Holomorphic Functions

RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL . 97 (2), 2003, pp. 243–255 An´alisis Matem´atico / Mathematical Analysis

(Ultra)distributions of Lp -growth as Boundary Values of Holomorphic Functions ´ ´ C. Fernandez, A. Galbis and M.C. Gomez-Collado

To the memory of I. Cioranescu and K. Floret Abstract. We study the representation of distributions (and ultradistributions of Beurling type) of Lp -growth, 1 ≤ p ≤ ∞, on RN as boundary values of holomorphic functions on (C \ R)N .

(Ultra)distribuciones de crecimiento Lp como valor frontera de funciones holomorfas Resumen. Estudiamos la representaci´on de distribuciones (y ultradistribuciones de tipo Beurling) en RN con crecimiento Lp , 1 ≤ p ≤ ∞, como valor frontera de funciones holomorfas en (C \ R)N .

1.

Introduction

Shortly after Schwartz introduced his theory of distributions, K¨othe represented distributions on the unit circle as boundary values of holomorphic functions on its complementary and his results were generalized by Tillmann. Since then, many authors have been concerned with the problem of representing several classes of distributions and ultradistributions as boundary values of holomorphic functions. Let us mention the work of Bengel [1], Carmichael [4], Luszczki and Zielezny [13], Meise [14], Petzsche and Vogt [17], Tillmann [20] and Vogt [22]. See also the section 4 of the recent paper [12]. In 1994, Carmichael and Pilipovi´c [6] represented each ultradistribution of Lp -growth (1 < p < ∞) in RN as the boundary value of a holomorphic function satisfying appropriate estimates and conversely, every such a function is shown to have an ultradistribution of Lp -growth as boundary value. The boundary value problem for distributions and ultradistributions of L∞ or L1 - growth is more involved. In fact, for p = 1 or p = ∞, Carmichael and Pilipovi´c only obtained partial results and their methods did not permit to prove the surjectivity of the corresponding boundary value operators. They worked in the context of ultradistributions as they were defined by Komatsu [11]. In [8] the authors completely solved the problem of representing bounded distributions and ultradistributions on R as boundary values of holomorphic functions in C \ R. The case of bounded distributions had not been treated previously in the literature. The lack of nice topological properties of the involved spaces does not permit us to apply the tensor techniques as in [16, 3.6] in order to extend the results obtained in [8] to the several variables setting. Presentado por Jos´e Bonet. Recibido: 5 de Febrero de 2003. Aceptado: 31 de Octubre de 2003. Palabras clave / Keywords: Distributions, ultradistributions, boundary values. Mathematics Subject Classifications: 46F05, 46F20. c 2003 Real Academia de Ciencias, Espa˜na.

243

C. Fern´andez, A. Galbis, M.C. G´omez-Collado

The results of Petzsche and Vogt [17] and the characterizations of ultradistributions of Lp -growth recently obtained in [2], which are in the spirit of the ones given by Cioranescu [7] and G´omez-Collado [9], allow us to define a class of holomorphic functions on (C \ R)N having ultradistributions (of Beurling type) of Lp -growth as boundary values and such that the boundary value operator is surjective. We work with ultradistributions in the sense of Braun, Meise and Taylor [3]. Our approach is different from the one of Carmichael and Pilipovi´c and permits a unified treatment of all values of p, including the limit cases p = 1 and p = ∞. Of course the most interesting cases are the extreme values p = 1, ∞. Our results also cover the classical spaces (DL1 (RN ))0 and (DL∞ (RN ))0 . We would like to emphasize that no satisfactory answer to the boundary value problem in these cases was previously known.

2.

Preliminaries and statement of the problem

First we introduce the spaces of functions and ultradistributions and most of the notation that will be used in the sequel. Definition 1 ([3]) A weight function is an increasing continuous function ω : [0, ∞[−→ [0, ∞[ with the following properties: (α) there exists L ≥ 1 with ω(2t) ≤ L(ω(t) + 1) for all t ≥ 0, Z ∞ ω(t) dt < ∞, (β) t2 1 (γ) log(t) = o(ω(t)) as t tends to ∞, (δ) ϕ : t → ω(et ) is convex. For most of the results of the paper we have to replace the condition (α) by the stronger condition (α1 ) sup limsup λ≥1 t→∞

ω(λt) < ∞. λω(t)

For a weight function ω we define ω e : CN −→ [0, ∞[ by ω e (z) = ω(|z|) and again call this function ω, PN by abuse of notation. Here |z| = j=1 |zj |. The function ϕ∗ : [0, ∞[→ R defined by ϕ∗ (s) := sup{st − ϕ(t)} t≥0

is called the Young conjugate of ϕ. There is no loss of generality to assume that ω vanishes on [0, 1]. Then ϕ∗ has only non-negative values and ϕ∗∗ = ϕ. Examples of weight functions can be found in [3]. Remark 1 If the weight ω(t) is concave for t large enough then every equivalent weight satisfies (α1 ). See [17, 1.1] for details.  Definition 2 ([3]) Let ω be a weight function. For a compact set K ⊂ RN we let D(ω) (K) := {f ∈ D(K) : k f kK,λ < ∞ where

244

f or every λ > 0},

   |α| . k f kK,λ := sup sup |f (α) (x)| exp −λϕ∗ λ x∈K α∈NN 0

(Ultra)distributions of Lp -growth as boundary values of holomorphic functions

Then D(ω) (K), endowed with its natural topology, is a Fr´echet space. For a fundamental sequence (Kj )j∈N of compact subsets of RN we let D(ω) (RN ) := ind D(ω) (Kj ). j→

0 D(ω) (RN )

0 The dual of D(ω) (R ) is endowed with its strong topology. The elements of D(ω) (RN ) are called ultradistributions of Beurling type. N

We denote by E(ω) (RN ) the set of all functions f ∈ C ∞ (RN ) such that k f kK,λ < ∞ for every compact K in RN and for every λ > 0. Definition 3 ([2]) For every 1 ≤ p ≤ ∞, k ∈ N and φ ∈ C ∞ (RN ), γk,p (φ) is defined as follows    |α| (α) ∗ , γk,p (φ) = sup kφ kp exp −kϕ k α∈NN 0 where k.kp denotes the usual norm in Lp (RN ). If 1 ≤ p < ∞ the space DLp ,(ω) (RN ) is the set of all C ∞ -functions φ on RN such that γk,p (φ) < ∞ for each k ∈ N. A function φ ∈ C ∞ (RN ) is in BL∞ ,(ω) (RN ) when γk,∞ (φ) < ∞, for every k ∈ N. We denote by DL∞ ,(ω) (RN ) the subspace of BL∞ ,(ω) (RN ) consisting of those functions φ ∈ BL∞ ,(ω) (RN ) for which lim |φ(α) (x)| = 0 for each α ∈ NN 0 . |x|→∞

The topology of DLp ,(ω) (RN ), 1 ≤ p ≤ ∞, is generated by the family of seminorms {γk,p }k∈N . Also, we consider on BL∞ ,(ω) (RN ) the topology associated with {γk,∞ }k∈N . Then DLp ,(ω) (RN ), 1 ≤ p ≤ ∞ and BL∞ ,(ω) (RN ) are Fr´echet spaces. For the definition of the spaces DLp (RN ), 1 ≤ p ≤ ∞, we refer to [19, VI,8]. Remark 2 (a) Clearly DLp ,(ω) (RN ) is continuously contained in DLp (RN ), 1 ≤ p ≤ ∞. Hence, if φ ∈ DLp ,(ω) (RN ), 1 ≤ p ≤ ∞, then lim |φ(α) (x)| = 0, for each α ∈ NN 0 . |x|→∞

(b) The inclusions D(ω) (RN ) ⊂ DLp ,(ω) (RN ) ⊂ E(ω) (RN ) are continuous and dense, and for 1 ≤ p ≤ q ≤ ∞, DLp ,(ω) (RN ) is continuously contained in DLq ,(ω) (RN ). (c) Although the paper [2] was written in the one variable setting, all the results in its section 2 also hold for N > 1 with the same proofs.  The dual of DLp ,(ω) (RN ) will be denoted by (DLp ,(ω) (RN ))0 and it will be endowed with the strong topology. Since D(ω) (RN ) is continuously and densely embedded in DLp ,(ω) (RN ) then (DLp ,(ω) (RN ))0 0 can be identified with a subspace of D(ω) (RN ). The elements of (DLp ,(ω) (RN ))0 are known as ultradistributions of Beurling type of Lp0 -growth where p0 is the conjugate exponent of p. The ultradistributions of L∞ -growth are called bounded ultradistributions of Beurling type. The classical case DLp (RN ) is formally not a particular case of what we present here since ω(t) = log(1 + t) does not satisfy property (γ). However, all our results also hold in this case after minor modifications. Let G ∈ H(CN ) be an entire function such that log |G(z)| = O(ω(|z|)) as |z| tends to infinity. Then TG (ϕ) :=

X α∈NN 0

(−i)|α|

G(α) (0) (α) ϕ (0) α!

0 defines an ultradistribution TG ∈ E(ω) (RN ). The operator 0 0 G(D) : D(ω) (RN ) → D(ω) (RN ),

G(D)ν := TG ∗ ν 245

C. Fern´andez, A. Galbis, M.C. G´omez-Collado

is called an ultradifferential operator of (ω)-class. We note that, for every f ∈ E(ω) (RN ), (G(D)f )(x) =

X

(i)|α|

α∈NN 0

G(α) (0) (α) f (x). α! (α)

∗ |α|

It can be easily shown that there are constants C > 0 and k ∈ N such that | G α!(0) | ≤ Ce−kϕ ( k ) . As in [9, Proposition 2.4] it can be shown that each ultradifferential operator G(D) of (ω)-class defines a continuous linear mapping from DLp ,(ω) (RN ) into itself, for every 1 ≤ p ≤ ∞ and also from BL∞ ,(ω) (RN ) into itself. Thus, G(D) is also a continuous linear operator from (DLp ,(ω) (RN ))0 into itself. The problem we are concerned with consists in finding a weighted space HωN∗ ,p of holomorphic functions on (C \ R)N such that the map T : HωN∗ ,p → (DLp0 ,(ω) (RN ))0 given by Z < T (f ), ϕ >:= lim+ →0

X

{

(

N Y

σj )f (x + iσ)} ϕ(x) dx

RN σ∈{−1,1}N j=1

is a well-defined, linear, continuous and surjective operator. The description of the appropriate space of holomorphic functions and its basic properties is the aim of the section 3, while the boundary value operator is investigated in section 4.

The spaces HωN∗ ,p and HpN

3.

In [8] we defined the weighted (LB)-spaces of holomorphic functions on C \ R, Hω∗ , and we showed that T : Hω∗ → (DL1 ,(ω) (R))0 given by Z < T (f ), ϕ >:= lim (f (x + iε) − f (x − iε))ϕ(x)dx ε→0+

R

is well-defined, continuous, linear and surjective. The natural extension of Hω∗ to the several variables case and for arbitrary 1 ≤ p ≤ ∞ leads to the following definition. In what follows, ω will be either a weight function or ω(t) = log(1 + t), t ≥ 0. Definition 4 For s > 0, ω ∗ (s) is defined by ω ∗ (s) := sup{ω(t) − st}. t≥0

The function ω ∗ is continuous, convex and decreasing. Since each weight function ω satisfies that ω(t) = o(t) where t tends to ∞, ω ∗ (s) < ∞ for all s > 0. Given y ∈ R

N

∗

with yj 6= 0 for all 1 ≤ j ≤ N , we denote ω (y) :=

N X

ω ∗ (|yj |).

j=1

Definition 5 For a given ω, 1 ≤ p ≤ ∞ and N ∈ N, we define HωN∗ ,p := {f ∈ H((C \ R)N ) : where

 y  |f |ω,p,k := sup k f (· + iy) kp exp −k|y| − kω ∗ ( ) . k y

Then HωN∗ ,p is an (LB)-space. 246

|f |ω,p,k < ∞ for some k ∈ N}

(Ultra)distributions of Lp -growth as boundary values of holomorphic functions

Remark 3 For ω(t) = log(1 + t), t ≥ 0, we have that ω ∗ (s) = 0 for s ≥ 1 while ω ∗ (s) = s − log(es) whenever 0 < s < 1. Therefore, it is easy to see that in this case HpN := HωN∗ ,p can be described as HpN = {f ∈ H((C \ R)N ) : |f |p,k < ∞ for some k ∈ N} where  |f |p,k := sup k f (· + iy) kp e−2k|y|  y

N Y

k |yj | . 

j=1

For any weight ω, HpN ⊂ HωN∗ ,p and given two weights σ ≤ ω we have HσN∗ ,p ⊂ HωN∗ ,p with continuous inclusions. Since we can represent (ultra)distributions of Lp -growth as a (infinite) linear combination of derivatives of functions in Lp , we will show that the spaces just defined are stable under (ultra)differential operators. Lemma 1

(α) (1) For each f ∈ HpN and each α ∈ NN ∈ HpN . 0 we have f

(2) For each f ∈ HωN∗ ,p and each ultradifferential operator G(D) of class (ω) we have G(D)f ∈ HωN∗ ,p . P ROOF. Let f ∈ Hω∗ ,p (RN ) be given. We fix y ∈ RN such that yj 6= 0 for every 1 ≤ j ≤ N and we put ρj := 21 |yj |. Let Dρ be the polidisc of poliradius ρ := (ρ1 , . . . , ρN ). For each x ∈ RN and each α ∈ NN 0 , by the Cauchy integral formula Z f (x + iy + ξ) α! f (α) (x + iy) = dξ. (2πi)N Dρ ξ α+1 The function gξ (x) :=

f (x + iy + ξ) belongs to Lp (RN ) and ξ α+1 Z α! f (α) (· + iy) = gξ (·) dξ. (2πi)N Dρ

Therefore k f (α) (· + iy) kp ≤ Since k gξ kp ≤

N Y α! max k g k (2πρj ). ξ p (2π)N ξ∈Dρ j=1

∗ y 1 |f |ω,p,k e2k|y|+kω ( 2k ) for some constant k ∈ N, ρα+1

k f (α) (· + iy) kp ≤ |f |ω,p,k α! e2k|y|+kω

∗

y ( 2k )

αj N  Y 2 . |yj | j=1

If ω(t) = log(1 + t) (t ≥ 0) this gives (1). To show (2), let G(D) be an ultradifferential operator of class PN X ∗ αj ∗ |α| (ω). Then G(D)g = aα g α where |aα | ≤ Ce−mϕ ( m ) ≤ Ce−m j=1 ϕ ( m ) for some C > 0 and α∈NN 0

some m ∈ N. Thus k G(D)f (· + iy) kp

≤

X

|aα | k f (α) (· + iy) kp

α∈NN 0

≤ C|f |ω,p,k

∗ αj (m

αj −mϕ N  Y 2 α! |yj | N j=1

X

)

e2k|y|+kω

∗

y ( 2k )

.

α∈N0

247

C. Fern´andez, A. Galbis, M.C. G´omez-Collado

That is, k G(D)f (· + iy) kp ≤ C |f |ω,p,k

N Y



∞ X

 j=1

αj ! e

α −mϕ∗ ( mj

αj =0

To finish it is enough to proceed as in [8, Prop.1].

)



2 |yj |

αj

  e2k|yj |+kω

∗

|yj | 2k




.



It is clear now that, given f ∈ HpN and y ∈ RN , yj 6= 0 for 1 ≤ j ≤ N, the function f (· + iy) belongs to DLp (RN ) for 1 ≤ p < ∞ and f (· + iy) ∈ BL∞ (RN ) for p = ∞. A similar result holds for arbitrary weight functions ω. Corollary 1

(1) For every 1 ≤ p < ∞ and each f ∈ HωN∗ ,p we have f (· + iy) ∈ DLp ,(ω) (RN ).

(2) Each f ∈ HωN∗ ,∞ satisfies f (· + iy) ∈ BL∞ ,(ω) (RN ). P ROOF. Let 1 ≤ p ≤ ∞ and f ∈ HωN∗ ,p be given. Then G(D)f (. + iy) ∈ Lp (RN ) for every ultradifferential operator G(D) of (ω)−class. Now it is enough to apply [2, Corollary 2.2] to conclude.  Now, it is easy to show that the spaces HωN∗ ,p increase with p. Corollary 2 For a fixed function ω and p < q we have HωN∗ ,p ⊂ HωN∗ ,q with continuous inclusion. P ROOF. Since DLp ,(ω) (RN ) ⊂ DLq ,(ω) (RN ) ⊂ BL∞ ,(ω) (RN ) with continuous inclusions ([2]), we deduce from Corollary 1 that for every f ∈ HωN∗ ,p there is a continuous seminorm γ on DLp ,(ω) (RN ) such that ||f (. + iy)||q ≤ γ(f (. + iy)) for every y ∈ RN with yj 6= 0 for 1 ≤ j ≤ N. Hence there is an ultradifferential operator G(D) of (ω)−class and there is a positive constant C satisfying ||f (. + iy)||q ≤ C||G(D)f (. + iy)||p for all y ∈ RN with yj 6= 0 for all 1 ≤ j ≤ N ([2, 2.0.4]). Since G(D)f ∈ HωN∗ ,p then it easily follows that f ∈ HωN∗ ,q . 

4.

Boundary values

In this section we will show that each function in HωN∗ ,p has an element of (DLp0 ,(ω) (RN ))0 , p1 + p10 = 1, as boundary value and that, conversely, each (ultra)distribution of Lp -growth can be obtained as the boundary value of a suitable f in HωN∗ ,p . From now on ω will be either a weight function satisfying (α1 ) or ω(t) = log(1 + t), t ≥ 0. We first observe that each f ∈ HωN∗ ,p belongs to HωN∗ ,∞ and, after applying [22] for ω(t) = log(1 + t) and [17] for ω a weight function satisfying (α1 ), we have the following result 0 Lemma 2 The boundary value operator T : HωN∗ ,p → D(ω) (RN ) given by

Z < T (f ), ϕ >:= lim+ →0

{

X

(

N Y

σj )f (x + iσ)} ϕ(x) dx

RN σ∈{−1,1}N j=1

is a well-defined, continuous and linear mapping. Moreover T (f ) ∈ (DL1 ,(ω) (RN ))0 for each f ∈ HωN∗ ,∞ and T is continuous as a map from HωN∗ ,∞ into (DL1 ,(ω) (RN ))0 . Therefore T is also continuous from HωN∗ ,p into (DL1 ,(ω) (RN ))0 . 248

(Ultra)distributions of Lp -growth as boundary values of holomorphic functions

P ROOF. In fact, T is nothing else but the restriction of the boundary value operator considered in [17] and [22]. Now it is enough to proceed as in the first part of the proof of Theorem 3 in [8].  Next, we show that the boundary value of a function in HωN∗ ,p is an (ultra)distribution of Lp -growth. Proposition 1 T (HωN∗ ,p ) is contained in (DLp0 ,(ω) (RN ))0 with X

T (f ) = lim+ →0

(

N Y

1 p

+

1 p0

= 1. Moreover,

σj )f (x + iσ)

σ∈{−1,1}N j=1

in the weak topology σ((DLp0 ,(ω) (RN ))0 , DLp0 ,(ω) (RN )). P ROOF. First we assume that N = 1 and that ω is a weight function. Given f ∈ Hω1 ∗ ,p we choose k ∈ N and C > 0 such that ∗ y max(k f (· + iy) k∞ , k f (· + iy) kp ) ≤ Cekω ( k ) (1) for 0 < y < 2. Without loss of generality we may assume that f ≡ 0 in the lower half-plane. We will show that {f (· + i) : 0 <  < 1} is a bounded set in (DLp0 ,(ω) (R))0 and that T (f ) ∗ ϕ ∈ Lp (R) for every ϕ ∈ D(ω) (R). We put fi (x) := f (x + i). Let ϕ ∈ D(ω) (R) be given and let b > 0 be such that supp ϕ ⊂] − b, b[. By [17, 3.4] we find φ ∈ D((−b, b) × (− 21 , 12 )) such that (i) φ|R = ϕ ∂ |y| kω ∗ ( k ) (ii) sup φ(x + iy) e < ∞. ¯ z∈C\R ∂ z Applying Stokes’ theorem to the function θx (ξ) := f (ξ + i)φ(x − ξ) in the rectangle Dx := [x − 2b, x + 2b] × [0, 1] we get that Z ∂ (fi ∗ ϕ)(x) = 2i f (x − t + i(v + )) φ(t − iv) d(t, v). ∂ z¯ D where D := [−2b, 2b] × [0, 1]. Therefore Z k fi ∗ ϕ kp ≤ 2 k f (· + i(v + )) kp D

∂ φ(t − iv) d(t, v), ∂ z¯

from where we conclude that {fi ∗ ϕ : 0 <  < 1} is a bounded set in Lp (R), which shows that {fi : 0 <  < 1} is bounded in (DLp0 ,(ω) (R))0 ([2]), hence equicontinuos. Moreover, for every null sequence of positive numbers (n )n one has (T (f ) ∗ ϕ)(x) = limn (fin ∗ ϕ)(x) pointwise and there is C > 0 with |(fin ∗ ϕ)(x)| ≤ C for every n ∈ N and each x ∈ R. Using Lebesgue’s dominated convergence theorem we get that {(T (f ) ∗ ϕ)χ[−n,n] : n ∈ N} is bounded in Lp (R), hence T (f ) ∗ ϕ ∈ Lp (R) and T (f ) ∈ (DLp0 ,(ω) (R))0 . Let us take a 0-neighbourhood V in DLp0 ,(ω) (R) such that T (f ) ∈ V o and fi ∈ V o for 0 <  < 1 and let τ denote the topology of pointwise convergence on the dense subspace D(ω) (R) of DLp0 ,(ω) (R). Then the weak topology and τ coincide on the equicontinuous set V o . Since R < T (f ), ϕ >= lim→0 R fi (x)ϕ(x)dx for every ϕ ∈ D(ω) (R) we get that T (f ) is the limit of (fi ) in the weak topology. If ω(t) = log(1 + t), given ϕ ∈ D(R) we choose k ∈ N satisfying (1) and we put φ(x, y) :=

Then

k X 1 (j) ϕ (x) (iy)j . j! j=0

1 ϕ(k+1) (x) ∂φ (x, y) = (iy)k and we proceed as above. See [16, 2.2]. ∂ z¯ 2 k! 249

C. Fern´andez, A. Galbis, M.C. G´omez-Collado

For N > 1 and ω a weight function, let σ ∈ {−1, 1}N and f ∈ HωN∗ ,p be given. We want to show that lim+ f (· + iσ) ∈ (DLp0 ,(ω) (RN ))0 and that {f (· + iσ) : 0 <  < 1} is bounded in (DLp0 ,(ω) (RN ))0 .

→0

To do this, let A be a real invertible matrix such that Ae1 = σ. We put g := f ◦ A and ϕ := Ψ ◦ A for Ψ ∈ D(ω) (RN ) and we denote fiσ (x) := f (x + iσ). Then, for each (x2 , . . . , xN ) ∈ RN −1 , g(·, x2 , . . . , xN ) ∈ H(C \ R) and Z (fiσ ∗ Ψ)(Ax) = |detA| g(x + t + ie1 )ϕ(−t)dt. RN

Let b > 0 satisfy supp ϕ ⊂ [−b, b]N . Since {ϕ(·, x2 , . . . , xN ) : (x2 , . . . , xN ) ∈ RN −1 } is bounded in D(ω) (R) we may find Φ(z, x2 , . . . , xN ), z ∈ C, x0 := (x2 , . . . , xN ) ∈ RN −1 having compact support such that (i) Φ(·, x2 , . . . , xN ) ∈ D((−b, b) × (− 21 , 21 )) and Φ(λ, x 2 , . . . , xN ) = ϕ(−λ, −x2 , . . . , −xN ), λ ∈ R. ∂ ∗ |Imz| ∂ Φ(z, x2 , . . . , xN ) is continuous and Φ(z, x0 ) ≤ Ce−kω ( k ) for every x0 ∈ RN −1 . (ii) ∂ z¯ ∂ z¯ As in the one-dimensional case, one can show that {(fiσ ∗ Ψ)(A·) : 0 <  < 1} is bounded in Lp (RN ), therefore {(fiσ ∗ Ψ) : 0 <  < 1} is bounded in Lp (RN ) and, again as for N = 1, the conclusion follows. In the case N > 1 and ω(t) = log(1 + t), we choose σ, A, f, Ψ, ϕ and b as above. For y ∈ R and k X (−1)j ∂ j ϕ(−x)(iy)j . x ∈ RN we put z = x1 + iy, x0 = (x2 , . . . , xN ) and φ(x1 , y, x2 , . . . , xN ) = j j! ∂x 1 j=0 Then ∂ 1 (−1)k+1 ∂ k+1 ϕ(−x)(iy)k . φ(z, x0 ) = ∂ z¯ 2 k! ∂xk+1 1 Now, we proceed as in the case of a weight function ω.



N Our next aim is to show that the boundary value map is surjective. Since Hω∗,p is closed under (ultra)differential operators and each (ultra)distribution of Lp -growth is essentially of the form G(D)f , where G(D) is a (ultra)differential operator and f ∈ Lp (RN ) ([2] and [19]), it suffices to show that N ). Lp (RN ) ⊂ T (HpN ) and, a fortiori, Lp (RN ) ⊂ T (Hω∗,p

Given N ∈ N we denote by GN the Banach space GN := {f ∈ H((C \ R)N ) :

k f k:= sup k f (. + iy) k1 e−|y| y

N Y

|yj | < ∞}.

j=1

Clearly G1 ⊗ GN ⊂ GN +1 and the canonical bilinear map G1 × GN −→ GN +1 induces a continuous ˆ π GN −→ GN +1 . linear map i : G1 ⊗ ˆ π D(RN ) such that its restricLemma 3 There exists a continuous linear map j : D(RN +1 ) −→ D(R)⊗ N tion to D(R) ⊗ D(R ) is the identity. P ROOF.

From a well known result of Grothendieck [10], for each compact K ⊂ R the bilinear map D(K) × D(K N ) −→ (φ, ψ) 

D(K N +1 ), φψ

ˆ π D(K N ) and D(K N +1 ), from where we coninduces an isomorphism of Fr´echet spaces between D(K)⊗ clude.  Proposition 2 For each N ∈ N there exists a continuous linear operator SN : D(RN ) −→ GN such that T ◦ SN gives the identity on D(RN ). 250

(Ultra)distributions of Lp -growth as boundary values of holomorphic functions

P ROOF. We proceed by induction on N . 0 0 For N = 1, each ϕ ∈ D(R) satisfies ϕ = −ie−ix (eix ϕ) + iϕ , hence ϕ = T (S1 (ϕ)), where 1 S1 (ϕ)(z) := 2π

Z R

e−iz ϕ(t) dt − (t − z)2 2π

Z R

eit ϕ(t) dt, (t − z)2

for Imz 6= 0 [8, 2.3]. For y 6= 0 we put Ky (t) = 1/(2π(t + iy)2 ). Then given z = x + iy, y 6= 0, we have S1 (ϕ)(z) = (ϕ ∗ Ky )(x) − e−iz (eit ϕ ∗ Ky )(x). Thus, k S1 (ϕ)(· + iy) k1 ≤ C

e|y| k ϕ k∞ . |y|

This shows that S1 : D(R) −→ G1 is well-defined, linear and continuous. b π SN ) ◦ j, which is linear and Let us assume that the claim is true for N . We define SN +1 := i ◦ (S1 ⊗ continuous. To show that T ◦SN +1 gives the identity on D(RN +1 ), it is enough to see that (T ◦SN +1 )(ϕ1 ⊗ ϕ2 ) = (ϕ1 ⊗ ϕ2 ) as distributions whenever ϕ1 ∈ D(R) and ϕ2 ∈ D(RN ), and again it is sufficient to check this equality on D(R) ⊗ D(RN ), which is very easy.  In the sequel, for each compact K ⊂ RN and m ∈ N, we denote by Dm (K) the set of all C m -functions f such that suppf ⊂ K. Corollary 3 For each compact K ⊂ RN , there are m ∈ N and a continuous linear map SN,K : Dm (K) −→ GN such that for every Γ ∈ Dm (K) and each ϕ ∈ D(RN ) we have that hT ◦ SN,K (Γ), ϕi = hΓ, ϕi. P ROOF. Let B be the closed unit ball in RN . We take an even function η ∈ D(B), η ≥ 0, with k η k1 = 1. For n ∈ N we define ηn (t) := nN η(nt) and we put K1 := K + B. Using the continuity of SN , we find ` ∈ N such that k SN (ϕ) k ≤ k ϕ kK1 ,` (2) for each ϕ ∈ D(K1 ). We put m = 2`. If Γ ∈ Dm (K), then Γ ∗ ηn ∈ D(K1 ), k Γ ∗ ηn kK1 ,` ≤ k Γ kK

(3)

and (Γ ∗ ηn )n converges to Γ in D` (K1 ). It follows from (2) that (SN (Γ ∗ ηn )) is a Cauchy sequence in GN , hence it converges and we may define SN,K (Γ) = lim SN (Γ ∗ ηn ). n→∞

Thus, SN,K : Dm (K) → GN is well defined, linear and by (2) and (3), it is continuous.



From now on, if f is a function, we denote by fˇ the map defined by fˇ(x) = f (−x). For g ∈ GN and f ∈ Lp (RN ) we put Z (g ∗ f )(x + iy) :=

g(x − t + iy)f (t)dt. RN

Then g ∗ f ∈ HpN . Moreover, we have the following result. Proposition 3 Let 1 ≤ p < ∞, ψ ∈ D(RN ) and f ∈ Lp (RN ) be given. Then SN (ψ) ∗ f ∈ HpN and T (SN (ψ) ∗ f ) = ψ ∗ f. 251

C. Fern´andez, A. Galbis, M.C. G´omez-Collado

P ROOF.

Given σ ∈ {−1, 1}N ,  > 0 and ϕ ∈ D(RN ), one has that fˇ ∗ ϕ ∈ DLp (RN ) and h(SN (ψ) ∗ f )(· + iσ), ϕi = hSN (ψ)(· + iσ), fˇ ∗ ϕi.

Since SN (ψ) ∈ H1N ⊂ HpN0 , p0 being the conjugate number of p, it suffices to apply Proposition 1 for ω(t) = log(1 + t) and Proposition 2.  The following lemma permit us to get a similar result for p = ∞. Lemma 4 Let K be a compact set in R and let ψ ∈ D(K) be given. Then S1 (ψ) ∈ H(C \ K) and, for some positive constants A and C, C |S1 (ψ)(x ± i)| ≤ |x|2 whenever |x| ≥ A and 0 <  < 1. P ROOF.

It is clear from the definition of S1 .



N Proposition 4 Given Γ ∈ D(RN ) and f ∈ L∞ (RN ) we have SN (Γ) ∗ f ∈ H∞ and T (SN (Γ) ∗ f ) = Γ ∗ f. N . To see that T (SN (Γ) ∗ f ) = Γ ∗ f we proceed P ROOF. We already know that SN (Γ) ∗ f ∈ H∞ in two steps. First, let K be a compact set in R and let ϕ1 , . . . , ϕN ∈ D(K) be given. We consider Γ := ϕ1 ⊗ · · · ⊗ ϕN and we put fj := S1 (ϕj ) ∈ H11 . Then F := SN (Γ) is given by F (z1 , . . . , zN ) = QN j=1 fj (zj ). Let us check that T (F ∗ f ) = Γ ∗ f. We observe that



N Y

X  σ∈{−1,1}N

 σj  F (x + iσ) =

N Y

(fj (xj + i) − fj (xj − i)) .

j=1

j=1

e a compact subset in R, K ⊂ K, e and C > 0 such that From Lemma 4, we choose K |fj (xj ± i)| ≤

C |x|2

e and 0 <  < 1. Let η ∈ D(R) be identically one on a neighborhood of K. e For each whenever x ∈ / K N ψ ∈ D(R ) and each 0 <  < 1 we have, h(

X

(

N Y

σj )F (x + iσ)) ∗ f, ψi

σ∈{−1,1}N j=1

= h

N Y

(fj (xj + i) − fj (xj − i)),

j=1

N Y

(1 − η(xj ) + η(xj ))(fˇ ∗ ψ)i

j=1

= I1 () + I2 () + I3 (), where I1 ()

:= h

N Y

(fj (xj + i) − fj (xj − i)), (

j=1

I2 ()

:= h

N Y

j=1

252

N Y

η(xj ))(fˇ ∗ ψ)i,

j=1

(fj (xj + i) − fj (xj − i)),

N Y j=1

(1 − η(xj ))(fˇ ∗ ψ)i,

(Ultra)distributions of Lp -growth as boundary values of holomorphic functions

and I3 () consists of the remaining terms. QN QN Since ( j=1 η(xj ))(fˇ ∗ ψ) ∈ D(RN ) and j=1 η(xj ) is identically one on a neighborhood of the support of Γ, lim I1 () = hΓ, fˇ ∗ ψi = hΓ ∗ f, ψi. →0+

e N and QN (1 − η(xj ))(fˇ∗ ψ) ∈ L∞ (RN ), and on has support in (R \ K) j=1 e we may apply Lebesgue’s convergence account of the given estimates for the functions fj0 s outside of K, theorem to get that lim+ I2 () = 0. Using that

QN

j=1 (1 − η(xj ))

→0

Now, we observe that I3 () is a sum of integrals of the form Z Z Y Y
(fj (xj +i)−fj (xj −i))(1−η(xj )) (fj (xj +i)−fj (xj −i))η(xj )(fˇ∗ψ)dx d˜ x, e m (R\K) j∈S

RN −mj ∈S /

m

m

where Sm is a proper subset of {1, . . . , N }, x denotes the coordinates corresponding to indices in Sm and x ˜ are the remaining coordinates. Without loss of generality we take Sm = {1, . . . , m}. Since fˇ ∗ ψ ∈ BL∞ (RN ), one has that {(

N Y

η(xj ))(fˇ ∗ ψ)(x1 , . . . , xm , . . . ) : (x1 , . . . , xm ) ∈ Rm }

j=m+1

is a bounded subset of D(RN −m ). Also N Y

{

(fj (· + i) − fj (· − i)) : 0 <  < 1}

j=m+1 0

is bounded in (DL1 (RN −m )) , hence   N  Z Y ( (fj (xj + i) − fj (xj − i))η(xj ))(fˇ ∗ ψ)dx : 0 <  < 1   RN −m j=m+1

is a bounded set in L∞ (Rm ). As before we conclude that lim I3 () = 0. Consequently T (F ∗ f ) = Γ ∗ f. →0+

To finish, we fix a compact K in R, f ∈ L∞ (RN ), and we consider the continuous linear map 0

R : D(K N ) −→ (DL1 (RN ))

given by R(Γ) = T (SN (Γ) ∗ f ) − Γ ∗ f. Since R vanishes on the dense subset D(K) ⊗ . . . ⊗ D(K), R must be identically zero, that is, T (SN (Γ) ∗ f ) = Γ ∗ f for every Γ ∈ D(K N ).  Our next result gives the surjectivity of the boundary value operator in the case of distributions of Lp growth. For 1 < p < ∞, this follows from Tillman [21], Luszczki and Zielezny [13] and Bengel [1]. However, their methods did not work for p = 1, ∞. Our approach permits a unified treatment of all values of p. Theorem 1 For 1 ≤ p ≤ ∞, the boundary value operator T : HpN −→ (DLp0 (RN ))0 is surjective. 253

C. Fern´andez, A. Galbis, M.C. G´omez-Collado

P ROOF. Since HpN is stable under differential operators and on account of [19, Th.XXV] it is enough to show that Lp (RN ) ⊂ T (HpN ). Let B be the closed unit ball in RN . Then, there are m ∈ N and SN,B : Dm (B) −→ GN continuous and linear such that T ◦ SN,B = Id. Now, we can find a differential operator P (D) and two functions Ψ ∈ Dm (B) and ϕ ∈ D(RN ) such that δ = P (D)Ψ + ϕ. Therefore, given f ∈ Lp (RN ) we have f = P (D)(Ψ ∗ f ) + ϕ ∗ f. We take F := P (D)(SN,B (Ψ) ∗ f ) + SN (ϕ) ∗ f. Then F ∈ HpN and T (F ) = P (D)T (SN,B (Ψ) ∗ f ) + T (SN (ϕ) ∗ f ). Clearly, T (SN (ϕ) ∗ f ) = ϕ ∗ f. On the other hand, SN,B (Ψ) = limn SN (Ψn ), where the convergence is in GN for a suitable sequence (Ψn ) ⊂ D(RN ) as in the proof of Corollary 3. Therefore, SN,B (Ψ) ∗ f = limn SN (Ψn ) ∗ f , where the convergence is in HpN , and, since T is continuous, T (SN,B (Ψ) ∗ f ) = limn (Ψn ∗ f ). Here the convergence is in D0 (RN ), but since (Ψn )n converges to Ψ in L1 (RN ) then (Ψn ∗ f )n converges to Ψ ∗ f in Lp (RN ) and we conclude T (F ) = f.  Theorem 2 Let ω be a weight function satisfying (α1 ) and 1 ≤ p ≤ ∞. Then the boundary value operator T : HωN∗ ,p −→ (DLp0 ,(ω) (RN ))0 is surjective. P ROOF. For every µ ∈ (DLp0 ,(ω) (RN ))0 there are an ultradifferential operator G(D) of (ω)−class and f ∈ Lp (RN ) such that µ = G(D)f ([2, 2.1]). By Theorem 1, there is F ∈ HpN ⊂ HωN∗ ,p with T (F ) = f. Then G(D)F ∈ HωN∗ ,p on account of Lemma 1 and T (G(D)F ) = µ.  Acknowledgement. The research of the authors was supported by MCYT and FEDER proyecto n. BFM 2001-2670.

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[19] L. Schwartz. (1966). Theorie des distributions, Hermann, Paris, 1966. [20] H. G. Tillmann. (1953). Randverteilungen analytischer Funktionen und Distributionen, Math. Z., 59, 61–83. [21] H. G. Tillmann. (1961). Distributionen als Randverteilungen analytischer Funktionen II, Math. Z. ,77, 106–124. [22] D. Vogt. (1973). Distributionen auf dem RN als Randverteilungen holomorpher Funktionen, J. Reine Angew. Math., 261, 134–145.

Carmen Fern´andez; Antonio Galbis Departamento de An´alisis Matem´atico Universidad de Valencia Doctor Moliner 50 E-46100 Burjasot (Valencia), Spain [email protected]; [email protected]

M.Carmen G´omez-Collado Departamento de Matem´atica Aplicada E.T.S. Arquitectura Camino de Vera E-46071 Valencia, Spain [email protected]

255