The modulus of continuity of analytic functions and CR- geometry

1 Introduction Contour and solid modulus of continuity. The one-dimensional case There has been an extensive study of the relation between the " contour " and the " solid " moduli of continuity for analytic functions in planar domains, starting with the following beautiful result of Hardy and Littlewood. For a domain G in C n (n ≥ 1) we denote as usual by A(G) the algebra of analytic functions in G which extend continuously to the closure ¯ G. The result of Hardy and Littlewood concerns the case when G equals D, the unit disc in the complex plane [HL]. Theorem 1. (Hardy, Littlewood) If f ∈ A(D) and its restriction to the unit circle is Hölder continuous of order α, α ∈ (0, 1), then f is Hölder continuous of the same order in the closed disc ¯ D. In other words, the estimate sup z,z ∈∂D |z−z |<δ |f (z) − f (z)| ≤ C 1 δ α implies sup z,z ∈ ¯ D |z−z |<δ |f (z) − f (z)| ≤ C 2 δ α for all δ ≥ 0 with constants C 1 , C 2. This result has been generalized and sharpened in various ways. Call a continuous non-negative, non-decreasing function µ on the positive half-axis [0, +∞) a modulus of continuity if µ(0) = 0 and µ is subadditive, i.e. µ(δ 1 + δ 2) ≤ µ(δ 1) + µ(δ 2) for δ 1 ≥ 0, δ 2 ≥ 0. Note that for any continuous function g on a compact convex set E ⊂ R n the function µ g (δ) = sup z,z ∈E |z−z |<δ |g(z) − g(z)| is a modulus of continuity. Note that for any continuous function f on an arbitrary compact subset of R n there exists a modulus of continuity µ such that sup z,z ∈E |z−z |<δ |f (z) − f (z)| ≤ µ(δ) for all δ ≥ 0. (1) Indeed, it is enough to extend f to a continuous function on a large closed ball and apply the preceding argument. Often it will be more convenient to use the following more general notion instead. Call a non-negative, non-decreasing function µ on the positive half-axis [0, +∞) a generalized modulus of continuity if µ(nδ) ≤ nµ(δ) (2) for all integers n ≥ 0 and δ ≥ 0. We do not require here that µ(0) = 0. …