Iernat. J. Math. Mah. Si.
(1980) 575-581
Vol. 3 No.
575
RESEARCH NOTES
A POINTWISE GROWTH ESTIMATE FOR ANALYTIC FUNCTIONS IN TUBES
RICHARD D. CARMICHAEL and ELMER K. HAYASHI
Department of Mathematics
Wake Forest University
Winston-Salem, North Carolina 27109
U.S.A.
(Received October 4, 1979)
ABSTRACT.
A class of analytic functions in tube domains T
C
n
+
n
dimensional complex space, where C is an open connected cone in 1t
been defined by V. S. Vladimirov is studied.
L
2
iC in
n-
which has
We show that a previously obtained
growth estimate concerning these functions can be replaced by a pointwise growth
estimate, and we obtain further new properties of these functions.
shows that these functions of Vladimirov are exactly the Hardy H
functions corresponding to the tube T
2
Our analysis
class of
C.
KEY WORDS AND PHRASES. Analytic Function in Tubes, Hardy H 2 Space, Cauchy Kernel
and Integral, Foi- Laplace Integral.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES:
i.
32A07, 32A10, 32A25, 2A5.
INTRODUCTION.
All notation in this note is that of Vladimirov
connected cone in I
n
i, p. i].
Let C be an open
and let C’ be an arbitrary compact subcone of C [i, p. 218].
R. D. CARMICHAEL AND E. K. HAYASHI
576
Let f(z) be analytic in T
I(x+i)l
g
C’
C.
25.4].
]R
n
C’
+ iC and for any
}f(x+iy)I 2 dx) 1/2
> 0 where the constant
for every
y
n
IL2
C
M,f(C’)
<
C C let
f(z) satisfy
M,f (c’) e 11 ,xsc’c,
depends
one,
Vladimirov has studied these analytic functions in
%n this note we
show that the L
2
growth estimate
C’
f, and
[i,
but not on
sections 25.3-
[i, p. 227, (74)] can
be
replaced by a pointwise growth estimate on the function with exactly the same
growth on the right of the estimate and obtain further new information concerning
Our analysis also shows that the analytic functions of Vladimirov
these functions.
defined above are in fact exactly the Hardy H
pp.
90-91];)
for tubes T
thus the growth of Bochner
C
[2, (13)]
2
<
Mf
which defines the Hardy H
4-5],
p. 227, lines
space, the Hardy H
2
y
S
2
space
(2)
C
(i), contrary to
is not a more restrictive condition than
2.
functi6ns ([2, section 3] or [3,
namely
L
[i,
2
in the sense that both
(i)
space corresponding to tubes T
and
the statement made in
(2)
characterize the same
C
RESULTS.
To obtain our results we need three lemmas.
that of
[i, p. 223, Lemma 2]
and is omitted.
Let C be an open (not necessarily connected) cone.
LEMMA I.
the convex envelope of C.
There exists a
yt
{t
for all t e C
cone of
0(C)
for all y g
yt > 0, y
there exists a 6
C’
LEMMA 2.
The proof of Lemma 1 is like
and all t
_>
6
Y
Let y e
0(C),
> 0 depending on y such that
lyl Itl
C}.
6(C’) >
Further, if
C’
is an arbitrary compact sub-
0 depending only on
C*.
Let C be an open connected cone.
We have
C’
such that
(3)
holds
GROWTH ESTIMATE FOR ANALYTIC FUNCTIONS IN TUBES
(Ic.(t) e izt]"
e L
p
I
for all p
!
P
!
n for arbitrary z TO(C) n + i
as a function of t e
577
O(C),
where
Ic.(t)
denotes the characteristic function of C*
Let z
PROOF.
O(C)
T
x+iy
Using Lemma 1 we have
llc* (t) e iztl= Ic,(t)
8
for some
(5) proves (4)
32],
> 0
y
and
and integration by parts
n
!
Ic, (t) exp(-lyl tl)
Now let i < p <
r
n-i
Using
(n-l) times,
lc* (t) e iztlp
n
where
-yt
n
holds for all t e ]R since
(5)
for p
n
e
dt
!
!
exp(-plylr)
()
Ic.(t)
0 if t
(5), [4,
p. 39, Theorem
we have
exp(-plyl tl)
dt
n
(6)
f2
dr
’0
is the volume of the unit sphere in IR
n
(n-l)!
n
(PlYl)-n
(6) proves (4)
for i < p <
For any open connected cone C the Cauchy kernel corresponding to T
p. 201] or
[1, p. 223, (61)])
!,
K(z-t)
LEMMA 3.
K(z-t)
where
(3);
n
exp(i(z-t)N) dN
<
]R
n
T
z
n.
(n-l)’.
6-n yl-n
I < p < 2, (l/p) + (l/q)
IK(z-e)II
< (f
L
where the L
q
q-
n
O(C)
K(z-t)
n
and
y
then
t
IR n
()
> 0 is the number of
i, we have
z
norm is with respect to the variable t e ]R
O(C)
n
is an analytic function
O (C)
x+iy g T
z
(n-l)!)i/P (p6)-r/p [yI-n/p
arbitrary compact subcone of
t e R
We have
is the volume of the unit sphere in IR
and for
0(C) ([5,
is
Let C be an open connected cone.
O(C) for fixed t
of z g T
@ C*
(7)
and
(8)
O(C)
x+iy g T
n.
Further, if
hold for z
(8)
C’
is an
C’ and
x+iy g T
578
t
_
n with
follows by
2 and (i/p) + (i/q)
[5, P. 202,
Inequality
(8)
O(C)
ILq IIc*
<
(9)
(5)
obtain
and
and not on y
holding for z
(6)
C’
for z g T
C’C O(C).
C’
T
x+iy
The result
[1,
for fixed t
]R
for p
I.
For
t]
in the L
(6)
e iz
0(C)
Lp
(6)
for i < p
now depends only on
where
This fact and the above analysis yields
p. 223,
depending only on
(62)]
n
_
sense
(9)
2.
If
we use the second part of Lemma 1 to
C’ 0(C),
with
q
O (C)
z g T
and a computation as in
is an arbitrary compact subcone of
= 0(C).
hence by the Parseval inequality
eizl
D
O(C)
--I[Ic,()
proof of Theorem i]
now follows from
T
the analysis of
K(z-t)
i,
C’
and not on y
is analytic in z
(7) follows by
[1, p. 223].
IK(z-t)
C’
K(z-t)
The fact that
as noted in
C’
depending only on
PROOF.
i < p
K. HAYASHI
R. C. CARMICHAEL AND E
C’
(7)
C’
C
O(C)
(8)
and
O(C).
is a special case of Lemma 3 for p
2.
We now obtain our result which adds information to [I, p. 227, Corollary]
[i,
and hence to the analytic functions considered in
note a misprint in the statement of
[i, p. 227,
line 2 of the
THEOREM.
satisfy
(i).
Then
f(z)
O(C)
PROOF.
First
(64)".
Let f(z) be analytic in T
H2(T O(C))
F(z)
to T
C
and
0(C)
such
O(C)
C’
IL2
!1 -n/2
z
is a constant which depends at most on
sup
yC
and if
read "inequality
has an analytic extension
boundary value of F(x+iy) as y
25.3-25.4].
Corollary]; "equality (63)" in
Let C be an open connected cone.
I(z)l <_ (c’) lhl
M(C’)
p. 227,
Corollary] should
that for any compact subcone
where
[I,
sections
0, y
/
I.If(x+iy)ll
L
2
0(C).
sup
C’
x+iy g T
C’
0(C)
C
and h
(o)
L
2
is the L
2
Further,
IF(x+iy)ll
yO(C)
contains an entire straight line then
L
F(z)
(--)
2
0.
From [i, pp. 225-226] we have the existence of a function g
L
2
579
GROWTH ESTIMATE FOR ANALYTIC FUNCTIONS IN TUBES
with support in
C*
almost
everwhere such that
f
f(z)
R
g(t)
e
izt
C
z g T
dt
(12)
and the Fourier-Laplace integral on the right of
the properties of
g(t)
(4).
and
The fact that
I
L
(13)
y
g
g(t) e izt
is analytic in T
0(C)
dt
0(C)
z g T
(13)
follows as a special case of
[6,
Theorem
Because of (4) and the properties on g(t), we have that (g(t) exp(-yt))
2.1].
L
F(z)
is well defined because of
Now put
nI
F(z)
(12)
n
2
n
as a function of t g ]R for y g
can be interpreted to be the L
0(C).
2
0(C);
g
hence the integral on the right of
Fourier transform of
(g(t) exp(-yt)),
The Parseval equality and the fact that the support of g is in C
almost everywhere now yield
IF(x+iy)II
L
and we conclude that
Theorem
3.1].) We
F(z)
g
e-Ytll L2 <
lg(t)
2
H2(T0(C)).
lgll
L
(This fact also follows by [3, p. i01,
[i, p. 227, Bochner’s formula]
now apply the proof of
103, Theorem 3.6] to obtain the existence of a function h
Fourier transform of g and is the L
2
(i)
2
g
L
2
or
[3,
which is the L
boundary value of F(x+iy) as y
/
0, y
p.
2
0(C),
such that
F(z)
Using
(15),
z s T
C’ C’
(2w)-n
I
h(t) K(z-t) dt, z
the Hider inequality, and the estimate
being an arbitrary compact subcone of
lF(z)l ! (2)-n
x+iy e T
C’
which proves
T0(C)
(8)
O(C),
for p
(5)
2 valid for
we have
llhllL2
! (2w)-n llhl[
for z
g
(I0)
L
with
2
(an (n-l)!)i/2 (26)-n/2
M(C’)
lYl-n/2
((2) -n (n (n-l)!) I/2 (26) -n/2)
R. D. CARMICHAEL AND E. K. HAYASHI
580
depending only on
C’
C 0(C) since
now yields (ii).
If
0(C)
of
[i, p. 222, Lemma I]
[3,
p. 93, Corollary
contains an entire straight line then
g(t)
and the fact that the support of
F(z)
0 because
C*
is in
2.4]
almost
The proof is complete.
everywhere.
H2(T C)
Since any
almost everywhere
satisfies
The proof of
does.
(15)
([6,
(12)
function satisfies
4.1]
Corollary
for z E T
C
or
with g E L
[3, P.
2
H2(T C)
3.1])
i01, Theorem
2
and some h g L by the proof of
proof of our Theorem shows that any
having support in
and hence
[i, p. 227, (72)],
function satisfies
C*
(i0).
the
As another
consequence of the proof of our Theorem, any function f(z) which is analytic in
T
C
and satisfies
hence is an H
2
TC
(I),
i.e.
[1,
p.
224, (64)],
functions.
and
(2)
to tubes T
[i,
The statement made in
more restrictive condition than
(I)
(14).
function because of analysis as in
functions considered by Vladimirov in
H2(T C)
has the representation
sections
lines
(i) is thus not correct in
and
Thus the analytic
25.3-25.4]
[i, p. 227,
(12)
are exactly the
4-5]
that
(2)
is a
the sense that both
characterize exactly the same space, the Hardy H
2
space corresponding
C
However, the growth (i) of Vladimirov has suggested to us
a way to define
analytic functions in tubes which do generalize the Hardy spaces.
The definitions
of these new spaces and our representations and analysis concerning them will
appear in
[6].
ACKNOWLEDGEMENT.
The research on this paper began while one of us, R.D.C., was
Visiting Associate Professor at lowa State University.
This author thanks the
Department of Mathematics of lowa State University for its support during 1978-
1979.
GROWTH ESTIMATE FOR ANALYTIC FUNCTIONS IN TUBES
581
REFERENCES
Methods of the Theory of Functions of Many Complex
Variables, M. I. T. Press, Cambridge, Mass., 1966.
i.
VLADIMIROV, V. S.
2.
BOCHNER, S.
Group Invariance of Cauchy’s Formula in Several Variables, Ann.
of Math.
45 (1944) 686-707.
3.
STEIN, E. M. and G. WEISS. Introduction to Fourier Analysis on Euclidean
Spaces, Princeton University Press, Princeton, N. J., 1971.
4.
SCHWARTZ, L.
Mass.,
Mathematics for the Physical Sciences, Addison-Wesley, Reading,
1966.
5.
CARMICHAEL, Richard D.
6.
CARMICHAEL, Richard D. and Elmer K. HAYASHI. Analytic Functions in Tubes
Which Are Representable by Fourier-Laplace Integrals, Pacific J. Math.
Generalized Cauchy and Poisson Integrals and
Distributional Boundary Values, SIAM J. Math. Anal. 4 (1973) 198-219.
(to appear).