William S. Cohn

A factorization theorem for the derivative of a function in HP

We show that a function G is the derivative of a function f in the Hardy space HP of the unit disk D for 0 1 there are maximal functions N, and A, associated with the approach regions IF(() = Iz E D: 11z-1 < a(,1-|Z12)1 Received by the editors May 28, 1997. 1991 Mathematics Subject Classification. Primary 32A35. ?)1999 American Mathematical Society 509 This content downloaded from 157.55.39.231 on Wed, 05 Oct 2016 04:05:55 UTC All use subject to http://about.jstor.org/terms

Some examples of cyclic vectors in the Dirichlet space

We consider the Hilbert space of analytic functions in the open unit disc that have a finite Dirichlet integral. For E, a closed subset of the unit circle with logarithmic capacity zero, we construct a function in this space which is uniformly continuous, vanishes on E, and is cyclic with respect to the shift operator. For A, the open unit disk in the complex plane, let D be the space of functions f analytic on A with finite Dirichlet integral J f12 dx dy < oo. If f has a Taylor expansion

Interpolation and multipliers on besov and sobolev spaces

In this note, analogues of the Carleson interpolation theorems for the Hardy spaces are proven for Besov spaces and Sobolev spaces of holomorphic functions on the unit disk.

Radial imbedding theorems for invariant subspaces

Let I denote an inner function on the unit disk in the complex plane. Associate with I the star- invariant subspace K p , (I) of the usual Hardy space H p , where 1<p<∞. We show that the condition is equivalent to the continuous imbedding of K2(I) in where 0<α<1 and 2≦q≦∞. If q=∞ this reduces to a theorem of Ahern and Clark. Similar embedding theorems are discossed for Kp(1) and Lq(dv) where u is a measure supported on the radius [O, 1].

A radial phragmén–lindelöf theorem for functions of slow growth

Let ω ≢ 0 be a modulus of continuity and H ω the Hausdorff measure defined on the unit circle C with generating function ω. Let u be a subharmonic function on the unit disk satisfying the growth condition M(r;u) = 0[(ω(1-r))/(1-r)] as r→1 where M(r;u) = max{u(rζ):ζ∈C}. We show that if E is a Borel subset of the unit circle with H ω(E) = 0 and lim supr→1u(rζ)⩽0 for each ζ∈C\E, then u(z)⩽0 for all z in the unit disk. We also obtain an analogous result where the “O” growth condition is replaced by the corresponding “o” condition and the requirement on E is that it be a Borel set which is σ-finite with respect to H ω. Both results are sharp. The latter generalizes an earlier theorem of Dahlberg where ω(t) = t α, 0 < α<1, and E was assumed to be a countable union of closed sets of finite H tα–measure.