Showing papers in "Memoirs of the American Mathematical Society in 2003"
TL;DR: In this article, Calderon-Zygmund singular integral operators have been studied in the context of discrete groups of dilations, and they have been shown to be an unconditional basis for the anisotropic Hardy space H A.
Abstract: In this paper, motivated in part by the role of discrete groups of dilations in wavelet theory, we introduce and investigate the anisotropic Hardy spaces associated with very general discrete groups of dilations. This formulation includes the classical isotropic Hardy space theory of Fefferman and Stein and parabolic Hardy space theory of Calderon and Torchinsky. Given a dilation A, that is an n × n matrix all of whose eigenvalues λ satisfy |λ| > 1, define the radial maximal function M φf(x) := sup k∈Z |(f ∗ φk)(x)|, where φk(x) = | detA|φ(Ax). Here φ is any test function in the Schwartz class with ∫ φ 6= 0. For 0 < p < ∞ we introduce the corresponding anisotropic Hardy space H A as a space of tempered distributions f such that M φf belongs to L (R). Anisotropic Hardy spaces enjoy the basic properties of the classical Hardy spaces. For example, it turns out that this definition does not depend on the choice of the test function φ as long as ∫ φ 6= 0. These spaces can be equivalently introduced in terms of grand, tangential, or nontangential maximal functions. We prove the Calderon-Zygmund decomposition which enables us to show the atomic decomposition of H A. As a consequence of atomic decomposition we obtain the description of the dual to H A in terms of Campanato spaces. We provide a description of the natural class of operators acting on H A, i.e., Calderon-Zygmund singular integral operators. We also give a full classification of dilations generating the same space H A in terms of spectral properties of A. In the second part of this paper we show that for every dilation A preserving some lattice and satisfying a particular expansiveness property there is a multiwavelet in the Schwartz class. We also show that for a large class of dilations (lacking this property) all multiwavelets must be combined minimally supported in frequency, and thus far from being regular. We show that r-regular (tight frame) multiwavelets form an unconditional basis (tight frame) for the anisotropic Hardy space H A. We also describe the sequence space characterizing wavelet coefficients of elements of the anisotropic Hardy space. 2000 Mathematics Subject Classification. Primary 42B30, 42C40; Secondary 42B20, 42B25.
213 citations
TL;DR: In this article, the authors study the classification problem of Polish metric spaces up to isometry and define the isometry groups of the metric spaces, including locally compact, ultrametric, zero-dimensional, homogeneous and ultrahomogeneous spaces.
Abstract: We study the classification problem of Polish metric spaces up to isometry and
the isometry groups of Polish metric spaces. In the framework of the descriptive
set theory of definable equivalence relations, we determine the exact complexity of
various classification problems concerning Polish metric spaces. We start with the
class of all Polish metric spaces and prove that it is Borel bireducible to the universal
orbit equivalence relation induced by Borel actions of Polish groups. We then turn
to special classes of Polish metric spaces, including locally compact, ultrametric,
zero-dimensional, homogeneous, and ultrahomogeneous spaces. In the investigation
of the classification problems we also obtain characterizations for isometry groups
of various classes of Polish metric spaces.
117 citations
TL;DR: The Hamiltonian Symplectic geometry and the splitting of invariant manifolds have been studied in this article, where the splitting matrix is estimated by the Hamilton-Jacobi method for a simple resonance.
Abstract: Introduction and some salient features of the model Hamiltonian Symplectic geometry and the splitting of invariant manifolds Estimating the splitting matrix using normal forms The Hamilton-Jacobi method for a simple resonance Appendix. Invariant tori with vanishing or zero torsion Bibliography.
109 citations
TL;DR: The $\ell$-adic formalism Stratifications Topoi Algebraic stacks Convergent complexes Bibliography as discussed by the authors is a formalism for algebraic stacks and convergent complexes.
Abstract: Introduction The $\ell$-adic formalism Stratifications Topoi Algebraic stacks Convergent complexes Bibliography
76 citations
TL;DR: In this article, the authors present a group theoretic exceptionality approach for arithmetic exceptionality, including Dickson polynomials and Redei functions with Euclidean signatures.
Abstract: Introduction Arithmetic-Geometric preparation Group theoretic exceptionality Genus 0 condition Dickson polynomials and Redei functions Rational functions with Euclidean signature Sporadic cases of arithmetic exceptionality Bibliography.
67 citations
TL;DR: The classical homotopy theory of self-smash products was studied in this paper. But the classical theory is not applicable to the case of loop spaces, as shown in Section 2.1.
Abstract: Preliminary and the classical homotopy theory Decompositions of self smash products Decompositions of the loop spaces The homotopy groups $\pi_{n+r}(\Sigma^n\mathbb{R}\mathrm{P}^2)$ for $n\geq 2$ and $r\leq8$ The homotopy theory of $\Sigma\mathbb{R}\mathrm{P}^2$ Bibliography.
52 citations
TL;DR: In this article, the Nussbaum Theorem for open convex cones with non-empty interior and positive definite functions on convex sets was proved for conelike semigroups.
Abstract: Part I. Preliminaries and Preparatory Results: Bounded and unbounded operators Cone-valued measures Measures on topological spaces Projective limits of cone-valued measures Holomorphic functions Involutive semigroups and their representations Positive definite kernels and functions $\boldmath{C^*}$-algebras associated with involutive semigroups Integral representations of positive definite functions Convex cones and their faces Examples of convex cones Conelike semigroups: definition and examples Representations of conelike semigroups I Fourier and Laplace transforms Generalized Bochner and Stone Theorems Part II. Main Results: Nussbaum Theorem for open convex cones Positive definite functions on convex cones with non-empty interior Positive definite functions on convex sets Associated Hilbert spaces and representations Nussbaum Theorem for generating convex cones Representations of conelike semigroups II Associated unitary representations Holomorphic extension of unitary representations Holomorphic extension of representations of nuclear groups References Index List of symbols.
33 citations
TL;DR: In this paper, a review of basic objects of $0$-geometry is given, including the small $0-calculus and the $0)-calculus with bounds The reduced normal operator Weighted Sobolev spaces.
Abstract: Part 1. Fredholm theory for $0$-pseudodifferential operators: Review of basic objects of $0$-geometry The small $0$-calculus and the $0$-calculus with bounds The $b$-$c$-calculus on an interval The reduced normal operator Weighted $0$-Sobolev spaces Fredholm theory for $0$-pseudodifferential operators Part 2. Algebras of $0$-pseudodifferential operators of order $0$: $C^*$-algebras of $0$-pseudodifferential operators $\Psi^*$-algebras of $0$-pseudodifferential operators Appendix A. Spaces of conormal functions Bibliography Notations Index.
26 citations
TL;DR: In this article, the moduli space of $N = 1$ superspheres with tubes and the sewing operation is introduced and a formal algebraic study of the operation is presented.
Abstract: Introduction An introduction to the moduli space of $N=1$ superspheres with tubes and the sewing operation A formal algebraic study of the sewing operation An analytic study of the sewing operation Bibliography.
22 citations
TL;DR: In this article, the authors present a review of Hilbert and symplectic space theory for elliptic differential operators with respect to global boundary conditions and modified Laplace operators, as well as a list of symbols and notations.
Abstract: Introduction: Organization of results Review of Hilbert and symplectic space theory GKN-theory for elliptic differential operators Examples of the general theory Global boundary conditions: Modified Laplace operators Appendix A. List of symbols and notations Bibliography Index.
22 citations