IJMMS International Journal of Mathematics and Mathematical Sciences 1687-0425 0161-1712 Hindawi Publishing Corporation 926790 10.1155/2014/926790 926790 Research Article Cauchy and Poisson Integral of the Convolutor in Beurling Ultradistributions of L p -Growth Sohn Byung Keun Kalla Shyam L. Department of Mathematics Inje University Gimhae 621-749, Gyeongnam Republic of Korea inje.ac.kr 2014 942014 2014 11 12 2013 28 02 2014 16 03 2014 9 4 2014 2014 Copyright © 2014 Byung Keun Sohn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

1. Introduction

Let C be a regular cone in n and let 𝒪 ( C ) denote its convex envelope. In  (or ) Carmichael defined the Cauchy and Poisson integrals for Schwartz distributions 𝒟 L p , 1 < p 2 , corresponding to tubular domain T 𝒪 ( C ) = n + i C n . Carmichael obtained the boundary values of these integrals in the distributional sense on the boundary of T 𝒪 ( C ) and found the relation between analytic functions with a specific growth condition in T 𝒪 ( C ) and the Cauchy and Poisson integrals of their distributional boundary values. In  Pilipović defined ultradistributions 𝒟 ( ( M p ) , L 2 ) of L 2 -growth, where M p , p = 0,1 , 2 , , is a certain sequence of positive numbers, and studied the Cauchy and Poisson integrals for elements of 𝒟 ( ( M p ) , L 2 ) in the case that the Cauchy and Poisson kernel functions are defined corresponding to the first quadrant { y n ; y j > 0 , j = 1,2 , , n } in n . Pilipović showed that elements in 𝒟 ( ( M p ) , L 2 ) are boundary values of suitable analytic functions with a certain L 2 -norm condition by means of the Cauchy integral representation and an analytic function with a certain L 2 -norm condition determines, as a boundary value, an element from 𝒟 ( ( M p ) , L 2 ) . In  Carmichael et al. defined ultradistributions of Beurling type 𝒟 ( ( M p ) , L s ) of L s -growth and of Roumieu type 𝒟 ( { M p } , L s ) of L s -growth, both of which generalize the Schwartz distributions 𝒟 L p , and studied the Cauchy and Poisson integrals for elements of both 𝒟 ( ( M p ) , L s ) and 𝒟 ( { M p } , L s ) for 2 s corresponding to the arbitrary tubes T C = n + i C n where C is an open connected cone in n of which the quadrants are special cases. They showed that the Cauchy integral for elements of both 𝒟 ( ( M p ) , L s ) and 𝒟 ( { M p } , L s ) for 2 s corresponding to T C is shown to be analytic in T C , 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 ( ( M p ) , L s ) and 𝒟 ( { M p } , L s ) for 2 s corresponding to T C 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  and for various kinds of distributions in .

In the meantime, Betancor et al.  introduced the spaces of Beurling ultradistributions of L p -growth, 𝒟 L p , ( 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 . Betancor et al. defined the convolutors in 𝒟 L p , ( ω ) , 1 p ; that is, the functionals U 𝒟 L p , ( ω ) such that U * ϕ 𝒟 L p , ( ω ) for every ϕ 𝒟 L p , ( ω ) , and studied the convolutors and the surjective convolution operators acting on 𝒟 L p , ( ω ) in .

In this paper we define the Cauchy and Poisson integrals for the convolutors in 𝒟 L p , ( ω ) , 1 p 2 . We will show that the Cauchy integrals for the convolutors in 𝒟 L p , ( ω ) , 1 p 2 , are analytic in a tubular domain T C = + i C where C is a regular cone in and satisfies a certain boundedness condition. We will give the representation of the convolutors in 𝒟 L p , ( 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 𝒟 L p , ( ω ) , 1 p 2 , attain U as boundary value in the distributional sense.

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

2. Beurling Ultradistributions of <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M86"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>L</mml:mi></mml:mrow> <mml:mrow> <mml:mi>p</mml:mi></mml:mrow> </mml:msub></mml:mrow> </mml:math></inline-formula>-Growth

In this section we will review Beurling ultradistributions of L p -growth in which is introduced by Betancor et al. in  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 .

Definition 1.

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

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

1 ( ω ( t ) / t 2 ) d t < ,

log ( t ) = o ( ω ( t ) ) as t tends to ,

ψ : t ω ( e t ) is convex.

By ( β ) ,

Lemma 2.

Consider the following: (1) lim t ω ( t ) t = 0 .

By ( δ ) , ψ ( 0 ) = 0 and lim x x / ψ ( x ) = 0 . Then we can define the Young conjugate ψ * of ψ by (2) ψ * : [ 0 , ) , ψ * ( y ) = sup x 0 ( x y - 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 lim y y / ψ * ( y ) = 0 .

(iii) ( ψ * ) * = ψ .

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

Let ω be a weight function. For a compact set K , we define the following: (3) 𝒟 ( ω ) ( K ) = { f 𝒟 ( K ) : f K , λ    < for every λ > 0 } , where f K , λ = sup x K sup n 0 | f ( n ) ( x ) | exp ( - λ ψ * ( n / λ ) ) .

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

We denote by ( 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 ( ω ) , we refer to .

A function ϕ C ( ) is in the space 𝒮 ( ω ) when, for every m , n , (4) p m , n = sup x e m ω ( x ) | ϕ ( n ) ( x ) | < , π m , n = sup x e m ω ( x ) | ϕ ^ ( n ) ( x ) | < .

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

For every 1 p , k and ϕ C ( ) , and γ k , p ( ϕ ) is defined as follows: (5) γ k , p ( ϕ ) = sup α ϕ ( α ) p e - k ψ * ( α / k ) , where · p denotes the usual norm in L p ( ) . ( f means ess sup | f ( t ) | .) If 1 p < , the space 𝒟 L p , ( ω ) is the set of all C functions ϕ on such that γ k , p ( ϕ ) < for each k . We denote by 𝒟 L , ( ω ) the set of all bounded C functions ϕ on such that γ k , ( ϕ ) < . The topology of 𝒟 L p , ( ω ) , 1 p , is generated by the family { γ k , p ( ϕ ) } k of seminorms.

The dual of 𝒟 L p , ( ω ) will be denoted by 𝒟 L p , ( ω ) and it will be endowed with the strong topology. The elements of 𝒟 L p , ( ω ) are called the Beurling ultradistributions of L p -growth. For more details about 𝒟 L p , ( ω ) , we refer to .

Lemma 4.

(i) 𝒟 L p , ( ω ) , 1 p , is Fréchet spaces.

(ii) 𝒟 L p , ( ω ) is continuously contained in the Schwartz distributions 𝒟 L p , 1 p .

(iii) 𝒟 L p , ( ω ) is continuously contained in 𝒟 L q , ( ω ) , when 1 p q .

(iv) 𝒟 ( ω ) 𝒟 L p , ( ω ) ( ω ) , 1 p , with continuous and dense inclusions.

(v) 𝒮 ω is continuously contained in 𝒟 L 1 , ( ω ) .

Proof.

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

Definition 5.

The convolutors in 𝒟 L p , ( ω ) , 1 p , are the functionals T 𝒟 L p , ( ω ) such that T * ϕ 𝒟 L p , ( ω ) for every ϕ 𝒟 L p , ( ω ) .

Remark 6.

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

If T 𝒟 L 2 , ( ω ) and there exists k such that T ^ e - k ω L , then T is a convolutor in 𝒟 L p , ( ω ) for 1 / 2 p 2 .

In particular, if T 𝒟 L 1 , ( ω ) 𝒟 L , ( ω ) , then T is a convolutor in 𝒟 L 1 , ( ω ) .

We will consider the convolutors in 𝒟 L p , ( 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 T G on ( ω ) is defined by (6) T G , ϕ = α = 0 ( - 1 ) α G ( α ) ( 0 ) α ! ϕ ( α ) ( 0 ) , ϕ ( ω ) .

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

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 ) | M e l ω ( | z | ) when | Im z | < M | Re z | .

Now we will obtain the characterization of convolutors in 𝒟 L p , ( ω ) , 1 p .

Lemma 8.

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

Proof.

Let 1 p . Since T is a convolutor in 𝒟 L p , ( ω ) , T * ϕ 𝒟 L p , ( ω ) 𝒟 L p L p for every ϕ 𝒟 ( ω ) . By the proof of Proposition 2.3 in , 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 * 𝒟 L p , ( ω ) 𝒟 L p , ( ω ) since T is a convolutor in 𝒟 L p , ( 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  by (9) γ k , p ( h * g ) = sup α ( h * g ) ( α ) p e - k ψ * ( α / k ) C sup α h ( α ) p       e - k ψ * ( α / k ) = C γ k , 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 𝒟 L p , ( ω ) is contained in G ( D ) ( 𝒟 L p , ( ω ) ) .

Combining Lemmas 8 and 9 and given the fact that 𝒟 L p , ( ω ) ( ω ) , we have the characterization of the convolutor in 𝒟 L p , ( ω ) , 1 p .

Theorem 10.

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

3. The Cauchy and Poisson Kernel Functions

Let C be a regular cone in , 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 T C = + i C is a tube in . If C is open, T C is called a tubular cone. If C is open and connected, T C is called a tubular radial domain. The set C * = { t : t , y 0       for all       y 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 .

Lemma 11.

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

Definition 12.

Let C be a regular cone in . The Cauchy kernel K ( z - t ) , z T C = + i C , t , corresponding to the tube T C is (12) K ( z - t ) = C * exp ( 2 π i z - t , η ) d η , z T C , t .

The Poisson kernel P ( z ; t ) corresponding to the tube T C is (13) P ( z ; t ) = K ( z - t ) K ( z - t ) ¯ K ( 2 i y ) = | K ( z - t ) | 2 K ( 2 i y ) , h h h h h h h z = x + i y T C , t .

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

In this section we will prove that K ( z - t ) and P ( z ; t ) are elements of 𝒟 L q , ( ω ) , 2 q , as a function of t for z T C .

Theorem 13.

Let C be a regular cone in and let 2 p . Then K ( z - t ) 𝒟 L q , ( ω ) as a function of t for z T C .

Proof.

Let z = x + i y T C be arbitrary but fixed and let α . Let I C * denote the characteristic function of C * . By the proof of Theorem 4.4 . 1 in , K ( z - t ) C , I C * ( η ) η α e 2 π i z , η L 1 L p , 1 < p 2 , as a function of t for z T C , and (14) D t α K ( z - t ) = - 1 [ I C * ( η ) η α e 2 π i z , η ; t ] , h h h h h h h h h z T C , t , where the inverse transform - 1 can be interpreted in both L 1 and L p sense. If 1 < p 2 and 1 / p + 1 / q = 1 , we have from (14) and the Parseval inequality the following: (15) D t α K ( z - t ) q I C * ( η ) η α e 2 π i z , η p < .

Using (15) and Lemmas 3 (iii), 11, and 2, we get that, for every k and z T C , there exist δ > 0 such that (16) sup α D t α K ( z - t ) q e - k ψ * ( α / k ) sup α ( C * | η | p α e - 2 π p y , η d η ) 1 / p e - k ψ * ( α / k ) sup α ( C * e p α log | η | e - p k ψ * ( α / k ) e - 2 π p y , η d η ) 1 / p ( C * e p k ψ * * ( log | η | ) e - 2 π p y , η d η ) 1 / p ( C * e p k ψ ( log | η | ) e - 2 π p δ | y η | d η ) 1 / p ( C * e p | η | ( k · ( ω ( η ) / | η | ) - 2 π δ | y | ) d η ) 1 / p < .

Let p = 1 . We get from the same method of estimation of integrand in (16) that, for every k and z T C , there exist δ > 0 such that (17) sup α D t α ( z - t ) e - k ψ * ( α / k ) sup α C * | η | α e - 2 π p y , η e - k ψ * ( α / k ) d η C * e | η | ( k · ( ω ( η ) / | η | ) - 2 π δ | y | )    d η < .

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

Theorem 14.

Let C be a regular cone in and let 2 q . Then P ( z ; t ) 𝒟 L q , ( ω ) as a function of t for z T C .

Proof.

By Theorem 4.4 . 4 in , P ( z ; t ) C as a function of t for z T C . 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 , D t γ K ( z - t ) ¯ is bounded on . Hence, we get from Lemma 3 and Theorem 13 that, for every k , z T C , and 2 q , (18) γ k , q ( P ( z ; t ) ) = sup α D t α P ( z - t ) q e - k ψ * ( α / k ) = sup α 1 K ( 2 i y ) β + γ = α α ! β ! γ ! D t β K ( z - t ) D t γ K ( z - t ) ¯ q h h h h h h h h h h h h h h h h h × e - k ψ * ( α / k ) 1 K ( 2 i y ) sup α β + γ = α C γ , K α ! β ! γ ! D t β K ( z - t ) q       h h h h h h h h h h h h h h h h × e - k ψ * ( β / k ) = 1 K ( 2 i y ) β + γ = α C γ , K α ! β ! γ ! γ k , q ( K ( z - t ) ) < .

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

Let C be a regular cone in and 1 p 2 . By Theorem 13, K ( z - t ) 𝒟 L q , ( ω ) , 1 / p + 1 / q = 1 , as a function of t for z T C . Hence U t , K ( z - t ) is well defined for U 𝒟 L p , ( ω ) , 1 p 2 .

Definition 15.

Let C be a regular cone in and let U be a convolutor in 𝒟 L p , ( ω ) , 1 p 2 . The Cauchy integral C ( U ; z ) of U corresponding to C is (19) C ( U ; z ) = U t , K ( z - t ) ,    z T C .

Theorem 16.

Let C be a regular cone in and let U be a convolutor in 𝒟 L p , ( ω ) , 1 p 2 . The Cauchy integral C ( U ; z ) of U corresponding to C is analytic in T C and for any compact subset C C and z = x + i y T C , 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 𝒟 L p , ( ω ) ( ω ) , 1 p 2 , such that U = G ( D ) f ; that is, (21) U ( x ) = ( G ( D ) f ) ( x ) = α = 0 i α G ( α ) ( 0 ) α ! f ( α ) ( x ) , x .

Then (22) C ( U ; z ) = G ( D ) f ( t ) , K ( z - t ) = α = 0 i α G ( α ) ( 0 ) α ! f ( α ) ( t ) C * e 2 π i z - t , η d η d t ; hence (23) D z β C ( U ; z ) = ( - 1 ) β G ( D ) f ( t ) , D z β K ( z - t ) = ( - 1 ) β α = 0 i α G ( α ) ( 0 ) α ! f ( α ) ( t ) h h h h h h h h h h h h × C * η β e 2 π i z - t , η d η d t .

Now we will apply the method in the proof of Proposition 2.4 of  to L p -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 .

If we let C R = { z ; | z | = e r ,    r > 0 } , we get that, by Cauchy's theorem and (24), (25) | G ( α ) ( 0 ) α ! | = 1 2 π | C R G ( z ) z α + 1 d z | 1 2 π C R e k e k ω ( | z | ) | z | α + 1 | d z | = 1 2 π e k e k ω ( e r ) e α ( r + 1 ) · e r = ( 2 π ) - 1 e k · e k ψ ( r ) - α r .

Let m and let L be as in Lemma 3 (iv). Take h max { k L , m } . By Lemma 4 (iii), we have the following: (26) f 𝒟 p , ( ω ) 𝒟 h , ( ω ) , 1 p 2 .

Since ψ * ( t ) / t is increasing by Lemma 3 (ii), we get that, for every α 0 , (27) α + h ψ * ( α h ) α + k L ψ * ( α k L ) k ψ * ( α h ) + k L .

Using (25), (26), and (27), (28) α = 0 | G ( α ) ( 0 ) α ! | p | f ( α ) ( t ) | p d t = α = 0 ( 2 π ) - 1 e p k e p ( k ψ ( r ) - α r ) γ h , p ( f ) e h ψ * ( α / h ) α = 0 γ h , p ( f ) ( 2 π ) - 1 e p k e - k ψ * ( α / k ) e h ψ * ( α / h ) α = 0 γ h , p ( f ) ( 2 π ) - 1 e p k e k L - α C h , k , p , L    γ h , p ( f ) .

We will consider the L q -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 , there exist δ > 0 as in Lemma 11 such that if 1 < p 2 and 1 / p + 1 / q = 1 , (29) D t β K ( z - t ) q = ( C * | η | β q e 2 π q z - t , η d η ) 1 / q = - 1 [ I C * ( η ) η β e 2 π i z , η ; t ] q I C * ( η ) η β e 2 π i z , η p ( 2 Γ ( p β + 1 ) ) 1 / p ( 2 π p δ | y | ) ( - p β - 1 ) / p .

Let K be an arbitrary compact subset of T C . For z K T C , 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 + i y K T C , 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 | β q e 2 π q z - t , η d η ) 1 / q ( 2 Γ ( p β + 1 ) ) 1 / p ( 2 π p δ | l | ) ( - p β - 1 ) / p , where Γ ( x ) is the gamma function. By (23), (28), and (30), consider the following: (31) | D z β C ( U ; z ) | α = 0 G ( α ) ( 0 ) α ! f ( α ) ( t ) p ( C * | η | β q e 2 π q z - t , η d η ) 1 / q C h , k , p , L γ h , p ( f ) ( 2 Γ ( p β + 1 ) ) 1 / p ( 2 π p δ l ) ( - p β - 1 ) / p for compact subcone C C and z = x + i y K T C . Equation (31) shows that any derivatives of C ( U ; z ) with respect to z converge absolutely and uniformly for all z K T C ; hence C ( U ; z ) is analytic in T C .

It remains to prove the analyticity of C ( U ; z ) for p = 1 . By Proposition 2.4 in , G ( D ) f 𝒟 L 1 , ( ω ) for f 𝒟 L 1 , ( ω ) . Since 𝒟 L 1 , ( ω ) 𝒟 L 1 L 1 , 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 𝒟 L 1 of Theorem 5.5 . 1 in .

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 ) | C h , k , p , L γ h , p ( f ) ( 2 Γ ( 1 ) ) 1 / p ( 2 π p δ ) - 1 / p | y | - 1 / p for compact subcone C C , z = x + i y T C , and δ = δ ( C ) . If p = 1 , we get that, by (29) with p = 1 and β = 0 , (33) | I C * e 2 π i z - t , η d η | 2 Γ ( 1 ) ( 2 π δ ) - 1 | y | - 1 ; hence (34) | C ( U ; z ) | C h , k , 1 , L γ h , 1 ( f ) 2 Γ ( 1 ) ( 2 π δ ) - 1 | y | - 1 for compact subcone C C , z = x + i y T C , 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 + i y T C .

Now we will represent a convolutor U in 𝒟 L p , ( 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 and let U be a convolutor in 𝒟 L p , ( 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 ) 𝒟 L p , ( ω ) by Theorem 13, (37) K y = K ( x + i y ) = C * e 2 π i x + i y , η d η 𝒟 L p , ( ω ) , y C .

Since U is a convolutor in 𝒟 L p , ( ω ) , ( U ; z ) = ( U * K y ) ( x ) 𝒟 L p , ( ω ) . Since 𝒟 L p , ( ω ) 𝒟 L p , ( ω ) and 𝒮 ( ω ) 𝒟 L p , ( ω ) , C ( U ; z ) , φ ( x ) = ( U * K y ) ( 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 𝒟 L p , ( ω ) ( ω ) , 1 p 2 , such that (38) C ( U ; z ) , φ ( x ) = U , K ( z - t ) , φ ( x ) = α = 0 ( - 1 ) α G ( α ) ( 0 ) α ! f ( α ) ( x ) K ( z - t ) d t φ ( x ) d x = α = 0 ( - 1 ) α G ( α ) ( 0 ) α ! f ( α ) ( x ) K ( z - t ) φ ( x ) d x d t = α = 0 ( - 1 ) α G ( α ) ( 0 ) α ! f ( α ) ( x ) K ( z - t ) φ ( x ) d x d t = 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 and φ 𝒮 ( ω ) . Then (39) lim y 0 K ( x + i y - t ) , φ ( x ) = - 1 [ I C * ( η ) φ ^ ( η ) ; t ] , h h h h h h h h h h h h y C , in 𝒟 L q , ( ω ) , 1 p 2 , 1 / p + 1 / q = 1 , where I C * ( η ) is the characteristic function in C * .

Proof.

Let 1 < p 2 and let 1 / p + 1 / q = 1 . For an arbitrary α , we get from (14) and (15) that (40) D t α K ( z - t ) , φ ( x ) - D t α - 1 [ I C * ( η ) φ ^ ( η ) ; t ] q = - 1 [ I C * ( η ) η α φ ^ ( η ) ( e - 2 π y , η - 1 ) ; t ] q I C * ( η ) η α φ ^ ( η ) ( e - 2 π y , η - 1 ) p .

For any k and q , 1 / p + 1 / q = 1 , we get that by (40) (41) γ k , q ( K ( z - t ) , φ ( x ) - - 1 [ I C * ( η ) φ ^ ( η ) ; t ] ) = sup α D t α K ( z - t ) , φ ( x ) - D t α - 1 [ I C * ( η ) φ ^ ( η ) ; t ] q × e - k ψ * ( α / k ) sup α I C * ( η ) | η | α φ ^ ( η ) ( e - 2 π y , η - 1 ) p e - k ψ * ( α / k ) = sup α I C * ( η ) e α log | η | e - k ψ * ( α / k ) φ ^ ( η ) ( e - 2 π y , η - 1 ) p = I C * ( η ) e k ψ * * ( log | η | ) φ ^ ( η ) ( e - 2 π y , η - 1 ) p = I C * ( η ) e k ω ( η ) φ ^ ( η ) ( e - 2 π y , η - 1 ) p .

Since (42) | I C * ( η ) e k ω ( η ) φ ^ ( η ) ( e - 2 π y , η - 1 ) | | e k ω ( η ) φ ^ ( η ) | and φ ^ ( η ) 𝒮 ( ω ) , (43) I C * ( η ) e k ω ( η ) φ ^ ( η ) ( e - 2 π y , η - 1 ) L 1 L p .

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 𝒟 L , ( ω ) . By the same method of estimation of integrand in (41), we get that (44) γ k , ( K ( z - t ) , φ ( x ) - - 1 [ I C * ( η ) φ ^ ( η ) ; t ] ) = sup α D t α K ( z - t ) , φ ( x ) - D t α - 1 [ I C * ( η ) φ ^ ( η ) ; t ] n n n n n n n n n - D t α - 1 [ I C * ( η ) φ ^ ( η ) ; t ] e - k ψ * ( α / k ) = sup α - 1 [ I C * ( η ) η α φ ^ ( η ) ( e - 2 π y , η - 1 ) ; t ] e - k ψ * ( α / k ) = sup α e i η , t I C * ( t ) t α φ ^ ( t ) ( e - 2 π y , t - 1 ) d t e - k ψ * ( α / k ) = I C * ( t ) e k ω ( t ) φ ^ ( t ) ( e - 2 π y , t - 1 ) d t .

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 𝒟 L p , ( ω ) , consider the following:

Lemma 20.

Let C be a regular cone in and let U be a convolutor in 𝒟 L p , ( ω ) , 1 p 2 . If φ 𝒮 ( ω ) , one has the following: (45) lim y 0 C ( U ; z ) , φ ( x ) = U , - 1 [ I C * ( η ) φ ^ ( η ) ; t ] , h h h h h h h h h y = Im ( z ) C .

Now we will give the representation of a convolutor U in 𝒟 L p , ( 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 𝒟 L p , ( ω ) , 1 p 2 , and let φ 𝒮 ( ω ) . Let C j , j = 1 , , r , be a finite number of regular cones whose dual cone satisfies the property that (46) j = 1 r C j * , C j * C k * , j k , j , k = 1 , , r , are sets of Lebesgue’s measure 0 . Then there exist functions U j which are analytic in + i C j , j = 1 , , r , and satisfy (47) | U j ( z ) | M ( p , C j ) | y | - 1 / p for an arbitrary compact subcone C j C j and a constant M ( p , C j ) depending on p and C j and (48) U , φ = j = r r lim y 0 U j ( x + i y ) , φ ( x ) , y C j .

Proof.

Let (49) U j = U , C j * e 2 π i z - t , η d η , z = x + i y + i C j .

The analyticity and growth of U j ( z ) , j = 1 , , r , are followed by Theorem 16. Using Lemma 20, the linearity of U , and (46), if y C j , (50) j = r r lim y 0 U j ( x + i y ) , φ ( x ) = j = r r U , - 1 [ I C j * ( η ) φ ^ ( η ) ; t ] = U , j = r r - 1 [ I C j * ( η ) φ ^ ( η ) ; t ] = U , - 1 [ φ ^ ( η ) ; t ] = U , φ .

The proof is complete.

Remark 22.

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

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Let C be a regular cone in and let 2 q . In Theorem 14, we showed that P ( z ; t ) 𝒟 L q , ( ω ) as a function of t for z T C . Hence, for U 𝒟 L p , ( ω ) , 1 p 2 , U t , P ( z ; t ) is a well-defined function of t for z T C .

Definition 23.

Let C be a regular cone in and let U be a convolutor in 𝒟 L p , ( ω ) , 1 p 2 . The Poisson integral PI ( U ; z ) of U corresponding to C is (51) PI ( U ; z ) = U t , P ( z ; t ) , z T C .

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

Definition 24 (see [<xref ref-type="bibr" rid="B1">6</xref>], Definition <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M616"> <mml:mrow> <mml:mn mathvariant="normal">3</mml:mn></mml:mrow> </mml:math></inline-formula>).

Let S 𝒟 L p , ( ω ) and T 𝒟 L q , ( 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 ~ ) .

Lemma 25 (see [<xref ref-type="bibr" rid="B5">2</xref>], Lemma <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M624"> <mml:mn>5.5</mml:mn> <mml:mo>.</mml:mo> <mml:mn>9</mml:mn></mml:math> </inline-formula>).

Let C be a regular cone and g L p , 1 p < . One has (52) lim y 0 g ( t ) P ( x + i y ; t ) d t = g ( t ) , y C , in L p .

Lemma 26.

Let C be a regular cone in and let U be a convolutor in 𝒟 L p , ( ω ) , 1 p 2 . For φ 𝒟 ( ω ) , (53) PI ( U ; z ) , φ ( x ) = U , P ( z ; t ) , φ ( x ) , h h h h h y = Im ( z ) C .

Proof.

For φ 𝒟 ( ω ) , (54) P ( x + i y ; t ) φ ( x ) d x = P ( x , y ) φ ( x + t ) d x ,    y C , where (55) P ( x , y ) = K ( x + i y ) K ( x + i y ) ¯ K ( 2 i y ) , x , y C .

Since P ( x , y ) 𝒟 L q , ( ω ) 𝒟 L q , ( ω ) , 2 q , as a function of x and y C , we get from Definition 24 that, for a convolutor U in 𝒟 L p , ( 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 𝒟 L p , ( ω ) ( ω ) , 1 p 2 , such that (58) U , P ( z ; t ) , φ ( x ) = α = 0 ( - 1 ) α G ( α ) ( 0 ) α ! f ( α ) ( x ) P ( z ; t ) φ ( x ) d x d t = φ ( x ) α = 0 ( - 1 ) α G ( α ) ( 0 ) α ! f ( α ) ( x ) P ( z ; t ) d t d x = PI ( U ; z ) , φ ( x ) .

Lemma 27.

Let C be a regular cone in and φ 𝒟 ( ω ) . Consider the following: (59) lim y 0 P ( x + i y ; t ) φ ( x ) d x = φ ( t ) , y C , in the topology of 𝒟 L q , ( ω ) , 1 q .

Proof.

Let α be an arbitrary integer. By (ii) and (iv) in Lemma 4, φ 𝒟 ( ω ) 𝒟 L 1 , ( ω ) 𝒟 L 1 L 1 . We get from (5.92) in the proof of Lemma 5.5 . 11 of  that, for 1 p , (60) lim y 0 D t α P ( x + i y ; t ) φ ( x ) d x - D t α φ ( t ) q = 0 , y C .

Thus for any k and 2 q (61) lim y 0 sup α D t α P ( x + i y ; t ) φ ( x ) d x - D t α φ ( t ) q × e - k ψ * ( k / α ) = 0 , y C .

Combining Lemmas 26 and 27, and the continuity of U 𝒟 L p , ( 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 and let U be a convolutor in 𝒟 L p , ( ω ) , 1 p 2 . For φ 𝒟 ( ω ) , (62) lim y 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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