#### Abstract

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

#### 1. Introduction

Let be a regular cone in and let denote its convex envelope. In [1] (or [2]) Carmichael defined the Cauchy and Poisson integrals for Schwartz distributions , , corresponding to tubular domain . Carmichael obtained the boundary values of these integrals in the distributional sense on the boundary of and found the relation between analytic functions with a specific growth condition in and the Cauchy and Poisson integrals of their distributional boundary values. In [3] Pilipović defined ultradistributions of -growth, where , , is a certain sequence of positive numbers, and studied the Cauchy and Poisson integrals for elements of in the case that the Cauchy and Poisson kernel functions are defined corresponding to the first quadrant ; , in . Pilipović showed that elements in are boundary values of suitable analytic functions with a certain -norm condition by means of the Cauchy integral representation and an analytic function with a certain -norm condition determines, as a boundary value, an element from . In [4] Carmichael et al. defined ultradistributions of Beurling type of -growth and of Roumieu type of -growth, both of which generalize the Schwartz distributions , and studied the Cauchy and Poisson integrals for elements of both and for corresponding to the arbitrary tubes where is an open connected cone in of which the quadrants are special cases. They showed that the Cauchy integral for elements of both and for corresponding to is shown to be analytic in , 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 and for corresponding to 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 -growth, , of which Schwartz distribution and ultradistributions of Beurling type are special cases. Here is a weight function in the sense of [7]. Betancor et al. defined the convolutors in , ; that is, the functionals such that for every , and studied the convolutors and the surjective convolution operators acting on in [6].

In this paper we define the Cauchy and Poisson integrals for the convolutors in , . We will show that the Cauchy integrals for the convolutors in , , are analytic in a tubular domain where is a regular cone in and satisfies a certain boundedness condition. We will give the representation of the convolutors in , , as boundary value of a finite sum of analytic functions in tubes. Also we will show that the Poisson integrals for the convolutor in , , attain as boundary value in the distributional sense.

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

#### 2. Beurling Ultradistributions of -Growth

In this section we will review Beurling ultradistributions of -growth in 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 with the following properties:

there exists with for all ,, as tends to , is convex.

By ,

Lemma 2. *Consider the following:
*

By , and . Then we can define the Young conjugate of by

Obviously we have the following:

Lemma 3 (see [7]). *(i) is convex and increasing and satisfies .**(ii) is increasing and .**(iii) .**(iv) There exists such that .*

Let be a weight function. For a compact set , we define the following: where .

. The elements of are called ultradistributions of Beurling type.

We denote by the set of all functions such that for every compact and every . For more details about and , we refer to [7].

A function is in the space when, for every ,

is endowed with the topology generated by the family , where , of seminorms. Thus is a Fréchet space and the Fourier transform defines an automorphism of . is a dense subspace of . For more details about , we refer to [9].

For every , and , and is defined as follows: where denotes the usual norm in . ( means ess sup .) If , the space is the set of all functions on such that for each . We denote by the set of all bounded functions on such that . The topology of , , is generated by the family of seminorms.

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

Lemma 4. *(i) , , is Fréchet spaces.**(ii) is continuously contained in the Schwartz distributions , .**(iii) is continuously contained in , when .**(iv) , , with continuous and dense inclusions.**(v) is continuously contained in .*

*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 , , are the functionals such that for every .

*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 and there exists such that , then is a convolutor in for .

In particular, if , then is a convolutor in .

We will consider the convolutors in for in Sections 4 and 5.

Assume that is an entire function such that , as . The functional on is defined by

The operator defined on by
is called an ultradistributional operator of -class. When is restricted to , is a continuous operator from into and, for every ,

*Definition 7. *An ultradistributional operator of -class is said to be* strongly elliptic* if there exist and such that when .

Now we will obtain the characterization of convolutors in , .

Lemma 8. *Let be a convolutor in , . Then there exists a strongly elliptic ultradistributional operator of -class and such that .*

*Proof. *Let . Since is a convolutor in , for every . By the proof of Proposition 2.3 in [6], there exists a strongly elliptic ultradistributional operator of -class and and such that . If we let and , then since is a convolutor in .

Only by replacing in the second to the last line of the proof of Lemma 2.4 in [6] by that we have the following:

Lemma 9. *Let . If is a strongly elliptic ultradistributional operator of -class, then is contained in .*

Combining Lemmas 8 and 9 and given the fact that , we have the characterization of the convolutor in , .

Theorem 10. *Let be a convolutor in , . Then there exists a strongly elliptic ultradistributional operator of -class and such that
*

#### 3. The Cauchy and Poisson Kernel Functions

Let be a regular cone in , that is, an open convex cone, such that does not contain any straight line. will denote the convex hull (envelop) of , and is a tube in . If is open, is called a tubular cone. If is open and connected, is called a tubular radial domain. The set is the dual cone of the cone . 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 be an open cone in and let . Then there exist a depending on such that
*

*Definition 12. *Let be a regular cone in . The Cauchy kernel , corresponding to the tube is

The Poisson kernel corresponding to the tube is

We note that , , by Lemma 1 in page 222 of [10]. For a regular cone , and are well defined for and ([1], Section 3).

In this section we will prove that and are elements of , , as a function of for .

Theorem 13. *Let be a regular cone in and let . Then as a function of for .*

*Proof. *Let be arbitrary but fixed and let . Let denote the characteristic function of . By the proof of Theorem in [2], , , , as a function of for , and
where the inverse transform can be interpreted in both and sense. If and , we have from (14) and the Parseval inequality the following:

Using (15) and Lemmas 3 (iii), 11, and 2, we get that, for every and , there exist such that

Let . We get from the same method of estimation of integrand in (16) that, for every and , there exist such that

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

Theorem 14. *Let be a regular cone in and let . Then as a function of for .*

*Proof. *By Theorem in [2], as a function of for . Since is contained in the spaces of infinitely differentiable functions which vanish at infinity together with each of their derivatives by Theorem in [2], is bounded on . Hence, we get from Lemma 3 and Theorem 13 that, for every , , and ,

#### 4. The Cauchy Integral of the Convolutor in

In this section we will define the Cauchy integral of the convolutor in , , and show that the Cauchy integral of the convolutor in , , is analytic in a tubular domain and satisfies certain growth properties.

Let be a regular cone in and . By Theorem 13, , , as a function of for . Hence is well defined for , .

*Definition 15. *Let be a regular cone in and let be a convolutor in , . The Cauchy integral of corresponding to is

Theorem 16. *Let be a regular cone in and let be a convolutor in , . The Cauchy integral of corresponding to is analytic in and for any compact subset and , there exists a constant depending on and such that
*

*Proof. *By Theorem 10, there exists a strongly elliptic ultradistributional operator of -class and , , such that ; that is,

Then
hence

Now we will apply the method in the proof of Proposition 2.4 of [11] to -norm estimation of the integral with respect to in (23). Since , as , there exists such that

If we let , we get that, by Cauchy's theorem and (24),

Let and let be as in Lemma 3 (iv). Take . By Lemma 4 (iii), we have the following:

Since is increasing by Lemma 3 (ii), we get that, for every ,

Using (25), (26), and (27),

We will consider the -norm estimation of the integral with respect to in (23). Using (14), (15), and (5.63) in the proof of Theorem in [2], there exist as in Lemma 11 such that if and ,

Let be an arbitrary compact subset of . For , for some fixed compact subcone and is bounded away from by for some . For this compact subcone and , if and , we get from (29) that there exist as in Lemma 11 such that
where is the gamma function. By (23), (28), and (30), consider the following:
for compact subcone and . Equation (31) shows that any derivatives of with respect to converge absolutely and uniformly for all ; hence is analytic in .

It remains to prove the analyticity of for . By Proposition 2.4 in [11], for . Since , the analyticity of for in this theorem can be obtained from the analyticity of for in the case of of Theorem in [2].

The growth of remains to be proved. If , we get that by, (29) with ,
for compact subcone , , and . If , we get that, by (29) with and ,
hence
for compact subcone , , and . From (32) and (34) we get that there is a constant such that
for compact subcone and .

Now we will represent a convolutor in , , as the boundary value of analytic functions in tube. We need several lemmas.

Lemma 17. *Let be a regular cone in and let be a convolutor in , . Let . For fixed ,
*

*Proof. *Let . Since by Theorem 13,

Since is a convolutor in , . Since and , is well defined for . We get from Theorem 10, (28), and Fubini's theorem that there exists a strongly elliptic ultradifferential operator of -class and , , such that
for .

*Remark 18. *We note that the reason for working with in Lemma 17 is that the symmetry of under the Fourier and inverse Fourier transform will be needed later on.

Lemma 19. *Let be a regular cone in and . Then
**
in , , , where is the characteristic function in .*

*Proof. *Let and let . For an arbitrary , we get from (14) and (15) that

For any and , we get that by (40)

Since
and ,

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

It remains to prove the convergence for , that is, in . By the same method of estimation of integrand in (41), we get that

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

Combining Lemmas 17 and 19 and using the continuity of a convolutor in , consider the following:

Lemma 20. *Let be a regular cone in and let be a convolutor in , . If , one has the following:
*

Now we will give the representation of a convolutor in , , as the distributional limit of a finite number of functions analytic in tubes.

Theorem 21. *Let be a convolutor in , , and let . Let , , be a finite number of regular cones whose dual cone satisfies the property that
**
are sets of Lebesgue’s measure . Then there exist functions which are analytic in , , and satisfy
**
for an arbitrary compact subcone and a constant depending on and and
*

*Proof. *Let

The analyticity and growth of , , are followed by Theorem 16. Using Lemma 20, the linearity of , and (46), if ,

The proof is complete.

*Remark 22. *As mentioned in page 246 of [2], under certain conditions on , the Cauchy integral of does have as boundary value. But we see from Lemma 20 that the Cauchy integral of , , corresponding to a regular cone in does not attain as boundary value as . We cannot find the conditions on , , such that the Cauchy integral of , , does attain as boundary value.

#### 5. Poisson Integral of the Convolutor in

Let be a regular cone in and let . In Theorem 14, we showed that as a function of for . Hence, for , , is a well-defined function of for .

*Definition 23. *Let be a regular cone in and let be a convolutor in , . The Poisson integral of corresponding to is

In this section we will show that, for the convolutor in , , the Poisson integral , has as boundary value distributionally as . We need several lemmas.

*Definition 24 (see [6], Definition ). *Let and be given, . Then the convolution is the ultradistribution given by , where .

Lemma 25 (see [2], Lemma ). *Let be a regular cone and , . One has
**
in .*

Lemma 26. *Let be a regular cone in and let be a convolutor in , . For ,
*

* Proof. *For ,
where

Since , , as a function of and , we get from Definition 24 that, for a convolutor in , , . Hence, for , is well defined and by the definition of convolution

From (54) and (56),
for . Hence is well defined for . Using Theorem 10 and (28), we get from change of order of integration that, for and , there exists a strongly elliptic ultradifferential operator of -class and , , such that