Vector-Valued Analytic Functions Having Vector-Valued Tempered Distributions as Boundary Values

: Vector-valued analytic functions in C n , which are known to have vector-valued tempered distributional boundary values, are shown to be in the Hardy space H p , 1 ≤ p < 2, if the boundary value is in the vector-valued L p , 1 ≤ p < 2, functions. The analysis of this paper extends the analysis of a previous paper that considered the cases for 2 ≤ p ≤ ∞ . Thus, with the addition of the results of this paper, the considered problems are proved for all p , 1 ≤ p ≤ ∞ .


Introduction
Historically, the analysis of tempered distributions as boundary values of analytic functions has found applications in mathematical physics, in the study of quantum field theory.An important reference in this study is Streater and Wightman [1].In field theory, the "vacuum expectation values" are tempered distributions that are boundary values in the tempered distribution topology of analytic functions, with the analytic functions being Fourier-Laplace transforms.In addition, a field theory can be recovered from its "vacuum expectation values" [1] (Chapter 3).A similar field theory analysis is contained in the work by Simon [2].
Of particular interest with respect to the contents of this paper is the work of Raina [3] in mathematical physics.In [3], Raina considered analytic functions in the upper half plane that satisfied a pointwise growth condition associated with the analytic functions that have tempered distributions as boundary value when Im(z) → 0+.The important mathematical result in [3] showed that if the tempered distributional boundary value was an element of L p (R 1 ), 1 ≤ p ≤ ∞, then the analytic function was in the Hardy space H p , 1 ≤ p ≤ ∞ of analytic functions in the upper half plane.A converse result was proved.Raina described the importance of the results of this type concerning tempered distributional boundary values and the Hardy spaces H p , 1 ≤ p ≤ ∞, which, in mathematical physics, are associated with "form factor bounds", including the use of Hardy spaces in general in related topics in mathematical physics.Several associated references are given in [3].Importantly, the tempered distributions are used in the analysis of the mathematical physics in [1][2][3].
The results in [3] have led the author to consider the results of the type in [3] for higher dimensions and for the analytic and L p functions being both scalar-valued and vector-valued.We have also desired to obtain representations of the analytic functions involved in terms of Fourier-Laplace transforms, Cauchy integrals, and Poisson integrals.Further, we have desired to obtain new results concerning both the scalar-valued and vector-valued Hardy functions in higher dimensions, including the growth properties of these functions.
Given our desires expressed in the previous paragraph, we first considered the scalarvalued case in [4] where we obtained the pointwise growth of scalar-valued H p functions on tubes in C n .In [4], we considered scalar-valued analytic functions on tubes in C n that had a specified pointwise growth, leading to the existence of tempered distributions as boundary values, and showed that if these boundary values were a L p function, 1 ≤ p ≤ ∞, then the scalar-valued analytic function was in H p .Related results for other spaces of distributions were obtained in [4].
Continuing to the vector-valued case and building upon the results of [4], in [5] we considered vector-valued analytic functions in tube domains in C n that have pointwise growth, leading to the existence of vector-valued tempered distributions as boundary values, and proved that if the boundary value is a vector-valued L p , 2 ≤ p ≤ ∞ function then the analytic function must be in the Hardy space H p , 2 ≤ p ≤ ∞.We obtained integral representations of the analytic functions and obtained pointwise growth of vector-valued H p functions in tubes, 1 ≤ p ≤ ∞.
In [6], we considered vector-valued analytic functions in tube domains without a defining pointwise growth so that any boundary value would be considered to be in the Schwartz vector-valued D space.We showed that if the analytic functions obtained a distributional boundary value in the vector-valued distribution D sense with the boundary value being a vector-valued function in L p , 1 ≤ p ≤ ∞, then the analytic function is in the vector-valued Hardy space.We obtained a Poisson integral representation of the analytic functions in this case.
The cases for 1 ≤ p < 2 in the setting of [5] as described above are missing from our analysis at this point.That is, we desire to consider vector-valued analytic functions in tube domains that have specified pointwise growth that leads to the existence of vector-valued tempered distributions as boundary values.We then desire to prove that if the boundary value is a vector-valued L p , 1 ≤ p < 2 function, then the analytic function is in the vectorvalued H p , 1 ≤ p < 2, space.This additional analysis is desirable in order to obtain the appropriate extension of the important Raina results to all of 1 ≤ p ≤ ∞ in our generalized setting.Thus, the analysis in this paper concerns the values of p in 1 ≤ p < 2.

Definitions and Notation
All notation and definitions needed in this paper are the same as described or referred to in [5].We mention and refer to several of the most frequently used definitions and notations here.
B will denote a Banach space, H will denote a Hilbert space, N will denote the norm of the specified Banach or Hilbert space, and Θ will denote the zero vector of the specified Banach or Hilbert space.C ⊂ R n is a cone with a vertex at 0 = (0, 0, ..., 0) in R n if y ∈ C implies λy ∈ C for all λ > 0. The intersection of a cone C with the unit sphere |y| = 1 is the projection of C and is denoted pr(C).A cone C such that pr(C ) ⊂ pr(C) is a compact subcone of C. The dual cone C * of C is defined as C * = {t ∈ R n :< t, y > ≥ 0 for all y ∈ C}.An open convex cone that does not contain any entire straight line is called a regular cone.Let v = (v 1 , v 2 , ..., v n ) be any of the 2 n n-tuples whose entries are 0 or 1.The 2 n n-rants C v = {y ∈ R n : (−1) v j y j > 0, j = 1, 2, ..., n} are examples of regular cones that will be useful in this paper.
The L p (R n , B) functions, 1 ≤ p ≤ ∞, with values in B and their norms |h| p , the Schwartz test spaces S(R n ) and S (m) (R n ), m ∈ N, and the spaces of tempered vectorvalued distributions with values in B, S (R n , B) and S (m) (R n , B), are all noted in ([5], Section 2).The reference for the L p (R n , B) functions is Dunford and Schwartz [7].The references for vector-valued distributions are Schwartz [8,9].
The Fourier transform on S (R n , B) and on ) is given in [5] (Section 2).The Fourier transform of U ∈ S (R n , B) comes from [8], and will be denoted F [U], with the inverse Fourier transform being denoted F −1 [U].Similarly, all Fourier (inverse Fourier) transforms on scalar-valued or vector-valued functions will be denoted F [φ(t); x] or φ (F −1 [φ(t); x]).Of particular importance in this paper are the Fourier and inverse Fourier transforms on the vector-valued L 2 functions; the results that we need for these functions are discussed and proved in [10] (Section 1.8).As stated in this reference and referenced in [10] (Section 1.11), the Plancherel theory is not valid for vector-valued functions except when B = H, a Hilbert space.That is, in order for the Fourier transform F to be an isomorphism of L 2 (R n , B) onto itself with the Parseval identity | f| 2 = |f| 2 holding, it is necessary and sufficient that B = H, a Hilbert space; this fact comes from Kwapie ń [11].The Plancherel theory is complete in the L 2 (R n , H) setting in that the inverse Fourier transform is the inverse mapping of the Fourier transform with F −1 F = I = F F −1 , with I being the identity mapping.As stated in [10] (Section 1.8), the Plancherel theory stated there is valid for functions of several variables with values in Hilbert space.In the analysis of this paper, we need the Plancherel theory holding on L 2 (R n , B), and thus where needed we take B = H, a Hilbert space.
Associated with the Fourier transform on vector-valued functions with values in Banach space is the concept of Banach space of type p, 1 ≤ p ≤ 2, discussed in [12] (Section 6).We note that every Banach space has Fourier type 1 and leave pursuit of this concept of Fourier type to the interested reader.
Let B be an open subset of R n .The Hardy space H p (T B , B), 0 < p < ∞, consists of those analytic functions f(z) on the tube where z = x + iy ∈ T B and the constant M > 0 is independent of y ∈ B; the usual modification is made for the case p = ∞.
Let C be an open convex cone in R n .E (R n ) will denote the set of all infinitely differentiable complex valued functions on R n .We define the function We define and state known results concerning the Cauchy and Poisson kernel functions corresponding to tubes T B = R n + iC ⊂ C n .Let C be a regular cone in R n and C * be the corresponding dual cone of C. The Cauchy kernel corresponding to T C is where C * is the dual cone of C as noted.The Poisson kernel corresponding to T C is Referring to [13] (Chapters 1 and 4) for details, we know for z ∈ T where * is Beurling (M p ) or Roumieu {M p }.These ultradifferentiable functions are contained in the Schwartz space D L p = D(L p , R n ).We also use the results [4] (Lemmas 3.1 and 3.2).Because of the combined properties of the Cauchy and Poisson kernels from [13,14], we know that the Cauchy and Poisson integrals respectively, where B is a Banach space.
We use [5] (Lemma 3.4) several times in this paper.For convenience to the reader, we state this result here to conclude this section.Throughout N(0, r) denotes the closed ball of radius r > 0 centered at 0 ∈ R n .Theorem 1.Let f be analytic in T C = R n + iC with values in a Banach space B, where C is a regular cone in R n , and have the Poisson integral representation for all compact subcones C ⊂ C, M(C ) being a constant depending on C ⊂ C and not on y ∈ C , while where M y is a constant depending on y ∈ C; and for for all compact subcones C ⊂ C and all r > 0, M(C , r) being a constant depending on C ⊂ C and on r > 0, but not on y ∈ (C \ (C ∩ N(0, r))), while where M y is a constant depending on y ∈ C.

Tempered Distributional Boundary Values
Let C be an open convex cone in R n and T C = R n + iC ⊂ C n .We denote the set of analytic functions on T C with values in a Banach space B by A(T C , B).As above, N(0, r) denotes the closed ball about 0 ∈ R n of radius r > 0.
In [5] (Theorem 4.1), we have stated the following result which we need here.
Theorem 2. Let C be an open convex cone.Let f ∈ A(T C , B).For every compact subcone C ⊂ C and every r > 0, let where M(C , r) is a constant depending on C ⊂ C and on r, R is a nonnegative integer, k is an integer greater than 1, and neither R nor k depend on C or r.There exists a positive integer m and a unique element In Theorem 2, and in the remainder of this paper, by y → 0, y ∈ C, we mean that y → 0, y ∈ C ⊂ C, for every compact subcone C of C.
In [5] (Theorem 4.4), we proved for C, a regular cone, and the boundary value In [5], we were not able to obtain this result for the cases 1 ≤ p < 2. We now have a proof for the cases 1 ≤ p < 2, and we obtain the result [5] (Theorem 4.4) for the cases 1 ≤ p < 2 here.
To obtain [5] (Theorem 4.4) for 1 ≤ p < 2, we follow some of the structure of [5] by first proving our result for the case that the cone C is a n-rant cone C v or is contained in a n-rant cone and then using this case to obtain the general result for the cone C being any regular cone.Because 1 ≤ p < 2, here the details of our proof in the case C ⊆ C v in Theorem 3 below are different in many instances than those of [5] (Theorems 4.2 and 4.3).The values of the functions and distributions in the remainder of this section will be in Hilbert space H because of the need for the Fourier transform properties on L 2 (R n , H), as described in Section 2 above.
We give an outline of the proof of Theorem 3 for the benefit of the reader.Given the assumed function f(z) in Theorem 3, we will divide it by a structured analytic function X (z), z ∈ T C , > 0, and put g (z) = f(z)/X (z), z ∈ T C , > 0. g (z) is represented as the Fourier transform involving a function G (t), which has support in C * .g (x + iy) is shown to have boundary value F [G ] in S (R n , H) as y → 0, y ∈ C, and then is shown to equal the Cauchy integral and the Poisson integral of a function involving the boundary value h of f(x + iy).After establishing some important limit analysis, we proceed to prove that f(z) equals the Poisson integral of the boundary value h, which will then yield the conclusions of Theorem 3. Theorem 3. Let C be an open convex cone which is contained in or is any of the 2 n n-rants C v ⊂ R n .Let H be a Hilbert space.Let f ∈ A(T C , H) and satisfy (1).Let the unique boundary value U of Theorem 2 be h ∈ L p (R n , H), 1 ≤ p < 2. We have f ∈ H p (T C , H), 1 ≤ p < 2, and Proof.As noted above, the proof has a structure similar to that of [5] (Theorems 4.2 and 4.3), but many details are different.We refer to [5] (Theorems 4.2 and 4.3) where appropriate.Put g (z) = f(z)/X (z), z ∈ T C , > 0, where g (z) satisfies ( 1).(By Theorem 2, there is a unique U ∈ S (R n , H) such that (2) holds for g (z) in S (R n , H), a fact that we use later in this proof.)By the same analysis as in the proof of [5] (Theorem 4.2), we obtain [5, (15)] here; that is, for all compact subcones C ⊂ C ⊆ C v and all r > 0 where M (C , r, ) is a constant.Put Using ( 4), the same proof as in the proof of [5] (Theorem 4.2) yields that G (t) is a continuous function of t ∈ R n for y ∈ C and > 0, is independent of y ∈ C, and has support in C * , the dual cone of C.
For any compact subcone C ⊂ C, any r > 0, and any > 0 Equation (4) yields from which e −2π y,t G (t) ∈ L p (R n , H) for y ∈ C and for all p, 1 ≤ p < ∞, by ([5], Lemma 3.1).From Equation ( 5), e −2π y,t G (t) = F −1 [g (x + iy); t], y ∈ C, with the transform holding in both the L 1 (R n , H) and L 2 (R n , H) cases, and in L 2 (R n , H): From the properties of G , the Fourier transform in ( 7) is in both the L 1 (R n , H) and L 2 (R n , H) cases, and (7) becomes Both G (t) and e −2π y,t G (t), y ∈ C, are elements of S (R n , H), and g (x as y → 0, y ∈ C. As noted above, by Theorem 2, there is a unique Recalling the function d y (t) defined in Section 2, we have is the characteristic function of C * , and the integral on the right of (10) is convergent, since H ∈ L ∞ (R n , H) and supp(H ) ⊆ C * almost everywhere.From (8), (10), and the fact that supp(G ) ⊆ C * , we have for z ∈ T We proceed to construct a Poisson integral representation for g (z) in addition to the Cauchy integral representation in (11).Let w be an arbitrary point of T C .Using [4] (Lemma 3.2), we have for z ∈ T C that K(z + w)g (z) = K(z + w)f(z)/X (z) is analytic in z ∈ T C and satisfies the growth (1) of f(z).Further, lim y→0,y∈C and 1/X (x) are bounded for x ∈ R n .The same proof leading to (11) applied to K(z + w)g (z), z ∈ T C , yields For z = x + iy ∈ T C , we choose w = −x + iy ∈ T C .Then, (12) combined with (11) becomes We now present some limited analyses, which we need to analyze the function, the Poisson integral of h, that we will show represents f(z) and from which the conclusion of the proof of this theorem will follow.Since |1/X (x)| ≤ 1, x ∈ R n , > 0, both h and h/X are in L p (R n , H), 1 ≤ p < 2. We have with the right side being independent of > 0. Further, lim By the Lebesgue dominated convergence theorem which proves h/X → h in L p (R n , H), 1 ≤ p < 2, as → 0+.
We now define and analyze the function which we desire to be the Poisson integral representation of f(z), as noted in the preceding paragraph; this function is Let z o be an arbitrary but fixed point of T C .Choose the closed neighborhood [5] (Lemma 3.3), and note that [5] (Lemma 3.3) holds for all p, 1 ≤ p ≤ ∞.Let the constant B(z o ) in ( 16) below be the constant obtained in [5] (Lemma 3.3).Using the Hölder inequality if 1 < p < 2 and the boundedness of Q(z; t) from the proof of [5] (Lemma 3.3) ([4] (Lemma 3.4)) if p = 1 and using ( 13) and (15), we have we have the pointwise bound on G(z) for y = Im(z) in any compact subcone C ⊂ C contained in Theorem 1. (see also ([5], ( 6)).) Thus, combining the bounds (1) on f(z) and the pointwise bound just noted on G(z) for y = Im(z) in any compact subcone C ⊂ C, we have the inequality on f(z) − G(z) for the cases 1 < p < 2 for any compact subcone C ⊂ C and any r > 0 where P(C , r) is a constant depending on C ⊂ C and on r > 0. If p = 1, by combining inequalities ( [4], (10) and ( 11)) in the proof of Theorem 1 given in ([5], Lemma 3.4), we have where δ depends only on C and not on C and Z n is the surface area of the unit sphere in R n .Combining this inequality on G(z) with inequality (1) on f(z), we again have that f(z) − G(z), z ∈ T C , also satisfies (17) for p = 1.In addition, we know from the boundary values of f(z) and G(z) that lim y→0,y∈C in S (R n , H) for 1 ≤ p < 2. Using (18), we now proceed to complete the proof by proving f 17) and (18) for each p, 1 ≤ p < 2. Consider g(z) = F(z)/X 1 (z), z ∈ T C , where X 1 (z) is defined at the beginning of this proof for = 1.As in obtaining (4) for = 1, we have for C ⊂ C ⊆ C v and r > 0 N (g(z)) ≤ P (C , r, 1)(1 + |z|) −n−2 , z = x + iy ∈ T(C , r) = R n + i(C \ (C ∩ N(0, r))), where P (C , r, 1) is a constant.Now putting as in ( 5) R n g(x + iy)e −2πi x+iy,t dx, y ∈ C, t ∈ R n , and proceeding with the proof from ( 5) to (8), we have that A(t) is continuous, is independent of y ∈ C; has support in C * ; satisfies a growth as in (6); satisfies e −2π y,t A(t) = F −1 [g(x + iy); t], t ∈ R n , y ∈ C, with the transform holding in both the L 1 (R n , H) and L 2 (R n , H) cases; and with e −2π y,t A(t) ∈ L p (R n , H) for all p, 1 ≤ p < ∞, y ∈ C; satisfies g(x + iy) = F [e −2π y,t A(t); x], x ∈ R n , y ∈ C; and satisfies With Theorem 3 proved for 1 ≤ p < 2, we now obtain this result for C being an arbitrary regular cone in R n ; this is our desired result, which extends [5] (Theorem 4.4) to the cases 1 ≤ p < 2. The proof of the following theorem for the cases 1 ≤ p < 2 is obtained using Theorem 3 by exactly the same proof that [5] (Theorem 4.4) was proved using [5] (Theorems 4.2 and 4.3); we ask the interested reader to follow the suggested proof if desired.
(14)o , ρ) ⊂ T C .Using(14)and (16) for 1 ≤ p < 2, we have∈ N(z o , ρ).Since g (z) is analytic in T C , > 0, we have that G(z) is analytic at z o ∈ T C ; hence G(z) is analytic in T C since z o is an arbitrary point in T C .Applying Theorem 1, we have G(z) ∈ H p (T C , H), 1 ≤ p < 2.Let φ ∈ S(R n ).Using Hölder's inequality, if 1 < p < 2 and the boundedness of φ p for z ∈