A POINTWISE GROWTH ESTIMATE FOR ANALYTIC FUNCTIONS IN TUBES

A class of analytic functions in tube domains T C n + iC in n- dimensional complex space, where C is an open connected cone in 1t n which has been defined by V. S. Vladimirov is studied. We show that a previously obtained L 2 growth estimate concerning these functions can be replaced by a pointwise growth estimate, and we obtain further new properties of these functions. Our analysis shows that these functions of Vladimirov are exactly the Hardy H 2 class of

Let f(z) be analytic in T C ]R n + iC and for any C' C C let f(z) satisfy I(x+i)l IL2 n }f(x+iy)I 2 dx) 1/2 < M,f (c') e 11 ,xsc'c, for every > 0 where the constant M,f(C') depends one, f, and C' but not on y g C' C. Vladimirov has studied these analytic functions in [i, sections 25. 3-  25.4].%n this note we show that the L 2 growth estimate [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 these functions.Our analysis also shows that the analytic functions of Vladimirov defined above are in fact exactly the Hardy H 2 functi6ns ([2, section 3] or [3, 2 pp.90-91];) thus the growth of Bochner [2, (13)] which defines the Hardy H space T C for tubes L is not a more restrictive condition than (i), contrary to the statement made in [i, p. 227, lines 4-5], in the sense that both (i) and (2)

RESULTS.
To obtain our results we need three lemmas.The proof of Lemma 1 is like that of [i, p. 223, Lemma 2] and is omitted.
LEMMA I. Let C be an open (not necessarily connected) cone.Let y e 0(C), the convex envelope of C.There exists a > 0 depending on y such that Y yt _> 6 lyl Itl for all t e C {t yt > 0, y C}.Further, if C' is an arbitrary compact sub- cone of 0(C) there exists a 6 6(C') > 0 depending only on C' such that (3) holds for all y g C' and all t C*.Using Lemma 1 we have llc* (t) e iztl= Ic,(t) e -yt !Ic, (t) exp(-lyl tl) () for some 8 > 0 and (5) holds for all t e ]R n since Ic.(t) 0 if t @ C* y (5) proves (4) for p Now let i < p < Using (5), [4, p. 39, Theorem 32], and integration by parts (n-l) times, we have lc* where is the volume of the unit sphere in IR n where is the volume of the unit sphere in IR n and > 0 is the number of n y (3); and for I < p < 2, (l/p) + (l/q) i, we have where the L q norm is with respect to the variable t e ]Rn.Further, if C' is an arbitrary compact subcone of O(C) then (7) and (8) hold for z x+iy g T C' and R. C. CARMICHAEL AND E K. HAYASHI n t with depending only on C' O(C) and not on y C' = 0(C).
PROOF.The fact that K(z-t) is analytic in z T O(C) for fixed t ]R n follows by [1, p. 223]. (7)follows by the analysis of (6) for p I. For iz i < p _ 2 and (i/p) + (i/q) i, K(z-t) --I[Ic,() e t] in the sense L q as noted in [5, P. 202, proof of Theorem i] hence by the Parseval inequality < IIc* D eizl Lp z g T O (C) IK(z-t) ILq ( 9) Inequality (8) now follows from (9) and a computation as in (6) for i < p _ 2. If C' is an arbitrary compact subcone of 0(C) we use the second part of Lemma 1 to obtain ( 5) and (6) for z g T C' C' 0(C), where now depends only on C' C O(C) and not on y C'C O(C).This fact and the above analysis yields (7) and (8) holding for z x+iy T C' with depending only on C' O(C).
We now obtain our result which adds information to [I, p. 227, Corollary] and hence to the analytic functions considered in [i, sections 25.3-25.4].First note a misprint in the statement of [I, p. 227, Corollary]; "equality (63)" in [i, p. 227, line 2 of the Corollary] should read "inequality (64)".
THEOREM.Let C be an open connected cone.Let f(z) be analytic in T C and satisfy (i).Then f(z) has an analytic extension F(z) H2(TO(C)) to T 0(C) such that for any compact subcone C' O(C) where M(C') is a constant which depends at most on C' C 0(C) and h L 2 is the L 2 boundary value of F(x+iy) as y / 0, y 0(C).and the Fourier-Laplace integral on the right of ( 12) is well defined because of the properties of g(t) and ( 4).Now put F(z) I g(t) e izt dt z g T 0(C) (13) n The fact that F(z) is analytic in T 0(C) follows as a special case of [6, Theorem 2.1].Because of (4) and the properties on g(t), we have that (g(t) exp(-yt)) g

L I L
2 ]R n as a function of t g for y g 0(C); hence the integral on the right of (13) can be interpreted to be the L 2 Fourier transform of (g(t) exp(-yt)), y g 0(C).The Parseval equality and the fact that the support of g is in C almost everywhere now yield and we conclude that F(z) g H2(T0(C)).(This fact also follows by [3, p. i01, Theorem 3.1].)We now apply the proof of [i, p. 227, Bochner's formula] or [3, p. 103, Theorem 3.6] to obtain the existence of a function h g L 2 which is the L 2 Fourier transform of g and is the L 2 boundary value of F(x+iy) as y / 0, y 0(C), such that T0(C) Using (15), the Hider inequality, and the estimate (8) for p 2 valid for z s T C' C' being an arbitrary compact subcone of O(C), we have lF(z)l ! (2)-n llhllL2 !(2w)-n llhl[ 2 (an (n-l)!)i/2 (26)-n/2 lYl-n/2 depending only on C' C 0(C) since does.The proof of [3, p. 93, Corollary 2.4] now yields (ii).If 0(C) contains an entire straight line then F(z) 0 because of [i, p. 222, Lemma I] and the fact that the support of g(t) is in C* almost everywhere.The proof is complete.
Since any H2(TC) function satisfies (12) with g E L 2 having support in C* almost everywhere ([6, Corollary 4.1] or [3, P. i01, Theorem 3.1]) and hence satisfies (15) for z E T C and some h g L 2 by the proof of [i, p. 227, (72)], the proof of our Theorem shows that any H2(TC) function satisfies (i0).As another consequence of the proof of our Theorem, any function f(z) which is analytic in T C and satisfies (I), i.e. [1, p. 224, (64)], has the representation (12) and  2) is a more restrictive condition than (i) is thus not correct in the sense that both (I) and (2) characterize exactly the same space, the Hardy H 2 space corresponding to tubes T 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].
576 R. D. CARMICHAEL AND E. K. HAYASHI

LEMMA 2 .
Let  C be an open connected cone.We have (Ic.(t)e izt]" e L p for all p I !P !TO(C) n + i O(C), where Ic.(t) as a function of t e n for arbitrary z denotes the characteristic function of C* PROOF.Let z x+iy T O(C)