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Interpolation Theory, Function Spaces, Differential Operators
26 Feb 1978-
About: The article was published on 1978-02-26 and is currently open access. It has received 4781 citations till now. The article focuses on the topics: Operator theory & Interpolation.
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01 Dec 1992TL;DR: In this paper, the existence and uniqueness of nonlinear equations with additive and multiplicative noise was investigated. But the authors focused on the uniqueness of solutions and not on the properties of solutions.
Abstract: Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.
4,042 citations
TL;DR: In this paper, an abstract Strichartz estimate for the wave equation (in dimension n ≥ 4) and for the Schrodinger equation (n ≥ 3) was proved.
Abstract: We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n ≥ 4) and the Schrodinger equation (in dimension n ≥ 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Strichartz-type estimates for more general dispersive equations and for the kinetic transport equation.
2,187 citations
Book•
01 Jan 1990TL;DR: In this article, the authors introduce the concept of bounded linear operators on the Bergman space and define a set of operators based on the bounded linear operator on the Hardy space, including the following operators:
Abstract: Bounded linear operators Interpolation of Banach spaces Integral operators on $L^p$ spaces Bergman spaces Bloch and Besov spaces The Berezin transform Toeplitz operators on the Bergman space Hankel operators on the Bergman space Hardy spaces and BMO Hankel operators on the Hardy space Composition operators Bibliography Index.
1,903 citations
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14 Jul 2009TL;DR: The main topics of interest about observation and control operators are admissibility, observability, controllability, stabilizability and detectability as discussed by the authors, which is a mature area of functional analysis, which is still very active.
Abstract: 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
1,174 citations
Book•
04 Oct 2007
TL;DR: In this article, the authors propose a model for solving the model elliptic problems and model parabolic problems. But their model is based on Equations with Gradient Terms (EGS).
Abstract: Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.
935 citations