By Nikolai A. Shirokov
This learn monograph matters the Nevanlinna factorization of analytic services soft, in a feeling, as much as the boundary. The atypical homes of any such factorization are investigated for the most typical periods of Lipschitz-like analytic capabilities. The ebook units out to create a passable factorization thought as exists for Hardy sessions. The reader will locate, between different issues, the theory on smoothness for the outer a part of a functionality, the generalization of the concept of V.P. Havin and F.A. Shamoyan additionally identified within the mathematical lore because the unpublished Carleson-Jacobs theorem, the whole description of the zero-set of analytic capabilities non-stop as much as the boundary, generalizing the classical Carleson-Beurling theorem, and the constitution of closed beliefs within the new wide selection of Banach algebras of analytic services. the 1st 3 chapters think the reader has taken a typical direction on one complicated variable; the fourth bankruptcy calls for supplementary papers brought up there. The monograph addresses either ultimate yr scholars and doctoral scholars commencing to paintings during this zone, and researchers who will locate right here new effects, proofs and techniques.
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Extra resources for Analytic Functions Smooth up to the Boundary
Therefore, ∞ δ(u) 2 D1,2 = (n + 1)(n + 1)! fn 2 L2 ([0,T ]n+1 ) n=0 ∞ ≤ 2n2 n! fn 2 L2 ([0,T ]n+1 ) + 2n! 22). Then we conclude that δ(u) ∈ D1,2 . By similar computations, T T u(s)δW (s) − Dt Dt um (s)δW (s) 0 ∞ 0 = (n + 1)In (fn (·, t)) 2 L2 (P ×λ) 2 L2 (P ×λ) n=m+1 ∞ T (n + 1)2 n! 33) 2 n n! fn 2L2 ([0,T ]n+1 ) + n! fn 2L2 ([0,T ]n+1 ) . n=m+1 That vanishes when m → ∞. 30). 19. 18 and assume in addition that u(s), s ∈ [0, T ], is F-adapted. Then T T u(s)dW (s) = Dt 0 Dt u(s)dW (s) + u(t). 13.
3. Closability of the Malliavin derivative. , such that (1) Fk −→ F , k → ∞, in L2 (P ) ∞ (2) Dt Fk k=1 converges in L2 (P × λ). Then F ∈ D1,2 and Dt Fk −→ Dt F , k → ∞, in L2 (P × λ). Proof Let F = by (1) ∞ n=0 In (fn ) and Fk = fn(k) −→ fn , (k) ∞ n=0 In (fn ), k → ∞, k = 1, 2, ... Then in L2 (λn ) for all n. By (2) we have ∞ nn! fn(k) − fn(j) 2 L2 (λn ) = Dt Fk − Dt Fj 2 L2 (P ×λ) −→ 0, j, k → ∞. n=1 Hence by the Fatou lemma, ∞ k→∞ ∞ nn! fn(k) − fn lim 2 L2 (λn ) n=1 ≤ lim lim k→∞ j→∞ nn! fn(k) − fn(j) n=1 This implies that F ∈ D1,2 and Dt Fk −→ Dt F, k → ∞, in L2 (P × λ).
Fn(k) − fn(j) n=1 This implies that F ∈ D1,2 and Dt Fk −→ Dt F, k → ∞, in L2 (P × λ). 2 L2 (λn ) = 0. 2 Computation and Properties of the Malliavin Derivative In this section we proceed presenting a collection of results that constitute the rules of calculus of the Malliavin derivatives. 1 Chain Rules for Malliavin Derivative We proceed to prove a useful chain rule for Malliavin derivatives. , tn ) = f (t1 ) · · · f (tn ). 4) T where f = f L2 ([0,T ]) , θ = 0 f (t)dW (t) and hn is the Hermite polynomial of order n.