Random Dirichlet Functions: Multipliers and Smoothness

We show that if f(z) = ∑ anz n is a holomorphic function in the Dirichlet space of the unit disk, then almost all of its randomizations ∑ ±anz are multipliers of that space. This parallels a known result for lacunary power series, which also has a version for smoothness classes: every lacunary Dirichlet series lies in the Lipschitz class Lip1/2 of functions obeying a Lipschitz condition with exponent 1/2. However, unlike the lacunary situation, no corresponding “almost sure” Lipschitz result is possible for random series: we exhibit a Dirichlet function with no randomization in Lip1/2. We complement this result with a “best possible” sufficient condition for randomizations to belong almost surely to Lip1/2. Versions of our results hold for weighted Dirichlet spaces, and much of our work is carried out in this more general setting. Introduction The Dirichlet space D of the open unit disc U of the complex plane is the set of holomorphic functions on U for which D(f) ≡ ∫ U ∣∣f ′(z)∣∣2 dA(z) <∞, where the measure dA is two-dimensional Lebesgue measure normalized so thatA(U) = 1. If f(z) = ∑ anz n then it is easily seen thatD(f) = ∑ n |an|. The Dirichlet space is a Hilbert space under the norm defined by: ‖f‖2 = D(f) + |f(0)|. A multiplier of the Dirichlet space is a holomorphic function φ on U such that the pointwise product φ(z)f(z) ∈ D whenever f ∈ D. If φ is a multiplier of D, then by the closed graph theorem the multiplication operator Mφ : f 7→ φf is a bounded linear operator on D. The study of such operators on D and its various weighted generalizations has been attracting considerable attention; see for example [2], [7], [11], [12], and [15]. Research supported in part by the National Science Foundation