Preface.- 1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions.- 2. The Uniform Boundedness Principle and the Closed Graph Theorem. Unbounded Operators. Adjoint. Characterization of Surjective Operators.- 3. Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity.- 4. L^p Spaces.- 5. Hilbert Spaces.- 6. Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators.- 7. The Hille-Yosida Theorem.- 8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension.- 9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions.- 10. Evolution Problems: The Heat Equation and the Wave Equation.- 11. Some Complements.- Problems.- Solutions of Some Exercises and Problems.- Bibliography.- Index.

Functional Analysis, Sobolev Spaces and Partial Differential Equations

This paper deals with the fractional Sobolev spaces W s;p . We analyze the relations among some of their possible denitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.

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Hitchhiker's guide to the fractional Sobolev spaces

This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.

A Posteriori Error Estimation in Finite Element Analysis

Why is Nonparametric Regression Important? * How to Construct Nonparametric Regression Estimates * Lower Bounds * Partitioning Estimates * Kernel Estimates * k-NN Estimates * Splitting the Sample * Cross Validation * Uniform Laws of Large Numbers * Least Squares Estimates I: Consistency * Least Squares Estimates II: Rate of Convergence * Least Squares Estimates III: Complexity Regularization * Consistency of Data-Dependent Partitioning Estimates * Univariate Least Squares Spline Estimates * Multivariate Least Squares Spline Estimates * Neural Networks Estimates * Radial Basis Function Networks * Orthogonal Series Estimates * Advanced Techniques from Empirical Process Theory * Penalized Least Squares Estimates I: Consistency * Penalized Least Squares Estimates II: Rate of Convergence * Dimension Reduction Techniques * Strong Consistency of Local Averaging Estimates * Semi-Recursive Estimates * Recursive Estimates * Censored Observations * Dependent Observations

http://www.gbv.de/dms/hebis-darmstadt/toc/106925350.pdf

A Distribution-Free Theory of Nonparametric Regression

The principal problem considered is the determination of all nonnegative functions, U(x), for which there is a constant, C, such that | [f*(x)rUix)dx g CJ \\f(x)\\'U(x) dx, where l<p<oo, J is a fixed interval, C is independent of f, and/* is the Hardy maximal function, f*(x) = sup -ii\" |/(/)| dt. yïx'.yej y — X Jx The main result is that U(x) is such a function if and only if [ f C/(x) ¿.xi Í f [Uixïï-uo-vdxV ' S K\\I\\\" where I is any subinterval of J, \\I\\ denotes the length of / and AT is a constant independent of /. Various related problems are also considered. These include weak type results, the problem when there are different weight functions on the two sides of the inequality, the case whenp = l orp = oo, a weighted definition of the maximal function, and the result in higher dimensions. Applications of the results to mean summability of Fourier and Gegenbauer series are also given.

/pdf/weighted-norm-inequalities-for-the-hardy-maximal-function-4cmd35jivw.pdf

J. L. B. Cooper

Papers

Singular integrals and differentiability properties of functions