Hyperbolic mean growth of bounded holomorphic functions in the ball
نویسندگان
چکیده
منابع مشابه
Hyperbolic Mean Growth of Bounded Holomorphic Functions in the Ball
We consider the hyperbolic Hardy class %Hp(B), 0 < p < ∞. It consists of φ holomorphic in the unit complex ball B for which |φ| < 1 and sup 0<r<1 ∫ ∂B {%(φ(rζ), 0)} dσ(ζ) < ∞, where % denotes the hyperbolic distance of the unit disc. The hyperbolic version of the Littlewood-Paley type g-function and the area function are defined in terms of the invariant gradient of B, and membership of %Hp(B) ...
متن کاملModuli of bounded holomorphic functions in the ball
We prove that there is a continuous non-negative function g on the unit sphere in C d, d ≥ 2, whose logarithm is integrable with respect to Lebesgue measure, and which vanishes at only one point, but such that no non-zero bounded analytic function m in the unit ball, with boundary values m⋆, has |m⋆| ≤ g almost everywhere. The proof analyzes the common range of co-analytic Toeplitz operators in...
متن کاملNon-constant bounded holomorphic functions of hyperbolic numbers - Candidates for hyperbolic activation functions
The Liouville theorem states that bounded holomorphic complex functions are necessarily constant. Holomorphic functions fulfill the socalled Cauchy-Riemann (CR) conditions. The CR conditions mean that a complex z-derivative is independent of the direction. Holomorphic functions are ideal for activation functions of complex neural networks, but the Liouville theorem makes them useless. Yet recen...
متن کاملBounded Holomorphic Functions on Bounded Symmetric Domains
Let D be a bounded homogeneous domain in C , and let A denote the open unit disk. If z e D and /: D —► A is holomorphic, then ß/(z) is defined as the maximum ratio \Vz(f)x\/Hz(x, 3c)1/2 , where x is a nonzero vector in C and Hz is the Bergman metric on D . The number ßf(z) represents the maximum dilation of / at z . The set consisting of all ß/(z), for z e D and /: D —► A holomorphic, is known ...
متن کاملWeak Topologies on the Bounded Holomorphic Functions
Let G be a region in the complex plane such that there is a nonconstant bounded holomorphic function on G, and denote the algebra of all such functions by BH{G). Let H^{G) denote the Banach algebra that arises when BH{G) is endowed with the supremum norm. In the case where G is the unit disc D, H*>(G) has been extensively studied, mostly by a real-variables analysis of the radial boundary value...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2002
ISSN: 0002-9947,1088-6850
DOI: 10.1090/s0002-9947-02-03169-0