Cauchy and Poisson Integral of the Convolutor in Beurling Ultradistributions of L p-Growth

Let C be a regular cone in R and let T = R + iC ⊂ C be a tubular radial domain. Let U be the convolutor in Beurling ultradistributions of L p -growth corresponding to T. We define the Cauchy and Poisson integral of U and show that the Cauchy integral of U is analytic in T 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 T attains U as boundary value in the distributional sense.


Introduction
Let  be a regular cone in R  and let O() denote its convex envelope.In [1] (or [2]) Carmichael defined the Cauchy and Poisson integrals for Schwartz distributions ∈ D    , 1 <  ≤ 2, corresponding to tubular domain  O() = R  +  ⊂ C  .Carmichael obtained the boundary values of these integrals in the distributional sense on the boundary of  O() and found the relation between analytic functions with a specific growth condition in  O() and the Cauchy and Poisson integrals of their distributional boundary values.In [3] Pilipović defined ultradistributions D  ((  ),  2 ) of  2 -growth, where   ,  = 0, 1, 2, . .., is a certain sequence of positive numbers, and studied the Cauchy and Poisson integrals for elements of D  ((  ),  2 ) in the case that the Cauchy and Poisson kernel functions are defined corresponding to the first quadrant { ∈ R  ;   > 0,  = 1, 2, . . ., } in R  .Pilipović showed that elements in D  ((  ),  2 ) are boundary values of suitable analytic functions with a certain  2 -norm condition by means of the Cauchy integral representation and an analytic function with a certain  2 -norm condition determines, as a boundary value, an element from D  ((  ),  2 ).In [4] Carmichael et al. defined ultradistributions of Beurling type D  ((  ),   ) of   -growth and of Roumieu type D  ({  },   ) of   -growth, both of which generalize the Schwartz distributions D    , and studied the Cauchy and Poisson integrals for elements of both D  ((  ),   ) and D  ({  },   ) for 2 ≤  ≤ ∞ corresponding to the arbitrary tubes   = R  +  ⊂ C  where  is an open connected cone in R  of which the quadrants are special cases.They showed that the Cauchy integral for elements of both D  ((  ),   ) and D  ({  },   ) for 2 ≤  ≤ ∞ 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 D  ((  ),   ) and D  ({  },   ) for 2 ≤  ≤ ∞ 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, D    ,() (1 ≤  ≤ ∞), of which Schwartz distribution and ultradistributions of Beurling type are special cases.Here  is a weight function in the sense of [7].Betancor    ,() , 1 ≤  ≤ ∞; that is, the functionals  ∈ D    ,() such that  *  ∈ D   ,() for every  ∈ D   ,() , and studied the convolutors and the surjective convolution operators acting on D    ,() in [6].In this paper we define the Cauchy and Poisson integrals for the convolutors in D    ,() , 1 ≤  ≤ 2. We will show that the Cauchy integrals for the convolutors in D    ,() , 1 ≤  ≤ 2, are analytic in a tubular domain   = R +  ⊂ C where  is a regular cone in R and satisfies a certain boundedness condition.We will give the representation of the convolutors in D   ,() , 1 ≤  ≤ 2, as boundary value of a finite sum of analytic functions in tubes.Also we will show that the Poisson integrals for the convolutor  in D   ,() , 1 ≤  ≤ 2, attain  as boundary value in the distributional sense.
Since D    ,() is the natural generalization of the space D    from Lemma 4 (ii) and we can find a weight function  such that D  ((  ),   ) = D    ,() 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 D   and D((  ),   ), respectively.

Beurling Ultradistributions of 𝐿 𝑝 -Growth
In this section we will review Beurling ultradistributions of   -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].
Let  be a weight function.For a compact set  ⊂ R, we define the following: where ).The elements of D  () are called ultradistributions of Beurling type.
We denote by E () the set of all  ∞ functions  such that ‖‖ , < ∞ for every compact  and every  > 0. For more details about D () and E () , we refer to [7].
A function  ∈  ∞ (R) is in the space S () when, for every ,  ∈ N, S () is endowed with the topology generated by the family { , ,  , }, where ,  ∈ N, of seminorms.Thus S () is a Fréchet space and the Fourier transform F defines an automorphism of S () .D () is a dense subspace of S () .For more details about S () , we refer to [9].
We denote by D  ∞ ,() the set of all bounded  ∞ functions  on R such that  ,∞ () < ∞.The topology of D   ,() , 1 ≤  ≤ ∞, is generated by the family { , ()} ∈N of seminorms.The dual of D   ,() will be denoted by D    ,() and it will be endowed with the strong topology.The elements of D    ,() are called the Beurling ultradistributions of   -growth.For more details about D   ,() , we refer to [6].
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].International Journal of Mathematics and Mathematical Sciences 3 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  ∈ D   2 ,() and there exists  ∈ N such that We will consider the convolutors in D   ,() for 1 ≤  ≤ 2 in Sections 4 and 5.

Now we will obtain the characterization of convolutors in
Then there exists a strongly elliptic ultradistributional operator () of ()-class and ,  ∈ D   ,() such that  = () + .
. By the proof of Proposition 2.3 in [6], there exists a strongly elliptic ultradistributional operator () of ()-class and  ∈ D Only by replacing ‖ℎ * ‖  ≤ ‖ℎ‖  in the second to the last line of the proof of Lemma 2.4 in [6] by that we have the following: Combining Lemmas 8 and 9 and given the fact that D   ,() ⊂ E () , we have the characterization of the convolutor in D   ,() , 1 ≤  ≤ ∞.

The Cauchy and Poisson Kernel Functions
Let  be a regular cone in R, that is, an open convex cone, such that  does not contain any straight line.() will denote the convex hull (envelop) of , and The Poisson kernel (; ) corresponding to the tube   is We note that (2) > 0,  ∈ , by Lemma 1 in page 222 of [10].For a regular cone , ( − ) and (; ) are well defined for  ∈   and  ∈ R ([1], Section 3).
In this section we will prove that ( − ) and (; ) are elements of D   ,() , 2 ≤  ≤ ∞, as a function of  ∈ R for  ∈   .Theorem 13.Let  be a regular cone in R and let 2 ≤  ≤ ∞.Then ( − ) ∈ D   ,() as a function of  ∈ R for  ∈   .
Hence we have from Lebesgue's dominated convergence theorem that the last line of (41) converges to 0 as  → 0 for  ∈ .
It remains to prove the convergence for  = 1, that is, in D  ∞ ,() .By the same method of estimation of integrand in (41), we get that (45) Now we will give the representation of a convolutor  in D   ,() , 1 ≤  ≤ 2, as the distributional limit of a finite number of functions analytic in tubes.
The analyticity and growth of   (),  = 1, . . ., , are followed by Theorem 16.Using Lemma 20, the linearity of , and ( 46 The proof is complete. Remark 22.As mentioned in page 246 of [2], under certain conditions on  ∈ D    , the Cauchy integral of  does have  as boundary value.But we see from Lemma 20 that the Cauchy integral of  ∈ D    ,() , 1 ≤  ≤ 2, corresponding to a regular cone  in R  does not attain  as boundary value as  → 0,  ∈ .We cannot find the conditions on  ∈ D    ,() , 1 ≤  ≤ 2, such that the Cauchy integral of  ∈ D    ,() , 1 ≤  ≤ 2, does attain  as boundary value.

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et al. defined the convolutors in Journal of Mathematics and Mathematical Sciences D