1
DECOMPOSITIONS
OF FUNCTIONS
AS
SUMS OF ELEMENTARY
FUNCTIONS
By F. F. BONSALL
[Received 8 January 1985]
THE purpose of this article is to demonstrate the usefulness of Banach's
closed range theorem for proving decomposition theorems. The author
learnt this method from D. H. Luecking, who has used it ([5], Prop. 4.5)
to give an elegant proof of the Coifman- Rochberg decomposition
theorem for the Bergman space L~, [2]. After a few decomposition
theorems had been proved in this way, it became apparent that it would
be economical to express the method as a decomposition theorem for
vectors in a Banach space. This is done in Theorem 1, in which certain
inequalities involving the norms of the elementary vectors and of linear
functionals give the decomposition with corresponding inequalities between norms. The proof of the theorem is no more than an exercise on
Banach's closed range theorem.
In Corollary 2, we show that, if {Zk} is a suitably chosen sequence of
points in the open unit disc D, then every integrable function on the
circle aD can be decomposed as an [I-sum of Poisson kernel functions
corresponding to the points Zk' Many analogues and generalizations of
this corollary are available, and in particular Corollary 3 is a decomposition theorem for L 1(G), with G a locally compact group, in terms of
elements of an approximate identity. Corollary 4 is Fefferman's atomic
decomposition theorem for ReHI in terms of simple atoms, that is atoms
taking only two values. In Corollary 5, we show that a suitably chosen
sequence of rank one operators suffices to decompose all trace class
operators.
Notation. Given a set
11(~) denotes as usual the Banach space of
complex valued functions A on ~ for which
~J
11).,111
= 2:
1).,(0)1
bEil
< 00.
THEOREM 1. Let X be a Banach space, let M1J 1\12 be positive constantsJ
and let u be a mapping of a set ~ into X such that
(i) lIu(<5)II~Ml
(bE~),
(ii) sup {) 'ljJ(u( (5)1: (5 E L\}
Quart.
J. Math.
? M2
II1/-'
Oxford (2), 37 (1986), 129-136
II
(1J1
E
X*).
© 1986 Oxford
University Press
130
F. F. BONSALL
The n every f
EX
is of the form
f= 2:
A(b)u(b)
E/\
(1)
<5
with
A E
11(~), and also
M2
inf IIAIiI ~
IIfli
~
inf IIAlll'
M1
(2)
with the infimum taken over all decompositions (1) off
Proof. Let T be the bounded linear mapping of 11(~) into X defined
by
TA =
~
A(b)u(b).
<5E/\
Given
TeE
EE~,
= u( E),
define ee on ~ by eJE)=l,
and so, for all W E X*, we have
~
M211wll
sup {Iw(u(b»l:
= sup
(b=/=E).
eE(b)=O
Then
b E~}
{!1jJ(TeE)I:
E~} ~
E
(3)
IIT*wll.
Thus T* has closed range and zero kernel, and therefore, by Banach's
closed range theorem ([1] p. 150), Tll(~) = X, every f E X is of the form
(1).
Let N = ker T, Y = 11(~)/N, and define S by Sy = TA (A EYE Y). Then
S is a bounded linear bijection of Y onto X, and therefore has a bounded
inverse. It follows that S* is a bounded linear bijection of X* onto Y*.
By (3), we have, for all1jJ E X*,
M211wll
~
IIT*1jJII
= sup
= sup {11jJ(TA)/:
{11jJ(Sy)l:
lIyll
~
IIAIII ~ I}
I}
= Ils*wll·
Therefore liS-III = II (S *)-111 ~ Mil, and the inequalities (2) are now
clear.
Our first corollary is exceedingly elementary and could have been
proved with ease at any time since 1932. However, to our surprise, it
does not seem to be known. I am indebted to R. Rochberg for a helpful
comment which has substantially improved the result.
Notation. Let D = {z E C: Izi < I}, aD = {z E C: Izi = I}, and, for
zED,
~ E aD, let
Pz(C)
so that
= (1-lzI2)/11-
zC12,
is the Poisson kernel Pz( 8). For
is defined in terms of the normalized
(2n)-1 d8.
U(aD)
pz(ei6)
1~ P
~ 00,
Lebesgue
the space
measure
COROLLARY
2. Let {Zk} be a sequence of points of D such that almost
every point of aD is the non-tangential limit of a subsequence of {zd.
DECOMPOSITIONS
Then
LIe aD)
131
OF FUNCTIONS
is the set of all f of the form
(4)
ex;
with
Ak
E
C and k=I
1: IAkl <
00.
Also
""
IIfl11 = inf
with the infimum
Proof.
2:
k=l
IAkl,
taken over all decompositions
(4) off.
Given gEL ""(aD), we have
2Jr
[Pz,
g] = ~1~
r g(ei8)pz(ei8)
~.;lt J
de
(z ED),
= g(z)
o
where g(z) denotes the harmonic extension of g. We have IIpz/l1 = 1;
and, by Fatou's theorem (see Hoffman [3], p. 38), g(~) is the limit of g(z)
as z ~ ~ non-tangentially for almost all ~ E aD. Therefore
Ilglioo = sup
and the usual identification
gIves
111J11!
for all1jJ E (L 1(aD»*.
{1[PZk'
g]l:
kEN},
of the dual space of L1(aD)
= sup
with L'XJ(aD)
kEN}
{11J1(PzJI:
Thus Theorem 1 applies with M1
= M2 = 1.
Remark. A similar corollary holds for L l(R) with D replaced by the
open upper half plane U, and with
1
pz(t)
= -:;r 1m zllz
-
tl2
(z
E
U, t
E
lR).
The Poisson kernel can be replaced by other positive kernels. A further
generalization is given in Corollary 3.
Since Pz(~) = (1 - Z~)-l + z~(l - z~)-l, the Fourier series of pz is
""
1+ 2:
n=l
(Znein8
+ Zne-in8).
Therefore every Fourier series can be decomposed as an Zl-sum of the
Fourier series of this form with Z = Zk' A similar result holds for Fourier
transforms.
Notation. Let G be a locally compact group, and let L 1(G) denote the
group algebra with respect to a left invariant Haar integral (Loomis [4]).
132
F. F. BONSALL
Thus, for J, g
Ll(G),
E
the productf*g
(f*g)(x)=
is given by
JI(y)g(y-1X)dY.
Let {vy: Y E f} be a right approximate identity of unit vectors in L \ G);
that is
is a directed set, IIvyll1 = 1, and, given IE L1(G) and c > 0,
there exists Yo E such that IIf - f * vy Ih < c whenever Y ~ Yo.
Let ~ = X G, and for b = (y, Y) E~, let
f
f
f
= Vy(y-1X)
uo(x)
COROLLARY
with
A E
3. Each f
l\~),
E
L\G)
(x E G).
is of the form
and
Ilflh = inf
IIAll1
with the infimum taken over all such decompositions off.
Proof. Let h
Iluoill
E L CO( G).
By left invariance of the Haar integral, we have
= 1, and so it is enough to prove that
Ilhllco~sup
For g
E
{I[uo,
let gt(x) =g(x-1).
L\G),
[f*g,
=J
h]
b
h]l:
f, g E L1(G),
Then, for
{J 1(y)g(y-1X)
(5)
E ~}.
we have
dy }h(X) dx
I
= f I(y){f
h(x)g(y-1x)
dx} dy
= (f, h * gt].
Given c
> 0,
we choose
and then choose
Yo
I
E
L\
G) with
I [I,
h] I > II h 1100
such that, for y ~
l(f*
vy'
h]1
11/111
-
= 1 and
c,
Yo,
> Ilhlloo -
c.
Then for y ~ Yo,
Ilh * v~ILYO ~ IU, h * vtll
= l(f*vy,
h]1
>
Ilhllc.o- E.
DECOMPOSITIONS
133
OF FUNCTIONS
But
11]
[U(y,y),
Therefore, for y ~
= f h(X)Vy(y-1X)
dx
= f h(x)vt(X-1y)
dx
= (h *vt)(y)·
Yo,
sup
{![u(y,y),
y E G}
h]l:
> IIhll",
-
E,
and the inequality (5) is proved.
Our next corollary is Fefferman's atomic decomposition of ReHl.
Several authors have remarked that this follows easily from Fefferman's
duality theorem, but without indicating a method. The present proof
gives the result with atoms of a special kind that we call simple atoms.
Notation. L~ will denote the real valued functions in LP(aD), and
ReHl the set of f E L1 such that the harmonic conjugate 1also belongs to
L1. (Plainly, ReHl is also the set of real parts of functions in HI.) With
the norm given by
IIfll = IIflll
+ 111111,
ReHl becomes a Banach space. Given
length
III
=1=
f
L1 and an arc I
E
c aD
with
0,
I(J)
= 111-1 fI f(
ei8)
de.
Then BMO denotes the space of functions
oscillation, that is with
Ilfll
* = sup
I I(IJ - 1(J)/)
f E L1 with
bounded mean
< co.
IIfllBMo is defined by II/l1BMo
= 11/11 * + ~(O)I·
A simple atom (on an arc 1) is a function a of the form
The norm
a
= 11\-2 (IQlxp
-lplxQ)'
where P, Q are measurable subsets of I with P U Q = I, P n Q = 0, and
XE denotes the characteristic function of a set E. Let A denote the set of
all simple atoms together with the function on aD with the constant value
(2n)-1.
COROLLARY
4. ReHl is the set of functions f of the form
'"
f= k=l
~
?O
with Ak E~,
.2:
k=l
/Akl
< 00,
and ak
EA.
Akab
134
F. F. BONSALL
Proof By Fefferman's duality theorem, there exists a positive constant
C such that, given 1jJ E (ReH1)*, there exists g E BMO with
7/J(f)
We note that if
= [f,
g]
(f E
ReH1),
1 and
a is a simple atom on
[a,
IIgllmvlo ~ C II'",II.
g
g] = (2n)-1 JI a (eie)(g(eie)
E
(6)
BMO, then
(7)
- leg»~ de;
and therefore
I[a, g]1 ~ (2n)-1IlgIlBMo
By (6), (8) and the Hahn-Banach
(a EA,
g E BMO).
(8)
theorem,
(a E A).
By taking Ll = A and u(a) = a, we obtain a mapping of ~ into a bounded
subset of ReH1, and it now suffices to prove the inequality
IIgllBMo ~ 6n sup {I[a, g]l:
a E A}
(g EBMO).
(9)
Let g E BMO, let 1 be an arc in aD, and let
a
= 111-2 {IQI Xp -ipi
XQ},
where
p = {e
E
1: g(ei8) - l(g);? O},
Q
= {e
E I: g(ei6)
-
leg)
< O}.
We note that
fP
and therefore,
[a, g]
Ig - l(g)1 de
==
Jig - l(g)1 de,
Q
by (7),
= (2n)-1111-2 {IQI
= (4n)-11(lg
f
P
Ig - l(g)1 de + IPI
fig
Q
- l(g)l
de}
- l(g)I),
and the inequality (9) follows easily.
Notation. Let C1 denote the Banach space of all trace class operators
on a Hilbert space H. It is well known (see Ringrose [6] p. 99) that the
dual space (C1)* is isometrically isomorphic to the space BL(H) of all
bounded linear operators by way of the mapping T ~ fT: BL(H)--;.
(C1)*, where
fT(S) = tr (ST)
r
5. Let
> 0, let be a set of unit vectors in H such that
unit vector x satisfies Ilx - v II ~ for some v E
and let Ll = x
COROLLARY
every
'YJ
'YJ
r,
r r.
DECOMPOSITIONS
(i) If
1]
<
t
then every 5
OF FUNCTIONS
135
CI is of the form
E
L
s=
A(V)V@V
VEr
with A E LIef), and
115
(ii) If
1]
< 2~-
III ~ (~-
1, then every 5
E
21] ) inf IIA Ik
CI is of the form
5= L
v)u @v,
A(U,
(U,V)E~
with A E P(!1), and
(1- 2Yf
IIs111~
inf IIAlk
1]2)
-
Proof. (i). Given T E BL(H) and E > 0, there exists a unit vector x
with I(Tx, x)1 >~ IITII- E. Choose v E such that x = v +XI with IlxI11 ~
1]. Then
f
(Tx, x)
and I(Tv,
Xl)
+ (Txv
= (Tv,
v)
Xl)
+ (Tx 1, x),
x)1 ~ 21] II TII. Therefore
> (! -
I(Tv, v)1
sup {1(Tv, v)l:
Since (v
+ (Tv,
® v)T = v ® T*v
v E
IITII-
217)
r} ~ (~-
E,
21]) II TII.
and tr (x @y) = (x, y), we have
tr «v @ v)T) = (v, T*v) = (Tv, v).
Thus, for every
1jJ
E
sup
(CI)*,
@v)l:
{11jJ(v
v E
f} ~ (~- 21])
II 1JJ II,
and (i) follows from Theorem 1.
(ii). Given T E BL(H) and E > 0, there exist unit vectors x, y with
I(Tx, y)1 > /ITIIE. Choose u, v E
with x = U +Xl' Y = v + YI, and
!lXIII,
"Ylll ~1]. Then
f
= (Tu,
(Tx, y)
I(Tx1'
v)
I(Tu,
v)1
Thus, as above,
+ (TXl' v) + (Tu, Yl) + (TX1' Yl),
Y1) + (TX1' Y1)1 ~ liT" (21] + 1]2),
v)
+ (Tu,
> (1- 217 -
for every
lfJ
sup {11jJ(u ® v)l:
172) II
E
TII-
E.
(CI)*,
(u, v) E ll} ~
(1- 217 -172)
IllfJll,
and (ii) follows from Theorem 1.
Remark. If
IISIl1?;~
r
is norm dense in the unit sphere
inf 11).111 in (i), and 115111 = inf IIAIII in (ii).
of H, we have
136
F. F. BONSALL
REFERENCES
1. S. Banach, Theorie des operations lineaires, (Hafner, 1932).
R. R. Coifman and R. Rochberg, 'Representation theorems for holomorphic and
harmonic functions in £P', Asterisque 77 (1980) 12-66.
3. 1<. Hoffman, Banach spaces of analytic functions, (Prentice-Hall, 1962).
4. L. H. Loomis, An introduction to abstract harmonic analysis, (Van Nostrand, 1953).
5. D. H. Luecking, 'Representation and duality in weighted spaces of analytic functions',
Indiana Univ. Math. J. 34 (1985), 319-336.
6. J. R. Ringrose, Compact non-self-adjoint operators, (Van Nostrand, 1971).
2.
School of Mathematics
University of Leeds
Leeds LS2 9fT .
"
.