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1.
Fully symmetric functionals on a Marcinkiewicz space are Dixmier traces   总被引:1,自引:0,他引:1  
As a consequence of the exposition of Dixmier type traces in the book of A. Connes (1994) [2], we were led to ask how general is this class of functionals within the space of all unitarily invariant functionals on the corresponding Marcinkiewicz ideal Mψ. In this paper we prove the surprising result that the set of all Dixmier traces on Mψ coincides with the set of all fully symmetric functionals on this space.  相似文献   

2.
LetX be a Riemannian symmetric space of the noncompact type. We prove the multiplier theorem for the Helgason-Fourier transform and the vector valued function spacesL p (X, l q ). As a consequence we get the inequalities of the Littlewood-Paley type forL p (X) spaces.Research supported by K.B.N. Grant 210519101 (Poland).  相似文献   

3.
The intersection body of a ball is again a ball. So, the unit ball BdRd is a fixed point of the intersection body operator acting on the space of all star-shaped origin symmetric bodies endowed with the Banach–Mazur distance. E. Lutwak asked if there is any other star-shaped body that satisfies this property. We show that this fixed point is a local attractor, i.e., that the iterations of the intersection body operator applied to any star-shaped origin symmetric body sufficiently close to Bd in Banach–Mazur distance converge to Bd in Banach–Mazur distance. In particular, it follows that the intersection body operator has no other fixed or periodic points in a small neighborhood of Bd. We will also discuss a harmonic analysis version of this question, which studies the Radon transforms of powers of a given function.  相似文献   

4.
To each groupN of Heisenberg type one can associate a generalized Siegel domain, which for specialN is a symmetric space. This domain can be viewed as a solvable extensionS =NA ofN endowed with a natural left-invariant Riemannian metric. We prove that the functions onS that depend only on the distance from the identity form a commutative convolution algebra. This makesS an example of a harmonic manifold, not necessarily symmetric. In order to study this convolution algebra, we introduce the notion of “averaging projector” and of the corresponding spherical functions in a more general context. We finally determine the spherical functions for the groupsS and their Martin boundary. Communicated by Guido Weiss  相似文献   

5.
A complex Radon measure μ on ℝ n is said to be of at most exponential-quadratic growth if there exist positive constants C and α such that . Let Xexp denote the space of all complex Radon measure on ℝ n of at most exponential-quadratic growth. Using elementary methods, we obtain injectivity sets for spherical means for Xexp. We also discuss similar results for symmetric spaces.  相似文献   

6.
《Mathematische Nachrichten》2018,291(14-15):2204-2228
Paley–Wiener type theorems describe the image of a given space of functions, often compactly supported functions, under an integral transform, usually a Fourier transform on a group or homogeneous space. In this article we proved a Paley–Wiener theorem for smooth sections f of homogeneous line bundles on a compact Riemannian symmetric space . It characterizes f with small support in terms of holomorphic extendability and exponential growth of their χ‐spherical Fourier transforms, where χ is a character of K. An important tool in our proof is a generalization of Opdam's estimate for the hypergeometric functions associated to multiplicity functions that are not necessarily positive. At the same time the radius of the domain where this estimate is valid is increased. This is done in an appendix.  相似文献   

7.
We consider interpolation methods defined by positive definite functions on a locally compact group G. Estimates for the smallest and largest eigenvalue of the interpolation matrix in terms of the localization of the positive definite function on G are presented, and we provide a method to get positive definite functions explicitly on compact semisimple Lie groups. Finally, we apply our results to construct well-localized positive definite basis functions having nice stability properties on the rotation group SO(3).  相似文献   

8.
We give a simple and direct proof of the Grobman–Hartman theorem for nonautonomous differential equations obtained from perturbing a nonuniform exponential dichotomy. In particular, we do not need to pass through discrete time and obtain the result as a consequence of a corresponding result for maps. To the best of our knowledge, this is the first direct approach for nonuniform exponential dichotomies. We also show that the conjugacies are continuous in time and Hölder continuous in space. In addition, we describe the dependence of the conjugacies on the perturbation, and we obtain a reversibility result for the conjugacies of reversible differential equations. We emphasize that the additional work required to consider nonuniform exponential dichotomies is substantial.  相似文献   

9.
We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of Type I κ-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman?s previous result on backward limits of κ-solutions in dimension 3, in which case the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23], where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.  相似文献   

10.
The Fourier coefficients of a smooth K-invariant function on a compact symmetric space M=U/K are given by integration of the function against the spherical functions. For functions with support in a neighborhood of the origin, we describe the size of the support by means of the exponential type of a holomorphic extension of the Fourier coefficients.  相似文献   

11.
Recently a new basis for the Hopf algebra of quasisymmetric functions QSym, called quasisymmetric Schur functions, has been introduced by Haglund, Luoto, Mason, van Willigenburg. In this paper we extend the definition of quasisymmetric Schur functions to introduce skew quasisymmetric Schur functions. These functions include both classical skew Schur functions and quasisymmetric Schur functions as examples, and give rise to a new poset LC that is analogous to Young's lattice. We also introduce a new basis for the Hopf algebra of noncommutative symmetric functions NSym. This basis of NSym is dual to the basis of quasisymmetric Schur functions and its elements are the pre-image of the Schur functions under the forgetful map χ:NSymSym. We prove that the multiplicative structure constants of the noncommutative Schur functions, equivalently the coefficients of the skew quasisymmetric Schur functions when expanded in the quasisymmetric Schur basis, are nonnegative integers, satisfying a Littlewood–Richardson rule analogue that reduces to the classical Littlewood–Richardson rule under χ.As an application we show that the morphism of algebras from the algebra of Poirier–Reutenauer to Sym factors through NSym. We also extend the definition of Schur functions in noncommuting variables of Rosas–Sagan in the algebra NCSym to define quasisymmetric Schur functions in the algebra NCQSym. We prove these latter functions refine the former and their properties, and project onto quasisymmetric Schur functions under the forgetful map. Lastly, we show that by suitably labeling LC, skew quasisymmetric Schur functions arise in the theory of Pieri operators on posets.  相似文献   

12.
In this note, we look at some hypoelliptic operators arising from nilpotent rank 2 Lie algebras. In particular, we concentrate on the diffusion generated by three Brownian motions and their three Lévy areas, which is the simplest extension of the Laplacian on the Heisenberg group H. In order to study contraction properties of the heat kernel, we show that, as in the case of the Heisenberg group, the restriction of the sub-Laplace operator acting on radial functions (which are defined in some precise way in the core of the paper) satisfies a non-negative Ricci curvature condition (more precisely a CD(0,∞) inequality), whereas the operator itself does not satisfy any CD(r,∞) inequality. From this we may deduce some useful, sharp gradient bounds for the associated heat kernel.  相似文献   

13.
The Kantorovich–Rubinstein theorem provides a formula for the Wasserstein metric W1 on the space of regular probability Borel measures on a compact metric space. Dudley and de Acosta generalized the theorem to measures on separable metric spaces. Kellerer, using his own work on Monge–Kantorovich duality, obtained a rapid proof for Radon measures on an arbitrary metric space. The object of the present expository article is to give an account of Kellerer’s generalization of the Kantorovich–Rubinstein theorem, together with related matters. It transpires that a more elementary version of Monge–Kantorovich duality than that used by Kellerer suffices for present purposes. The fundamental relations that provide two characterizations of the Wasserstein metric are obtained directly, without the need for prior demonstration of density or duality theorems. The latter are proved, however, and used in the characterization of optimal measures and functions for the Kantorovich–Rubinstein linear programme. A formula of Dobrushin is proved.  相似文献   

14.
Let M be a topological G2-manifold. We prove that the space of infinitesimal associative deformations of a compact associative submanifold Y with boundary in a coassociative submanifold X is the solution space of an elliptic problem. For a connected boundary ∂Y of genus g, the index is given by Yc1(νX)+1−g, where νX denotes the orthogonal complement of TY in TX|∂Y and c1(νX) the first Chern class of νX with respect to its natural complex structure. Further, we exhibit explicit examples of non-trivial index.  相似文献   

15.
In this paper we introduce some new sequences of positive linear operators, acting on a sufficiently large space of continuous functions on the real line, which generalize Gauss–Weierstrass operators.We study their approximation properties and prove an asymptotic formula that relates such operators to a second order elliptic differential operator of the form Lu?αu′′+βu+γu.Shape-preserving and regularity properties are also investigated.  相似文献   

16.
After initial treatment of the Fourier analysis and operator ergodic theory of strongly continuous decomposable one-parameter groups of operators in the Banach space setting, we show that in the setting of a super-reflexive Banach space X these groups automatically transfer from the setting of R to X the behavior of the Hilbert kernel, as well as the Fourier multiplier actions of functions of higher variation on R. These considerations furnish one-parameter groups with counterparts for the single operator theory in Berkson (2010) [4]. Since no uniform boundedness of one-parameter groups of operators is generally assumed in the present article, its results for the super-reflexive space setting go well beyond the theory of uniformly bounded one-parameter groups on UMD spaces (which was developed in Berkson et al., 1986 [13]), and in the process they expand the scope of vector-valued transference to encompass a genre of representations of R that are not uniformly bounded.  相似文献   

17.
In this short note we prove that a Banach space X is reflexive if, and only if, the Eisenfeld–Lakshmikantham measure of nonconvexity in X satisfies the Cantor property. Using this characterization, some results in best approximation and fixed point theory for reflexive Banach spaces are generalized by removing convexity requirements.  相似文献   

18.
Let A be a complex, commutative Banach algebra and let MA be the structure space of A. Assume that there exists a continuous homomorphism h : L1(G) → A with dense range, where L1(G) is the group algebra of a locally compact abelian group G. The main results of this paper can be summarized as follows: (a) If the dual space A* has the Radon-Nikodym property, then MA is scattered (i.e., it has no nonempty perfect subset) and . (b) If the algebra A has an identity, then the space A* has the Radon-Nikodym property if and only if . Furthermore, any of these conditions implies that MA is scattered. Several applications are given. Received: 29 September 2005  相似文献   

19.
A polytope is integral if all of its vertices are lattice points. The constant term of the Ehrhart polynomial of an integral polytope is known to be 1. In previous work, we showed that the coefficients of the Ehrhart polynomial of a lattice-face polytope are volumes of projections of the polytope. We generalize both results by introducing a notion of k-integral polytopes, where 0-integral is equivalent to integral. We show that the Ehrhart polynomial of a k-integral polytope P has the properties that the coefficients in degrees less than or equal to k are determined by a projection of P, and the coefficients in higher degrees are determined by slices of P. A key step of the proof is that under certain generality conditions, the volume of a polytope is equal to the sum of volumes of slices of the polytope.  相似文献   

20.
We study the global in time existence of small solutions to the nonlinear Schrödinger equation with quadratic interactions (0.1) We prove that if the initial data u0 satisfy smallness conditions in the weighted Sobolev norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore, we prove the existence of the usual scattering states and find the large time asymptotics of the solutions.  相似文献   

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