Internat. J. Math. & Math. Sci.
(1987) 1-8
Vol. i0 No.
HOLOMORPHIC EXTENSION OF GENERALIZATIONS
OF HP FUNCTIONS. II
RICHARD D. CARMICHAEL
Department of Mathematics and Computer Science
Wake Forest University
Winston-Salem, North Carolina 27109, U.S.A.
(Received July 28, 1986)
In a previous article we have obtained a holomorphic extension theorem (edge
ABSTRACT.
theorem) concerning holomorphic functions in tubes in
of the wedge
p
functions for the cases 1 < p < 2.
the Hardy H
n
which generalize
.
In this paper we obtain a similar
holomorphic extension theorem for the cases 2 < p <
of H p Funons in Tube Domains, Holomorphic
Extension, FoYer-Laplace Transform, Edge of the Wave Theorem.
KEY WORDS AND PHRASES.
Generzation
1980 AMS SUBJECT CLASSIFICATION CODE.
32A07, 32AI0, 32A25, 32A35, 32A40, 46F20.
INTRODUCTION.
i.
This paper is a continuation of Carmichael
(0, 0,
with vertex at the origin
cone are all contained in Carmichael
trix function
Uc(t)
*
C
of the cone C.
C
En
(convex envelope)
En with
cone C c
lekl
C,
PC
the definitions of the indica-
which characterizes the nonconvexity
(Vladimirov [2, p. 218]).
of a cone C.
Following Vladimirov
0(C) will denote the convex
[3, p. 930], we say that a
interior points has an admissible set of vectors if there are vectors
n
1,2,
,n, which form a basis for
k
that such a set of n vectors in C is admissible for the cone C.
e
i, k
( Z2’
{y e
be any of the 2
Wn
Z
n
(-1)
quadrants.
Let
yj
En
regular cone, and projection of a
<y,t> > 0 for all y e C} is the dual cone of the cone C;
is always closed and convex
hull
The definitions of cone C in
,0) in
[i, p. 417] as are
of the cone C and the number
{t
[i].
En,
> O, j
i
n
equivalently we say
Let
n-tuples whose entries are 0 or i.
n}
We note that any quadrant C
is a quadrant in
n
and there are 2
is a regular cone in
denote the space of tempered distributions.
,
C
n
such
n.
of
The subspace
n
such that
is define to be the set of all measurable functions g(-), t e
-b
p
g(t)) e L (Carmichael
there exists a real number b _> 0 for which ((i +
i
< p <
ItlP)
[4 p. 83]).
R.D. CARMICHAEL
2
We now
define the norm growth and the space of functions which are of interest
n
Let B denote a proper open subset in
in this paper.
associated tube in
n
n.
y e B to the complement of B in
the set of all functions
f(z),
satisfy
z
I
lf(x+iy)llLP
<
,
f(x+iY)IP
l+(dCy) )-r) s
M
S()
The space
x+iy e
iB be the
Let d(y) denote the distance from
and A > 0
Let 0 < p <
n+
and let
which
dx
(Carmichael [4, pp. 80-81]) is
B
in T and which
are
holomorphic
]l/p
exp(
<
AI Yl ),
y e B,
for some constants r > 0 and s > 0 which can depend on f, p, and A but not on y E B
M(f,p,A,r,s)
and for some constant M
on y e B.
boundary of C.
which can depend on f, p,
A, r, and s but not
C, a cone, then d(y) in (1.1) is the distance from y e C to the
If B
SAP()
We have defined and studies the functions
in Carmichael
[i]
[4-7] and have stated our motivation in studying these functions in
[4, p. 81].
In Carmichael [1, Theorem] we proved a holomorphic extension theorem (edge of
C
C
wedge theorem) for holomorphic functions in T which satisfy (1.1) for y
and Carmichael
Carmichael
the’
where C is a finite union of open convex cones in
We did not consider the cases 2 < p <
did not know whether elements of
for any base C of the tube
sAP(TC),
structure of
TC;
n
and for the cases 1 < p
in Carmichael
SAP(TC),
2 < p <
,
[1]
<_
2.
because at that time we
had distributional boundary values
in fact we did not know anything about the basic
2 < p <
SAP(TC),
In Carmichael [6] we have recently proved that, indeed, elements of
2 < p <
do have distributional boundary values in the strong topology of
where C is a polygonal cone or a regular cone. A polygonal cone is a more general
,
cone than a regular cone (Carmichael
[6,
section
’
2]);
a polygonal cone is a finite
union of open convex cones which satisfy a certain intersection property and each
of which is the image of the first quadrant
formation.
[6]
C under
a nonsingular linear trans-
For our purposes here the main importance of the results in Carmichael
is that we now know that elements of
sAP(TC),
2 < p <
,
for regular cones C do
have distributional boundary values in the strong topology of
some technical details from Carmichael
[6]
%
and we also use
here.
The purpose of this paper is to prove a holomorphic extension theorem like that
[1, Theorem]
in Carmichael
for holomorphic functions in a tube T
union of open convex cones, which satisfy
In so doing we complete
alize the H
p
(1.1)
this type of result for 1 < p < 2 in Carmichael
,
.
where C is a finite
this holomorphic extension problem for functions that gener-
functions in tubes to the values 2 < p <
values 2 < p <
C,
C and for the values 2 < p <
for y
[1] as
;
we have already obtained
we have noted above,
For the
the analysis to obtain our basic restult is somewhat different than
for the cases 1 < p < 2; although there is some overlap in the technical details,
We
obtain our general holomorphic extension theorem here by first proving a special case
corresponding to special cones and then use this special case to
prove
the general case.
HOLOMORPHIC EXTENSION OF GENERALIZATIONS OF Hp FUNCTIONS
HOLOMORPHIC EXTENSION THEOREM.
2.
m
Let C
j=l
T
C
3
and satisfy
C. where each C. is a regular cone in
J
(1.1)
.
for y e C and 2 < p <
En.
Let f(z) be holomorphic in
Cj, J
For any y e
,m, the
l,
distance from y to the boundary of C is larger than or equal to the distance from y to
[6,
Carmichael
sP(TCJ),
f(z) e
the boundary of C. from which we have
Corollary l] for each j
,m, 2 < p <
l,
j
,m there is an element
l,
.
By
e
Vj
such
that
im
f(x+iy)
[Vj]
yaC.
A
in the strong topology of
O, y
independently of how y
(2.)
,m,
l,
j
with this boundary value being unique and being obtained
,m.
l,
j
e,Cj,
Here
’
[Vj]
means the
,
(Schwartz 8, Chapter 7 ])
7], by y
in the papers Carmichael [1] and Carmichael [h
y e C, for
mean y
y e C’, for every compact subcone C’ of C.)
one-one and onto
Fourier transform which maps
(As
’,
e
,
usual
cone C we
a
Before stating our theorem we first need to make a technical discussion which is
needed for the theorem. Let the open cone C be the union of a finite number of regular
m
C
as in the preceding paragraph. For each of the regular cones
cones, C
j,
j=l
n
Let
C for the 2 quadrants C in
C
,m, consider
J 1
Cj
j’
l,
k
Cj,
be an enumeration of the intersections
,rj,
J
and for each
Cj
0C
Cj ,k’
which are nonempty;
as an upper bound.
,m, r. is a positive integer with 2
l,
l,
k, k
n
W
Each
,m, is an open convex cone that is contained in or is
1
,rj,
j
En.
Now put F
r.
m
We have F C and r is the finite
k.
k=l
We
union of open convex cones each of which is contained in or is a quadrant in
CW
a quadrant
in
Cj,
j=l
n.
(Vladimirov [2, p. 220]).
We now state our holomorphic extension theorem (edge of the wedge theorem); in
this theorem F is the cone constructed from the cone C as in the preceding paragraph.
have that
0(C)
0(F)
and 1
<_
OC
<_ O F
<
En which
Let C be an open cone in
THEOREM.
m
regular cones, C
j=l
admissible set of vectors.
(1.1)
.-
given in
T0’C)
(2.1)
x+iy, be holomorphic in T
z
F(z)
C
f(z),
P(z) H(z),
l, for all u, 1 < u
<_
z e T
z e
C
where
j, J
F(z) which is holomorphic
F(z) is of the form
H(z) e s
2
AP F
2])
in
(2.2)
(T 0(C)
S
q
AP F
(T0(C)),
2.
Cj, J
i,
Theorem are regular cones instead of the more general polygonal cones
section
,m, of C
1
TO(C)
For our purposes here we have assumed that the
[6,
and satisfy
Let the boundary values of f(x+iy) in the strong
being a polynomial in z and
(l/u) + (l/q)
contains interior points and has an
There is a function
be equal in
and which satisfies
P(z)
f(z),
is the union of a finite number of
corresponding to each connected component C
F(z)
with
.
Let
for y e C and 2 < p <
topology of
(0(C))
such that
Cj,
.
because the hypothesis that
(O(C))
m, in the
(Carmichael
contain interior points and an
R.D. CARMICHAEL
4
admissible set of vectors cannot be true for polygonal cones that are not regular
By comparing the statement of the Theorem with the statement of Carmichael
cones.
[i, Theorem] we see
that these two results have the same type of conclusion; this is
We thus have been able to extend the result Carmichael [1, Theorem]
what we desired.
.
to the cases 2 < p <
We will prove the Theorem by first proving a special case of it corresponding to
the cones C., j
,m, being contained in quadrants.
l,
case we will then use it to give a proof of the Theorem.
special case of the Theorem.
LEMMA.
Let C be an open cone in
En
After proving this special
We now present the desired
m
C. where
of the form C
each
J=l
C,
,m, is an open convex cone that is contained in or is any of the 2
l,
j
quadrants C
in
En
.
y e C and 2 < p <
’
(0(C))
and let
Let f(z), z
set of vectors.
a polynomial in
2.
By this discussion we have f(z) e
Vje
!
A}
,-’’
S(TCJ),
Vj,
Vj
e
0, J
radius
_
V
I
V2
[vm]
’
..... Vm
{t:
Ucj(t)
(2.)
< A},
j
by V, and V e
<A
c}
Since the support of
+ N(; AD
c)
Vladimirov
[3,
we ob-
(Vladimirov [3, Lemma i, p. 936]) with
(O(C))
n
centered at the origin
5
in
Theorem I, p.
n
with
is closed and convex; and by hypothesis in this
contains interior points and has an admissible set of vectors.
has finite order; we denote the order of V e
element of
h20]
on p.
ADC
(O(C))*
The dual cone
(0(C))
onto
,m, then by exactly the same proof
1
being the closure of the open ball in
ADC.
theorem,
(Vj)
(2.3)
[i, equation (2.h) on p. h19 through equation (2.6)
{t:
tain that supp(V)
Uo(c) (t) < }; and
Uo(c)(t)
In addition,
,m.
i,
[v]
[v]
we denote the common value in
N(; AQ c)
all u,
and we have
have supp
Since the Fourier transform is a topological isomorphism of
as in Carmichael
(t:
,
,m, 2 < p <
these elements
exp(2gi<z,t>)>, z e T
is contained in
Vj
being
C.
we then have
in
_
and which
(2.2) with P(z)
+
(l/u)
(l/q)
i, for
form given in
,m, such that (2.1) holds.
By hypothesis the boundary values in (2.1) satisfy
=...
each
O(C)
is holomorphic in T
and
f(z) =<
--’.
m, of C given in (2.1)
l,
l,
j
l,
j
[6, Lemma i]
by the proof of Carmichael
in
for
Recall the discussion in the first paragraph of section
is a regular cone.
Ucj(t)
J
(T0(C)),
S
SA0 c
the existence of elements
{t:
(1.1)
and satisfy
n
An open convex cone that is contained in or is any of the 2 quadrants
PROOF.
En
Cj,
F(z) which
f(z), z e T C, where F(z) has the
2
(T 0(C)
z and H(z)
1 < u < 2.
in
C
Let the boundary values of f(x+iy) in the strong topology of
There is a function
F(z)
satisfies
C
contain interior points and have an admissible
x+iy, be holomorphic in T
corresponding to each connected component
be equal in
n
930] we have
by m0.
By
Any
_
HOLOMORPHIC EXTENSION OF GENERALIZATIONS OF Hp FUNCTIONS
n
H
V
<e
k=l
where
{ek}nk=l
of V e
so
IG(t)I
gradient>me +2 G(t)
k’
(2.5)
(O(C))*, G(t) is a
is unique corresponding to ek}
k=in and the
Uo(c)(t) < ADC} (O(C)) + N(5;Oc), and
is an admissible set of vectors for the cone
,
n
continuous function of t
order
5
supp(G)
< K (i +
which
{t:
+ 1
Itl) 3me
(2.6)
p. 930]
t e
n.
[3, Theorem i,
((O(C))
O(C) (Vladimirov
[2, p. 218]) should have non-empty interior (Vladimirov [3, p. 930]) which is certainly the case in this Theorem.
Since G(t) is continuous on
then supp(G)
{t" Uo(c)(t) <Apc} as a function (Schwartz [8, Chapter l, sections 1 and 3]).
(This fact is also obtained in the proof of Vladimirov [3, Theorem i], and the
{t:
containment supp(G)
< AD c} gives the support of G(t) as a function
here as well as a distribution.) Choose a function k(t) e C
t e En such that
where the constant K is independent of t e
the term
"acute"
(In
Vladimirov
in our present situation means that
n
Uo(c)(t)
for any n-tuple a of nonnegative integers
Dak(t)I
_<
Me,
n
t e
where
Me
is a
E > 0, k(t) 1 for t on an
neighbor2g
t
not
for
e
on
a
0
k(t)
neighborhood
but
AC}
< AO C} (Carmichael [i, p. 420] and [4, p. 94]). For z e T O(C) we
of {t:
have (k(t) exp(2wi<z,t>))
a function of t g
Recalling that supp(V)
(t: u
(t)
we put
O(c)
constant which depends only on ; and for
Uo(c)(t)
Uo(c)(t)
hood of
{t:
<
En
and
n.
as
<Ac
TO(C)
F(z)
(2.7)
<V,k(t) exp(2i<z,t>)>, z e
<V, exp(2i<z,t>)>
{t- Uo(c) (t)
From (2.5) and supp(G)
as a function we have (Vladimirov
<APc
[3, (3.1), p. 931])
F(z)
1
<ek, _2iz>me+2
H(z),
(2.8)
z e
where
O!C)(t)
H(z)
{t: u
Since
O(t)
is continuous on
for all u, 1 < u <
,
n
G(t) exp(2i<z,t>) dt,
<
z e T
O(C).
(2.9)
AO c}
and satisfies
(2.6)
for all t e
n
we have
G(t) e
’
u
3me+ 3 in the definition
of $i (section 1). Combining this fact with the support of G(t) as a function,
{t:
< AOC} and Carmichael [h, Theorem 6.1, p. 98] yield
which is supp(G)
as can easily be seen by choosing b
Uo(c)(t)
(exp(-21<y,t>) G(t))e L
u,
y e O(C), 1
<_
u <
=,
(2.10)
and
lexp(-2<y,t>) G(t)
llLu <M
(i +
(d(y))-r)s
exp(2Acly
), y e O(C),
1 < u <
,
(.)
r(G,u,A) > 0, s s(G,u,A) >_ 0, and M M(G,u,A,r,s) > 0 which are
independent of y e O(C); and we emphasize that (2.10) and (2.11) hold for all u,
Now (2.10), (2.11), and Carmichael [4, Theorem 5.1, p. 97] combine to
i < u <
q (TO(C)),
(i/u) + (l/q)
i, for all
prove that H(z) in (2.9) satisfies H(z) e S
for constants r
.
Ac
R.D. CARMICHAEL
6
2
u, i < u < 2; and in particular H(z) e S
_
A0C
TO(C)
F(z),
then so is
desired representation
P(z)
nomial
which is defined in
(2.2)
F(z)
of
n
(2.7),
H(z)
Since
is holomorphic in
(2.8)
because of
and
is the
<e k, -2iz>m+2
H
k=l
(TO(C)),
q
S
(i/u) + (i/q)
H(z) e S 2 (T O(C)
i, i < u < 2,
A0 C
A0 C
{t:
< AOC} in
(2.9). From (2.h), the fact that supp(V)
definition of (t) preceding (2.7), we can write (2.3) as
and
<V, k(t) exp(2i<z,t>)>
<V, exp(2i<z,t>)>, z e T
f(z)
(2.8)
in the statement of the Lemma where the poly-
is
P(z)
(TO(C)).
is given in
’, and the
Uo(c)(t)
cj,
J
,m.
i,
(2.7) prove that F(z) is the
F(z)
f(z), z e T C. The proof
These identities and
desired holomorphic extension of
f(z) to T 0(C)
of the Lemma is complete.
and
We can obtain a holomorphic extension result like that in the Lemma without the
assumption that (O(C)) contains interior points and has an admissible set of vectors.
But we loose the detailed information concerning the holomorphic extension function as
see in the following corollary to the Lemma.
we
=
m
COROLLARY i.
Let C
C
J=l
j
where each C
that is contained in or is any of the 2
holomorphic in T
of
f(x+iy) in
given in
TO(C)
C
and satisfy
n
(2.1)
,m, i an open convex cone
I,
j
quadrants C
in
En;
be equal in
f(z),
F(z)
z e T
F(z)
By the proof of the
common value.
j, J
,m,
i,
which is holomorphic in
C
(2.4)
Proceeding as in the proof of the Lemma, obtain
PROOF.
x+iy, be
Let the boundary values
corresponding to each C
There is a function
f(z),z
and let
(I.i) for y e C and 2 < p <
the strong topology of
and which satisfies
j,
Lemma, supp(V) __(t:
and call V the
Uo(c)(t) !Oc}.
Define
F(z), z e TO(C) as in (2.7). By the necessity of Vladimirov [2, Theorem 2, p. 239],
C
F(z) is holomorphic in TO(C); and F(z) f(z), z e T because of (2.3), (2.4), and
the definition (2.7) of F(z) as in the Lemma.
Using the Lemma we can now give a proof of the Theorem.
PROOF OF THE THEOREM.
m
r
From the cone C construct the cone F
C
k=l
J=l
We have that each
in the second paragraph of this section.
J
Further we have F
j, k
__-C C,
O(F)
to the boundary of
boundary of C, we have
O(C)
Cj,K.
f(z) e
the m boundary values given in
nent
Cj, J
I,
y e C
j,
[V],
and 1
_< 0C _< 0F
,r
i,
S(TCj’k),
(2.1)
of
1
in
En.
Since the distance from
k
rj, J
i,
,m.
i,
f(x+iy) corresponding to
,m, of C are equal in
for some V e
e
j
<
j,
as
is less than or equal to the distance from y to the
By hypothesis
each connected compo-
we denote the common value of the
[V]
boundary values as
[Vj]
k
,m, is an open convex cone that is contained in or is a quadrant C
i,
y e C
.
Cj,k,
j, k
Since each boundary value
m, is obtained uniquely and independently of how y
5,
then we have
y+
yeCj ,k
f(x+iy)
[Vj]
[V],
k
1
rj, J
i,
m,
(2.12)
HOLOMORPHIC EXTFSION OF GENERALIZATIONS OF H P FUNCTIONS
O(C),
O(F)
Since we have
in
(O(C)) contains
(O(C))
In
(O(F))
than
interior points and
f(z) is
F
from the hypothesis
holomorphic in T and satisfies (i.i) for y e F and 2 < p <
C and the
on f(z) in this Theorem corresponding to the cone C and the facts that F
from
distance
the
to
or
than
equal
is
less
F
of
distance from y e F to the boundary
addition
has an admissible set of vectors by the hypothesis on
y to the boundary of C.
f(z),
F(z)
z
g
P(z)
(l/q)
f(z)
and
F(z)
F(z),
H(z)
g
S
2
AOI.,
(T O(C)
Now consider each
cj
T
TCj,k
z e
z
T
C
j
0(C)
and which satisfies
with
@
q
AO r
(TO(C)),
For each
TCj
(l/u) +
,m; we have that
i,
,m.
J
,m, we
I,
TCJ
k
It thus follows
k=l
F(z), z e T cj
f(z)
that
S
Cj,
i,
k=l
(Vladimirov [2, p. 39])
T
To(r) TO(C)
are holomorphic in
f(z)
further have that
and the Lemma, we have the
0(r)
of the form
being a polynomial in z and
i, for all u, i < u < 2.
both
is holomorphic in T
P(z) H(z), z
F(z)
with
(2.12),
Thus by these facts,
F(z) which
where F(z) is
existence of a function
j
F(z),
,m; hence f(z)
i,
The proof of the Theorem is complete
We have the following corollary to the Theorem which is similar to the Corollary
The proof of the following corollary is obtained by the construction
i of the Lemma.
of the proof of the Theorem and the use of Corollary i in place of the use of the
We leave the obvious details to the reader.
Lemma.
m
COROLLARY 2.
z
Let C
C
J=l J
x+iy, be holomorphic in T
C
where each
and satisfy
Cj
is a regular cone in
(i.i)
C and 2 < p <
for y
J
,m, given in (2.1) be equal in
holomorphic in T
0(C)
and which satisfies
There is a function
F(-z)
f(z),
and let
.
T
z
F(z)
f(z),
Let the
corresponding to each
boundary values of f(x+iy) in the strong topology of
l,
n
Cj,
which is
C.
An additional fact concerning the holomorphic extension function F(z) in the
From the proof of the Theorem and the con-
Theorem can be observed as we now note.
struction of the Lemma we have that the analytic extension function
F(z)
in the
Theorem has the form
F(z)
with V
e
<V, l(t) exp(2i<z,t>)>, z e
{t:
(t)
(t) <_ &o
<V, exp(2i<z,t>)>
%
and
{t
supp(V)
r}
Uo(r
Uo(c)
To(r TO( C
<_ &or}. We
thus
have
im
[[v]
(x+i)
y0(c)
by the boundary value proof in Carmichael
in the strong topology of
Corollary
4.1, p. 93].
distinguished boundary
Thus
1n
from each connected component
3.
F(x+iy) has the same
+ +/-
of T
Cj
T
j
O(C)
l,
as the original function
,m, of T
[4,
boundary value on the
f(x+iy)
does
C
ACKNOWLEDGEMENT.
This material is based upon work supported by the National Science Foundation
under Grant No.
DMS-8418435.
R.D. CARMICHAEL
8
REFERENCES
1.
2.
3.
h.
5.
CARMICHAEL, Richard D. Holomorphic Extension of Generalizations of Rp Functions,
Internat. J. Math. Math. Sci. 8 (1985),
VLADIMIROV, V. S. Methods of the Theory of Functions of Many Complex VaFi.ables,
M.I.T. Press, Cambridge, Mass., 1966.
VLADIMIROV, V. S. Generalized Functions with Supports Bounded by the Side of an
Acute Convex Cone, Siberian Math. J.
(1968), 930-937.
CARMICHAEL, Richard D. Generalization of Hp Functions in Tubes. I, Complex
Variables Theory Appl. 2 (1983), 79-101.
CARMICHAEL, Richard D. Generalization of H p Functions in Tubes. II, Comple.x
Variable.s ..Theorv
Appl. 2
(198h), 2h3-259.
6.
CARMICHAEL, Richard D. Boundary Values of Generalizations of H p Functions in
Tubes, Complex Variables Theory .Appl. (1986), to appear.
CARMICHAEL, Richard D. Canchy and Poisson Integral Representations of
Generalizations of H p Functions, Complex Variables Theory Appl. (1986),
to appear.
8.
SCHWARTZ, L. Thorie des Distributions, Hermann, Paris,
1966.