Functions of one complex variable II

Overview Preliminaries The Cauchy transform as a function The Cauchy transform as an operator Topologies on the space of Cauchy transforms Which functions are Cauchy integrals? Multipliers and divisors The distribution function for Cauchy transforms The backward shift on $H^2$ Clark measures The normalized Cauchy transform Other operators on the Cauchy transforms List of symbols Bibliography Index.

The Cauchy Transform

Let ${\bold D}$ denote the unit disk in the complex plane and let $m$ be the area Lebesgue measure on ${\bold D}$. Given a positive integrable function $w$ (a weight) on ${\bold D}$, let $L^p_{\rm a}(w)$ denote the collection of analytic functions $f$ on ${\bold D}$ such that $|f|^pw$ is integrable. Given an analytic function $g$ on ${\bold D}$, the operator $T_g$ is defined by $T_g f(z) = \int_0^{z} f(\zeta)g'(\zeta)\,d\zeta$. The authors consider conditions on $g$ such that $T_g$ is bounded on $L^p_{\rm a}(w)$. In many cases the derivative $D$ is an isomorphism between the subspace of $L^p_{\rm a}(w)$ consisting of functions vanishing at $0$ and some space $L^p_{\rm a}(v)$ with another weight $v$. Thus, the question of boundedness or compactness of $T_g$ becomes the corresponding question for the operator of multiplication by $g'$ acting from $L^p_{\rm a}(w)$ to $L^p_{\rm a}(v)$. The authors consider only $w$ which are radial: $w(re^{i\theta}) = w(r)$. In the first part of the paper it is shown that $\int |f|^pw\,dm \le C\int |f'(z)|^p v(|z|) \,dm(z)$, $p \ge 1$, where $v(r) = \int_r^1 w(u)\,du$. Under the assumption that $v(r) \le C(1 - r)w(r)$, which is valid in particular for $w(r) \equiv (1 - r)^\alpha$, $\alpha > -1$, it follows that $T_g$ is bounded when $g'(z)(1 - |z|)$ is bounded. The converse can be proved in a more general setting. It is obtained by estimating the norm of the linear functional $D_\lambda\colon f \mapsto f'(\lambda)$ in terms of that of the evaluation functional $L_\lambda\colon f \mapsto f(\lambda)$. Rather general hypotheses on a Banach space of analytic functions are obtained in order that $\| D_\lambda \|(1 - |\lambda|) \le C\| L_\lambda \|$. This leads immediately to the converse: if the operator $T_g$ is bounded, then $g'(z)(1 - |z|)$ is bounded. Several classes of weights are shown to satisfy the hypotheses required for both the necessity and the sufficiency of the condition. In all cases, the problem of compactness of $T_g$ is also considered, and the solution involves the little-oh versions of the same conditions for boundedness. In the special case where $p = 2$ and $w(r) = (1 - r)^\alpha$ the Schatten class of $T_g$ is determined. The techniques are briefly applied to the weight $w(r) = \exp[ -\beta(1 - r)^{-\alpha}]$, $\alpha > 0$. In this case $v(r) \le C(1 - r)^{\alpha + 1}w(r)$, and that leads to the conclusion that $T_g$ is bounded when $g'(z)(1 - |z|)^{\alpha + 1}$ is bounded. The necessity of this condition is left open. A theorem of V. L. Oleinik [Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 47 (1974), 120--137, 187, 192--193; MR0369705 (51 #5937) (Theorem 3.3)] shows that it is necessary at least when $\alpha > 1$. (Less)

Integration operators on Bergman spaces

Introduction Preliminaries Linear-fractional composition operators Linear-fractional models The hyperbolic and parabolic models Cyclicity: Parabolic nonautomorphism case Endnotes References.

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Cyclic Phenomena for Composition Operators

An integral operator on $Hsp p$ and Hardy's inequality.

Let g be an analytic function on the unit disk D . We study the operator on the Hardy spaces Hp . We show that Tg is bounded on Hp , 1 ≤ p < ∞ it and only if g ∊ BMOA and compact if and only if g ∊ VOMA. Further on the Hilbert space H2 Tg is in the Schattcn ρ-class it and only if g is in the Besov space Bp , 1 < p < ∞. A relation of Tg with integration operators is also discussed.

An integral operator on $H\sp p$

Received 16 April 1985.1980 Mathematics Subject Classification 47D05.The content of this paper is part of the author's Ph.D. thesis written under the supervision of ProfessorE. Berkson at the University of Illinois. I wish to express my gratitude to Professor Berkson for hisencouragement and help during the preparation of this work.J. London Math. Soc. (2) 36 (1987) 153-164

Composition Semigroups and the Cesàro Operator OnHp

A one way flow and return system for vapor cooled transformer heat exchangers employs an inlet coupled with an entrance manifold and an outlet coupled with a return manifold. The incoming vaporized coolant enters the entrance manifold in a first direction of flow and the condensed vapor droplets leave the exit manifold in an opposite flow direction. The arrangement of separate entrance and exit manifolds prevents interference between the incoming vaporized coolant and the returning condensed coolant.

Semigroups of composition operators on spaces of analytic functions, a review

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Aristomenis G. Siskakis

Papers

Integration operators on Bergman spaces

An integral operator on $H\sp p$

Composition Semigroups and the Cesàro Operator OnHp

Semigroups of composition operators on spaces of analytic functions, a review

Composition operators and the Hilbert matrix