ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 30, Number 3, Fall 2000 SOME RESULTS ON MEAN LIPSCHITZ SPACES OF ANALYTIC FUNCTIONS DANIEL GIRELA AND CRIST ´ OBAL GONZ ´ ALEZ ABSTRACT. If f is a function which is analytic in the unit disk Δ and has a nontangential limit f (e iθ ) at almost every e iθ ∈ ∂Δ and 1 ≤ p ≤∞, then ω p (·,f ) denotes the integral modulus of continuity of order p of the boundary values f (e iθ ) of f . If ω : [0,π] → [0, ∞) is a continuous and increasing function with ω(0) = 0 and ω(t) > 0 if t> 0 then, for 1 ≤ p ≤∞, the mean Lipschitz space Λ(p, ω) consists of those functions f which belong to the classical Hardy space H p and satisfy ω p (δ, f )= O(ω(δ)) as δ → 0. If, in addition, ω satisfies the so-called Dini condition and the condition b 1 , we say that ω is an admissible weight. If 0 <α ≤ 1 and ω(δ)= δ α , we shall write Λ p α instead of Λ(p, ω), that is, we set Λ p α = Λ(p, δ α ). In this paper we obtain several results about the Taylor coefficients and the radial variation of the elements of the spaces Λ(p, ω). In particular, if ω is an admissible weight, then we give a complete characterization of the power series with Hadamard gaps which belong to Λ(p, ω). If f is an analytic function in Δ and θ ∈ [-π,π), we let V (f,θ) denote the radial variation of f along the radius [0,e iθ ). We also define the exceptional set E(f ) associated to f as E(f )= {e iθ ∈ T : V (f,θ)= ∞}. For any given p ∈ [1, ∞], we obtain a characterization of those admissible weights ω for which the implication f ∈ Λ(p, ω)= ⇒ E(f )= ∅, holds. We also obtain a number of results about the “size” of the exceptional set E(f ) for f ∈ Λ p α . 1. Introduction. Let Δ denote the unit disk {z ∈ C : |z| < 1} and T the unit circle {ξ ∈ C : |ξ | =1}. If 0 <r< 1 and g is a function Received by the editors on January 20, 1999, and in revised form on June 21, 1999. 1991 AMS Mathematics Subject Classification. 30D55, 30D50. This research has been supported in part by a grant from “El Ministerio de Educaci´on y Cultura, Spain,” (PB97-1081) and by a grant from “La Junta de Andaluc´ ıa.” Copyright c 2000 Rocky Mountain Mathematics Consortium 901