Holomorphic Functions

(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