David Ullrich

Bloch-to-BMOA pullbacks on the disk

For a given holomorphic self map (P of the unit disk, we consider the Bloch-to-BMOA composition property (pullback property) of (p. Our results are (1) (p cannot have the pullback property if (p touches the boundary too smoothly, (2) while (p has the pullback property if (p touches the boundary rather sharply. One of these results yields an interesting consequence completely contrary to a higher dimensional result which has been known. These results resemble known results concerning the compactness of composition operators on the Hardy spaces. Some remarks in that direction are included.

An extension of the Kahane-Khinchine inequality in a Banach space

We show that the geometric mean of the norm of a linear combination of the Steinhaus variables with “coefficients” in a Banach space is equivalent to the variance of the norm. This extends a result of Kahane, who established the corresponding inequality for theLp means.

On the behavior of harmonic functions near a boundary point

Several results on the behavior of harmonic functions at an individual boundary point are obtained. The results apply to positive harmonic functions as well as to Poisson integrals of functions in BMO.

A Simple Elementary Proof of Hilbert's Inequality

Abstract We give a very simple proof of Hilberts inequality.

The Ascoli-Arzelà Theorem via Tychonoff's Theorem

The purpose of this note is to point out that the Ascoli-Arzelà theorem may be derived as an immediate consequence of the Tychonoff theorem. Suppose then that K is a compact Hausdorff space. Let C(K ) denote the space of continuous complex-valued functions on K . We say that a subfamily F of C(K ) is pointwise bounded if for every x in K there exists r(x) > 0 such that | f (x)| ≤ r(x) for all f in F , and that F is equicontinuous if for each x in K and > 0 there exists U , a neighborhood of x , such that | f (x) − f (y)| < for all f in F whenever y belongs to U .