Uniform Algebras and Jensen Measures

Abstract

These lecture notes are devoted to an area of current research interest that bridges functional analysis and function theory. The unifying theme is the notion of subharmonicity with respect to a uniform algebra. The topics covered include the rudiments of Choquet theory, various classes of representing measures, the duality between abstract sub-harmonic functions and Jensen measures, applications to problems of approximation of plurisubharmonic functions of several complex variables, and Cole's theory of estimates for conjugate functions. Many of the results are published here for the first time in monograph form.

Inequalities for the truncated Hilbert transform and the segment multiplier

“…For the remaining p's, the norms of H T acting on real L p spaces were found by Pichorides [18] and, independently, by Cole (unpublished; consult Gamelin [8]):…”

Hankel Operators on Bounded Analytic Functions

“…(See II.1 or VIII.10 of [12], or Section 2 of [15].) Since the convolution of a locally integrable function with a compactly supported bounded Borel function is continuous everywhere, the integral on the right is in C(U ).…”

“…Choose z ± in the bottom half of ∆ near to ±ε respectively such that |f ε (z ± ) − (±1)| < ε respectively. Since R(U ) = A(U) (see VIII.13 of [12]), we may use the Cauchy integral formula over ∂U \ [−1/4, +1/4] to recapture values of g on U ,…”

Uniform Algebras Generated by Holomorphic and Pluriharmonic Functions

“…Consequently, there are disjoint open sets t/n and Ux of C" such that Fn C C/0 and Ex c Ux. Hence the function h that is 0 on Fn and 1 on Ex extends to be holomorphic on the neighborhood UoUU\ of E, and therefore, by the Oka-Weil approximation theorem [G,Theorem IH.5.1] can be approximated uniformly on E by polynomials. It follows that h is in A(Bn)[fix, ... , f"]\E which contradicts the fact that F is a set of antisymmetry for A(Bn)[fix, ... , /"].…”