Holomorphic Functions

(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