- Email: [email protected]
A Factorization Theorem for the Real Hardy Spaces
Proceedings of the Analysis Conference. Singapore 1986 S.T.L. Choy, J.P. Jesudason, P.Y. Lee (Editors) 0 Elsevier Science Publishers B.V. (North-Holland). 1988
I67
Factorization Theorem for the Real Hardy Spaces
A
Akihiko MIYACHI Department of Mathematics, Hitotsubashi University Kunitachi, Tokyo, 186 Japan * ) The purpose of this article is to give a generalization of the factorization theorem for the real Hardy spaces and its application to the majorant property of the real Hardy spaces.
1.
INTRODUCTION First we shall recall the factorization theorem for the classical Hardy
spaces. Let
gp,
0 < p <
m,
denotes the classical Hardy space over the upper half
plane, i.e., this is the set of those functions F
which are holomorphic in
the upper half plane and for which
0 < p , q, r
The factorization theorem for these spaces reads as follows: Let < a
l / p = l/q
and
plane belongs to that
F = GH
gp
+ l/r
; then, a holomorphic function
if and o n l y if there exist
(pointwise product of
H ).
and
G
G
E
F
on the upper half
gq and H
E
Er
such
As for this theorem, see e.g.
Zygmund [ Z O ; Chapt. VII, 171, Duren [8; esp. Chapters 2 and 111 , or Koosis [ 1 2 ; esp. Chapters IV and VIJ. We can restate this theorem in terms of the Hilbert transform and the boundary values of the holomorphic functions. We define Re Hp, 0 C p < follows:
f
belongs to
Re Hp
if
real line) and if there exists an
f
F
Ep
(the
such that
=
function F
-f
is uniquely determined by
relation, then we define
E
as
is a tempered distribution on in
lim Re F( + iy), Y + O where the limit is taken in the sense of tempered distribution. f
m,
f.
If
f
and
The above
F have the above
by
lim Im F ( * + iy), Y + O where the limit is taken, again, in the sense of tempered distribution. This Y
f
=
is called the Hilbert transform of
f.
Now, taking the boundary values of
* ) Partly supported by the Japan Society for the Promotion of Science and by
the Grant-in-Aid for Scientific Research (C 61540088), the Ministry of Education, Japan.
A. Miyachi
168
the real parts of
F = GH, we can restate the factorization theorem as follows:
Let
be the same as before; then, a tempered distribution
p, q, and
R
r
belongs to Re He
if and only if there exist
g
Re Hq
E
and
h
E
f on
Re Hr
such that N
N
g h - gh.
f
(1)
(This is n o t a precise statement unless we determine the meaning of the right hand side of (1); the terms
gh
and
meanings as tempered distributions if F
=
GH, we use the formula F =
N N
g h , each by itself, may n o t have the p < 1.)
-6 GH,
If, instead of the formula
then we obtain another restatement
of the factorization theorem, i.e., we see that the factorization theorem for the spaces Re He
holds if we replace (1) by
f = g x +:ha Coifman-Rochberg-Weiss [ 61 , Uchiyama [ 181 , [ 191 , Chanillo [ 51, Komori [ 111, and the author [14], [15] obtained generalizations, in a weak form, of the above factorization theorem for Re Hp Remark (iv) in the next section).
to the case of the Hp
spaces over
En
(see
The purpose of the present article is to give
a further generalization of these results and, as an application, to give a proof of the majorant property of the Hp 2.
spaces.
PRELIMINARIES Hereafter, we fix a Euclidean space
En ; the letter
n
always denotes the
dimension of this space.
5
We denote by functions on
En
and
5'
the Schwartz class of rapidly decreasing smooth
and the space of tempered distributions on
En
-
respectively.
The Fourier transform and the inverse Fourier transform are denoted by
and
F - ~respectively. For p > 0, we denote by
He
the HP-space given by C . Fefferman and E. M.
Stein [ 9 ; 1111, i.e., this is the set of those f
in
2'
for which the func-
tions f*(X)
=
sup t > 0
Lp, where $
belong to For
f
d.) * f)(x) I
is a fixed function in
5
in Hp, we set
IIf II
belongs to
HP
=
I f*
LP
and He
Re Hp
such that
s
$(x)dx
#
0.
'
then Hp = Lp with equivalent norms.
If 1 < p < between He Re Hp.
(t-n
If n = 1, the relation
is as follows: A tempered distribution f
on
E
if and only if both its real part and imaginary part belong to
Factorization Theorem for the Real Hardy Spaces For
f
0
with
h
En \
on
2
h < n, we denote by
G(A)
169
the set of those smooth functions
such that
{O}
for all multi-indices a.
En
integrable functions on We denote by
G'(n)
We shall regard the elements of and the set G(A)
G(A)
2'.
as a subset of
the set of those smooth functions f
as locally
En \
on
{O}
such that
and
If
f
G'(n),
E
then there exists a sequence
{a.} J
= 0 and the limit
f' =
+
G(n)
+ c6, where
the above way,
1x1 > aj 1
a
>
j
0, lim
j + o aj
If
m
S',where
exists in
We denote by
f'
x[ I x I
lim j
such that
x[E]
E.
denotes the characteristic function of the set
the set of all those tempered distributions of the form
f' c
is the tempered distribution arising from is a complex number, and
Proposition 1. Let
0 5- A 5- n.
Then
f
6 E
f
E
G'(n)
in
denotes Dirac's distribution.
G(A)
if and only if
f
E
G(n-A).
We can prove this proposition by elementary calculations (the integration by parts). If
m
with
G(h)
E
0
2
A < n, then an operator
T
from
3 to
s'
is
defined by
We call T
the operator associated with m, and m the multiplier correspond-1 If we set k = F m, then the operator T associated with m is
T.
ing to given by
k
Tf where
*
For
h
f E
s,
05 h < n, we denote by
with m
E
K(h)
the set of the operators associ-
G(h).
m c G(A),
We define %I
$.
f,
denotes the convolution.
ated with Let
*
by
0 5- h < n, and let T be the operator associated with m. g(c) = m ( - c ) and denote by T' the operator associated with
We call T'
the conjugate of
T. The operator T
satisfy
=
for all
f, g
E
5.
and its conjugate T'
A. Miyachi
170
Suppose T c K ( A ) ,
Proposition 2. = A/n.
2
0
Then there exists a constant C
n, p > 0, q > 0, and
<
depending only on
l/p
T, p, q, and
-
l/q
n
for which the inequality HP holds for all
f
5
in
n Hp.
A s for a proof of this proposition, see e.g. [ 4 ; 141.
2
If p
1, then we
can also easily prove it by using the atomic decomposition for Hp.
( A s for
the atomic decomposition, see [ 1 3 ] . )
where
m
in K ( A ) .
j
is the multiplier corresponding to If
J
is the empty set, t h e n m j
T
This product is an operator
j'
Tj
~
identity operator.
shall be understood as the
MAIN RESULT
3.
The following is the main result of this article. Theorem 1. Let
..., N, j = 1,
N
A=xjfl,
be a positive integer and let
*.*,
N.
g
For
and
h
respect to and
N}
11, . a * ,
l/p = l/q
+
l/r
-
.
J , and
Jc
5
E
n Hq
of
{ 1,
*..,
m,
m
and all h
j
where
corresponding to
j
depending only on
5
c
T
j
is a homogeneous function of degree
-A
t'O, m.(tt;) = t j m j ( c ) , J j = 1, .*., N; ( b ) For every 5 E
Rn \ { O }
such that
T. J
N 1,
j = 1,
K(A.), J
IJI
J
l/q > A/n, T1,
with
l/r > A/n,
3
TN, P, q, r,
n Hr.
(ii) In addition t o the above assumptions, assume further that the multipliers m
where
Let
+ N/n.
(i) Then, there exists a constant C1 and n for which the inequality
holds for all g
Aj
denotes the complement of
Suppose 0 < p, q, r <
A/n < 1
and
2, we set
in
where the summation is taken over all subsets J denotes the cardinality of
A
A j < n.
be nonnegative numbers satisfying
p 5- 1 and that
have the following properties: (a)
-A
j'
i.e.,
5 # 0 ,
En \
{O},
there exists an
n
E
Factorization Theorem for the Real Hardy Spaces
1 and Then, every
f
# 0 for j with A.
m.(n) J
J
in He
where
ak
> 0.
can be decomposed as
> :ak P(T~, k=1 m
f =
171
, TN;
are complex numbers, gk
t
\I,
gk,
5
n Hq, hk
n Hr,
i S
and
Here
C2
and
C3
are constants depending only on
T1,
* * *
, TN, P,
q , r, and
n. g
Remark. (i) For
and
h
in
5,
the right hand side of (2) is well defined
since, by Proposition 2 and HGlder's inequality, each term in the summation in (2) belongs to
Ls
for all sufficiently large
s
and hence, a fortiori, to
S'.
(ii) In terms of the Fourier transform, the product in the theorem can be redefined as follows:
If n = 1 and
..., TN;
U Y
gh
(if
T1 =
g , h)
N
..*
= TN
is equal to
the Hilbert transform, then pi;
+
gh
(if
is an odd integer) o r
N
is an even integer) multiplied by a nonzero constant.
(iv) Theorem 1 for some special cases have been known. Coifman-Rochberg-Weiss [ 6 ] gave the theorem for the case
N = 1, A = 0, and
p = 1
(they gave (ii)
for T the Riesz transforms). Uchiyama [18], [19] treated the case N = 1, j A = 0, and p > n/(n+l). Chanillo [5] treated the case N = 1, 0 < A < n, and
p
1. Komori [ll] treated the case N = 1, 0
The author [14], [15] treated the case N 4.
21
and
<
A
< n, and
p
>
n/(n+l).
= 0.
SKETCH OF THE PROOF We can prove Theorem 1 by only slightly modifying the arguments in [14] and So, we shall give only a sketch of the proof.
[15].
First we shall sketch the proof of Theorem 1 (i).
The basic idea of the
proof of this part is due to Uchiyama [ 1 9 1 . Let
0
2v
< n,
x
E
En, and
k be a nonnegative integer. We define the set
172 T'(x)
k
A. Miyachi a s follows:
g
belongs t o
P
Tk(X)
if
g
Rn
i s a smooth f u n c t i o n on
I n o r d e r t o prove Theorem 1 ( i ) , w e u s e t h e f o l l o w i n g lemmas.
Lemma
1.
If
0 < p, q <
-,
nonnegative i n t e g e r s a t i s f y i n g
If
tion.
0
2
p < n,
l / p - l / q = p / n , and i f
is a
k
k > n / p - n , then
p 5 - 1, then we can prove t h e above lemma by u s i n g t h e a t o m i c decomposiIf
p > 1, t h e n the lemma i s a c o r o l l a r y t o t h e lemma below. If
Lemma 2.
0
then
5-
p < n,
0 < s < p <
m,
0 < q <
m,
and
l/p - l/q
P/n,
T h i s lemma i s due t o C h a n i l l o [ 5 ; Lemma 21. Now w e s h a l l prove Theorem 1 ( i ) .
For s u b s e t s
J
{ 1,
of
9 * * ,
1
N
we use
t h e following notations:
By s l i g h t l y modifying t h e arguments i n [ 1 4 ] , w e can prove t h e f o l l o w i n g : There exist
C , k , k ' , u , and
holds f o r a l l
g, h c
5
large positive integers, i n g only on
T1,
* * * ,
v
such t h a t t h e i n e q u a l i t y
and a l l
x
E
0 < u < q,
En;
here
k
and
0 < v < r , and
TN, p. q , r , k , k', u , and
v.
k' C
are s u f f i c i e n t l y
i s a c o n s t a n t depend-
From t h i s i n e q u a l i t y , w e
can e a s i l y deduce Theorem 1 (i) w i t h t h e a i d of P r o p o s i t i o n 2 , Lemmas 1 and 2 , and H B l d e r ' s i n e q u a l i t y . Next we s h a l l s k e t c h t h e proof of Theorem 1 ( i i ) . For a p o s i t i v e i n t e g e r M and f o r 2 L - f u n c t i o n s f on
t h e set of t h o s e
p
En
with
0 < p
2
-
1, w e d e n o t e by
whose F o u r i e r t r a n s f o r m s
f
A
P J
satisfy
173
Factorization Theorem for the Real Hardy Spaces
and
5
t > 0.
f o r some
0 < p 5 - 1 and decomposed as f o l l o w s : Lemma 3 .
ak
where
Here
If
a r e complex numbers,
M > n/p
fk
t
A
-
5 6 En,
'
P,M
i s a c o n s t a n t depending o n l y on
C
n/2, then every
M , n , and
f
in
He
can be
and
p.
A s f o r a proof,
Hp.
This i s a modification of t h e atomic decomposition f o r see [ 1 5 ] .
By s l i g h t l y m o d i f y i n g t h e argument i n [ 1 5 ] , we c a n p r o v e t h e f o l l o w i n g : F o r every
f
in
A
P,M
and e v e r y
0, t h e r e e x i s t
E >
g
and
h
in
5
such t h a t
and
where
CE
is a c o n s t a n t depending o n l y on
.-.,TN,
T1,
P , q , r , M, n , and
E.
Combining t h i s w i t h Lemma 3 , w e can e a s i l y p r o v e Theorem 1 ( i i ) . 5.
AN APPLICATION We s h a l l g i v e a proof o f t h e f o l l o w i n g theorem.
Theorem 2. in
Here
Let
such t h a t
He
C
0 < p
2
1.
g(5) 2- I ? ( S ) (
Then, f o r e v e r y
5
for all
i s a c o n s t a n t depending o n l y on
p
E
f
in
En
and
and
Hp, t h e r e e x i s t s a
g
n.
T h i s theorem h a s a l r e a d y been proved by s e v e r a l methods.
Proof f o r t h e c a s e
n = p = 1 can be found i n Zygmund's book 120; Chapt. V I I , Proof o f Theorem ( 8 . 7 ) , p.2871.
Coifman and Weiss [ 7 ; p.5841 used t h e a t o m i c d e c o m p o s i t i o n t o
g i v e a new proof ( a l s o f o r t h e c a s e
n = p = 1 1.
B a e r n s t e i n and Sawyer [ 2 ] ,
[ 3 ; 881, Aleksandrov [ l ] , and t h e a u t h o r [16] e x t e n d e d t h e method of Coifman
A. Miyachi
174
The a u t h o r [ 1 6 ] a l s o gave 1 two o t h e r d i f f e r e n t p r o o f s , one of which i s based on t h e d u a l i t y between H
and Weiss t o prove t h e theorem i n t h e g e n e r a l c a s e . and
and is v a l i d f o r t h e c a s e
BMO
p = 1, and t h e o t h e r i s s i m i l a r t o t h e one
t o be given below.
Here we s h a l l g i v e a proof of Theorem 2 u s i n g our f a c t o r i z a t i o n theorem; t h i s i s an e x t e n s i o n of one of t h e p r o o f s g i v e n i n [ 1 6 ] . nology: We s a y t h a t f o r every all
5
f
in
En
E
0 < p 5 - 2 , h a s t h e lower majorant p r o p e r t y i f g i n Hp such t h a t ^ g ( S ) 2 i ? ( S ) I f o r
H p , where
Hp,
We s h a l l i n t r o d u c e a termi-
there exists a
and
(Thus, Theorem 2 asserts t h a t
Hp
with
0 < p 5- 1 h a s t h e lower majorant
property*).) Proof of Theorem 2. t h e two f a c t s : ( a ) p
5
1,
2
l/p
In o r d e r t o prove t h e theorem, i t i s s u f f i c i e n t t o prove h a s t h e lower m a j o r a n t p r o p e r t y ; ( b ) I f
H2
l / q + 1 / 2 , and i f
Hq
h a s t h e lower m a j o r a n t p r o p e r t y , t h e n
P l a n c h e r e l ' s theorem.
We s h a l l prove ( b ) .
t h e r e and suppose
h a s t h e lower m a j o r a n t p r o p e r t y .
+
l/q
112 - x/n.
l/p < 1
for a l l if
N
Since E
and t a k e
X
j
mj
p
and
q
be as mentioned
E
G(Xj),
j = 1,
X
Define
. a * ,
N,
N
by
l/p =
satisfying
such t h a t
A =
j
and
in
ri
En \
f and
in
Hp
{O}.
(This condition ( 4 ) i s c e r t a i n l y s a t i s f i e d
...
m = = 51, and m i s r e a l v a l u e d . ) Take an 1 j and decompose i t a s i n Theorem 1 ( i i ) ( t a k e r = 2 ) .
i s an even i n t e g e r , Hq
and
Hp
m ' s s a t i s f y t h e c o n d i t i o n s i n Theorem 1 ( i i ) , and
5
arbitrary hi
+ N/n
hj,
l!&
Let
Take a p o s i t i v e i n t e g e r
0 5 - X < n/2.
Then
2,
The f a c t ( a ) i s obvious by v i r t u e of
a l s o h a s t h e lower majorant p r o p e r t y . Hq
2
0 < p < q
€I2 have t h e lower m a j o r a n t p r o p e r t y , w e can f i n d
gi
E
Hq
and
2
I?(S) I .
such t h a t
H2
,
2 Define
g
C llh, II 2 . H
by
Using ( 4 ) and t h e formula i n Remark ( i i ) ( § 3 ) , we e a s i l y see t h a t
g(S)
On t h e o t h e r hand, by v i r t u e of Theorem 1 ( i ) , t h e i n e q u a l i t y ( 3 ) h o l d s . completes t h e p r o o f .
This
(To be p r e c i s e , some l i m i t i n g arguments are n e c e s s a r y
*) A s f o r t h i s property f o r
Hp = Lp
with
p > 1, see [ l o ] and [ 1 7 ] .
Factorization Tlieorervi for the Real Hardy Spaces since g i
and
hi may not belong to
5; we
175
omitted the limiting arguments.)
REFERENCES [l] A. B. Aleksandrov, The majorant property for the multi-dimensional HardyStein-Weiss classes (in Russian), Vestnik Leningrad Univ. 13 (1982), 97-98. [2] A. Baernstein I1 and E. T. Sawyer, Fourier transforms of He spaces, Abstracts Amer. Math. Sac., Val. 1, No. 5 (1980), 779-42-8, p. 444. [3] A. Baernstein I1 and E. T. Sawyer, Embedding and multiplier theorems for Hp(Rn), Mem. Amer. Math. Sac. 53 (1985), no. 318. [4] A. P. Calder6n and A . Torchinsky, Parabolic maximal functions associated with a distribution, 11, Advances in Math. 24 (1977), 101-171. [5] S . Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982), 7-16. [6] R. R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. 171 R. R. Coifman and G . Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Sac. 83 (1977), 569-645. [8] P. L. Duren, Theory o f Hp spaces, Academic Press, New York-San FranciscoLondon, 1970. [91 C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137-193. [lo] E. T. Y. Lee and G . Sunouchi, On the majorant properties in Lp(G), Tchoku Math. J. 31 (1979), 41-48. [ll] Y. Komori, The factorization of Hp and the commutators, Tokyo J . Math. 6 (1983), 435-445. Spaces, London Math. SOC. Lecture Note [12] P. Koosis, Introduction to Hp Series 40, Cambridge Univ. Press, Cambridge, 1980. [13] R. H. Latter, A characterization of Hp(En) in terms o f atoms, Studia Math. 62 (1978), 93-101. [I41 A. Miyachi, Products of distributions in Hp spaces, TBhoku Math. J. (2) 35 (1983), 483-498. [15] A. Miyachi, Weak factorization of distributions in Hp spaces, Pacific J. Math. 115 (1984), 165-175. [16] A . Miyachi, Majorant properties in Hardy spaces, Research Reports Dept. Math. Hitotsubashi Univ., 1983. [17] M. Rains, Majorant problems in harmonic analysis, Ph. D. dissertation, Univ. of British Columbia, Vancouver, 1976. [18] A. Uchiyama, On the compactness of operators of Hankel type, TBhoku Math. J . 30 (1978), 163-171. [19] A. Uchiyama, The factorization of Hp on the space of homogeneous type, Pacific J. Math. 92 (1981), 453-468. [ZO] A. Zygmund, Trigonometric Series, 2nd ed., Vols I, 11, Cambridge Univ. Press, Cambridge, 1959.
I67
Factorization Theorem for the Real Hardy Spaces
A
Akihiko MIYACHI Department of Mathematics, Hitotsubashi University Kunitachi, Tokyo, 186 Japan * ) The purpose of this article is to give a generalization of the factorization theorem for the real Hardy spaces and its application to the majorant property of the real Hardy spaces.
1.
INTRODUCTION First we shall recall the factorization theorem for the classical Hardy
spaces. Let
gp,
0 < p <
m,
denotes the classical Hardy space over the upper half
plane, i.e., this is the set of those functions F
which are holomorphic in
the upper half plane and for which
0 < p , q, r
The factorization theorem for these spaces reads as follows: Let < a
l / p = l/q
and
plane belongs to that
F = GH
gp
+ l/r
; then, a holomorphic function
if and o n l y if there exist
(pointwise product of
H ).
and
G
G
E
F
on the upper half
gq and H
E
Er
such
As for this theorem, see e.g.
Zygmund [ Z O ; Chapt. VII, 171, Duren [8; esp. Chapters 2 and 111 , or Koosis [ 1 2 ; esp. Chapters IV and VIJ. We can restate this theorem in terms of the Hilbert transform and the boundary values of the holomorphic functions. We define Re Hp, 0 C p < follows:
f
belongs to
Re Hp
if
real line) and if there exists an
f
F
Ep
(the
such that
=
function F
-f
is uniquely determined by
relation, then we define
E
as
is a tempered distribution on in
lim Re F( + iy), Y + O where the limit is taken in the sense of tempered distribution. f
m,
f.
If
f
and
The above
F have the above
by
lim Im F ( * + iy), Y + O where the limit is taken, again, in the sense of tempered distribution. This Y
f
=
is called the Hilbert transform of
f.
Now, taking the boundary values of
* ) Partly supported by the Japan Society for the Promotion of Science and by
the Grant-in-Aid for Scientific Research (C 61540088), the Ministry of Education, Japan.
A. Miyachi
168
the real parts of
F = GH, we can restate the factorization theorem as follows:
Let
be the same as before; then, a tempered distribution
p, q, and
R
r
belongs to Re He
if and only if there exist
g
Re Hq
E
and
h
E
f on
Re Hr
such that N
N
g h - gh.
f
(1)
(This is n o t a precise statement unless we determine the meaning of the right hand side of (1); the terms
gh
and
meanings as tempered distributions if F
=
GH, we use the formula F =
N N
g h , each by itself, may n o t have the p < 1.)
-6 GH,
If, instead of the formula
then we obtain another restatement
of the factorization theorem, i.e., we see that the factorization theorem for the spaces Re He
holds if we replace (1) by
f = g x +:ha Coifman-Rochberg-Weiss [ 61 , Uchiyama [ 181 , [ 191 , Chanillo [ 51, Komori [ 111, and the author [14], [15] obtained generalizations, in a weak form, of the above factorization theorem for Re Hp Remark (iv) in the next section).
to the case of the Hp
spaces over
En
(see
The purpose of the present article is to give
a further generalization of these results and, as an application, to give a proof of the majorant property of the Hp 2.
spaces.
PRELIMINARIES Hereafter, we fix a Euclidean space
En ; the letter
n
always denotes the
dimension of this space.
5
We denote by functions on
En
and
5'
the Schwartz class of rapidly decreasing smooth
and the space of tempered distributions on
En
-
respectively.
The Fourier transform and the inverse Fourier transform are denoted by
and
F - ~respectively. For p > 0, we denote by
He
the HP-space given by C . Fefferman and E. M.
Stein [ 9 ; 1111, i.e., this is the set of those f
in
2'
for which the func-
tions f*(X)
=
sup t > 0
Lp, where $
belong to For
f
d.) * f)(x) I
is a fixed function in
5
in Hp, we set
IIf II
belongs to
HP
=
I f*
LP
and He
Re Hp
such that
s
$(x)dx
#
0.
'
then Hp = Lp with equivalent norms.
If 1 < p < between He Re Hp.
(t-n
If n = 1, the relation
is as follows: A tempered distribution f
on
E
if and only if both its real part and imaginary part belong to
Factorization Theorem for the Real Hardy Spaces For
f
0
with
h
En \
on
2
h < n, we denote by
G(A)
169
the set of those smooth functions
such that
{O}
for all multi-indices a.
En
integrable functions on We denote by
G'(n)
We shall regard the elements of and the set G(A)
G(A)
2'.
as a subset of
the set of those smooth functions f
as locally
En \
on
{O}
such that
and
If
f
G'(n),
E
then there exists a sequence
{a.} J
= 0 and the limit
f' =
+
G(n)
+ c6, where
the above way,
1x1 > aj 1
a
>
j
0, lim
j + o aj
If
m
S',where
exists in
We denote by
f'
x[ I x I
lim j
such that
x[E]
E.
denotes the characteristic function of the set
the set of all those tempered distributions of the form
f' c
is the tempered distribution arising from is a complex number, and
Proposition 1. Let
0 5- A 5- n.
Then
f
6 E
f
E
G'(n)
in
denotes Dirac's distribution.
G(A)
if and only if
f
E
G(n-A).
We can prove this proposition by elementary calculations (the integration by parts). If
m
with
G(h)
E
0
2
A < n, then an operator
T
from
3 to
s'
is
defined by
We call T
the operator associated with m, and m the multiplier correspond-1 If we set k = F m, then the operator T associated with m is
T.
ing to given by
k
Tf where
*
For
h
f E
s,
05 h < n, we denote by
with m
E
K(h)
the set of the operators associ-
G(h).
m c G(A),
We define %I
$.
f,
denotes the convolution.
ated with Let
*
by
0 5- h < n, and let T be the operator associated with m. g(c) = m ( - c ) and denote by T' the operator associated with
We call T'
the conjugate of
T. The operator T
satisfy
=
for all
f, g
E
5.
and its conjugate T'
A. Miyachi
170
Suppose T c K ( A ) ,
Proposition 2. = A/n.
2
0
Then there exists a constant C
n, p > 0, q > 0, and
<
depending only on
l/p
T, p, q, and
-
l/q
n
for which the inequality HP holds for all
f
5
in
n Hp.
A s for a proof of this proposition, see e.g. [ 4 ; 141.
2
If p
1, then we
can also easily prove it by using the atomic decomposition for Hp.
( A s for
the atomic decomposition, see [ 1 3 ] . )
where
m
in K ( A ) .
j
is the multiplier corresponding to If
J
is the empty set, t h e n m j
T
This product is an operator
j'
Tj
~
identity operator.
shall be understood as the
MAIN RESULT
3.
The following is the main result of this article. Theorem 1. Let
..., N, j = 1,
N
A=xjfl,
be a positive integer and let
*.*,
N.
g
For
and
h
respect to and
N}
11, . a * ,
l/p = l/q
+
l/r
-
.
J , and
Jc
5
E
n Hq
of
{ 1,
*..,
m,
m
and all h
j
where
corresponding to
j
depending only on
5
c
T
j
is a homogeneous function of degree
-A
t'O, m.(tt;) = t j m j ( c ) , J j = 1, .*., N; ( b ) For every 5 E
Rn \ { O }
such that
T. J
N 1,
j = 1,
K(A.), J
IJI
J
l/q > A/n, T1,
with
l/r > A/n,
3
TN, P, q, r,
n Hr.
(ii) In addition t o the above assumptions, assume further that the multipliers m
where
Let
+ N/n.
(i) Then, there exists a constant C1 and n for which the inequality
holds for all g
Aj
denotes the complement of
Suppose 0 < p, q, r <
A/n < 1
and
2, we set
in
where the summation is taken over all subsets J denotes the cardinality of
A
A j < n.
be nonnegative numbers satisfying
p 5- 1 and that
have the following properties: (a)
-A
j'
i.e.,
5 # 0 ,
En \
{O},
there exists an
n
E
Factorization Theorem for the Real Hardy Spaces
1 and Then, every
f
# 0 for j with A.
m.(n) J
J
in He
where
ak
> 0.
can be decomposed as
> :ak P(T~, k=1 m
f =
171
, TN;
are complex numbers, gk
t
\I,
gk,
5
n Hq, hk
n Hr,
i S
and
Here
C2
and
C3
are constants depending only on
T1,
* * *
, TN, P,
q , r, and
n. g
Remark. (i) For
and
h
in
5,
the right hand side of (2) is well defined
since, by Proposition 2 and HGlder's inequality, each term in the summation in (2) belongs to
Ls
for all sufficiently large
s
and hence, a fortiori, to
S'.
(ii) In terms of the Fourier transform, the product in the theorem can be redefined as follows:
If n = 1 and
..., TN;
U Y
gh
(if
T1 =
g , h)
N
..*
= TN
is equal to
the Hilbert transform, then pi;
+
gh
(if
is an odd integer) o r
N
is an even integer) multiplied by a nonzero constant.
(iv) Theorem 1 for some special cases have been known. Coifman-Rochberg-Weiss [ 6 ] gave the theorem for the case
N = 1, A = 0, and
p = 1
(they gave (ii)
for T the Riesz transforms). Uchiyama [18], [19] treated the case N = 1, j A = 0, and p > n/(n+l). Chanillo [5] treated the case N = 1, 0 < A < n, and
p
1. Komori [ll] treated the case N = 1, 0
The author [14], [15] treated the case N 4.
21
and
<
A
< n, and
p
>
n/(n+l).
= 0.
SKETCH OF THE PROOF We can prove Theorem 1 by only slightly modifying the arguments in [14] and So, we shall give only a sketch of the proof.
[15].
First we shall sketch the proof of Theorem 1 (i).
The basic idea of the
proof of this part is due to Uchiyama [ 1 9 1 . Let
0
2v
< n,
x
E
En, and
k be a nonnegative integer. We define the set
172 T'(x)
k
A. Miyachi a s follows:
g
belongs t o
P
Tk(X)
if
g
Rn
i s a smooth f u n c t i o n on
I n o r d e r t o prove Theorem 1 ( i ) , w e u s e t h e f o l l o w i n g lemmas.
Lemma
1.
If
0 < p, q <
-,
nonnegative i n t e g e r s a t i s f y i n g
If
tion.
0
2
p < n,
l / p - l / q = p / n , and i f
is a
k
k > n / p - n , then
p 5 - 1, then we can prove t h e above lemma by u s i n g t h e a t o m i c decomposiIf
p > 1, t h e n the lemma i s a c o r o l l a r y t o t h e lemma below. If
Lemma 2.
0
then
5-
p < n,
0 < s < p <
m,
0 < q <
m,
and
l/p - l/q
P/n,
T h i s lemma i s due t o C h a n i l l o [ 5 ; Lemma 21. Now w e s h a l l prove Theorem 1 ( i ) .
For s u b s e t s
J
{ 1,
of
9 * * ,
1
N
we use
t h e following notations:
By s l i g h t l y modifying t h e arguments i n [ 1 4 ] , w e can prove t h e f o l l o w i n g : There exist
C , k , k ' , u , and
holds f o r a l l
g, h c
5
large positive integers, i n g only on
T1,
* * * ,
v
such t h a t t h e i n e q u a l i t y
and a l l
x
E
0 < u < q,
En;
here
k
and
0 < v < r , and
TN, p. q , r , k , k', u , and
v.
k' C
are s u f f i c i e n t l y
i s a c o n s t a n t depend-
From t h i s i n e q u a l i t y , w e
can e a s i l y deduce Theorem 1 (i) w i t h t h e a i d of P r o p o s i t i o n 2 , Lemmas 1 and 2 , and H B l d e r ' s i n e q u a l i t y . Next we s h a l l s k e t c h t h e proof of Theorem 1 ( i i ) . For a p o s i t i v e i n t e g e r M and f o r 2 L - f u n c t i o n s f on
t h e set of t h o s e
p
En
with
0 < p
2
-
1, w e d e n o t e by
whose F o u r i e r t r a n s f o r m s
f
A
P J
satisfy
173
Factorization Theorem for the Real Hardy Spaces
and
5
t > 0.
f o r some
0 < p 5 - 1 and decomposed as f o l l o w s : Lemma 3 .
ak
where
Here
If
a r e complex numbers,
M > n/p
fk
t
A
-
5 6 En,
'
P,M
i s a c o n s t a n t depending o n l y on
C
n/2, then every
M , n , and
f
in
He
can be
and
p.
A s f o r a proof,
Hp.
This i s a modification of t h e atomic decomposition f o r see [ 1 5 ] .
By s l i g h t l y m o d i f y i n g t h e argument i n [ 1 5 ] , we c a n p r o v e t h e f o l l o w i n g : F o r every
f
in
A
P,M
and e v e r y
0, t h e r e e x i s t
E >
g
and
h
in
5
such t h a t
and
where
CE
is a c o n s t a n t depending o n l y on
.-.,TN,
T1,
P , q , r , M, n , and
E.
Combining t h i s w i t h Lemma 3 , w e can e a s i l y p r o v e Theorem 1 ( i i ) . 5.
AN APPLICATION We s h a l l g i v e a proof o f t h e f o l l o w i n g theorem.
Theorem 2. in
Here
Let
such t h a t
He
C
0 < p
2
1.
g(5) 2- I ? ( S ) (
Then, f o r e v e r y
5
for all
i s a c o n s t a n t depending o n l y on
p
E
f
in
En
and
and
Hp, t h e r e e x i s t s a
g
n.
T h i s theorem h a s a l r e a d y been proved by s e v e r a l methods.
Proof f o r t h e c a s e
n = p = 1 can be found i n Zygmund's book 120; Chapt. V I I , Proof o f Theorem ( 8 . 7 ) , p.2871.
Coifman and Weiss [ 7 ; p.5841 used t h e a t o m i c d e c o m p o s i t i o n t o
g i v e a new proof ( a l s o f o r t h e c a s e
n = p = 1 1.
B a e r n s t e i n and Sawyer [ 2 ] ,
[ 3 ; 881, Aleksandrov [ l ] , and t h e a u t h o r [16] e x t e n d e d t h e method of Coifman
A. Miyachi
174
The a u t h o r [ 1 6 ] a l s o gave 1 two o t h e r d i f f e r e n t p r o o f s , one of which i s based on t h e d u a l i t y between H
and Weiss t o prove t h e theorem i n t h e g e n e r a l c a s e . and
and is v a l i d f o r t h e c a s e
BMO
p = 1, and t h e o t h e r i s s i m i l a r t o t h e one
t o be given below.
Here we s h a l l g i v e a proof of Theorem 2 u s i n g our f a c t o r i z a t i o n theorem; t h i s i s an e x t e n s i o n of one of t h e p r o o f s g i v e n i n [ 1 6 ] . nology: We s a y t h a t f o r every all
5
f
in
En
E
0 < p 5 - 2 , h a s t h e lower majorant p r o p e r t y i f g i n Hp such t h a t ^ g ( S ) 2 i ? ( S ) I f o r
H p , where
Hp,
We s h a l l i n t r o d u c e a termi-
there exists a
and
(Thus, Theorem 2 asserts t h a t
Hp
with
0 < p 5- 1 h a s t h e lower majorant
property*).) Proof of Theorem 2. t h e two f a c t s : ( a ) p
5
1,
2
l/p
In o r d e r t o prove t h e theorem, i t i s s u f f i c i e n t t o prove h a s t h e lower m a j o r a n t p r o p e r t y ; ( b ) I f
H2
l / q + 1 / 2 , and i f
Hq
h a s t h e lower m a j o r a n t p r o p e r t y , t h e n
P l a n c h e r e l ' s theorem.
We s h a l l prove ( b ) .
t h e r e and suppose
h a s t h e lower m a j o r a n t p r o p e r t y .
+
l/q
112 - x/n.
l/p < 1
for a l l if
N
Since E
and t a k e
X
j
mj
p
and
q
be as mentioned
E
G(Xj),
j = 1,
X
Define
. a * ,
N,
N
by
l/p =
satisfying
such t h a t
A =
j
and
in
ri
En \
f and
in
Hp
{O}.
(This condition ( 4 ) i s c e r t a i n l y s a t i s f i e d
...
m = = 51, and m i s r e a l v a l u e d . ) Take an 1 j and decompose i t a s i n Theorem 1 ( i i ) ( t a k e r = 2 ) .
i s an even i n t e g e r , Hq
and
Hp
m ' s s a t i s f y t h e c o n d i t i o n s i n Theorem 1 ( i i ) , and
5
arbitrary hi
+ N/n
hj,
l!&
Let
Take a p o s i t i v e i n t e g e r
0 5 - X < n/2.
Then
2,
The f a c t ( a ) i s obvious by v i r t u e of
a l s o h a s t h e lower majorant p r o p e r t y . Hq
2
0 < p < q
€I2 have t h e lower m a j o r a n t p r o p e r t y , w e can f i n d
gi
E
Hq
and
2
I?(S) I .
such t h a t
H2
,
2 Define
g
C llh, II 2 . H
by
Using ( 4 ) and t h e formula i n Remark ( i i ) ( § 3 ) , we e a s i l y see t h a t
g(S)
On t h e o t h e r hand, by v i r t u e of Theorem 1 ( i ) , t h e i n e q u a l i t y ( 3 ) h o l d s . completes t h e p r o o f .
This
(To be p r e c i s e , some l i m i t i n g arguments are n e c e s s a r y
*) A s f o r t h i s property f o r
Hp = Lp
with
p > 1, see [ l o ] and [ 1 7 ] .
Factorization Tlieorervi for the Real Hardy Spaces since g i
and
hi may not belong to
5; we
175
omitted the limiting arguments.)
REFERENCES [l] A. B. Aleksandrov, The majorant property for the multi-dimensional HardyStein-Weiss classes (in Russian), Vestnik Leningrad Univ. 13 (1982), 97-98. [2] A. Baernstein I1 and E. T. Sawyer, Fourier transforms of He spaces, Abstracts Amer. Math. Sac., Val. 1, No. 5 (1980), 779-42-8, p. 444. [3] A. Baernstein I1 and E. T. Sawyer, Embedding and multiplier theorems for Hp(Rn), Mem. Amer. Math. Sac. 53 (1985), no. 318. [4] A. P. Calder6n and A . Torchinsky, Parabolic maximal functions associated with a distribution, 11, Advances in Math. 24 (1977), 101-171. [5] S . Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982), 7-16. [6] R. R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. 171 R. R. Coifman and G . Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Sac. 83 (1977), 569-645. [8] P. L. Duren, Theory o f Hp spaces, Academic Press, New York-San FranciscoLondon, 1970. [91 C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137-193. [lo] E. T. Y. Lee and G . Sunouchi, On the majorant properties in Lp(G), Tchoku Math. J. 31 (1979), 41-48. [ll] Y. Komori, The factorization of Hp and the commutators, Tokyo J . Math. 6 (1983), 435-445. Spaces, London Math. SOC. Lecture Note [12] P. Koosis, Introduction to Hp Series 40, Cambridge Univ. Press, Cambridge, 1980. [13] R. H. Latter, A characterization of Hp(En) in terms o f atoms, Studia Math. 62 (1978), 93-101. [I41 A. Miyachi, Products of distributions in Hp spaces, TBhoku Math. J. (2) 35 (1983), 483-498. [15] A. Miyachi, Weak factorization of distributions in Hp spaces, Pacific J. Math. 115 (1984), 165-175. [16] A . Miyachi, Majorant properties in Hardy spaces, Research Reports Dept. Math. Hitotsubashi Univ., 1983. [17] M. Rains, Majorant problems in harmonic analysis, Ph. D. dissertation, Univ. of British Columbia, Vancouver, 1976. [18] A. Uchiyama, On the compactness of operators of Hankel type, TBhoku Math. J . 30 (1978), 163-171. [19] A. Uchiyama, The factorization of Hp on the space of homogeneous type, Pacific J. Math. 92 (1981), 453-468. [ZO] A. Zygmund, Trigonometric Series, 2nd ed., Vols I, 11, Cambridge Univ. Press, Cambridge, 1959.