There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The evolution of the state of many systems modeled by linear partial diﬁerentialequations (PDEs) or linear delay-diﬁerential equations can be described by operatorsemigroups. The state of such a system is an element in an inﬂnite-dimensionalnormed space, whence the name \inﬂnite-dimensional linear system".The study of operator semigroups is a mature area of functional analysis, which isstill very active. The study of observation and control operators for such semigroupsis relatively more recent. These operators are needed to model the interactionof a system with the surrounding world via outputs or inputs. The main topicsof interest about observation and control operators are admissibility, observability,controllability, stabilizability and detectability. Observation and control operatorsare an essential ingredient of well-posed linear systems (or more generally systemnodes). Inthisbookwedealonlywithadmissibility, observabilityandcontrollability.We deal only with operator semigroups acting on Hilbert spaces.This book is meant to be an elementary introduction into the topics mentionedabove. By \elementary" we mean that we assume no prior knowledge of ﬂnite-dimensional control theory, and no prior knowledge of operator semigroups or ofunbounded operators. We introduce everything needed from these areas. We doassume that the reader has a basic understanding of bounded operators on Hilbertspaces, diﬁerential equations, Fourier and Laplace transforms, distributions andSobolev spaces on

Observation and Control for Operator Semigroups

Harmonic mappings in the plane are univalent complex-valued harmonic functions of a complex variable. Conformal mappings are a special case where the real and imaginary parts are conjugate harmonic functions, satisfying the Cauchy-Riemann equations. Harmonic mappings were studied classically by differential geometers because they provide isothermal (or conformal) parameters for minimal surfaces. More recently they have been actively investigated by complex analysts as generalizations of univalent analytic functions, or conformal mappings. Many classical results of geometric function theory extend to harmonic mappings, but basic questions remain unresolved. This book is the first comprehensive account of the theory of planar harmonic mappings, treating both the generalizations of univalent analytic functions and the connections with minimal surfaces.  Essentially self-contained, the book contains background material in complex analysis and a full development of the classical theory of minimal surfaces, including the Weierstrass-Enneper representation. It is designed to introduce non-specialists to a beautiful area of complex analysis and geometry.

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Harmonic Mappings in the Plane

A function of bounded mean oscillation is said to have vanish- ing mean oscillation if, roughly speaking, its mean oscillation is locally small, in a uniform sense. In the present paper the class of functions of vanishing mean oscil- lation is characterized in several ways. This class is then applied to answer two questions in analysis, one involving stationary stochastic processes satisfying the strong mixing condition, the other involving the algebra H + C.

https://www.ams.org/journals/tran/1975-207-00/S0002-9947-1975-0377518-3/S0002-9947-1975-0377518-3.pdf

Functions of vanishing mean oscillation

We prove regularity results for weak solutions to systems modelling electro-rheological fluids in the stationary case, as proposed in [27, 31]; a particular case of the system we consider is

/pdf/regularity-results-for-stationary-electro-rheological-fluids-4paa2azehn.pdf

Regularity Results for Stationary Electro-Rheological Fluids

Theory of H[p] spaces

Theory of Hp Spaces

Carleson's proof of this theorem involves a difficult covering argument and the consideration of a certain quadratic form (see also [ l ] ) . L. Hörmander later found a proof which appeals to the Marcinkiewicz interpolation theorem and avoids any discussion of quadratic forms. The main difficulty in this approach is to show that a certain sublinear operator is of weak type (1, 1). Here a covering argument reappears which is similar to Carleson's but apparently easier (see [4]). We wish to point out that Hörmander's argument, with appropriate modifications, actually proves the theorem in the following extended form.

/pdf/extension-of-a-theorem-of-carleson-2vk63trwp4.pdf

Extension of a theorem of Carleson

Generalizing a recent result of H. Hedenmalm for p = 2, a contractive zero-divisor is found in the Bergman space Ap over the unit disk for 1 < p < oo. This is a function G £ Ap with \\G\\P = 1 and a prescribed zero-set {ζj} , uniquely determined by the contractive property \\f/G\\p < \\f\\p for all f e Ap which vanish on {ζj} . The proof uses the positivity of the biharmonic Green function of the disk. For a finite zero-set, the canonical divisor G is represented explicitly in terms of the Bergman kernel of a certain weighted A2 space. It is then shown that G has an analytic continuation to a larger disk. 0. Introduction. It is well known that the zero-sets {ζj} of functions / in the Hardy space Hp are characterized by the Blaschke condition ]Γ}(1 — \ζj\) < oo, which guarantees the convergence of the Blaschke product H,ζj), where *(z,O = ψy ^ if ζ Φ 0, and b(z, 0) = z. A basic theorem of F. Riesz (see [3], Ch. 2) asserts that f/B is a nonvanishing function in Hp with norm equal to that of /. This simple fact plays an important role in the theory of Hp spaces, and it would be desirable to find an analogue for the Bergman spaces. We shall see in this paper that while isometric zerodivisors are not available in the Bergman spaces, there is an essentially unique contractive divisor of unit norm associated with every zero-set. A function / analytic in the unit disk D is said to belong to the Bergman space Ap , where 0 < p < oo, if

/pdf/contractive-zero-divisors-in-bergman-spaces-4k8o7oj9w7.pdf

Peter Duren

Papers

Theory of H[p] spaces

Theory of Hp Spaces

Harmonic Mappings in the Plane

Extension of a theorem of Carleson

Contractive zero-divisors in Bergman spaces