Holomorphic Functions

(Ultra)distributions of L p-growth as boundary values of holomorphic functionsThen D (ω) (K), endowed with its natural topology, is a Fréchet space. For a fundamental sequence (K j ) j∈Nof compact subsets of R N we letD (ω) (R N ) := indj→ D (ω)(K j ).The dual D ′ (ω) (RN ) of D (ω) (R N ) is endowed with its strong topology. The elements of D ′ (ω) (RN ) arecalled ultradistributions of Beurling type.We denote by E (ω) (R N ) the set of all functions f ∈ C ∞ (R N ) such that ‖ f ‖ K,λ < ∞ for every compactK in R N and for every λ > 0.Definition 3 ([2]) For every 1 ≤ p ≤ ∞, k ∈ N and φ ∈ C ∞ (R N ), γ k,p (φ) is defined as follows( ( )) |α|γ k,p (φ) = sup ‖φ (α) ‖ p exp −kϕ ∗ ,α∈N kN0where ‖.‖ p denotes the usual norm in L p (R N ).If 1 ≤ p < ∞ the space D Lp,(ω)(R N ) is the set of all C ∞ -functions φ on R N such that γ k,p (φ) < ∞for each k ∈ N. A function φ ∈ C ∞ (R N ) is in B L∞,(ω)(R N ) when γ k,∞ (φ) < ∞, for every k ∈ N. Wedenote by D L∞,(ω)(R N ) the subspace of B L∞,(ω)(R N ) consisting of those functions φ ∈ B L∞,(ω)(R N ) forwhich lim|x|→∞ |φ(α) (x)| = 0 for each α ∈ N N 0 .The topology of D Lp,(ω)(R N ), 1 ≤ p ≤ ∞, is generated by the family of seminorms {γ k,p } k∈N . Also, weconsider on B L∞,(ω)(R N ) the topology associated with {γ k,∞ } k∈N . Then D Lp,(ω)(R N ), 1 ≤ p ≤ ∞ andB L∞,(ω)(R N ) are Fréchet spaces.For the definition of the spaces D Lp (R N ), 1 ≤ p ≤ ∞, we refer to [19, VI,8].Remark 2 (a) Clearly D Lp,(ω)(R N ) is continuously contained in D Lp (R N ), 1 ≤ p ≤ ∞. Hence, ifφ ∈ D Lp,(ω)(R N ), 1 ≤ p ≤ ∞, then lim|x|→∞ |φ(α) (x)| = 0, for each α ∈ N N 0 .(b) The inclusions D (ω) (R N ) ⊂ D Lp,(ω)(R N ) ⊂ E (ω) (R N ) are continuous and dense, and for 1 ≤ p ≤q ≤ ∞, D Lp,(ω)(R N ) is continuously contained in D Lq,(ω)(R N ).(c) Although the paper [2] was written in the one variable setting, all the results in its section 2 also holdfor N > 1 with the same proofs. The dual of D Lp,(ω)(R N ) will be denoted by (D Lp,(ω)(R N )) ′ and it will be endowed with the strongtopology. Since D (ω) (R N ) is continuously and densely embedded in D Lp,(ω)(R N ) then (D Lp,(ω)(R N )) ′can be identified with a subspace of D ′ (ω) (RN ). The elements of (D Lp,(ω)(R N )) ′ are known as ultradistributionsof Beurling type of L p ′-growth where p ′ is the conjugate exponent of p. The ultradistributions ofL ∞ -growth are called bounded ultradistributions of Beurling type.The classical case D Lp (R N ) 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(C N ) be an entire function such that log |G(z)| = O(ω(|z|)) as |z| tends to infinity. ThenT G (ϕ) := ∑α∈N N 0defines an ultradistribution T G ∈ E ′ (ω) (RN ). The operator(−i) |α| G(α) (0)ϕ (α) (0)α!G(D) : D ′ (ω) (RN ) → D ′ (ω) (RN ),G(D)ν := T G ∗ ν245

(Ultra)distributions of L p-growth as boundary values of holomorphic functionsRemark 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 := Hω N ∗ ,p can be described asH N p = {f ∈ H((C \ R) N ) : |f| p,k < ∞ for some k ∈ N}where|f| p,k := supy⎛ ⎞N∏‖ f(· + iy) ‖ p e −2k|y| ⎝ |y j | ⎠j=1k. For any weight ω, HpN ⊂ Hω N ∗ ,p and given two weights σ ≤ ω we have Hσ N ∗ ,p ⊂ Hω N ∗ ,p with continuousinclusions.Since we can represent (ultra)distributions of L p -growth as a (infinite) linear combination of derivativesof functions in L p , we will show that the spaces just defined are stable under (ultra)differential operators.Lemma 1 (1) For each f ∈ H N p and each α ∈ N N 0 we have f (α) ∈ H N p .(2) For each f ∈ H N ω ∗ ,p and each ultradifferential operator G(D) of class (ω) we have G(D)f ∈ H N ω ∗ ,p.PROOF. Let f ∈ H ω ∗ ,p(R N ) be given. We fix y ∈ R N such that y j ≠ 0 for every 1 ≤ j ≤ N and we putρ j := 1 2 |y j|. Let D ρ be the polidisc of poliradius ρ := (ρ 1 , . . . , ρ N ). For each x ∈ R N and each α ∈ N N 0 ,by the Cauchy integral formulaf (α) (x + iy) = α! ∫f(x + iy + ξ)(2πi) N D ρξ α+1 dξ.The function g ξ (x) :=Thereforef(x + iy + ξ)ξ α+1belongs to L p (R N ) andf (α) (· + iy) = α!(2πi) N ∫‖ f (α) (· + iy) ‖ p ≤ α!(2π) ND ρg ξ (·) dξ.maxξ∈D ρ‖ g ξ ‖ pSince ‖ g ξ ‖ p ≤ 1ρ α+1 |f| ω,p,k e 2k|y|+kω∗ ( y2k ) for some constant k ∈ N,‖ f (α) (· + iy) ‖ p ≤ |f| ω,p,k α! e 2k|y|+kω∗ ( y2k )N∏j=1N∏j=1(2πρ j ).( ) αj2.|y j |If ω(t) = log(1 + t) (t ≥ 0) this gives (1). To show (2), let G(D) be an ultradifferential operator of class(ω). Then G(D)g = ∑a α g α where |a α | ≤ Ce −mϕ∗ ( |α|m ) ≤ Ce −m ∑ Nj=1 ϕ∗ ( α jm ) for some C > 0 andsome m ∈ N. Thusα∈N N 0‖ G(D)f(· + iy) ‖ p≤ ∑|a α | ‖ f (α) (· + iy) ‖ pα∈N N 0≤ C|f| ω,p,k∑α∈N N 0α!N∏j=1( ) αj−mϕ 2∗ ( α jm ) e 2k|y|+kω∗ ( y2k ) .|y j |247

C. Fernández, A. Galbis, M.C. Gómez-ColladoThat is,‖ G(D)f(· + iy) ‖ p ≤ C |f| ω,p,kN ∏⎛⎝∞∑j=1 α j=0To finish it is enough to proceed as in [8, Prop.1].⎞( ) αjα j ! e −mϕ∗ ( α j 2m ) |y j |( |yj⎠ e 2k|yj|+kω∗ |2k).It is clear now that, given f ∈ HpN and y ∈ R N , y j ≠ 0 for 1 ≤ j ≤ N, the function f(· + iy) belongsto D Lp (R N ) for 1 ≤ p < ∞ and f(· + iy) ∈ B L∞ (R N ) for p = ∞. A similar result holds for arbitraryweight functions ω.Corollary 1 (1) For every 1 ≤ p < ∞ and each f ∈ H N ω ∗ ,p we have f(· + iy) ∈ D Lp,(ω)(R N ).(2) Each f ∈ H N ω ∗ ,∞ satisfies f(· + iy) ∈ B L∞,(ω)(R N ).PROOF. Let 1 ≤ p ≤ ∞ and f ∈ Hω N ∗ ,p be given. Then G(D)f(. + iy) ∈ L p (R N ) for every ultradifferentialoperator 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 H N ω ∗ ,p increase with p.Corollary 2 For a fixed function ω and p < q we have H N ω ∗ ,p ⊂ H N ω ∗ ,q with continuous inclusion.PROOF. Since D Lp,(ω)(R N ) ⊂ D Lq,(ω)(R N ) ⊂ B L∞,(ω)(R N ) with continuous inclusions ([2]), we deducefrom Corollary 1 that for every f ∈ Hω N ∗ ,p there is a continuous seminorm γ on D Lp,(ω)(R N ) suchthat ||f(. + iy)|| q ≤ γ(f(. + iy)) for every y ∈ R N with y j ≠ 0 for 1 ≤ j ≤ N. Hence there is anultradifferential operator G(D) of (ω)−class and there is a positive constant C satisfying||f(. + iy)|| q ≤ C||G(D)f(. + iy)|| pfor all y ∈ R N with y j ≠ 0 for all 1 ≤ j ≤ N ([2, 2.0.4]). Since G(D)f ∈ Hω N ∗ ,p then it easily followsthat f ∈ Hω N ,q. ∗4. Boundary valuesIn this section we will show that each function in Hω N ∗ ,p has an element of (D Lp ′ ,(ω)(R N )) ′ , 1 p + 1 p= 1, as′boundary value and that, conversely, each (ultra)distribution of L p -growth can be obtained as the boundaryvalue of a suitable f in Hω N ,p. From now on ω will be either a weight function satisfying (α ∗ 1 ) or ω(t) =log(1 + t), t ≥ 0.We first observe that each f ∈ H N ω ∗ ,p belongs to H N ω ∗ ,∞ and, after applying [22] for ω(t) = log(1 + t)and [17] for ω a weight function satisfying (α 1 ), we have the following resultLemma 2 The boundary value operator T : H N ω ∗ ,p → D ′ (ω) (RN ) given by∫< T (f), ϕ >:= lim {ɛ→0 + R N∑(σ∈{−1,1} N j=1N∏σ j )f(x + iσɛ)} ϕ(x) dxis a well-defined, continuous and linear mapping. Moreover T (f) ∈ (D L1,(ω)(R N )) ′ for each f ∈ H N ω ∗ ,∞and T is continuous as a map from H N ω ∗ ,∞ into (D L1,(ω)(R N )) ′ . Therefore T is also continuous from H N ω ∗ ,pinto (D L1,(ω)(R N )) ′ .248

(Ultra)distributions of L p-growth as boundary values of holomorphic functionsPROOF. 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 Hω N ∗ ,p is an (ultra)distribution of L p -growth.Proposition 1 T (H N ω ∗ ,p) is contained in (D Lp ′ ,(ω)(R N )) ′ with 1 p + 1 p ′= 1. Moreover,T (f) = lim∑(ɛ→0 +σ∈{−1,1} N j=1in the weak topology σ((D Lp ′ ,(ω)(R N )) ′ , D Lp ′ ,(ω)(R N )).N∏σ j )f(x + iσɛ)PROOF. First we assume that N = 1 and that ω is a weight function. Given f ∈ Hω 1 ∗ ,p we choose k ∈ Nand C > 0 such thatmax(‖ f(· + iy) ‖ ∞ , ‖ f(· + iy) ‖ p ) ≤ Ce kω∗ ( y k ) (1)for 0 < y < 2. Without loss of generality we may assume that f ≡ 0 in the lower half-plane. We willshow that {f(· + iɛ) : 0 < ɛ < 1} is a bounded set in (D Lp ′ ,(ω)(R)) ′ and that T (f) ∗ ϕ ∈ L p (R) forevery ϕ ∈ D (ω) (R). We put f iɛ (x) := f(x + iɛ). Let ϕ ∈ D (ω) (R) be given and let b > 0 be such thatsupp ϕ ⊂] − b, b[. By [17, 3.4] we find φ ∈ D((−b, b) × (− 1 2 , 1 2)) such that(i) φ |R = ϕ∣ (ii) sup∂∣∂¯z φ(x + iy) ( ∣∣∣ |y|ekω∗ k ) < ∞.z∈C\RApplying Stokes’ theorem to the function θ x (ξ) := f(ξ + iɛ)φ(x − ξ) in the rectangle D x := [x − 2b, x +2b] × [0, 1] we get that∫(f iɛ ∗ ϕ)(x) = 2i f(x − t + i(v + ɛ)) ∂ φ(t − iv) d(t, v).∂¯zDDwhere D := [−2b, 2b] × [0, 1]. Therefore∫∣ ∣ ∣∣∣ ∂ ∣∣∣‖ f iɛ ∗ ϕ ‖ p ≤ 2 ‖ f(· + i(v + ɛ)) ‖ p∂¯z φ(t − iv) d(t, v),from where we conclude that {f iɛ ∗ ϕ : 0 < ɛ < 1} is a bounded set in L p (R), which shows that {f iɛ :0 < ɛ < 1} is bounded in (D Lp ′ ,(ω)(R)) ′ ([2]), hence equicontinuos. Moreover, for every null sequenceof positive numbers (ɛ n ) n one has (T (f) ∗ ϕ)(x) = lim n (f iɛn ∗ ϕ)(x) pointwise and there is C > 0with |(f iɛn ∗ ϕ)(x)| ≤ C for every n ∈ N and each x ∈ R. Using Lebesgue’s dominated convergencetheorem we get that {(T (f) ∗ ϕ)χ [−n,n] : n ∈ N} is bounded in L p (R), hence T (f) ∗ ϕ ∈ L p (R)and T (f) ∈ (D Lp ′ ,(ω)(R)) ′ . Let us take a 0-neighbourhood V in D Lp ′ ,(ω)(R) such that T (f) ∈ V o andf iɛ ∈ V o for 0 < ɛ < 1 and let τ denote the topology of pointwise convergence on the dense subspaceD (ω) (R) of D Lp ′ ,(ω)(R). Then the weak topology and τ coincide on the equicontinuous set V o . Since< T (f), ϕ >= lim ɛ→0∫R f iɛ(x)ϕ(x)dx for every ϕ ∈ D (ω) (R) we get that T (f) is the limit of (f iɛ ) inthe weak topology.If ω(t) = log(1 + t), given ϕ ∈ D(R) we choose k ∈ N satisfying (1) and we putThen ∂φ∂¯z (x, y) = 1 2φ(x, y) :=k∑j=01j! ϕ(j) (x) (iy) j .ϕ (k+1) (x)(iy) k and we proceed as above. See [16, 2.2].k!249

C. Fernández, A. Galbis, M.C. Gómez-ColladoPROOF.Given σ ∈ {−1, 1} N , ɛ > 0 and ϕ ∈ D(R N ), one has that ˇf ∗ ϕ ∈ D Lp (R N ) and〈(S N (ψ) ∗ f)(· + iσɛ), ϕ〉 = 〈S N (ψ)(· + iσɛ), ˇf ∗ ϕ〉.Since S N (ψ) ∈ H1N ⊂ Hp N, ′ p′ being the conjugate number of p, it suffices to apply Proposition 1 forω(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 S 1 (ψ) ∈ H(C \ K) and, forsome positive constants A and C,|S 1 (ψ)(x ± iɛ)| ≤ C|x| 2whenever |x| ≥ A and 0 < ɛ < 1.PROOF. It is clear from the definition of S 1 . Proposition 4 Given Γ ∈ D(R N ) and f ∈ L ∞ (R N ) we have S N (Γ) ∗ f ∈ H N ∞ and T (S N (Γ) ∗ f) =Γ ∗ f.PROOF. We already know that S N (Γ) ∗ f ∈ H∞. N To see that T (S N (Γ) ∗ f) = Γ ∗ f we proceedin two steps. First, let K be a compact set in R and let ϕ 1 , . . . , ϕ N ∈ D(K) be given. We considerΓ := ϕ 1 ⊗ · · · ⊗ ϕ N and we put f j := S 1 (ϕ j ) ∈ H1. 1 Then F := S N (Γ) is given by F (z 1 , . . . , z N ) =∏ Nj=1 f j(z j ). Let us check that T (F ∗ f) = Γ ∗ f. We observe that⎛∑⎝σ∈{−1,1} NN∏j=1σ j⎞⎠ F (x + iσɛ) =N∏(f j (x j + iɛ) − f j (x j − iɛ)) .From Lemma 4, we choose ˜K a compact subset in R, K ⊂ ˜K, and C > 0 such thatj=1|f j (x j ± iɛ)| ≤ C|x| 2whenever x /∈ ˜K and 0 < ɛ < 1. Let η ∈ D(R) be identically one on a neighborhood of ˜K. For eachψ ∈ D(R N ) and each 0 < ɛ < 1 we have,〈(∑(σ∈{−1,1} N j=1N∏σ j )F (x + iσɛ)) ∗ f, ψ〉N∏= 〈 (f j (x j + iɛ) − f j (x j − iɛ)),j=1= I 1 (ɛ) + I 2 (ɛ) + I 3 (ɛ),N∏(1 − η(x j ) + η(x j ))( ˇf ∗ ψ)〉j=1whereN∏N∏I 1 (ɛ) := 〈 (f j (x j + iɛ) − f j (x j − iɛ)), ( η(x j ))( ˇf ∗ ψ)〉,j=1j=1N∏N∏I 2 (ɛ) := 〈 (f j (x j + iɛ) − f j (x j − iɛ)), (1 − η(x j ))( ˇf ∗ ψ)〉,j=1j=1252

(Ultra)distributions of L p-growth as boundary values of holomorphic functions[11] H. Komatsu. (1972). Ultradistributions I. Structure theorems a characterization, J. Fac. Sci. Univ. Tokyo, 20,25–105.[12] M. Langenbruch. (2003). A general approximation theorem of Whitney type, this volume.[13] Z. Luszczki and Z. Zielezny. (1961). Distributionen der Räume D ′ L pund Randverteilungen analytischer Funktionen,Colloq. Math., 8, 125–131.[14] R. Meise. (1977). Representation of distributions and ultradistributions by holomorphic functions, FunctionalAnalysis: Surveys and Recent Results, K. D: Bierstedt, R. Fuchssteiner (eds.) North Holland Pub. Co..[15] R. Meise and B.A. Taylor. (1988). Whitney’s extension theorem for ultradifferentiable functions of Beurling type,Ark. Mat. 26 (2), 265–287.[16] H.J. Petzsche. (1984). Generalized functions and the boundary values of holomorphic functions, J. Fac. Sci. Univ.Tokyo, Sect. IA, Math. 31, 391–431.[17] H.J. Petzsche and D. Vogt. (1984). Almost Analytic Extension of Ultradifferentiable Functions and the BoundaryValues of Holomorphic Functions, Math. Ann., 267, 17–35.[18] S. Pilipović. (1996). Elements of D ′ (M p)L s and {M p}D′ L s as Boundary Values of Holomorphic Functions, J. Math.Anal. Appl., 203, 719–737.[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 R N als Randverteilungen holomorpher Funktionen, J. Reine Angew.Math., 261, 134–145.Carmen Fernández; Antonio GalbisM.Carmen Gómez-ColladoDepartamento de Análisis MatemáticoDepartamento de Matemática AplicadaUniversidad de ValenciaE.T.S. ArquitecturaDoctor Moliner 50Camino de VeraE-46100 Burjasot (Valencia), Spain E-46071 Valencia, Spainfernand@uv.es; Antonio.Galbis@uv.es cgomezc@mat.upv.es255