The Continuity of Commutators on Triebel-Lizorkin Spaces

The purpose of this paper is to study the continuity in the context of Triebel-Lizorkin spaces for some commutators related to certain convolution operators. The operators include Littlewood-Paley operator, Marcinkiewicz integral and Bochner-Riesz operator.     

Estimates of Marcinkiewicz Integrals with Bounded Homogeneous Kernels of Degree Zero

Under the cancellation property and a certain Dini-type condition on kernels, we prove that Marcinkiewicz integrals with kernels which are homogeneous functions of degree zero, are bounded from to , from to , and from to for .     

On the congruence modularity conjecture

A new condition of compatibility with projections, applicable to some Maltsev filters, is defined and shown to hold, among others, for the filter of congruence-modular varieties. As a consequence, it is shown that there exist no simple counterexamples (in a specified sense) to the modularity conjecture. This paper is dedicated to Walter Taylor. Received November 5, 2005; accepted in final form April 3, 2006.     

Essential Norms of Composition Operators

We obtain simple estimates for the essential norm of a composition operator acting from the Hardy space H ^p to H ^q , p > q, in one or several variables. When p = and q = 2 our results give an exact formula for the essential norm.     

Normality of Products of <Emphasis Type="Italic">p</Emphasis>-Hyponormal Operators and Their Aluthge Transforms

Let B(H) denote the algebra of operators on a complex separable Hilbert space H, and let A $\in$ B(H) have the polar decomposition A = U|A|. The Aluthge transform is defined to be the operator . We say that A $\in$ B(H) is p-hyponormal, . Let . Given p-hyponormal , such that AB is compact, this note considers the relationship between denotes an enumeration in decreasing order repeated according to multiplicity of the eigenvalues of the compact operator T (respectively, singular values of the compact operator T). It is proved that is bounded above by and below by for all j = 1, 2, . . . and that if also is normal, then there exists a unitary U¹ such that for all j = 1, 2, . . ..     

Commutators on Half-Spaces

We study the boundedness and compactness of commutators on , where and are defined by and respectively. If satisfies some upper and lower estimates, then we obtain a necessary and sufficient condition for to be bounded or compact on for . The reproducing kernel of the harmonic Bergman space of can be shown to satisfy all the required estimates. Our result is the real variable analogue of the complex variable one for commutators associated with an analytic reproducing kernel.     

Few subpowers, congruence distributivity and near-unanimity terms

We prove that for any variety , the existence of an edge-term (defined in [1]) and Jónsson terms is equivalent to the existence of a near-unanimity term. We also characterize the idempotent Maltsev conditions which are defined by a system of linear absorption equations and which imply congruence distributivity. The first author was supported by the grant no. 144011G of the Ministry of Science and Environment of Serbia. The work of the second author was supported by US NSF grant no. DMS 0245622.     

ON THE LEFT-DEFINITENESS OF S-HERMITIAN SYSTEMS OF DIFFERENTIAL EQUATIONS

Abstract

Initiated by Schneider and Niessen's conjecture that left-definite S-Hermitian systems are essentially canonical, a certain class of transformations of such systems is considered, and criteria for the left-definiteness are derived. The results seem to disprove the conjecture.     

Price Operators Analysis in L p -Spaces

An integral type representation and various extension theorems for monotone linear operators in L _p -spaces are considered in relation to market price modelling. As application, a characterization of the existence of a risk-neutral probability measure equivalent to the applied underlying one is provided in terms of the given prices. These results are in the line of the fundamental theorem of asset pricing. Here, in particular, the risk-neutral probability measure considered has the advantage of having its density laying in pre-considered upper and lower bounds.     

Bibliographie zur geometrischen optimierung

A probabilistic scheduling period inventory system is considered for items that deteriorate continuously in time. The demand is assumed to occur instantaneously at the beginning of the scheduling period. Shortages are not allowed. A model with a non-zero lead time is developed first and, then, its special case is considered in which the lead time is a multiple of the scheduling period. The inventory models are developed using a general deterioration function. Their particular cases, when the rate of deterioration is constant, are also developed. Numerical examples are given to illustrate the models.     

Applications of generalized perron trees to maximal functions and density bases

In this article we give some new necessary conditions for subsets of the unit circle to give collections of rectangles (by means of orientations) which differentiate L^p-functions or give Hardy-Littlewood type maximal functions which are bounded on L^p, p>1. This is done by proving that a well-known method, the construction of a Perron Tree, can be applied to a larger collection of subsets of the unit circle than was earlier known. As applications, we prove a partial converse of a well-known result of Nagel et al. [6] regarding boundedness of maximal functions with respect to rectangles of lacunary directions, and prove a result regarding the cardinality of subsets of arithmetic progressions in sets of the type described above. Acknowledgements and Notes. This research was partially supported by NSERC.     

Centralisers of spaces of symmetric tensor products and applications

We show that the centraliser of the space of n-fold symmetric injective tensors, n≥2, on a real Banach space is trivial. With a geometric condition on the set of extreme points of its dual, the space of integral polynomials we obtain the same result for complex Banach spaces. We give some applications of this results to centralisers of spaces of homogeneous polynomials and complex Banach spaces. In addition, we derive a Banach-Stone Theorem for spaces of vector-valued approximable polynomials. This project was supported in part by Enterprise Ireland, International Collaboration Grant – 2004 (IC/2004/009). The second author was also partially supported by PIP 5272,UBACYTX108 and PICT 03-15033     

The structure of branched transportation networks

The transportation problem can be formalized as the problem of finding the optimal paths to transport a measure μ ⁺ onto a measure μ ⁻ with the same mass. In contrast with the Monge–Kantorovich formalization, recent approaches model the branched structure of such supply networks by an energy functional whose essential feature is to favor wide roads. Given a flow s in a road or a tube or a wire, the transportation cost per unit length is supposed to be proportional to s ^α with 0 < α < 1. For the Monge–Kantorovich energy α = 1 so that it is equivalent to have two roads with flow 1/2 or a larger one with flow 1. If instead 0 < α < 1, a road with flow is preferable to two individual roads s ₁ and s ₂ because . Thus, this very simple model intuitively leads to branched transportation structures. Such a branched structure is observable in ground transportation networks, in draining and irrigation systems, in electric power supply systems and in natural objects like the blood vessels or the trees. When such structures can irrigate a whole bounded open set of . The aim of this paper is to give a mathematical proof of several structure and regularity properties empirically observed in transportation networks. It is first proven that optimal transportation networks have a tree structure and can be monotonically approximated by finite graphs. An interior regularity result is then proven according to which an optimal network is a finite graph away from the irrigated measure. It is also proven that the branching number of optimal networks has everywhere a universal explicit bound. These results answer questions raised in two recent papers by Xia.     

Positive Periodic Solutions of Systems of Second Order Ordinary Differential Equations

We establish the existence, multiplicity and nonexistence of positive periodic solutions of systems of second order ordinary differential equations.     

Infimal convolution and Muckenhoupt A p (.) condition in variable L p spaces

We study the Hardy-Littlewood maximal operator M on . Under the assumptions that the exponent p satisfies and is constant outside some large ball, we prove that if and only if . Received: 2 June 2006 Revised: 28 November 2006     

Invariant Subspaces of Positive Quasinilpotent Operators on Ordered Banach Spaces

In this paper we find invariant subspaces of certain positive quasinilpotent operators on Krein spaces and, more generally, on ordered Banach spaces with closed generating cones. In the later case, we use the method of minimal vectors. We present applications to Sobolev spaces, spaces of differentiable functions, and C*-algebras.      

Validity of nonlinear geometric optics for entropy solutions of multidimensional scalar conservation laws

Nonlinear geometric optics with various frequencies for entropy solutions only in L^∞ of multidimensional scalar conservation laws is analyzed. A new approach to validate nonlinear geometric optics is developed via entropy dissipation through scaling, compactness, homogenization, and L¹-stability. New multidimensional features are recognized, especially including nonlinear propagations of oscillations with high frequencies. The validity of nonlinear geometric optics for entropy solutions in L^∞ of multidimensional scalar conservation laws is justified.