Some results on mean Lipschitz spaces of analytic functions

If f is a function which is analytic in the unit disk a and has a nontangen tial limit f(e(i theta)) at almost every e(i theta) is an element of partia l derivative Delta and 1 less than or equal to p less than or equal to infi nity, then omega (p)((.), f) denotes the integral modulus of continuity of order p of the boundary values f(e(i theta)) of f. If omega : [0, pi] --> [ 0, infinity) is a continuous and increasing function with omega (0) = 0 and omega (t) > 0 if t > 0 then, for 1 less than or equal to p less than or eq ual to infinity, the mean Lipschitz space Lambda (p, omega) consists of tho se functions f which belong to the classical Hardy space H-p and satisfy om ega (p)(delta, f) = O(omega(delta)) as delta --> 0. If, in addition, w sati sfies the so-called Dini condition and the condition b(1), we say that omeg a is an admissible weight. If 0 < <alpha> less than or equal to 1 and omega (delta) = delta (alpha), we shall write Lambda (p)(alpha) instead of (p,ome ga), that is, we set Lambda (p)(alpha), = Lambda (p, delta (alpha)). In this paper we obtain several results about the Taylor coefficients and t he radial variation of the elements of the spaces Lambda (p,omega). In part icular, if omega is an admissible weight, then we give a complete character ization of the power series with Hadamard gaps which belong to Lambda (p, o mega). If f is an analytic function in Delta and theta is an element of [-pi,pi), we let V(f, theta) denote the radial variation of f along the radius [0, e( i theta)). We also define the exceptional set E(f) associated to f as E(f) = {e(i theta) is an element of T : V(f, theta) = infinity}. For any given p is an element of [1, infinity], we obtain a characterization of those admi ssible weights omega for which the implication f is an element of Lambda (p, omega) double right arrow E(f) = empty set, holds. We also obtain a number of results about the "size" of the exception al set E(f) for f is an element of Lambda (p)(alpha).