HOLOMORPHIC EXTENSION OF GENERALIZATIONS OF HP FUNCTIONS

In recent analysis we have defined and studied holomorphic functions in tubes in ℂn which generalize the Hardy Hp functions in tubes. In this paper we consider functions f(z), z=x

The purpoue of this paper is to prove a holomorphic extension theorem (edge of the wedge theorem) for functions which are holomorphic in a tube in n and which satisfy a norm growth condition that generalizes the norm growth for H p functions in tubes.The basis for the analysis presented here is the analysis in our papers Carmichael We begin by stating some needed definitions.A set C Rn is a cone (with is always closed and convex (Vladlmirov [3, p. 218]).The intersection of the cone C with the unit sphere in n is called the projection of C and is denoted pr(C).The function Uc(t sup (-<t,y>) ygpr(C) is the indicatrix of .thecone C, and we note that C* {t t Rn Uc(t) _< 0}.The set T C n + iC is a tube In n.The convex hull (convex envelope) of a cone C will be denoted by O(C), and O(C) is also a cone.Put C, Rnc * the number sup Uo(c) (t) OC tC, Uc(t) characterizes the nonconvextty of the cone C (Vladimirov [3, [.. . .Following Vladtmtrov [4, p. 930] we say that a cone C :; 1( n with interior points has an admissi- ble set of vectors if there are vectors e k C, lekl 1, k 1,2 ,n, which form a basis for Rn; equivalently we say that such a set of n vectors in C is admissible for the cone C. _< M (l+ (d(y))-r) exp(2Alyl), y B, for some constants r _> 0 and s _> 0 which can depend on f, p, and A but not on y B and for some constant M M(f,p,A,r,s) which can depend on f, p, A, r, and s but not on y E B. We defined and studied the functions sP(T B) in Carmichael [1-2].The spaces S(T ) were defined to generalize the H p functions in tubes ( C, an open convex cone in Rn, then f(x+ly) has a unique boundary value as y/, y E C, in the strong topology of ', the space of tempered distributions.
In this paper we prove a holomorphic extension theorem (edge of the wedge theore for holomorphic functions in T C which satisfy (i.I) for y E C where C is a finite union of open convex cones in n; the extended function is holomorphic in T O(C) where O(C) is the convex hull of C. To obtain our extension theor we use the information from Carmlchael [1] which is contained in the preceding paragraph.
We proceed to the result of this paper after making the following definition; the subspace of I <_ p < , is defined to be the set of all measurable P functions g(t), t n, such that there exists a real number b 0 for which ).All subsequent notation and terminology in thls paper are that of Carmlchael [1-2].open convex cones n n and m is a positive integer.Let f(z) be holomorphic in the tubular cone T C n + iC and satisfy (i.i) for y E C and for i < p 2. For ay y g Cj, j I, m, the distance from y to the boundary of C is larger than or equal to the distance from y to the boundary of Cj from which it follows that f(z) S(TCJ),- Thus by Carmchael [I, Corollary 4.1, p. 93] there exist measurable functions gj(t) E q, (l/p)+ (l/q) i, with supp(gj) _ 2) be equal in Then there is a function F(z) which is holomor- phic in T O(C) and which satisfies V(z) f(z), z e TC, where F(z) is of the form S, (TO(C)), (l/p) + with P(z) being a polynomial in z and H(z) e $2 A PC 0 c (l/q) I.
PROOF.By hypothesis the boundary values in (2.2) above are equal in Since the Fourier transform is a topological isomorphism of onto we have that the elements gj(t) (l/p) + (l/q) I I m, obtained in the q first paragraph of this section satisfy gl(t) g2(t) gm(t) (2.3)   in .We call this common value g(t) and have g(t) e (l/p) + (l/q) i.Now q' max Uc.(t), t e In. (2.4)

Uc(t) j=l m
We have Uc(t) Uo(c)(t), t e C*, (Vladimirov [3,p. 219, (54)]); and from the defni- tlon of 0 c we have U0(c) (t) <_ 0 C Uc(t), t e C, Rn C*.Since i !0 C < (Vladimirov [3, p. 220]) here we have Uo(c)(t) < 0 C Uc(t) t e n.From (2.4) we now obtain max Uc.(t) t e 1 n (2 5) Uo(c) (t) _< 0 C j=l m From (2.3) and the fact that supp(gj) _ {t: Ucj(t) _< A} almost everywhere, j =l m, m we have that g e q C vanishes on jlo= {t: Ucj(t)> A} as a distribution.Now let t {t: Uo(c)(t) > A 0 C }; for such a point t we have by (2.Thus if t e {t: Uo(c)(t) > A O C then t j--i {t: Ucj(t) > A} and on this latter set g vanishes.Since {t: Uo(c)(t) <_ A0 C is a closed set in n we thus have supp(g) in ' and {t: Uo(c)(t) < A O C (O(C))* + N(;A0 C) (Vladimirov [4, Lemma i, p. 936]) with N(;A 0 C) being the closure of the open ball in n centered at and with radius A 0 C Recall from section 1 that the dual cone (O(C))* is closed and convex and by hypothesis in this Theorem (0(C))* contains interior points and has an admissible set of vectors.Since g q = has order 0 then by Vladimlrov [4, Theorem i, p. 930] n g(t) <ek, gradient> 2 G(t) (2.7) k=l where {ek}= I is an admissible set of vectors for the cone (O(C))*, G(t) is a con- n tinuous function of t e n which is unique corresponding to {ek}k= 1 and the order 0 of g e supp(G) where the constant K is independent of t n. (In Vladlmlrov [4, Theorem i, p. 930] the term "acute" in our present situation means that ((O(C))*) * O(C) (Vladlmlrov [3, p. 218]) should have non-empty interior (Vladlmlrov [4, p. 930]) which is certain- ly the case in this Theorem.)Since G(t) is continuous on n, then supp(G) {t: U0(c)(t) _< A0 C as a function (Schwartz [8, Chapter i, sections 1 and 3]). (This fact is also obtained in the proof of Vladlmlrov [4, Theorem I], and the containment supp(G) _ {t: Uo(c)(t) < A PC which is stated preceding to (2.8) gives the support of G(t) as a function.)We now choose a function A(t) C t e n, such that for any n-tuple of nonnegatlve integers ID%(t) <_ M, t e n, where M is a constant which depends only on ; and for > 0, %(t) 1 for t on an neighborhood of {t: Uo(c)(t) _< A PC and %(t) 0 for t e n but not on a 2 neighborhood of {t: u0(c)(t) _< A 0 C (Carmlchael [i, p. 94]).We have that (%(t) exp(2i<z,t>)) as a function of t n for z e T 0(C) Recalling (2.6) we now put F(z) fn g(t) exp(2i<z,t>) dt (n g(t) (t) exp(2l<z,t>) dt, z E T O(C) (2.9) From (2.7) and supp(C,) _ {t: Uo(c)(t) <_ A O c as a function we have (Vladlmirov [4, (3.1), p. 931]) PROOF.Proceeding as in the proof of the Theorem we obtain the common value g __' (l/p) + (l/q) I from (2.3) and supp(g)C {t: Uo(c)(t) < A 0 C in '.

g(t)
By our assumption supp(g) _ {t: Uo(c)(t) <_ A 0 C almost everywhere; thus by Carmichael [I, Theorem 6.1, p. 98], g(t) satisfies (exp(-2<y,t>) g(t)) c L q, y O(C), lexp(-2<Y, t>) g(t) llLq for constants r r(g,q,A) >_ O, s s(g,q,A) >_ 0, and M M(g,q,A,r,s) > 0 which are independent of y E O(C).Then by Carmichael [I, Theorem 3.1, pp. 84-85]the function F(z) n g(t) exp(2i<z t>) dt g(t) X(t) exp(2i<z t>) dt z (2.16) is holomorphic in T O(C) where %(t) g C is the function defined in the proof of the Theorem.As in the proof of the Theorem F(z) is the desired holomorphic extension of f(z) to T O(C) If p 2 then q 2; in this case (2.14), (2.15), and Carmichael [I, A (TO(C)).The proof is complete.

0C
We have a more general holomorphic extension theore than either the Theorem or Corollary i.Here 0(C) is as general as possible and we make no assumption on the constructed g(t) in (2.3).We lose the explicit information on F(z) being in an S (T0(C)) space however.given in (2.2) be equal in '.Then there is a holomorphic function F(z) in TO(C) such that F(z) f(z), z TC.
PROOF.Define F(z), z E TO(C), as in (2.16) where g C (l/p) + (l/q)= is the common value in (2.3) in and supp(g)_ {t :Uo(c) (t) <_ AO C in ' i, from the proof of the Theorem.Then F(z) is holomorphic in T O(C) by the necessity of Vladlmlrov [3, Theorem 2, p. 239] and is the desired holomorphlc extension of f(z) to T O(C) because of (2.3) and (2.1).(Recall the proof of the Theorem.)The proof is complete.
Notice from Vladimlrov [3, Theorem 2, p. 239] that F(z) in Corollary 2 does satisfy a polntwise growth estimate; but we cannot conclude that F(z) is in an S (T O(C) space for any p in Corollary 2.

Oc
In the Theorem and Corollaries i and 2 the holomorphic extension function F(z), z E T0(C), is defined by (2.9) (i.e.(2.16)) where g(t) : (l/p) + (l/q)= i, and supp(g) {t: Uo(c)(t) <_ A 0 C in .Since O(C) is an open convex cone then in n) if y e C implies Ay C for all positive scalars A. A regular cone ls an open convex cone C such that does not contain any entire straight llne.The dual cone C* of a cone C is defined as C* {t n: <t, Let B denote a proper open subset of Rn.Let 0 < p < and A > O. Let d(y) denote the distance from y c B to the complement of B in In.The space sP(TB) (Carmtchael [1, pp.80-81]), T B 1( n + tB, is the set of all functions f(z), z x+ ty TB, which are holomorphtc in T B and whlch satisfy l[f(x+iy)[ILp If(x+iy) lP dx _< (I.1)

2 (
t) exp(2i<z,t>) dr, z T O(C) (2.11) {t:U0(c) (t)<_ 0 c} where p, <_ p From the continuity of G(t) and (2.8) we easily have G(t) P for all i < this combined with the support of G(t) as a function and Carmichael [I, Theorem 6.], p. 98] yield (exp(-2y,t>) G(t))Lp, y e O(C), C given in (2.2) be equal in and let this comon value g(t) hay, in {t: Uo(c)(t) _< A0 C almost everywhere (as well as in ').Then there i support a function F(z) which is holomorphic in T O(C) and which satisfies F(z) f(z), z E TC; i + (d(y))-r) s exp(2nA0 C IYl) Y O(C) (2.15)

COROLLARY 2 .
Let the open cone C and the function f(z) be as in the hypothesis of Corollary i with I < p 2. Let the boundary values of f(x+ ly) in the strong topology of corresponding to each connected component ] i, m, of C We now state and prove the main result of this paper.TttEOREM.Let C be an open cone in R n which is the union of a finite number of open convex cones, C .=Cj such that (O(C))* contains interior points and has an admissible set of vectors.Let f(z), z x+iy, be holomorphtc in the tubular cone T C and satisfy (i.i) for y e C and i < p _< 2. Let the boundary values of f(x+iy) in the strong topology of ,.' corresponding to each connected component Cj, l, m, t: Ucj(t) _< A} of C given in(2.