Let C be a regular cone in ℝ and let TC=ℝ+iC⊂ℂ be a tubular radial domain. Let U be the convolutor in Beurling ultradistributions of Lp-growth corresponding to TC. We define the Cauchy and Poisson integral of U and show that the Cauchy integral of U is analytic in TC and satisfies a growth property. We represent U as the boundary value of a finite sum of suitable analytic functions in tubes by means of the Cauchy integral representation of U. Also we show that the Poisson integral of U corresponding to TC attains U as boundary value in the distributional sense.

1. Introduction

Let C be a regular cone in ℝn and let 𝒪(C) denote its convex envelope. In [1] (or [2]) Carmichael defined the Cauchy and Poisson integrals for Schwartz distributions ∈𝒟Lp′, 1<p≤2, corresponding to tubular domain T𝒪(C)=ℝn+iC⊂ℂn. Carmichael obtained the boundary values of these integrals in the distributional sense on the boundary of T𝒪(C) and found the relation between analytic functions with a specific growth condition in T𝒪(C) and the Cauchy and Poisson integrals of their distributional boundary values. In [3] Pilipović defined ultradistributions 𝒟′((Mp),L2) of L2-growth, where Mp, p=0,1,2,…, is a certain sequence of positive numbers, and studied the Cauchy and Poisson integrals for elements of 𝒟′((Mp),L2) in the case that the Cauchy and Poisson kernel functions are defined corresponding to the first quadrant {y∈ℝn; yj>0, j=1,2,…,n} in ℝn. Pilipović showed that elements in 𝒟′((Mp),L2) are boundary values of suitable analytic functions with a certain L2-norm condition by means of the Cauchy integral representation and an analytic function with a certain L2-norm condition determines, as a boundary value, an element from 𝒟′((Mp),L2). In [4] Carmichael et al. defined ultradistributions of Beurling type 𝒟′((Mp),Ls) of Ls-growth and of Roumieu type 𝒟′({Mp},Ls) of Ls-growth, both of which generalize the Schwartz distributions 𝒟Lp′, and studied the Cauchy and Poisson integrals for elements of both 𝒟′((Mp),Ls) and 𝒟′({Mp},Ls) for 2≤s≤∞ corresponding to the arbitrary tubes TC=ℝn+iC⊂ℂn where C is an open connected cone in ℝn of which the quadrants are special cases. They showed that the Cauchy integral for elements of both 𝒟′((Mp),Ls) and 𝒟′({Mp},Ls) for 2≤s≤∞ corresponding to TC is shown to be analytic in TC, to satisfy a growth property and to obtain an ultradistribution boundary value, which leads to an analytic representation for the ultradistributions. They also showed that the Poisson integrals for elements of both 𝒟′((Mp),Ls) and 𝒟′({Mp},Ls) for 2≤s≤∞ corresponding to TC are shown to have an ultradistribution boundary value. We can find the works of the Cauchy and Poisson integrals for ultradistributions of compact support in [5] and for various kinds of distributions in [2].

In the meantime, Betancor et al. [6] introduced the spaces of Beurling ultradistributions of Lp-growth, 𝒟Lp,(ω)′(1≤p≤∞), of which Schwartz distribution and ultradistributions of Beurling type are special cases. Here ω is a weight function in the sense of [7]. Betancor et al. defined the convolutors in 𝒟Lp,(ω)′, 1≤p≤∞; that is, the functionals U∈𝒟Lp,(ω)′ such that U*ϕ∈𝒟Lp,(ω) for every ϕ∈𝒟Lp,(ω), and studied the convolutors and the surjective convolution operators acting on 𝒟Lp,(ω)′ in [6].

In this paper we define the Cauchy and Poisson integrals for the convolutors in 𝒟Lp,(ω)′, 1≤p≤2. We will show that the Cauchy integrals for the convolutors in 𝒟Lp,(ω)′, 1≤p≤2, are analytic in a tubular domain TC=ℝ+iC⊂ℂ where C is a regular cone in ℝ and satisfies a certain boundedness condition. We will give the representation of the convolutors in 𝒟Lp,(ω), 1≤p≤2, as boundary value of a finite sum of analytic functions in tubes. Also we will show that the Poisson integrals for the convolutor U in 𝒟Lp,(ω), 1≤p≤2, attain U as boundary value in the distributional sense.

Since 𝒟Lp,(ω)′ is the natural generalization of the space 𝒟Lp′ from Lemma 4 (ii) and we can find a weight function κ such that 𝒟′((Mp),Ls)=𝒟Ls,(κ)′ from Remark 3.11 in [8], our results in this paper extend the results in [1, 2, 4] under a condition of U to be convolutor in 𝒟Lp and 𝒟((Mp),Ls), respectively.

In this section we will review Beurling ultradistributions of Lp-growth in ℝ which is introduced by Betancor et al. in [6] and established some of their properties which will be needed later on. Firstly, we will review Beurling ultradistributions which are introduced by Braun et al. in [7].

Definition 1.

A weight function is an increasing continuous function ω:[0,∞)→[0,∞) with the following properties:

there exists L≥0 with ω(2t)≤L(ω(t)+1) for all t≥0,

∫1∞(ω(t)/t2)dt<∞,

log(t)=o(ω(t)) as t tends to ∞,

ψ:t→ω(et) is convex.

By (β),

Lemma 2.

Consider the following:
(1)limt→∞ω(t)t=0.

By (δ), ψ(0)=0 and limx→∞x/ψ(x)=0. Then we can define the Young conjugate ψ* of ψ by
(2)ψ*:[0,∞)⟶ℝ,ψ*(y)=supx≥0(xy-ψ(x)).

Obviously we have the following:

Lemma 3 (see [<xref ref-type="bibr" rid="B3">7</xref>]).

(i) ψ* is convex and increasing and satisfies ψ*(0)=0.

(ii) ψ*(y)/y is increasing and limy→∞y/ψ*(y)=0.

(iii) (ψ*)*=ψ.

(iv) There exists L≥1 such that Lψ*(t/L)+t≤ψ*(t)+L,t≥0.

Let ω be a weight function. For a compact set K⊂ℝ, we define the following:
(3)𝒟(ω)(K)={f∈𝒟(K):∥f∥K,λ∞<for everyλ>0},
where ∥f∥K,λ=supx∈Ksupn∈ℕ0|f(n)(x)|exp(-λψ*(n/λ)).

𝒟(ω)=indn𝒟(ω)([-n,n]). The elements of 𝒟(ω)′ are called ultradistributions of Beurling type.

We denote by ℰ(ω) the set of all C∞ functions f such that ∥f∥K,λ<∞ for every compact K and every λ>0. For more details about 𝒟(ω) and ℰ(ω), we refer to [7].

A function ϕ∈C∞(ℝ) is in the space 𝒮(ω) when, for every m,n∈ℕ,
(4)pm,n=supx∈ℝemω(x)|ϕ(n)(x)|<∞,πm,n=supx∈ℝemω(x)|ϕ^(n)(x)|<∞.

𝒮(ω) is endowed with the topology generated by the family {pm,n,πm,n}, where m,n∈ℕ, of seminorms. Thus 𝒮(ω) is a Fréchet space and the Fourier transform ℱ defines an automorphism of 𝒮(ω). 𝒟(ω) is a dense subspace of 𝒮(ω). For more details about 𝒮(ω), we refer to [9].

For every 1≤p≤∞, k∈ℕ and ϕ∈C∞(ℝ), and γk,p(ϕ) is defined as follows:
(5)γk,p(ϕ)=supα∈ℕ∥ϕ(α)∥pe-kψ*(α/k),
where ∥·∥p denotes the usual norm in Lp(ℝ). (∥f∥∞ means ess sup |f(t)|.) If 1≤p<∞, the space 𝒟Lp,(ω) is the set of all C∞ functions ϕ on ℝ such that γk,p(ϕ)<∞ for each k∈ℕ. We denote by 𝒟L∞,(ω) the set of all bounded C∞ functions ϕ on ℝ such that γk,∞(ϕ)<∞. The topology of 𝒟Lp,(ω), 1≤p≤∞, is generated by the family {γk,p(ϕ)}k∈ℕ of seminorms.

The dual of 𝒟Lp,(ω) will be denoted by 𝒟Lp,(ω)′ and it will be endowed with the strong topology. The elements of 𝒟Lp,(ω)′ are called the Beurling ultradistributions of Lp-growth. For more details about 𝒟Lp,(ω), we refer to [6].

Lemma 4.

(i) 𝒟Lp,(ω), 1≤p≤∞, is Fréchet spaces.

(ii) 𝒟Lp,(ω) is continuously contained in the Schwartz distributions 𝒟Lp, 1≤p≤∞.

(iii) 𝒟Lp,(ω) is continuously contained in 𝒟Lq,(ω), when 1≤p≤q≤∞.

(iv) 𝒟(ω)⊂𝒟Lp,(ω)⊂ℰ(ω), 1≤p≤∞, with continuous and dense inclusions.

(v) 𝒮ω is continuously contained in 𝒟L1,(ω).

Proof.

(i), (iii), and (iv) can be found in Proposition 2.1 of [6]. (ii) is obvious and (v) can be found in the proof of [6, Proposition 2.9].

Definition 5.

The convolutors in 𝒟Lp,(ω), 1≤p≤∞, are the functionals T∈𝒟Lp,(ω)′ such that T*ϕ∈𝒟Lp,(ω) for every ϕ∈𝒟Lp,(ω).

Remark 6.

From Proposition 3.2, Theorem 3, and Proposition 3.6 in [6], we can find a necessary condition for an ultradistribution to be a convolutor.

If T∈𝒟L2,(ω)′ and there exists k∈ℕ such that T^e-kω∈L∞, then T is a convolutor in 𝒟Lp,(ω) for 1/2≤p≤2.

In particular, if T∈𝒟L1,(ω)′∩𝒟L∞,(ω)′, then T is a convolutor in 𝒟L1,(ω).

We will consider the convolutors in 𝒟Lp,(ω) for 1≤p≤2 in Sections 4 and 5.

Assume that G is an entire function such that log|G(z)|=O(ω(|z|)), as |z|→∞. The functional TG on ℰ(ω) is defined by
(6)〈TG,ϕ〉=∑α=0∞(-1)αG(α)(0)α!ϕ(α)(0),ϕ∈ℰ(ω).

The operator G(D) defined on 𝒟(ω)′ by
(7)G(D):𝒟(ω)⟶𝒟(ω),μ⟶G(D)μ=μ*TG,
is called an ultradistributional operator of (ω)-class. When G(D) is restricted to ℰ(ω), G(D) is a continuous operator from ℰ(ω) into ℰ(ω) and, for every ϕ∈ℰ(ω),
(8)(G(D)ϕ)(x)=∑α=0∞(-1)αG(α)(0)α!ϕ(α)(x),x∈ℝ.

Definition 7.

An ultradistributional operator G(D) of (ω)-class is said to be strongly elliptic if there exist M>0 and l>0 such that |G(z)|≥Melω(|z|) when |Imz|<M|Rez|.

Now we will obtain the characterization of convolutors in 𝒟Lp,(ω), 1≤p≤∞.

Lemma 8.

Let T be a convolutor in 𝒟Lp,(ω), 1≤p≤∞. Then there exists a strongly elliptic ultradistributional operator G(D) of (ω)-class and f,g∈𝒟Lp,(ω) such that T=G(D)f+g.

Proof.

Let 1≤p≤∞. Since T is a convolutor in 𝒟Lp,(ω), T*ϕ∈𝒟Lp,(ω)⊂𝒟Lp⊂Lp for every ϕ∈𝒟(ω). By the proof of Proposition 2.3 in [6], there exists a strongly elliptic ultradistributional operator G(D) of (ω)-class and χ∈𝒟(ω) and Γ∈𝒟(ω) such that T=G(D)(T*χ)+(T*Γ). If we let f=T*χ and g=T*Γ, then f,g∈T*𝒟(ω)⊂T*𝒟Lp,(ω)⊂𝒟Lp,(ω) since T is a convolutor in 𝒟Lp,(ω).

Only by replacing ∥h*g∥p≤C∥h∥p in the second to the last line of the proof of Lemma 2.4 in [6] by
(9)γk,p(h*g)=supα∈ℕ∥(h*g)(α)∥pe-kψ*(α/k)≤Csupα∈ℕ∥h(α)∥pe-kψ*(α/k)=Cγk,p(h)
that we have the following:

Lemma 9.

Let 1≤p≤∞. If G(D) is a strongly elliptic ultradistributional operator of (ω)-class, then 𝒟Lp,(ω) is contained in G(D)(𝒟Lp,(ω)).

Combining Lemmas 8 and 9 and given the fact that 𝒟Lp,(ω)⊂ℰ(ω), we have the characterization of the convolutor in 𝒟Lp,(ω), 1≤p≤∞.

Theorem 10.

Let T be a convolutor in 𝒟Lp,(ω), 1≤p≤∞. Then there exists a strongly elliptic ultradistributional operator G(D) of (ω)-class and f∈𝒟Lp,(ω) such that
(10)T(x)=G(D)f(x)=∑α=0∞(-1)αG(α)(0)α!f(α)(x).

3. The Cauchy and Poisson Kernel Functions

Let C be a regular cone in ℝ, that is, an open convex cone, such that C¯ does not contain any straight line. O(C) will denote the convex hull (envelop) of C, and TC=ℝ+iC⊂ℂ is a tube in ℂ. If C is open, TC is called a tubular cone. If C is open and connected, TC is called a tubular radial domain. The set C*={t∈ℝ:〈t,y〉≥0forally∈C} is the dual cone of the cone C. We will give a very important lemma concerning cones and their dual cones, which is proved in Lemma 2 in page 223 of [10].

Lemma 11.

Let C be an open cone in ℝ and let y∈O(C). Then there exist a δ=δy>0 depending on y such that
(11)〈y,t〉≥δ|y||t|,t∈C*.

Definition 12.

Let C be a regular cone in ℝ. The Cauchy kernel K(z-t),z∈TC=ℝ+iC,t∈ℝ, corresponding to the tube TC is
(12)K(z-t)=∫C*exp(2πi〈z-t,η〉)dη,z∈TC,t∈ℝ.

The Poisson kernel P(z;t) corresponding to the tube TC is
(13)P(z;t)=K(z-t)K(z-t)¯K(2iy)=|K(z-t)|2K(2iy),hhhhhhhz=x+iy∈TC,t∈ℝ.

We note that K(2iy)>0, y∈C, by Lemma 1 in page 222 of [10]. For a regular cone C, K(z-t) and P(z;t) are well defined for z∈TC and t∈ℝ ([1], Section 3).

In this section we will prove that K(z-t) and P(z;t) are elements of 𝒟Lq,(ω), 2≤q≤∞, as a function of t∈ℝ for z∈TC.

Theorem 13.

Let C be a regular cone in ℝ and let 2≤p≤∞. Then K(z-t)∈𝒟Lq,(ω) as a function of t∈ℝ for z∈TC.

Proof.

Let z=x+iy∈TC be arbitrary but fixed and let α∈ℕ. Let IC* denote the characteristic function of C*. By the proof of Theorem 4.4.1 in [2], K(z-t)∈C∞, IC*(η)ηαe2πi〈z,η〉∈L1∩Lp, 1<p≤2, as a function of t∈ℝ for z∈TC, and
(14)DtαK(z-t)=ℱ-1[IC*(η)ηαe2πi〈z,η〉;t],hhhhhhhhhz∈TC,t∈ℝ,
where the inverse transform ℱ-1 can be interpreted in both L1 and Lp sense. If 1<p≤2 and 1/p+1/q=1, we have from (14) and the Parseval inequality the following:
(15)∥DtαK(z-t)∥q≤∥IC*(η)ηαe2πi〈z,η〉∥p<∞.

Using (15) and Lemmas 3 (iii), 11, and 2, we get that, for every k∈ℕ and z∈TC, there exist δ>0 such that
(16)supα∈ℕ∥DtαK(z-t)∥qe-kψ*(α/k)≤supα∈ℕ(∫C*|η|pαe-2πp〈y,η〉dη)1/pe-kψ*(α/k)≤supα∈ℕ(∫C*epαlog|η|e-pkψ*(α/k)e-2πp〈y,η〉dη)1/p≤(∫C*epkψ**(log|η|)e-2πp〈y,η〉dη)1/p≤(∫C*epkψ(log|η|)e-2πpδ|y∥η|dη)1/p≤(∫C*ep|η|(k·(ω(η)/|η|)-2πδ|y|)dη)1/p<∞.

Let p=1. We get from the same method of estimation of integrand in (16) that, for every k∈ℕ and z∈TC, there exist δ>0 such that
(17)supα∈ℕ∥Dtα(z-t)∥∞e-kψ*(α/k)≤supα∈ℕ∫C*|η|αe-2πp〈y,η〉e-kψ*(α/k)dη≤∫C*e|η|(k·(ω(η)/|η|)-2πδ|y|)dη<∞.

From (16) and (17), the proof is complete.

Theorem 14.

Let C be a regular cone in ℝ and let 2≤q≤∞. Then P(z;t)∈𝒟Lq,(ω) as a function of t∈ℝ for z∈TC.

Proof.

By Theorem 4.4.4 in [2], P(z;t)∈C∞ as a function of t∈ℝ for z∈TC. Since K(z-t) is contained in the spaces of infinitely differentiable functions which vanish at infinity together with each of their derivatives by Theorem 4.4.1 in [2], DtγK(z-t)¯ is bounded on ℝ. Hence, we get from Lemma 3 and Theorem 13 that, for every k∈ℕ, z∈TC, and 2≤q≤∞,
(18)γk,q(P(z;t))=supα∈ℕ∥DtαP(z-t)∥qe-kψ*(α/k)=supα∈ℕ∥1K(2iy)∑β+γ=αα!β!γ!DtβK(z-t)DtγK(z-t)¯∥qhhhhhhhhhhhhhhhhh×e-kψ*(α/k)≤1K(2iy)supα∈ℕ∑β+γ=αCγ,Kα!β!γ!∥DtβK(z-t)∥qhhhhhhhhhhhhhhhh×e-kψ*(β/k)=1K(2iy)∑β+γ=αCγ,Kα!β!γ!γk,q(K(z-t))<∞.

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In this section we will define the Cauchy integral of the convolutor in 𝒟Lp,(ω), 1≤p≤2, and show that the Cauchy integral of the convolutor in 𝒟Lp,(ω), 1≤p≤2, is analytic in a tubular domain TC and satisfies certain growth properties.

Let C be a regular cone in ℝ and 1≤p≤2. By Theorem 13, K(z-t)∈𝒟Lq,(ω), 1/p+1/q=1, as a function of t∈ℝ for z∈TC. Hence 〈Ut,K(z-t)〉 is well defined for U∈𝒟Lp,(ω)′, 1≤p≤2.

Definition 15.

Let C be a regular cone in ℝ and let U be a convolutor in 𝒟Lp,(ω), 1≤p≤2. The Cauchy integral C(U;z) of U corresponding to C is
(19)C(U;z)=〈Ut,K(z-t)〉,z∈TC.

Theorem 16.

Let C be a regular cone in ℝ and let U be a convolutor in 𝒟Lp,(ω), 1≤p≤2. The Cauchy integral C(U;z) of U corresponding to C is analytic in TC and for any compact subset C′⊂C and z=x+iy∈TC′, there exists a constant M(p,C′) depending on p and C′ such that
(20)|C(U;z)|≤M(p,C′)|y|-1/p.

Proof.

By Theorem 10, there exists a strongly elliptic ultradistributional operator G(D) of (ω)-class and f∈𝒟Lp,(ω)⊂ℰ(ω), 1≤p≤2, such that U=G(D)f; that is,
(21)U(x)=(G(D)f)(x)=∑α=0∞iαG(α)(0)α!f(α)(x),x∈ℝ.

Then
(22)C(U;z)=〈G(D)f(t),K(z-t)〉=∫∑α=0∞iαG(α)(0)α!f(α)(t)∫C*e2πi〈z-t,η〉dηdt;
hence
(23)DzβC(U;z)=(-1)β〈G(D)f(t),DzβK(z-t)〉=(-1)β∫∑α=0∞iαG(α)(0)α!f(α)(t)hhhhhhhhhhhh×∫C*ηβe2πi〈z-t,η〉dηdt.

Now we will apply the method in the proof of Proposition 2.4 of [11] to Lp-norm estimation of the integral with respect to t in (23). Since log|G(z)|=O(ω(|z|)), as |z|→∞, there exists k>0 such that
(24)log|G(z)|≤k(1+ω(|z|)),z∈ℂ.

If we let CR={z;|z|=er,r>0}, we get that, by Cauchy's theorem and (24),
(25)|G(α)(0)α!|=12π|∫CRG(z)zα+1dz|≤12π∫CRekekω(|z|)|z|α+1|dz|=12πekekω(er)eα(r+1)·er=(2π)-1ek·ekψ(r)-αr.

Let m∈ℕ and let L be as in Lemma 3 (iv). Take h≥max{kL,m}. By Lemma 4 (iii), we have the following:
(26)f∈𝒟p,(ω)⊂𝒟h,(ω),1≤p≤2.

Since ψ*(t)/t is increasing by Lemma 3 (ii), we get that, for every α∈ℕ0,
(27)α+hψ*(αh)≤α+kLψ*(αkL)≤kψ*(αh)+kL.

Using (25), (26), and (27),
(28)∑α=0∞|G(α)(0)α!|p∫|f(α)(t)|pdt=∑α=0∞(2π)-1epkep(kψ(r)-αr)γh,p(f)ehψ*(α/h)≤∑α=0∞γh,p(f)(2π)-1epke-kψ*(α/k)ehψ*(α/h)≤∑α=0∞γh,p(f)(2π)-1epkekL-α≤Ch,k,p,Lγh,p(f).

We will consider the Lq-norm estimation of the integral with respect to η in (23). Using (14), (15), and (5.63) in the proof of Theorem 5.5.1 in [2], there exist δ>0 as in Lemma 11 such that if 1<p≤2 and 1/p+1/q=1,
(29)∥DtβK(z-t)∥q=(∫C*|η|βqe2πq〈z-t,η〉dη)1/q=∥ℱ-1[IC*(η)ηβe2πi〈z,η〉;t]∥q≤∥IC*(η)ηβe2πi〈z,η〉∥p≤(2Γ(pβ+1))1/p(2πpδ|y|)(-pβ-1)/p.

Let K be an arbitrary compact subset of TC. For z∈K⊂TC, y=Im(z)∈C′ for some fixed compact subcone C′⊂C and y is bounded away from 0 by l>0 for some l. For this compact subcone C′⊂C and z=x+iy∈K⊂TC, if 1<p≤2 and 1/p+1/q=1, we get from (29) that there exist δ>0 as in Lemma 11 such that
(30)(∫C*|η|βqe2πq〈z-t,η〉dη)1/q≤(2Γ(pβ+1))1/p(2πpδ|l|)(-pβ-1)/p,
where Γ(x) is the gamma function. By (23), (28), and (30), consider the following:
(31)|DzβC(U;z)|≤∥∑α=0∞G(α)(0)α!f(α)(t)∥p(∫C*|η|βqe2πq〈z-t,η〉dη)1/q≤Ch,k,p,Lγh,p(f)(2Γ(pβ+1))1/p(2πpδl)(-pβ-1)/p
for compact subcone C′⊂C and z=x+iy∈K⊂TC. Equation (31) shows that any derivatives of C(U;z) with respect to z converge absolutely and uniformly for all z∈K⊂TC; hence C(U;z) is analytic in TC.

It remains to prove the analyticity of C(U;z) for p=1. By Proposition 2.4 in [11], G(D)f∈𝒟L1,(ω) for f∈𝒟L1,(ω). Since 𝒟L1,(ω)⊂𝒟L1⊂L1, the analyticity of C(U;z) for p=1 in this theorem can be obtained from the analyticity of C(U;z) for p=1 in the case of U∈𝒟L1 of Theorem 5.5.1 in [2].

The growth of C(U;z) remains to be proved. If 1<p≤2, we get that by, (29) with β=0,
(32)|C(U;z)|≤Ch,k,p,Lγh,p(f)(2Γ(1))1/p(2πpδ)-1/p|y|-1/p
for compact subcone C′⊂C, z=x+iy∈TC′, and δ=δ(C′). If p=1, we get that, by (29) with p=1 and β=0,
(33)|∫IC*e2πi〈z-t,η〉dη|≤2Γ(1)(2πδ)-1|y|-1;
hence
(34)|C(U;z)|≤Ch,k,1,Lγh,1(f)2Γ(1)(2πδ)-1|y|-1
for compact subcone C′⊂C, z=x+iy∈TC′, and δ=δ(C′). From (32) and (34) we get that there is a constant M(p,C′) such that
(35)|C(U;z)|≤M(p,C′)|y|-1/p
for compact subcone C′⊂C and z=x+iy∈TC′.

Now we will represent a convolutor U in 𝒟Lp,(ω), 1≤p≤2, as the boundary value of analytic functions in tube. We need several lemmas.

Lemma 17.

Let C be a regular cone in ℝ and let U be a convolutor in 𝒟Lp,(ω), 1≤p≤2. Let φ∈𝒮(ω). For fixed y=Im(z)∈C,
(36)〈C(U;z),φ(x)〉=〈U,〈K(z-t),φ(x)〉〉.

Proof.

Let 1≤p≤2. Since K(z-t)∈𝒟Lp,(ω) by Theorem 13,
(37)Ky=K(x+iy)=∫C*e2πi〈x+iy,η〉dη∈𝒟Lp,(ω),y∈C.

Since U is a convolutor in 𝒟Lp,(ω), (U;z)=(U*Ky)(x)∈𝒟Lp,(ω). Since 𝒟Lp,(ω)⊂𝒟Lp,(ω)′ and 𝒮(ω)⊂𝒟Lp,(ω), 〈C(U;z),φ(x)〉=〈(U*Ky)(x),φ(x)〉=〈〈U,K(z-t)〉,φ(x)〉 is well defined for φ∈𝒮(ω). We get from Theorem 10, (28), and Fubini's theorem that there exists a strongly elliptic ultradifferential operator G(D) of (ω)-class and f∈𝒟Lp,(ω)⊂ℰ(ω), 1≤p≤2, such that
(38)〈C(U;z),φ(x)〉=〈〈U,K(z-t)〉,φ(x)〉=∬∑α=0∞(-1)αG(α)(0)α!f(α)(x)K(z-t)dtφ(x)dx=∬∑α=0∞(-1)αG(α)(0)α!f(α)(x)K(z-t)φ(x)dxdt=∫∑α=0∞(-1)αG(α)(0)α!f(α)(x)∫K(z-t)φ(x)dxdt=〈U,〈K(z-t),φ(x)〉〉
for y=Im(z)∈C.

Remark 18.

We note that the reason for working with φ∈𝒮(ω) in Lemma 17 is that the symmetry of 𝒮(ω) under the Fourier and inverse Fourier transform will be needed later on.

Lemma 19.

Let C be a regular cone in ℝ and φ∈𝒮(ω). Then
(39)limy→0〈K(x+iy-t),φ(x)〉=ℱ-1[IC*(η)φ^(η);t],hhhhhhhhhhhhy∈C,
in 𝒟Lq,(ω), 1≤p≤2, 1/p+1/q=1, where IC*(η) is the characteristic function in C*.

Proof.

Let 1<p≤2 and let 1/p+1/q=1. For an arbitrary α∈ℕ, we get from (14) and (15) that
(40)∥Dtα〈K(z-t),φ(x)〉-Dtαℱ-1[IC*(η)φ^(η);t]∥q=∥ℱ-1[IC*(η)ηαφ^(η)(e-2π〈y,η〉-1);t]∥q≤∥IC*(η)ηαφ^(η)(e-2π〈y,η〉-1)∥p.

For any k∈ℕ and q,1/p+1/q=1, we get that by (40)
(41)γk,q(〈K(z-t),φ(x)〉-ℱ-1[IC*(η)φ^(η);t])=supα∈ℕ∥Dtα〈K(z-t),φ(x)〉-Dtαℱ-1[IC*(η)φ^(η);t]∥q×e-kψ*(α/k)≤supα∈ℕ∥IC*(η)|η|αφ^(η)(e-2π〈y,η〉-1)∥pe-kψ*(α/k)=supα∈ℕ∥IC*(η)eαlog|η|e-kψ*(α/k)φ^(η)(e-2π〈y,η〉-1)∥p=∥IC*(η)ekψ**(log|η|)φ^(η)(e-2π〈y,η〉-1)∥p=∥IC*(η)ekω(η)φ^(η)(e-2π〈y,η〉-1)∥p.

Since
(42)|IC*(η)ekω(η)φ^(η)(e-2π〈y,η〉-1)|≤|ekω(η)φ^(η)|
and φ^(η)∈𝒮(ω),
(43)IC*(η)ekω(η)φ^(η)(e-2π〈y,η〉-1)∈L1∩Lp.

Hence we have from Lebesgue’s dominated convergence theorem that the last line of (41) converges to 0 as y→0 for y∈C.

It remains to prove the convergence for p=1, that is, in 𝒟L∞,(ω). By the same method of estimation of integrand in (41), we get that
(44)γk,∞(〈K(z-t),φ(x)〉-ℱ-1[IC*(η)φ^(η);t])=supα∈ℕ∥Dtα〈K(z-t),φ(x)〉-Dtαℱ-1[IC*(η)φ^(η);t]nnnnnnnnn-Dtαℱ-1[IC*(η)φ^(η);t]∥∞e-kψ*(α/k)=supα∈ℕℱ-1[IC*(η)ηαφ^(η)(e-2π〈y,η〉-1);t]e-kψ*(α/k)=supα∈ℕ∫ei〈η,t〉IC*(t)tαφ^(t)(e-2π〈y,t〉-1)dte-kψ*(α/k)=∫IC*(t)ekω(t)φ^(t)(e-2π〈y,t〉-1)dt.

By (42), (43), and Lebesgue’s dominated convergence theorem, the last line of (44) converges to 0 as y→0 for y∈C.

Combining Lemmas 17 and 19 and using the continuity of a convolutor U in 𝒟Lp,(ω), consider the following:

Lemma 20.

Let C be a regular cone in ℝ and let U be a convolutor in 𝒟Lp,(ω), 1≤p≤2. If φ∈𝒮(ω), one has the following:
(45)limy→0〈C(U;z),φ(x)〉=〈U,ℱ-1[IC*(η)φ^(η);t]〉,hhhhhhhhhy=Im(z)∈C.

Now we will give the representation of a convolutor U in 𝒟Lp,(ω), 1≤p≤2, as the distributional limit of a finite number of functions analytic in tubes.

Theorem 21.

Let U be a convolutor in 𝒟Lp,(ω), 1≤p≤2, and let φ∈𝒮(ω). Let Cj, j=1,…,r, be a finite number of regular cones whose dual cone satisfies the property that
(46)ℝ∖∪j=1rCj*,Cj*∩Ck*,j≠k,j,k=1,…,r,
are sets of Lebesgue’s measure 0. Then there exist functions Uj which are analytic in ℝ+iCj, j=1,…,r, and satisfy
(47)|Uj(z)|≤M(p,Cj′)|y|-1/p
for an arbitrary compact subcone Cj′⊂Cj and a constant M(p,Cj′) depending on p and Cj′ and
(48)〈U,φ〉=∑j=rrlimy→0〈Uj(x+iy),φ(x)〉,y∈Cj.

Proof.

Let
(49)Uj=〈U,∫Cj*e2πi〈z-t,η〉dη〉,z=x+iy∈ℝ+iCj.

The analyticity and growth of Uj(z), j=1,…,r, are followed by Theorem 16. Using Lemma 20, the linearity of U, and (46), if y∈Cj,
(50)∑j=rrlimy→0〈Uj(x+iy),φ(x)〉=∑j=rr〈U,ℱ-1[ICj*(η)φ^(η);t]〉=〈U,∑j=rrℱ-1[ICj*(η)φ^(η);t]〉=〈U,ℱ-1[φ^(η);t]〉=〈U,φ〉.

The proof is complete.

Remark 22.

As mentioned in page 246 of [2], under certain conditions on U∈𝒟Lp′, the Cauchy integral of U does have U as boundary value. But we see from Lemma 20 that the Cauchy integral of U∈𝒟Lp,(ω)′, 1≤p≤2, corresponding to a regular cone C in ℝn does not attain U as boundary value as y→0¯,y∈C. We cannot find the conditions on U∈𝒟Lp,(ω)′, 1≤p≤2, such that the Cauchy integral of U∈𝒟Lp,(ω)′, 1≤p≤2, does attain U as boundary value.

Let C be a regular cone in ℝ and let 2≤q≤∞. In Theorem 14, we showed that P(z;t)∈𝒟Lq,(ω) as a function of t∈ℝ for z∈TC. Hence, for U∈𝒟Lp,(ω)′, 1≤p≤2, 〈Ut,P(z;t)〉 is a well-defined function of t∈ℝ for z∈TC.

Definition 23.

Let C be a regular cone in ℝ and let U be a convolutor in 𝒟Lp,(ω), 1≤p≤2. The Poisson integral PI(U;z) of U corresponding to C is
(51)PI(U;z)=〈Ut,P(z;t)〉,z∈TC.

In this section we will show that, for the convolutor U in 𝒟Lp,(ω), 1≤p≤2, the Poisson integral PI(U;z),z∈TC, has U as boundary value distributionally as y→0,y∈C. We need several lemmas.

Let S∈𝒟Lp,(ω)′ and T∈𝒟Lq,(ω)′ be given, 1/p+1/q=1. Then the convolution S*T is the ultradistribution S*T∈𝒟(ω) given by 〈S*T,ϕ〉=〈S,T~*ϕ〉, where T~(ϕ)=T(ϕ~).

Let C be a regular cone and g∈Lp, 1≤p<∞. One has
(52)limy→0∫g(t)P(x+iy;t)dt=g(t),y∈C,
in Lp.

Lemma 26.

Let C be a regular cone in ℝ and let U be a convolutor in 𝒟Lp,(ω), 1≤p≤2. For φ∈𝒟(ω),
(53)〈PI(U;z),φ(x)〉=〈U,〈P(z;t),φ(x)〉〉,hhhhhy=Im(z)∈C.

Proof.

For φ∈𝒟(ω),
(54)∫P(x+iy;t)φ(x)dx=∫P(x,y)φ(x+t)dx,y∈C,
where
(55)P(x,y)=K(x+iy)K(x+iy)¯K(2iy),x∈ℝ,y∈C.

Since P(x,y)∈𝒟Lq,(ω)⊂𝒟Lq,(ω)′, 2≤q≤∞, as a function of x∈ℝ and y∈C, we get from Definition 24 that, for a convolutor U in 𝒟Lp,(ω), 1≤p≤2, U*P(x,y)∈𝒟(ω). Hence, for φ∈𝒟(ω), 〈U*P(x,y),φ〉 is well defined and by the definition of convolution
(56)〈U*P(x,y),φ〉=〈U,〈P(x,y),φ(x+t)〉〉,y∈C.

From (54) and (56),
(57)〈U,〈P(z;t),φ(x)〉〉=〈U,〈P(x,y),φ(x+t)〉〉=〈U*P(x,y),φ〉
for φ∈𝒟(ω). Hence 〈U,〈P(z;t),φ(x)〉〉 is well defined for y∈C. Using Theorem 10 and (28), we get from change of order of integration that, for y∈C and φ∈𝒟(ω), there exists a strongly elliptic ultradifferential operator G(D) of (ω)-class and f∈𝒟Lp,(ω)⊂ℰ(ω), 1≤p≤2, such that(58)〈U,〈P(z;t),φ(x)〉〉=∫∑α=0∞(-1)αG(α)(0)α!f(α)(x)∫P(z;t)φ(x)dxdt=∫φ(x)∫∑α=0∞(-1)αG(α)(0)α!f(α)(x)P(z;t)dtdx=〈PI(U;z),φ(x)〉.

Lemma 27.

Let C be a regular cone in ℝ and φ∈𝒟(ω). Consider the following:
(59)limy→0∫P(x+iy;t)φ(x)dx=φ(t),y∈C,
in the topology of 𝒟Lq,(ω), 1≤q≤∞.

Proof.

Let α be an arbitrary integer. By (ii) and (iv) in Lemma 4, φ∈𝒟(ω)⊂𝒟L1,(ω)⊂𝒟L1⊂L1. We get from (5.92) in the proof of Lemma 5.5.11 of [2] that, for 1≤p≤∞,
(60)limy→0∥Dtα∫P(x+iy;t)φ(x)dx-Dtαφ(t)∥q=0,y∈C.

Thus for any k∈ℕ and 2≤q≤∞(61)limy→0supα∈ℕ∥Dtα∫P(x+iy;t)φ(x)dx-Dtαφ(t)∥q×e-kψ*(k/α)=0,y∈C.

Combining Lemmas 26 and 27, and the continuity of U∈𝒟Lp,(ω)′, we can obtain the convergence of PI(U;z) to U distributionally as y→0,y∈C, as follows.

Theorem 28.

Let C be a regular cone in ℝ and let U be a convolutor in 𝒟Lp,(ω), 1≤p≤2. For φ∈𝒟(ω),
(62)limy→0〈PI(U;z),φ(x)〉=〈U,φ〉,y∈C.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author thanks academic editor and reviewer for the valuable comments and suggestions on this paper.

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