HOLOMORPHIC MEAN LIPSCHITZ FUNCTIONS ON THE UNIT BALL OF ℂn

Abstract

Abstract. On the unit ball of C n , the space of those holomorphic func-tions satisfying the mean Lipschitz conditionZ 10 ω p (t,f) q dtt 1+αq <∞is characterized by integral growth conditions of the tangential derivativesas well as the radial derivatives, where ω p (t,f) denotes the L p modulusof continuity defined in terms of the unitary transformations of C n . 1. IntroductionLet B = B n be the open unit ball of C n and S be the boundary of B. Letv be the Lebesgue volume measure on C n = R 2n and σ be the surface areameasure on S normalized to be σ(S) = 1. We denote by H p (B), 1 ≤ p < ∞,the Hardy space on B. We use the customary notationkfk H p (B) = sup 0<r<1 M p (r,f) and M (r,g) =Z S |g(rζ)| p dσ(ζ) 1/p respectively for holomorphic f and measurable g on B. We will denote B 1 byD.Concerning the boundary smoothness of H p (D) functions, it is known thatthe growth rate ofω p (t,f) = sup |h|≤t Z π−π |f(e i(θ+h) ) −f(e iθ )| p dθ! 1/p Received February 21, 2012; Revised April 24, 2012.2010 Mathematics Subject Classification. 32A30, 30H25.Key words and phrases. mean Lipschitz condition, Besov space, mean modulus ofcontinuity.The first author was supported by (NRF-2010-0021986). The second author was sup-ported by the National Research Foundation of Korea(NRF) grant funded by the Koreagovernment (KRF-2009-0073976). The third author was supported by (NRF-2012000705).