fourier spectrum characterizations of spaces on tubes over ... · 1194 h.-c. li et al. keywords...

Complex Anal. Oper. Theory (2018) 12:1193–1218https://doi.org/10.1007/s11785-017-0737-6

Complex Analysisand Operator Theory

Fourier Spectrum Characterizations of H p Spaceson Tubes Over Cones for 1 ≤ p ≤ ∞Hai-Chou Li1 · Guan-Tie Deng2 · Tao Qian3

Received: 22 March 2017 / Accepted: 12 October 2017 / Published online: 25 October 2017© Springer International Publishing AG 2017

Abstract We give Fourier spectrum characterizations of functions in the Hardy H p

spaces on tubes for 1 ≤ p ≤ ∞.For F ∈ L p(Rn),weshow that F is the non-tangentialboundary limit of a function in a Hardy space, H p(T�),where� is an open cone ofR

n

and T� is the related tube inCn, if and only if the classical or the distributional Fourier

transform of F is supported in �∗, where �∗ is the dual cone of �. This generalizesthe results of Stein and Weiss for p = 2 in the same context, as well as those of Qianet al. in one complex variable for 1 ≤ p ≤ ∞. Furthermore, we extend the Poissonand Cauchy integral representation formulas to the H p spaces on tubes for p ∈ [1,∞]and p ∈ [1,∞), with, respectively, the two types of representations.

Communicated by Irene Sabadini.

Guan-Tie Deng: This work was partially supported by NSFC (Grant 11271045) and by SRFDP (Grant20100003110004). Tao Qian: The work was partially supported by Research Grant of University ofMacau, MYRG115(Y1-L4)-FST13-QT, Macao Government FDCT 098/2012/A3.

B Tao [email protected]

Hai-Chou [email protected]; [email protected]

Guan-Tie [email protected]

1 College of Mathematics and informatics, South China Agricultural University, Guangzhou, China

2 The First Institution, School of Mathematical Sciences, Beijing Normal University,Beijing 100875, China

3 Department of Mathematics, University of Macau, Macao (Via Hong Kong), China

http://crossmark.crossref.org/dialog/?doi=10.1007/s11785-017-0737-6&domain=pdf

1194 H.-C. Li et al.

Keywords Hardy spaces · Fourier transform · Tube domain · Fourier spectrum ·Integral representation

1 Introduction

The Paley–Wiener Theorem in relation to the Hardy H2(C+) asserts that for f ∈L2(R), f is the non-tangential boundary limit of some function in the Hardy spaceH2(C+) if and only if supp f ⊂ [0,∞) (see, for instance, [24]). We recall it in thefollowing.

Theorem A (Paley–Wiener) [13], ([24]) F ∈ H2(C+) if and only if there exists afunction f ∈ L2(0,∞), such that

F(z) =∫ ∞

0f (t)e2π i t zdt,

for z ∈ C+, and, furthermore,

∫ ∞

0| f (t)|2dt = ‖F‖2H2 .

Related generalizations were subsequently obtained in [17], [9], [20] and [21]. Thementioned literature proves the following Fourier spectrum characterization result: Forf ∈ L p(R), 1 ≤ p ≤ ∞, f is the non-tangential boundary limit of a function in theHardy H p(C+) if andonly if, in the pointwise or distributional sense, supp f ⊂ [0,∞).

Especially, Qian et al. obtained the following Theorems B and C.

Theorem B ([20]) Let f ∈ H p(C+), 1 ≤ p ≤ ∞. Then, as a tempered distribution,f is supported in [0,∞), denoted as d-supp f ⊂ [0,∞). That amounts to

( f , ϕ) = 0,

for all ϕ ∈ S(R) with supp ϕ ⊂ (−∞, 0]. Moreover, in the range 1 ≤ p ≤ 2 thereholds f (x) = 0 for almost all x ∈ (−∞, 0].In 2009, Qian et al. proved the converse result of Theorem B (see [21]) as follows.

Theorem C ([21]) For 1 ≤ p ≤ ∞, if f ∈ L p(R) and d-supp f ⊂ [0,∞), then f isthe boundary limit of a function in H p(C+).

It is certainly a natural question to ask whether in higher dimensional spaces, forinstance, involving several complex variables, analogous results hold. To the authors’knowledge, in the Euclidean space R

n context, there have been no general Fourierspectrum characterization results for all range of the index p. In [26] of Stein andWeiss, it is shown that the characterization result is valid for the Hardy spaces H2(T�)

on tubes over open cones � : They show that a square-integrable function is the non-tangential boundary limit of a function in H2(T�) if and only if its Fourier transformvanishes outside �∗, the latter being the dual cone of �. The following related resultof Stein and Weiss [26] is restricted to p = 2.

Fourier Spectrum Characterizations of H p Spaces on… 1195

Theorem D ([26]) Let B be an open connected subset of Rn. F belongs to H2(TB)

if and only if it has the form

F(z) =∫Rn

e2π i z·t f (t) dt, z ∈ T�,

where f is a function satisfying

supy∈B

∫Rn

e−2πy·t | f (t)|2 dt ≤ A2 < ∞.

When the base B is an open cone one obtains the following representation theoremof H2−functions sharper than that is obtained in the above Theorem D.

Theorem E ([26]) Let � be a regular open cone in Rn. F ∈ H2(T�) if and only if

F(z) =∫

�∗e2π i z·t f (t) dt, z ∈ T�,

where f (t)X�∗(t) is the Fourier transform of

F(ξ) = limη→0, η∈�

F(ξ + iη),

X�∗(t) is the characteristic function of �, that is, X�∗(t) = 1, for t ∈ �∗; andotherwise X�∗(t) = 0. Moreover,

‖F‖H2 = ‖F‖2 =(∫

�∗| f (t)|2 dt

)1/2

.

Moreover, for all range p, Stein and Weiss [26] obtained the fundamental result asgiven in the following Theorem F which is an important result of the theory of Hardyspaces on tubes.

Theorem F ([26]) Let � be a regular open cone in Rn. Suppose that F ∈ H p(T�).

Then ([26], Section 5, Chapter 3),

(a) for 1 ≤ p ≤ ∞,

limy∈�, y→0

F(x + iy) = F(x)

holds for almost all x ∈ Rn . F(x) is called the restricted (non-tangential) boundary

limit of F(z);(b) for 1 ≤ p < ∞,

limy∈�, y→0

‖F(x + iy) − F(x)‖p = 0;


(c) for 1 ≤ p ≤ ∞, ([26], page 120)

F(z) =∫Rn

P(x − t, y)F(t)dt.

In [15] the author proved the one-way result that for 1 ≤ p < ∞ and f ∈ H p(T�),

where � is an irreducible symmetric cone, there holds supp f ⊂ �. In [18] Hörman-der proved some results corresponding to the Paley–Wiener Theorem for bandlimitedfunctions involving entire functions in several complex variables. The complete col-lection of Fourier characterization results for Hardy space functions like what havebeen established in one-dimension is missing.

The aim of this paper is to prove the analogous Fourier spectrum characterizations(two-way results), as well as the related Poisson and Cauchy integral representationformulas for Hardy H p(T�) spaces on tubes over a wider class of bases � and for allthe indices p in [1,∞]. For the classical one dimensional case, the following resultsrelated integral representation formulas of Hardy spaces on the half plane are wellknown.

Theorem G ([7], [13]) Let F(x) ∈ L p(R), 1 < p < ∞, and

C(F)(z) = 1

2π i

∫ ∞

−∞F(t)

t − zdt, Im z > 0.

Then C(F)(z) ∈ H p(C+), and there exists a constant C p < ∞ such that

∫ ∞

−∞|C(F)(x + iy)|p dx ≤ C p

∫ ∞

−∞|F(x)|p dx .

Theorem H ([7], [13]) If F(z) ∈ H p(C+), 1 ≤ p < ∞, then F(z) is the Cauchyintegral of its boundary limit F(x), that is,

F(z) = 1

2π i

∫ ∞

−∞F(t)

t − zdt, Im z > 0.

In order to better understand the present paper, we introduce notation and termi-nology as follows. Let B be an open subset of R

n. Then the tube TB in Rn × iRn = C

n

with base B is the subset

TB = {z = x + iy ∈ C

n : x ∈ Rn, y ∈ B

}.

For example, for n = 1, the classical upper-half plane C+ and the lower-half

plane C− are the tubes in R × iR = C with the bases B+ = {y ∈ R; y > 0} and

B− = {y ∈ R; y < 0}, respectively, i.e., C+ = {z = x + iy : x ∈ R, y > 0} and

C− = {z = x + iy : x ∈ R, y < 0}. Obviously, tubes TB are generalizations of C

+and C

−.


A function F(z) holomorphic in the tube TB is said to belong to the spaceH p(TB), 0 < p < ∞, if

‖F‖H p = sup

{(∫Rn

|F(x + iy)|pdx

) 1p : y ∈ B

}< ∞.

For 1 ≤ p < ∞, the number ‖F‖H p is defined to be the H p-norm of the functionF(z). Under this norm H p(TB), 1 ≤ p < ∞, is a Banach space. For 0 < p < 1,under the distance function d(F, G) = ‖F − G‖p

H p the space H p(TB) becomes acomplete metric space.

When p = ∞, the Hardy space H∞(TB), as a Banach space, is defined by

H∞(TB) = {F : F holomorphic on TB and ‖F‖H∞ = supz∈TB

|F(z)| < ∞}.

For any open and connected subset B of Rn, Stein and Weiss ([12,25,26,28])

studied the space H2(TB) and obtained a fundamental representation theorem ([26])restricted to only H2(TB). They commented that the theory of H2(TB) becomes richerif the bases B are with more restrictions. In this paper, base on the existing results,we will continue to study the theory of the spaces H p(T�), 1 ≤ p ≤ ∞, on tubesover open cones �. What we call open cones are those satisfying the following twoconditions:

(1) 0 does not belong to �;(2) For any x, y ∈ �, and any α, β > 0, there holds αx + βy ∈ �.

We note that a cone � is a convex set. In this paper, we will mainly study the H p(T�)

spaces on tubes over open cones.The dual cone of �, denoted by �∗, is defined as

�∗ = {y ∈ Rn : y · x ≥ 0, for any x ∈ �}.

A cone � is said to be regular if the interior of its dual cone �∗ is nonempty.For instance, when n = 1, there are only two open cones, viz., the open half-lines{x ∈ R : x > 0} and {x ∈ R : x < 0}. Their dual cones are the closed half-lines{x ∈ R : x ≥ 0} and {x ∈ R : x ≥ 0}, both having non-empty interiors. When n = 2,the open cones are the angular regions between two rays meeting at the origin. Sucha cone is regular if and only if the corresponding angle is strictly less than π . The firstoctant of R

n is a particular open convex regular cone whose dual cone is the closureof itself. There are two particular types of regular cones, that is, polygonal cones andcircular cones. They, too, receive great attentions from researchers ([3–5]). Part of theresults on symmetric cones can be found in the book of [10] and the lecture notes [1].

An open subset � ofRn is called a polygonal cone if it is the interior of the convex hull

of a finite number of rays meeting at the origin among which one can find at least nsides that are linearly independent.We call a polygonal cone an n-sided polygonal coneif the minimum number of linearly independent rays convex-spanning � is exactly n.


The set

�c ={(x, xn) ∈ R

n : |x | ≤ cxn, x ∈ Rn−1, c > 0

},

is called a canonical circular cone. If through a rotation � can become identical witha canonical circular cone, then � is called a circular cone.

There are two important kernels related to the Hardy spaces H p(T�), viz., theCauchy kernels and the Poisson kernels ([2], [5], [26], and [14]). Below we recall thedefinitions and main properties of the Cauchy and the Poisson kernels.

If � is a regular open convex cone of Rn , then the Cauchy kernel and the Poisson

kernel of T� are, respectively, the functions defined by

K (z) =∫

�∗e2π i z·t dt and P(x, y) = |K (z)|2/K (2iy), (1)

where z = x + iy ∈ T� ([26]).As mentioned previously, if � is the first octant of R

n , then its dual cone is theclosure of �, namely, �∗ = �. The Cauchy kernel ([26]) associated with the firstoctant of R

n can be computed explicitly, being identical with the tensor product of theone-dimensional Cauchy kernel as follows,

K (z) =n∏

j=1

−1

2π i z j, z ∈ T�, (2)

and the Poisson kernel of the tube T� is the tensor product of n copies of the one-dimensional Poisson kernel in the upper-half plane:

P(x, y) = |K (z)|2/K (2iy) =n∏

j=1

1

π

y j

x2j + y2j, z ∈ T�. (3)

Let F ∈ (S(Rn))′ be a tempered distribution. If there exists a function f (x + iy)

analytic in T� such that for any ϕ in the Schwarz class S(Rn),

F(ϕ) = ( f, ϕ) = limy∈�, y→0

∫Rn

f (x + iy)ϕ(x)dx,

then we say that F is a holomorphic distribution and f (x + iy) is an analytic repre-sentation of F . We note that here and below the self-explanatory notation lim

y∈�, y→0means the restricted limit along �.

Let f ∈ L1(Rn). The Fourier transform of f, denoted f , is defined by

f (x) =∫Rn

f (t)e−2π i x ·t dt

for all x ∈ Rn .


Let T be any tempered distribution. If (T, ϕ) = 0, for any ϕ ∈ S(Rn) withsupp ϕ ⊂ C , where C is a closed subset of R

n , then we say that the distributionalsupport of T, denoted as d-supp T, is contained in Rn\C, or, d-supp T ⊂ Rn\C .

In what following, the second section of this paper states our main results. The thirdsection contains main technical Lemmas. The last section provides proofs of the maintheorems.

2 Main Results

We note that the following results have been announced, without proofs, in our recentsurvey paper [8]. Theorems 2.1–2.4 given below are higher-dimensional generaliza-tions of the Paley–Wiener Theorem to p �= 2.

Theorem 2.1 Let � be a regular open cone in Rn and F(x) ∈ L p(Rn), 1 ≤ p ≤ 2.

Then F(x) is the boundary limit function of some F(x + iy) ∈ H p(T�) if and only ifsuppF ⊂ �∗. Moreover, if the condition is satisfied, then

F(z) =∫Rn

X�∗(t)e2π i z·t F(t) dt =∫Rn

F(t)K (z − t) dt =∫Rn

P(x − t, y)F(t) dt,

(4)where K (z) and P(x, y) are, respectively, the Cauchy kernel and the Poisson kernelfor the tube T�.

Theorem 2.2 Let � be a regular open cone in Rn, and F(z) ∈ H p(T�), 2 < p ≤ ∞.

Then F(x), as the boundary limit of F(z), is the tempered holomorphic distributionrepresented by the function F(z). Moreover, d-suppF ⊂ �∗, that is (F, ϕ) = 0 for allϕ ∈ S(Rn) with suppϕ ⊂ Rn\�∗.

In order to prove the main Theorem 2.4, we have to prove first Theorem 2.3, whichitself is an important result in the Hardy space theory.

Theorem 2.3 Let � be a regular open cone in Rn. Suppose that F(z) ∈ H p(T�), 1 ≤

p ≤ ∞, and F(x) is the boundary limit function of the function F(z). If F(x) ∈Lq(Rn), 1 ≤ q ≤ ∞, then F(z) ∈ Hq(T�).

The following converse result of Theorem 2.2 holds.

Theorem 2.4 Let � be a regular open cone in Rn. If F ∈ L p(Rn), 2 < p ≤ ∞, and

(F, ϕ) = 0, for all ϕ ∈ S(Rn) with suppϕ ⊂ Rn\�∗, then F is the boundary value ofa function in H p(T�).

Theorems 2.2 and 2.4 show that we also have the spectrum characterization for func-tions in the Hardy spaces H p(T�), 2 < p ≤ ∞.

We note that the proofs of Theorem 2.4 and the L1-results of Theorem 2.1 use somemethodologies different from those for the analogous results in the one-dimensioncase given in [21]. In the latter one uses the Calderón-Zygmund decomposition forL1-functions, while in the present paper we take advantage of the Poisson integral


representation together with the “lifting up” method for the Hardy space functions(see [26] and [7]).

Up to this point, we have considered H p(T�) spaces on tubes over open cones.In what follows, we will restrict ourselves to consider tubes over the first octant ofR

n, and more generally, tubes over the polygonal cones, and obtain more delicateresults. In particular, for the first octant and the polygonal cones the counterparts ofTheorems G and H are obtained for higher dimensions.

Theorem 2.5 Let K (z) be the Cauchy kernel associated with the tube T� , where � isthe first octant of R

n, F(x) ∈ L p(Rn), 1 < p < ∞, and

C(F)(z) =∫Rn

F(t)K (z − t) dt.

Then C(F)(z) ∈ H p(T�), and there exists a finite constant C p, depending only on pand the dimension n such that

∫Rn

|C(F)(x + iy)|p dx ≤ C p

∫Rn

|F(x)|p dx .

Theorem 2.5 has a counterpart for tubes over the polygonal cones, as given in thefollowing Theorem 2.5*.

Theorem 2.5* Let K (z) be the Cauchy kernel for the tube domain T� , where � is apolygonal cone in R

n , F(x) ∈ L p(Rn), 1 < p < ∞, and

C(F)(z) =∫Rn

F(t)K (z − t) dt.

Then C(F)(z) ∈ H p(T�), and there exists a finite number N depending on �, and aconstant C p, the same as in Theorem 2.5, such that

∫Rn

|C(F)(x + iy)|p dx ≤ NC p

∫Rn

|F(x)|p dx .

Theorem 2.5 does not hold for general cones except for p = 2.We recall the famousresult of Fefferman ([11]) asserting that the characteristic function of the unit ball inR

n is not a bounded Fourier multiplier for L p(Rn)when p �= 2. This result, combinedwith the loose theorem of Stein ([27]) and one by de Leeuw ([6], [19]), implies thatthe counterpart result of Theorem 2.5 in the circular-cone case do not hold for p �= 2.

Next, we have the following result.

Theorem 2.6 Let � be the fist octant of Rn. If F(z) ∈ H p(T�), 1 ≤ p < ∞, then

F(z) is the Cauchy integral of its boundary limit F(x), that is,

F(z) =∫Rn

F(t)K (z − t) dt, z ∈ T�, (5)

where K (z) is the Cauchy kernel of the tube T� .


Like Theorem 2.5*, Theorem 2.6 may be extended to the polygonal case:

Theorem 2.6* Let � be a polygonal cone in Rn . If F(z) ∈ H p(T�), 1 ≤ p < ∞,

then F(z) is the Cauchy integral of its boundary limit F(x), that is,

F(z) =∫Rn

F(t)K (z − t) dt, z ∈ T�, (6)

where K (z) is the Cauchy kernel of the tube T� .We note that, when 1 ≤ p ≤ 2, as concluded in Theorem 2.1, the Cauchy integral

representation formulas of the Hardy space functions given in Theorem 2.6, in fact,are valid for more general cones other than the polygonal ones, since they, in fact, arevalid for at least all regular open cones. In the case 2 < p < ∞, however, we donot know whether the same representation formula holds for a general regular opencone. We are aware of some Cauchy integral representation results for holomorphicfunctions in several complex variables ([16,22]). For the Hardy space functions ourresults for the polygonal cones and the range 2 < p < ∞ are new.

The results of this study are summarized as follows.

Theorem 2.7 Let � be a regular open cone in Rn, and F(x) ∈ L p(Rn), 1 ≤ p ≤ ∞.

Then F(x) is the restricted boundary limit function of some F(x + iy) ∈ H p(T�) ifand only if d-supp F ⊂ �∗. In such case, we have

(1) for 1 ≤ p ≤ 2, F is locally integrable, and

F(z) =∫Rn

X�∗(t)e2π i z·t F(t) dt =∫Rn

F(t)K (z − t) dt;

(2) for 1 ≤ p ≤ ∞,

F(z) =∫Rn

P(x − t, y)F(t) dt.

Moreover, if � is the first octant of Rn, or a polygonal cone, we have

(3) for 2 < p < ∞,

F(z) =∫Rn

F(t)K (z − t) dt,

where K (z) and P(x, y) are, respectively, the Cauchy and the Poisson kernelsassociated with the tube T�.

Theorem 2.8 Let K (z) be the Cauchy kernel associated with the tube T� , where �

is a polygonal cone in Rn. Then F(z) ∈ H p(T�), 1 < p < ∞, if and only of there

exists G(x) ∈ L p(Rn) such that

F(z) = C(G)(z) =∫Rn

G(t)K (z − t) dt,


and there exists a finite constant C p, depending only on p and the dimension n suchthat

∫Rn

|F(x + iy)|p dx ≤ C p

∫Rn

|G(x)|p dx .

In the “only if” part, the range of p can be extended to 1 ≤ p < ∞, and G(x) canbe taken as the restricted boundary limit F(x) of the Hardy space function F(z).

3 Main Technical Lemmas

In order to prove our theorems, we need the following lemmas.

Lemma 3.1 ([1], p. 103) Let � be an open convex cone. Suppose F ∈ H p(T�), 0 <

p ≤ ∞,and�0 ⊂ � satisfies d(�0, �c) = inf{|y1−y2|; y1 ∈ �0, y2 ∈ �c} ≥ ε > 0,then there exists a constant C = C(ε), depending on (ε, n) but not on F, such that

supz∈T�0

|F(z)| ≤ C‖F‖H p .

Lemma 3.2 ([1], p. 105) If 1 ≤ q ≤ ∞, then Py(x) = P(x, y) ∈ Lq(Rn) for eachy ∈ �, where � is an open convex cone and P(x, y) is the Poisson kernel associatedwith the tube T� .

Lemma 3.3 If f ∈ L p(Rn), ϕ ∈ L p(Rn) ∩ L1(Rn), 1 ≤ p ≤ 2, then

∫Rn

f (x)ϕ(x) dx =∫Rn

f (x)ϕ(x) dx .

As generalization of a corresponding result in [26], we have

Lemma 3.4 Let F(z) ∈ H p(T�n ), 1 ≤ p ≤ ∞, where �n denotes the first octant ofR

n. Then, for any n = n1 + n2, 1 ≤ n1, n2 < n, and fixed (zn1+1, . . . , zn) ∈ T�n2,

we have

F(. . . , zn1+1, . . . , zn) ∈ H p(T�n1).

Lemma 3.5 ([5]) Let � be a polygonal cone, K (z) the Cauchy kernel associated withT� . Then Ky(x) = K (x + iy) ∈ L p(Rn), p > 1, for any y ∈ �, and there holds:

‖Ky‖p ≤ C(n, p, �)K (iy)1−1/p.

Lemma 3.6 (1) If � is an n−sided polygonal cone in Rn, then its dual cone �∗ is an

n−sided polygonal cone in Rn.


(2) If � is a polygonal cone in Rn, then � = ⋃N

j=1 � j , � j ∩ �k = ∅, j �= k, whereeach � j is an n-sided polygonal cone, j = 1, 2, . . . , N. The dual cone �∗ is alsoa polygonal cone in R

n . Moreover, �∗ = ⋂Nj=1 �∗

j , where �∗j is the dual cone of

� j , j = 1, 2, . . . , N .

Proofs of the lemmas are easy exercises ([10]).

4 Proofs of Theorems

4.1 Proof of Theorem 2.1

Proof To prove the “only if” part of Theorem 2.1, we need to show that, if F(x +iy) ∈H p(T�), then d-suppF ⊂ �∗.

Let 1 ≤ p ≤ 2. For any fixed y0 ∈ �,

Fy0(z) = F(z + iy0), �0 = {y + y0 : y ∈ �} ⊂ �.

Obviously, �0 ⊂ �, and, there exists ε > 0 such that

d(�0, �c) = inf{|y1 − y2|; y1 ∈ �0, y2 ∈ �c} ≥ ε > 0.

Then by Lemma 3.1, there exists a positive constant C = C(y0, n) such that

supz∈T�

|Fy0(z)| ≤ C‖F‖H p < ∞. (7)

From the last inequality, we have, for any y ∈ �,

∫Rn

|Fy0(x + iy)|2 dx =∫Rn

|Fy0(x + iy)|p|Fy0(x + iy)|2−p dx

≤∫Rn

|Fy0(x + iy)|p(C‖F‖H p )2−p dx

≤ (C‖F‖H p )2−p‖Fy0‖pp < ∞,

so that,

Fy0(z) ∈ H2(T�).

The Paley–Wiener Theorem for the H2-functions (Theorem E) implies that

suppFy0 ⊂ �∗. (8)

When p = 1, the relation for L1 functions holds

‖F‖∞ ≤ ‖F‖1.


When 1 < p ≤ 2, by Hausdorff–Young’s inequality, for 1q + 1

p = 1, we have

||F ||q ≤ ||F ||p.

Hence, for 1 ≤ p ≤ 2, by invoking Theorem E and the above two inequalities, wehave

||Fy0 − F ||q ≤ ||Fy0 − F ||p → 0, y0 → 0, (9)

where 1q + 1

p = 1.

Due to (8) and (9), for any ϕ ∈ S(Rn) satisfying suppϕ ⊂ Rn\�∗, we have∣∣∣∣∫Rn

F(x)ϕ(x) dx

∣∣∣∣ =∣∣∣∣∫Rn

(F(x) − Fy0(x))ϕ(x) dx +∫Rn

Fy0(x)ϕ(x) dx

∣∣∣∣=

∣∣∣∣∫Rn

(F(x) − Fy0(x))ϕ(x) dx

∣∣∣∣≤ ||F − Fy0 ||q ||ϕ||p → 0, y0 → 0.

This implies that d-suppF ⊂ �∗.Nowwe turn to prove the “if part” of Theorem 2.1. We have to show that if F ∈ L p

and d-suppF ⊂ �∗, then the relation (4) holds, and, as consequence, F(x + iy) ∈H p(T�).

Let F(x) ∈ L p, 1 ≤ p ≤ 2, and d-suppF ⊂ �∗. First, we can easily show that forany fixed y ∈ � there exists a δ = δy > 0 such that y · t ≥ δ|t | for all t in �∗ (see[26]).

As consequence, for any fixed y ∈ �,

|X�∗(t)e2π i z·t F(t)| = X�∗(t)|F(t)|e−2πy·t

≤ X�∗(t)|F(t)|e−2πδ|t | ∈ L1(Rn).

Restricted to a neighborhood of y ∈ �, the Lebesgue Dominated Convergence Theo-rem can be used to conclude that the function

G(z) =∫Rn

e2π i z·t F(t) dt =∫Rn

X�∗(t)e2π i z·t F(t) dt (10)

is holomorphic at y. Therefore, G is holomorphic in the whole T� .Next, we show G(z) ∈ H p(T�). Fix z ∈ T�. Let

gz(t) = X�∗(t)e2π i z·t , gz(t) = gz(−t).

Then gz ∈ L1(Rn)⋂

L2(Rn). By (1), gz(s) = K (z − s).For z, w ∈ T� , let

I (z, w) =∫Rn

K (z − t)F(t)K (t + w) dt. (11)


Then

I (z, w) =∫Rn

gz(t)F(t)gw(−t) dt.

Since, for z, w ∈ T� , gz(t)gw(−t) = gz ∗ gw(t), where

(gz ∗ gw)(t) =∫Rn

gz(ξ)gw(t − ξ)dξ =∫Rn

gz(ξ)gw(ξ − t) dξ

=∫Rn

X�∗(ξ)e2π i z·ξX�∗(ξ − t)e2π iw·(ξ−t) dξ,

we have

I (z, w) =∫Rn

F(s)X�(s)(gz ∗ gw)(s) ds

=∫Rn

F(s)X�∗(s)∫Rn

X�∗(ξ)e2π i z·ξX�∗(ξ − s)e2π iw·(ξ−s) dξds.

Let t = ξ −s.By (1), as well as the relationX�∗(t)X�∗(t +s)X�∗(s) = X�∗(t)X�∗(s),the last expression for I (z, w) coincides with

I (z, w) =∫Rn

F(s)X�∗(s)∫Rn

X�(t)X�(t + s)e2π i z·(s+t)e2π iw·t dtds

=∫Rn

F(s)X�∗(s)e2π i z·s∫Rn

X�∗(t)e2π i(z+w)·t dtds

= G(z)K (z + w).

For z ∈ T� , we have −z ∈ T� , and

I (z,−z) = G(z)K (z − z) = G(z)K (2iy). (12)

Hence, by the formulas (11) and (12), we have

G(z) =∫Rn

F(t)K (z − t)K (t − z)

K (2iy)dt =

∫Rn

F(t)P(x − t, y) dt, (13)

where Py = P(·, y) is the Poisson kernel associated with the tube T� . Due to (13), byinvoking the corresponding result in [26] (Theorem 2.1 in Chapter 1) we have

supy∈�

∫Rn

|G(x + iy)|p dx < ∞.

G(z) proven before, we have G(x + iy) ∈ H p(T�).


By Theorem E, the above formula (13) assures that the limit of G(z) as y → 0is F(x). Defining F(z) = G(z), we therefore have a function in the Hardy spaceH p(T�) having F(x) as its restricted boundary limit.

Moreover, by Lemma 3.3 and (10), we have

F(z) = G(z) =∫Rn

X�∗(t)e2π i z·t F(t) dt

=∫Rn

(X�∗(·)e2π i z·(·)) (t)F(t) dt

=∫Rn

F(t)K (z − t) dt.

The proof of Theorem 2.1 is complete. �


Proof We are to prove that if F(z) ∈ H p(T�), 2 < p ≤ ∞, then F(x), as theboundary limit of function F(z), is the tempered holomorphic distribution representedby F(z), and, moreover, d-suppF ⊂ �∗.

For 2 < p < ∞, by Holder’s inequality and (b) of Theorem F, for any ϕ ∈ S(Rn),

we have

|〈Fy, ϕ〉 − 〈F, ϕ〉| = |∫Rn

(F(x + iy) − F(x))ϕ(x) dx |≤ ||Fy − F ||p||ϕ||q → 0, y → 0,

where Fy(x) = F(x + iy), 1p + 1

q = 1.For p = ∞, under the same notation, by (a) and (c) of Theorem F and the Lebesgue

Dominated Convergence Theorem, we have the same result. Therefore, F(x) is thetempered holomorphic distribution represented by F(z) ∈ H p(T�).

By using the same method as in the proof of the inequality (7), we have

|Fy0(z)| ≤ C‖F‖p < ∞, (14)

where C is a constant only depending on n and y0.Let φ be a continuous function with compact support in �∗ satisfying∫

Rn φ(t) dt = 1. Set

ψ(z) =∫Rn

e2π i z·tφ(t) dt,

and ψδ(z) = ψ(δz), δ > 0.By the definition ofψ(z), and taking into account suppφ ⊂ �∗, Theorem E assures

that ψ ∈ H2(T�).Consequently, due to the boundedness of Fy0 given by (14), we have, for all δ > 0,

Fy0(z)ψδ(z) ∈ H2(T�). (15)


For z ∈ T�, the Lebesgue Dominated Convergence Theorem implies

limδ→0

ψδ(z) = limδ→0

∫Rn

e2π iδz·tφ(t) dt = 1.

Moreover, |ψδ(z)| ≤ M1,where M1 is a constant. In fact, for y ∈ �, by supp φ ⊂ �∗,we have

|ψδ(z)| ≤∫Rn

|e2π iδz·tφ(t)| dt =∫Rn

e−2πδy·t |φ(t)| dt

≤∫Rn

|φ(t)| dt ≤ M1 < ∞. (16)

Together with (14), this gives

|ψδ(z)Fy0(z)ϕ(t)| ≤ M |ϕ(t)| ∈ L1,

where M = M1C‖ f ‖p is a constant, and ϕ ∈ S(Rn) with suppϕ ⊂ Rn\�∗. Therelation

limδ→0

ψδ(z)Fy0(z)ϕ(t) = Fy0(z)ϕ(t) (17)

implies, by invoking the Lebesgue Dominated Convergence Theorem,

limδ→0

∫Rn

ψδ(t + iy0)Fy0(t)ϕ(t) dt =∫Rn

Fy0(t)ϕ(t) dt.

Therefore,

(F, ϕ) = (F, ϕ) = limy0→0

∫Rn

F(t + iy0)ϕ(t) dt

= limy0→0

limδ→0

∫Rn

ψδ(t + iy0)Fy0(t)ϕ(t) dt

= limy0→0

limδ→0

∫Rn

(ψδ(· + iy0)Fy0(·))(t)ϕ(t) dt. (18)

For y0 ∈ �, ψδ(x)Fy0(x) is the boundary limit of Fy0(z)ψδ(z) ∈ H2(T�) [byformula (15)]. Theorem D then assures that the distributional support of the Fouriertransform of ψδ(x)Fy0(x) is contained in �∗, that is,

supp(ψδ(·)Fy0(·)) ⊂ �∗. (19)

From (18), (19), and the condition suppϕ ⊂ Rn\�∗, we obtain

(F, ϕ) = 0.




Proof To prove Theorem 2.3 is to show that if F(z) ∈ H p(T�), 1 ≤ p ≤ ∞, andF(x) ∈ Lq(Rn), 1 ≤ q ≤ ∞, where F(x) is the boundary limit of F(z), thenF(z) ∈ Hq(T�).

Let F(z) ∈ H p(T�), 1 ≤ p ≤ ∞. By Theorem E, we have

F(z) =∫Rn

P(x − t, y)F(t) dt = (F ∗ Py)(x),

for all z = x + iy ∈ T� , where F(x) = limy→0, y∈�

F(x + iy), Py(x) = P(x, y) is the

Poisson kernel of the tube T�.

With the notation Fy(x) = F(x + iy), we have

‖Fy‖q = ‖F ∗ Py‖q ≤ ‖F‖q‖Py‖1 = ‖F‖q < ∞,

for all y ∈ �. Thus,

supy∈�

∫Rn

|F(x + iy)|q dx < ∞.

Since F(z) is analytic in T�, we have F(z) ∈ Hq(T�). �


Proof We have to show that if F ∈ L p(Rn), 2 < p ≤ ∞, and (F, ϕ) = 0 for allϕ ∈ S(Rn) with suppϕ ⊂ Rn\�∗, then F(x) is the boundary limit of a function F(z)in H p(T�).

We recall that if F ∈ L p(Rn), 2 < p ≤ ∞, then F is a tempered distribution TF

defined through

(F, ϕ) =∫Rn

F(t)ϕ(t) dt

for ϕ in the Schwartz class S(Rn). The Fourier transform of F viewed as a tempereddistribution is also a tempered distribution, F , given by

(F, ϕ) = (F, ϕ)

for ϕ in the Schwartz class S(Rn).Let φ ∈ S(Rn) be a continuous function with compact support in �∗ that satisfies∫

Rn φ(t)dt = 1. Set

ψ(z) =∫Rn

e2π i z·tφ(t) dt.


Then ψ(x) ∈ S(Rn) and φ(−x) = ψ(x). For δ > 0, let ψδ(z) = ψ(δz). Theorem Dassures that

ψδ(z) ∈ H2(T�). (20)

Moreover, ψδ(x) = φ(−δx) ∈ S(Rn) and

ψδ(x) ∈ Lq(Rn) (21)

for all q ≥ 1. Therefore, for 2 ≤ p < ∞, by Holder’s inequality, for 1p/2 + 1

q ′ = 1,we have

∫Rn

|F(x)ψδ(x)|2dx ≤(∫

Rn(|F(x)|2)p/2 dx

)2/p

‖ψδ‖22q ′ < ∞.

Hence,

F(x)ψδ(x) ∈ L2(Rn).

For p = ∞, the facts that F(x) ∈ L∞ and ψδ(x) ∈ Lq(Rn) for all q ≥ 1 show that

F(x)ψδ(x) ∈ L2(Rn).

Under the simplified notation h(x) = ψδ(x), we have

h(−s) = 1

δnφ

(−s

δ

)= 1

δnφ

(−s

δ

)X�∗(−s).

Now,

(Fh, ϕ) = (Fh, ϕ) =∫Rn

F(x)h(x)ϕ(x) dx

=∫Rn

F(x)h ∗ ϕ(x) dx

= (F,h ∗ ϕ) = (F, h ∗ ϕ), (22)

where

(h ∗ ϕ)(t) =∫Rn

h(−s)ϕ(t − s)X�∗(−s)XRn\�∗(t − s) ds. (23)

Noting that t ∈ �∗ and −s ∈ �∗ imply that t − s ∈ �∗, the relation (23) gives thatt ∈ �∗ implies (h ∗ ϕ)(t) = 0, viz.,

supp(h ∗ ϕ) ⊂ Rn\�∗. (24)

The condition d-suppF ⊂ �∗, being combined with (22), (24), implies

(Fh, ϕ) = 0.


Due to the proven fact that F(t)h(t) ∈ L2(Rn), Theorem 2.1 forces that F(t)h(t) =F(t)ψδ(t) be the boundary limit of a function, say, Gδ(z), in H2(T�), viz.,

Gδ(x) = limy∈�, y→0

Gδ(x + iy) = F(x)ψδ(x),

where, in particularly, Gδ(z) can be written as

Gδ(z) =∫Rn

P(x − t, y)F(t)ψδ(t) dt.

Since ψδ ∈ L∞(Rn), the assumption F(x) ∈ L p(Rn) implies that

Gδ(x) = F(x)ψδ(x) ∈ L p(Rn).

Therefore, from Theorem 2.3,

Gδ(z) ∈ H p(T�).

Let

G(z) =∫Rn

P(x − t, y)F(t) dt,

which implies that

supy∈�

∫Rn

|G(x + iy)|p dt ≤ ‖F‖pp.

Wefurther show thatG(z) is holomorphic inT� .Onceweachieve that,we can concludethat G(z) ∈ H p(T�). Now, since G(z) is harmonic, we denote the left-hand-side ofthe above inequality by ‖G‖H p .

Let z0 = x0 + iy0 ∈ T�0 be a fixed point, where �0 ⊂ �, satisfying

d(�0, �c) = inf{|y1 − y2|; y1 ∈ �0, y2 ∈ �c} ≥ ε > 0.

let Sε = {z ∈ Cn; |z−z0| < ε}. If�ε = {z ∈ R

n; |y−y0| < ε}, then Sε ⊂ T�ε ⊂ T� .Since Gδ(z) and G(z) both are harmonic functions, |Gδ(z) − G(z)|p is subharmonicas a function of 2n real variables ([26]). Therefore, we have

|Gδ(z0) − G(z0)|p ≤ 1

�2nε2n

∫Sε

|Gδ(x + iy) − G(x + iy)|p dxdy

≤ 1

�2nε2n

∫T�ε



≤ 1

�2nε2n

∫�ε

∫Rn


≤ �n

�2nεn‖Gδ − G‖h p ≤ �n

�2nεn‖F(ψδ − 1)‖p

L p . (25)

From the above inequality, and together with ‖F(ψδ −1)‖L p → 0, δ → 0, we have

|Gδ(z0) − G(z0)|p → 0, δ → 0. (26)

The above estimate (25), in fact, is valid for z in a neighborhood of z0.Hence the limit(26) is uniform in a neighborhood of z0. Since Gδ is holomorphic in T�, we concludethat G(z) is holomorphic in T� . The proof is complete. �


Proof The proof of Theorem 2.5 amounts to showing that the Cauchy integral of afunction in L p(Rn) is a function in H p(T�), 1 < p < ∞.

Let n = n1+n2, 1 ≤ n1, n2 < n, and fix (tn1+1, . . . , tn) ∈ Rn2 . Nowwe consider

the function f of n1 real variables defined by

f (x1, x2 . . . , xn1) = F(x1, . . . , xn1 , tn1+1, . . . , tn),

where (x1, x2 . . . , xn1) ∈ Rn1 .

The Fubini Theorem ensures that f (x1, x2 . . . , xn1) belongs to L p(Rn1) for almostall (tn1+1, . . . , tn) ∈ R

n2 . Below we will prove Theorem 2.5 by mathematical induc-tion. When n = 1, by Theorem G, Theorem 2.5 holds.

Next, we assume n > 1. The induction hypothesis is that Theorem 2.5 holds forn −1. Take n1 = n −1. Since f (x1, x2 . . . , xn−1) ∈ L p(Rn−1),we have, as the n −1dimensional Cauchy integral,

Cn−1( f )(z1, . . . , zn−1)

= 1

(2π i)n−1

∫Rn−1

f (t1, . . . , tn−1)∏n−1j=1(t j − z j )

dt1 · · · dtn−1 ∈ H p(T�n−1),

where �n−1 is the first octant of Rn−1. In an alternative notation, we have,

Cn−1(F)(z1, . . . , zn−1, tn)

= 1

(2π i)n−1

∫Rn−1

F(t1, . . . , tn)∏n−1j=1(t j − z j )

dt1 · · · dtn−1 ∈ H p(T�n−1), (27)

for (z1, . . . , zn−1) ∈ T�n−1 and a fixed tn ∈ R.


By the induction hypothesis, there exists a finite constant Cn−1, depending only onn − 1 and p, such that

∫Rn−1

|Cn−1(F)(x1 + iy1, . . . , xn−1 + iyn−1, tn)|p dx1 · · · dxn−1

≤ Cn−1

∫Rn−1

|F(x1, . . . , xn−1, tn)|p dx1 · · · dxn−1. (28)

Since F(x) ∈ L p(Rn), we have

∫Rn

|Cn−1(F)(x1 + iyi , . . . , xn−1 + iyn−1, tn)|p dx1 · · · dxn−1dtn

≤ Cn−1

∫Rn

|F(x1, . . . , xn−1, tn)|p dx1 · · · dxn−1dtn < ∞.

This implies thatCn−1(F)(x1+iy1, . . . , xn−1+iyn−1, tn) ∈ L p(Rn).As consequence,Cn−1(F)(x1 + iy1, . . . , xn−1 + iyn−1, tn) ∈ L p(R) as a function of one variable tn foralmost all (x1 + iy1, . . . , xn−1 + iyn−1) ∈ T�n−1 .

Using Theorem G again, we obtain, by fixing (z1, . . . , zn−1) ∈ T�n−1 ,

C(F)(z1, . . . , zn) = C1(Cn−1F)(z1, . . . , zn−1, zn)

= 1

2π i

∫R

Cn−1(F)(z1, . . . , zn−1, tn)

tn − zndtn

= 1

(2π i)n

∫Rn

F(t1, . . . , tn)∏nj=1(t j − z j )

dt1 · · · dtn

=∫Rn

F(t)K (z − t) dt ∈ H p(C+) (29)

for zn ∈ C+, and

∫R

|C1(Cn−1F)(z1, . . . , zn−1, xn + iyn)|p dxn

≤ C1

∫R

|Cn−1(F)(z1, . . . , zn−1, xn)|p dxn . (30)

Therefore, by (28) and (30), we have

∫Rn

|C(F)(x + iy)|p dx

=∫Rn−1

(∫R

|C1(Cn−1F)(x1 + iy1, . . . , xn + iyn)|p dxn

)dx1 · · · dxn−1

≤ C1

∫R

(∫Rn−1

|Cn−1(F)(z1, . . . , zn−1, xn)|p dx1 · · · dxn−1

)dxn


≤ C1

∫R

(Cn−1

∫Rn−1

|F(x1, . . . , xn−1, xn)|p dx1 · · · dxn−1

)dxn

= Cn‖F‖p < ∞, (31)

where Cn = C1Cn−1 depending only on n and p. Thus, C(F)(z1, . . . , zn) ∈ H p(T�).


4.6 Proof of Theorem 2.5∗

Proof If � is a polygonal cone in Rn , then, according to Lemma 3.6, the dual cone

�∗ is also a polygonal cone in Rn . Applying Lemma 3.6 (2) to the polygonal cone

�∗, �∗ can be decomposed into the union of a finite number of n-sided polygonalcones �∗

j , j = 1, 2, . . . , N , with disjoint interior parts. Using such decomposition,we denote

K�(z) =N∑

j=1

K� j (z),

where � j is the interior of (�∗j )

∗, j = 1, 2, . . . , N , being also n−sided polygonalcones (by Lemma 3.6). K�, in fact, is the Cauchy kernel associated with the tube T�.

This can be seen from the general definition of the Cauchy kernel:

K�(z) =∫

�∗e2π i z·t dt =

∫⋃N

j=1 �∗j

e2π i z·t dt

=N∑

j=1

∫�∗

j

e2π i z·t dt =N∑

j=1

K� j (z). (32)

Moreover, each � j , j = 1, 2, . . . , N , can be mapped onto the first octant by alinear transformation ([26], [23]). Now we work out the linear transformation. Each� j is the interior of the convex hull of n linearly independent rays meeting at theorigin. By selecting vectors η j1, η j2 , . . . , η jn as the unit vectors in the directions ofthose rays, the set � j is expressed as

� j = {λ1η j1 + λ2η j2 + · · · + λnη jn ∈ Rn; λ1 ≥ 0, . . . , λn ≥ 0}.

For each j, consider the linear transformations A j , j = 1, 2, . . . , N , that maps thestandard basis vectors ek onto η jk , k = 1, 2, . . . , n. Then A j maps the first octant �0onto the cone � j . That is, � j = A j�0. We further extend the mapping A j to T�0 bydefining A−1

j (z) = x + i A−1j y ∈ T�0 , where z = x + iy, y ∈ � j , j = 1, 2, . . . , N .

Let {η∗j1, . . . , η∗

jn} be the dual base of {η j1, . . . , η jn } such that (η jl , η

∗jk) = δkl ,

where δkl = 1 if l = k; and otherwise zero. This implies that {η∗j1, . . . , η∗

jn} is the base

of �∗j , the dual cone of � j . Let B j be the linear transformation mapping the standard


basis vectors ek onto η∗jk, k = 1, 2, . . . , n, or �∗

j = B j�0, j = 1, 2, . . . , N . Thenwe have

(η jl , η∗jk ) = (A j el , B j ek) = (A j el)

T (B j ek) = eTl AT

j (B j ek) = δkl ,

where δkl is the same function as above. Therefore, ATj B j = I , or B j = (AT

j )−1.Now, together with (32), we have

K�(z) =N∑

j=1

K� j (z) =N∑

j=1

∫�∗

j

e2π i z·t dt =N∑

j=1

∫B j �0

e2π i z·t dt

=N∑

j=1

∫�0

e2π i z·B j s | det B j | ds =N∑

j=1

∫�0

e2π i z·(ATj )−1s 1

| det A j | ds

=N∑

j=1

∫�0

e2π i(A j )−1z·s 1

| det A j | ds

=N∑

j=1

1

| det A j | K�0(A−1j z).

Thus, for any z ∈ T�,

C(F)(z) =∫Rn

F(t)K�(z − t) dt =∫Rn

F(t)N∑

j=1

1

| det A j | K�0(A−1j z − A−1

j t) dt

=N∑

j=1

∫Rn

F(A j s)K�0(A−1j z − s) ds =

N∑j=1

∫Rn

Fj (s)K�0(A−1j z − s) ds

=N∑

j=1

C0(Fj )(A−1j z) :=

N∑j=1

C0(Fj )(ζ ), (33)

where Fj (x) = F(A j x) ∈ L p(Rn), ζ = A−1j z ∈ T�0 , and

C0(Fj )(ζ ) =∫Rn

Fj (t)K�0(ζ − t) dt.

ByTheorem 2.5,C0(Fj )(ζ ) ∈ H p(T�0), and there exists a finite constantC p, depend-ing only on p and the dimension n such that, for any ζ = ξ + iη = A−1

j z ∈ T�0

∫Rn

|C0(Fj )(ξ + iη)|p dξ ≤ C p‖Fj‖pp.


Those imply that the function C0(Fj )(ζ ) = C0(Fj )(A−1j z) as a function of z ∈ T� j

is holomorphic in tube T� j , j = 1, 2, . . . , N , and

∫Rn

|C0(Fj )(ξ + i A−1j y)|p dξ ≤ C p‖Fj‖p

p.

Therefore, for any z ∈ T�, together with the relation �∗ = ⋃Nj=1 �∗

j , which implies

that � = ∩Nj=1� j (by Lemma 3.6), we get that C(F)(z) is holomorphic in tube T�.

Moreover, by the inequality and (33), we have

(∫Rn

|C(F)(x + iy)|p dx

)1/p

=⎛⎝

∫Rn

∣∣∣∣∣∣N∑

j=1

C0(Fj )(

A−1j (x + iy)

)∣∣∣∣∣∣p

dx

⎞⎠

1/p

≤N∑

j=1

(∫Rn

|C0(Fj )(

A−1j (x + iy)

)|p dx

)1/p

=N∑

j=1

(∫Rn

∣∣∣C0(Fj )(ξ+i A−1

j y)∣∣∣p | det A j | dξ

)1/p

≤N∑

j=1

| det A j |1/pC p

(∫Rn

|Fj (ξ)|p dξ

)1/p

=N∑

j=1

C p

(∫Rn

|F(A jξ)|p| det A j | dξ

)1/p

=N∑

j=1

C p

(∫Rn

|F(x)|p dx

)1/p

= NC p‖F‖L p .

Thus,

supy∈�

∫Rn

|C(F)(x + iy)|p dx ≤ NC p‖F‖pL p < ∞.

Hence, the proof of Theorem 2.5* is complete. �


The Theorem 2.6 asserts that any function in H p(T�) can be represented as the Cauchyintegral of its restricted boundary limit. Below we provide a proof using a densityargument combined with Theorem 2.5.


Proof The assertion for the case 1 ≤ p ≤ 2 is proved in Theorem 2.1. We only needto prove the case 2 < p < ∞.

Let F(z) ∈ H p(T�), 2 < p < ∞, F(x) ∈ L p(Rn) be the boundary limit offunction F(z). We adopt the function ψδ(z) in the proof of Theorem 2.4. We willshow

F(z)ψδ(z) ∈ H2(T�).

In fact, on one hand, F(z) ∈ H p(T�), and (20) imply that F(z)ψδ(z) is holo-morphic in T� . On the other hand, by (20) and (21), Theorem 2.3 shows thatψδ(z) ∈ H2r (T�), 2r > 1. Therefore, for 2 ≤ p < ∞, by Holder’s inequality,for 1

p/2 + 1r = 1, we have

∫Rn

|F(x + iy)ψδ(x + iy)|2dx

≤(∫

Rn(|F(x + iy)|2)p/2 dx

)2/p (∫Rn

|ψδ(x + iy)|2r dx

)1/r

≤ ‖F‖2Hp‖ψδ‖2H2r

. (34)

Therefore,

supy∈T�

∫Rn

|F(x + iy)ψδ(x + iy)|2dx ≤ ‖F‖2Hp‖ψδ‖2H2r

< ∞.

Due to the Cauchy integral representation in the Theorem 2.1, we have

F(z)ψδ(z) =∫Rn

F(t)ψδ(t)K (z − t) dt.

By Theorem E, we have

limδ→0

F(z)ψδ(z) = F(z), z ∈ T�. (35)

Let, for any z ∈ T�,

C(F)(z) =∫Rn

F(t)K (z − t) dt,

where K (z) is the Cauchy kernel associated with the tube T� . Then, by Theorem 2.5,we have C(F)(z) ∈ H p(T�).


Let q satisfy the equation 1p + 1

q = 1. For any fixed z, by Lemma 3.5, we have

|F(z)ψδ(z) − C(F)(z)|≤

∫Rn

|(F(t)ψδ(t) − F(t))K (z − t)| dt

≤ ‖Fψδ − F‖p‖K (z − ·)‖q → 0, δ → 0. (36)

(35) and (36) conclude that G(z) = C(F)(z). This completes the proof. �The proof of Theorem 2.6* is similar to the proof of Theorem 2.6.

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