By Nikolai A. Shirokov

This study monograph matters the Nevanlinna factorization of analytic capabilities soft, in a feeling, as much as the boundary. The abnormal homes of this sort of factorization are investigated for the most typical sessions of Lipschitz-like analytic capabilities. The ebook units out to create a passable factorization conception as exists for Hardy periods. The reader will locate, between different issues, the concept 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of the concept of V.P. Havin and F.A. Shamoyan additionally identified within the mathematical lore because the unpublished Carleson-Jacobs theorem, the entire description of the zero-set of analytic services non-stop as much as the boundary, generalizing the classical Carleson-Beurling theorem, and the constitution of closed beliefs within the new wide variety of Banach algebras of analytic capabilities. the 1st 3 chapters suppose the reader has taken a regular direction on one advanced variable; the fourth bankruptcy calls for supplementary papers brought up there. The monograph addresses either ultimate yr scholars and doctoral scholars starting to paintings during this sector, and researchers who will locate the following new effects, proofs and techniques.

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**Additional info for Analytic Functions Smooth up to the Boundary**

**Example text**

The absolute value of the right-hand side is maximal at a = f 1 or z = f 1 . 7), we see that rk Ck (x &/r) is a harmonic polynomial and is homogeneous of degree k . Making cv run over Sn-1 , we have infinitely many harmonic polynomials. We show, in the next two propositions, that they span all homogeneous harmonic polynomials of degree k . 1. Let Hn, k be the vector space of harmonic polynomials of n independent variables and homogeneous of degree k. Set , t (n , k) _ dim Hn, k . Then, (2,O)=1, ,u(n,k)=(2k+n-2) (nk, y(2, k) = 2 fork > 1 ; 3 n + k)!

Therefore, the assertion is true for n . 2. 12) . PROOF. 13) where y = pw (w E Sn-1) and dS(p) = pi-1dSw. Homogeneity implies h(y) = pI h(w) and 8h(y)lap =1p1-1h(6v) . 13), we have 2v S"-I h x = rkC` x (2v + k + 1 )p'-k c)o h (604) d . 12). 3 (Orthogonality relation). v + k vlsn-ll f -CSklCk(Co n_1 ' CV k,l>0. 14) PROOF. 12) to h(x) = rl Cl (x - w'/r) and r = I. In this way, we have shown that the integral operator h(x) rkC U v vISn-1I Sn-1 k D x w h(w)dS w r N is equal to the identity on H n , k and to zero on the other Hn ,1's.

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### Analytic Functions Smooth up to the Boundary by Nikolai A. Shirokov

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