 It is interesting to note that
Theorem
4.3
implies the classical
F. and M. Riesz theorem for measures defined on the real line.
To see this, consider the representation
of
![$ \mathbb{R}$](img5.png)
acting by translation on the Banach space
of complex regular
Borel measures on
![$ \mathbb{R}$](img5.png)
.
It is easy to see that a regular Borel measure is analytic
if and only if its Fourier-Stieltjes transform is supported in
![$ [0,\infty)$](img262.png)
.
In this case, each term in
(
28) belongs to
![$ L^1(\mathbb{R})$](img277.png)
,
being the convolution of an
![$ L^1(\mathbb{R})$](img277.png)
function with a regular Borel measure.
Thus, the unconditional convergence of the
series in (
28) implies that the
measure
![$ \mu$](img86.png)
is absolutely continuous.
This argument provides a
new proof of the F. and M. Riesz Theorem,
based on Littlewood-Paley theory and the result of
Bessaga-Pe
czynski [2]. Also, it can be used to
prove the following version of Bochner's generalization of the F. and M. Riesz Theorem.
Theorem 4.5
Suppose that
is a locally compact abelian group with
dual group
, and
is a
continuous homomorphism. Suppose that
is such that,
for every real number
,
is compact.
Then
is absolutely continuous with
respect to Haar measure on
. That is,
.
Proof.
Let
denote the continuous adjoint homomorphism
of
. Define a representation
of
on
the regular Borel measures
by
for all
and all Borel subsets
.
By Example 2.2,
is sup path attaining,
and every measure
is weakly measurable.
Moreover,
is weakly analytic (equivalently,
-analytic)
if and only if
(see
[6]).
To prove the theorem, we can, without loss of generality,
suppose that
.
Otherwise, we consider the measure
, where
.
Let
. Then
is a
-set. Applying Theorem 4.3, we see that
![$\displaystyle \nu=h*_T\nu +\sum_{n=1}^\infty m_n*_T \nu,$](img306.png) |
(33) |
unconditionally in
.
For
, a
straightforward calculation shows that
for all
.
Since
is compact
for every
, it follows that
and
are compact. Thus
and
are in
, and hence they belong to
.
As a consequence, (33) implies that
.
With Theorem 4.3 in hand, we can derive with ease
several fundamental properties of analytic measures
that were obtained previously by
de Leeuw-Glicksberg
[6], and
Forelli [9]. We note however, that the techniques
in [6] and [9] do not apply
in our more general settings.
Theorem 4.6
Let
be a representation of
in
that is sup path attaining,
and let
be a
weakly analytic measure in
.
Then the mapping
is
continuous from
into
.
Proof. Using the uniform continuity
of translation in
, it is a simple matter to show that
for any function
, and any weakly measurable
, the mapping
is continuous.
Now use Theorem 4.3 to complete the proof.
Theorem 4.6 is very specific to
representations of
or
, in the sense that
no similar result holds on more general groups.
To see this, consider the
group
with a lexicographic order on the dual
group
. Let
denote the normalized Haar
measure on the subgroup
, and consider the measure
. Its spectrum is supported on
the coset
and thus it is
analytic with respect to the
regular action of
by translation in
. Clearly, the measure
does not translate continuously, and so
a straightforward analog of
Theorem 4.6
fails in this setting.
The following application concerns bounded operators
from
into
that commute with
in the following sense:
for all
.
Theorem 4.7
Suppose that
is a representation of
that is sup path
attaining,
and that
commutes with
.
Let
be weakly analytic.
Then
is also weakly analytic.
Proof.
First note that by Theorem 4.6, the mapping
is continuous, and hence measurable.
Now suppose that
. Again, by Theorem 4.6,
the
map
is Bochner integrable.
Let
Then by properties of the Bochner integral,
and since
is weakly analytic, we have that
for all
Hence
.
Therefore, for all
we have
Since this is true for all
, it follows that
is weakly analytic.
Definition 4.8
Let
be a sup path attaining
representation of
in
.
A weakly measurable
in
is
called quasi-invariant if
and
are mutually absolutely continuous for all
. Hence
if
is quasi-invariant
and
, then
if and only if
for all
.
We can use Theorem 4.7 to generalize a result of
de Leeuw-Glicksberg [6]
and Forelli [9], concerning quasi-invariant measures. In this application, it
is necessary to restrict
to sup path attaining representations given by isometries of
.
We need a lemma.
Lemma 4.9
Suppose that
is a linear isometry of
onto itself.
Let
,
. Then,
(a)
and
are mutually singular (in symbols,
)
if and only if
;
(b)
if and only if
Proof. For (a), simply recall that two measures
and
are mutually singular if and only if
, and
.
For (b), it is clearly enough to prove the implication in one direction.
So suppose that
and write
where
and
.
Then
. Hence
, and hence
.
But
implies that
. So
. Thus
.
Theorem 4.10
Suppose that
is a sup path attaining representation
of
by isometries of
. Suppose
that
is weakly analytic, and
is quasi-invariant. Write
for the Lebesgue decomposition of
with respect to
. Then both
and
are weakly analytic. In particular,
the spectra of
and
are
contained in
.
Proof. Let
.
Since
is quasi-invariant, the operator
commutes with
by Lemma 4.9.
Now apply Theorem
4.7.
Let us finish with an example to show that the hypothesis of
sup path attaining is required in these results. The
next example is a variant of Example 2.5.
Example 4.11 Let
![$ \Sigma_1$](img354.png)
denote the sigma algebra of countable
and co-countable subsets of
![$ \mathbb{R}$](img5.png)
, let
![$ \Sigma_2$](img355.png)
denote
the Borel subsets of
![$ \mathbb{R}$](img5.png)
, and let
![$ \Sigma = \Sigma_1 \otimes
\Sigma_2$](img356.png)
denote the product sigma algebra on
![$ \mathbb{R}\times \mathbb{R}$](img357.png)
.
Let
![$ \nu_1:\Sigma_1 \to \{0,1\}$](img358.png)
be the measure that takes countable sets
to 0 and co-countable sets to
![$ 1$](img274.png)
, let
![$ \delta_t:\Sigma_1\to\{0,1\}$](img359.png)
be the measure that takes sets to
![$ 1$](img274.png)
if they contain
![$ t$](img153.png)
, and to 0 otherwise,
and let
![$ \nu_2$](img360.png)
denote the measure on
![$ \Sigma_2$](img355.png)
given by
Let
![$ \nu = \nu_1\otimes \nu_2$](img362.png)
,
let
![$ \theta = \delta_0\otimes\nu_2$](img363.png)
, and
let
![$ \mu = \nu-\theta$](img364.png)
.
Finally, let
![$ T_t$](img35.png)
be the representation given by
![$ T_t(x,y) = (x+t,y+t)$](img365.png)
.
Then, we see that
is quasi-invariant, and that
and
are mutually singular.
Arguing as in
Example 2.5, we see that
is weakly analytic.
However, the singular part of
with respect to
is
, and it may be readily seen that this is
not weakly analytic, for example
is not in
![$ H^\infty(\mathbb{R})$](img261.png)
.
We end this section by proving Theorem 4.2.
We have
![$\displaystyle \sum^\infty_{n=-\infty} \widehat{m_n}(s) =\left\{ \begin{array}{ll} 1 & \mbox{if $s>0$;}\\ 0 & \mbox{if $s\leq 0$.} \end{array} \right.$](img369.png) |
(34) |
Recall the Fejér kernels
, where
![$\displaystyle \widehat{k_a}(s) =\left\{ \begin{array}{ll} 1-\frac{\vert s\vert}{a} & \mbox{if $\vert s\vert<a$;}\\ 0 & \mbox{otherwise.} \end{array} \right.$](img371.png) |
(35) |
By Fourier inversion, we see that
![% latex2html id marker 4972
$\displaystyle m_n(x) =\exp(i 2^n x) k_{2^{n-1}}(x)+ \frac{1}{2} \exp (i 3\ 2^{n-1} x) k_{2^{n-1}} (x).$](img372.png) |
(36) |
Proof.
The proof of (ii) is immediate from (i) and (34),
by Fourier inversion.
For part (i),
use (36),
to write
Hence, to prove (37)
it is enough to show that there is a positive constant
, independent of
such that
, for
.
Appealing to [15, Theorem 3, p. 114], we will be done once we
establish that:
![$\displaystyle \left\vert\widehat{K_j}\right\vert\leq A,$](img385.png) |
(38) |
and the Hörmander condition
![$\displaystyle \sup_{y>0} \int_{\vert x\vert>2y} \vert K_j(x-y)-K_j(x)\vert dx\leq B,$](img386.png) |
(39) |
where
and
are absolute constants.
Inequality (38)
holds with
, since the
Fourier transforms of the summands defining the
kernels
have disjoint supports and are
bounded by 1. Condition (39), is well-known.
For a proof, see [8, pp. 138-140, and 7.2.2, p. 142].
Proof of Theorem 4.2.
Parts (i) and (ii) follow
as in Theorem 4.12,
so we only prove (iii).
For notational convenience, let
![$\displaystyle \kappa_N(x)=\sum_{n=0}^N m_n(x),\quad N=0,\,1,\, 2\, \ldots$](img389.png) |
(40) |
Let
denote the de la Vallée
Poussin kernels on
of order
.
Its Fourier transform is
continuous, piecewise linear,
and satisfies
![$\displaystyle \widehat{V_{2^N}}(s)=\left\{ \begin{array}{ll} 0 & \mbox{if $s\no...
...2^{n+1},2^{n+1}]$;}\\ 1 & \mbox{if $\vert s\vert\leq 2^n$.} \end{array} \right.$](img392.png) |
(41) |
It is well-known that
is a summability kernel for
and, in particular, that
converges pointwise almost everywhere to
for all
, for
.
Thus, we will be done, if we can show that for
,
![$\displaystyle V_{2^N}*f=h*f+ \sum^N_{n=0} m_n*f.$](img398.png) |
(42) |
Write
,
where
is as in (40).
For
, the Fourier transform of
,
vanishes on
.
Thus, for
, we have
, and so
(42) follows, and the proof
is complete.
Acknowledgements The work of the second author was supported
by a grant from the National Science Foundation (U.S.A.).
Next: Bibliography
Up: Transference in Spaces of
Previous: Proof of the Main
Stephen Montgomery-Smith
2002-10-30