Weighted norm inequalities for convolution operators and links with the Weiss Conjecture

We study convolution operators on weighted Lebesgue spaces and obtain weight characterisations for boundedness of these operators with certain kernels. Our main result is Theorem 3 which enables us to obtain results for certain kernel functions supported on bounded intervals; in particular we get a direct proof of the known characterisations for Steklov operators in Section 3 by using the weighted Hardy inequality. Our methods also enable us to obtain new results for other kernel functions in Section 4. In Section 5 we demonstrate that these convolution operators are related to operators arising from the Weiss Conjecture (for scalar-valued observation functionals) in linear systems theory, so that results on convolution operators provide elementary examples of nearly bounded semigroups not satisfying the Weiss Conjecture. Also we apply results on the Weiss Conjecture for contraction semigroups to obtain boundedness results for certain convolution operators.     

A direct approach to the Weiss conjecture for bounded analytic semigroups

We give a new proof of the Weiss conjecture for analytic semigroups. Our approach does not make any recourse to the bounded H ^∞-calculus and is based on elementary analysis.     

Applications of the Discrete Weiss Conjecture in Operator Theory

In this paper, we study a discrete version of the Weiss Conjecture. In Section 1 we discuss the Reproducing Kernel Thesis and in Section 2 we introduce the operators which concern us. Section 3 shows how to relate these operators to Carleson embeddings and weighted composition operators, so that we can apply the Carleson measure theorem to obtain conditions for boundedness and compactness of many weighted composition operators. Section 4 contains Theorem 4.4 which is a discrete version of the Weiss Conjecture for contraction semigroups, and finally Section 5 shows how the usual (continuous time) Weiss Conjecture is related to the discrete version studied here; in fact they are equivalent (for scalar valued observation operators). The main advantage of the discrete version is that it is technically simpler – the observation operators are automatically bounded and the functional calculus can be achieved using power series.     

The Weiss conjecture and weak norms

In this note, we show that for analytic semigroups the so-called Weiss condition of uniform boundedness of the operators $$Re(\lambda)^{1/2}C(\lambda+A)^{-1}, \quad Re(\lambda) > 0$$ on the complex right half plane and weak Lebesgue L ^2,∞ admissibility are equivalent. Moreover, we show that the weak Lebesgue norm is best possible in the sense that it is the endpoint for the ’Weiss conjecture’ within the scale of Lorentz spaces L ^p,q .     

Local Central Limit Theorem for Multi-group Curie–Weiss Models

Journal of Theoretical Probability - We study a multi-group version of the mean-field Ising model, also called Curie–Weiss model. It is known that, in the high-temperature regime of this...     

The Weiss constant and its analytical solution for the effect of internal demagnetization

The paper describes a solution to the Weiss effective field, in closed form, by using an analytical approximation to the Everett integral. Its application is demonstrated on one of the magnetism’s difficult problems, the presence of molecular interaction and its consequence, the internal demagnetisation. The solution is based on the Weiss effective field theory and fits within the framework of the Classic Scalar Preisach hysteresis modeling.     

The dynamics of critical fluctuations in asymmetric Curie–Weiss models

We study the dynamics of fluctuations at the critical point for two time-asymmetric version of the Curie–Weiss model for spin systems that, in the macroscopic limit, undergo a Hopf bifurcation. The fluctuations around the macroscopic limit reflect the type of bifurcation, as they exhibit observables whose fluctuations evolve at different time scales. The limiting dynamics of fluctuations of slow observable is obtained via an averaging principle.     

Geometry and Hardy spaces on spaces of homogeneous type in the sense of Coifman and Weiss

It is known that the space of homogeneous type introduced by Coifman and Weiss (1971) provides a very natural setting for establishing a theory of Hardy spaces. This paper concentrates on how the geometrical conditions of the space of homogeneous type play a crucial role in building a theory of Hardy spaces via the Littlewood-Paley functions.     

The Weiss conjecture on admissibility of observation operators for contraction semigroups

We prove the conjecture of George Weiss for contraction semigroups on Hilbert spaces, giving a characterization of infinite-time admissible observation functionals for a contraction semigroup, namely that such a functionalC is infinite-time admissible if and only if there is anM>0 such that for alls in the open right half-plane. HereA denotes the infinitesimal generator of the semigroup. The result provides a simultaneous generalization of several celebrated results from the theory of Hardy spaces involving Carleson measures and Hankel operators.     

The weighted Weiss conjecture and reproducing kernel theses for generalized Hankel operators

The weighted Weiss conjecture states that the system theoretic property of weighted admissibility can be characterized by a resolvent growth condition. For positive weights, it is known that the conjecture is true if the system is governed by a normal operator; however, the conjecture fails if the system operator is the unilateral shift on the Hardy space ${H^2(\mathbb{D})}$ (discrete time) or the right-shift semigroup on ${L^2(\mathbb{R}_+)}$ (continuous time). To contrast and complement these counterexamples, in this paper, positive results are presented characterizing weighted admissibility of linear systems governed by shift operators and shift semigroups. These results are shown to be equivalent to the question of whether certain generalized Hankel operators satisfy a reproducing kernel thesis.     

Hardy–Littlewood–Sobolev and Stein–Weiss inequalities on homogeneous Lie groups

In this note, we prove the Stein–Weiss inequality on general homogeneous Lie groups. The obtained results extend previously known inequalities. Special properties of homogeneous norms play a key role in our proofs. Also, we give a simple proof of the Hardy–Littlewood–Sobolev inequality on general homogeneous Lie groups.     

Smooth parametrized torsion: A manifold approach

We present a construction of a torsion invariant of bundles of smooth manifolds which is based on the work of Dwyer, Weiss and Williams on smooth structures on fibrations.     

Analysis of a simple Markovian re-entrant line with infinite supply of work under the LBFS policy

We consider a two machine 3 step re-entrant line, with an infinite supply of work. The service discipline is last buffer first served. Processing times are independent exponentially distributed. We analyze this system, obtaining steady state behavior and sample path properties. AMS Subject Classifications 60K25 · 90B22 I. Adan and G. Weiss: Research supported in part by Network of Excellence Euro-NGI. G. Weiss: Research supported in part by Israel Science Foundation Grant 249/02.     

The Weiss Conjecture for Bounded Analytic Semigroups

New results concerning the so-called Weiss conjecture on admissibleoperators for bounded analytic semigroups are given. Let be a bounded analytic semigroup withgenerator –A on some Banach space X. It is proved thatif A^1/2 is admissible for A, that is, if there is an estimate then any continuous mappingC : D(A) Y valued in a Banach space Y is admissible for A providedthat there is an estimate .for , Re()<0. This holds in particular if is a contractive (analytic) semigroup on Hilbertspace. In the converse direction, it is shown that this mayhappen for a bounded analytic semigroup on Hilbert space thatis not similar to a contractive one. Applications in non-HilbertianBanach spaces are also given.     

Eigenfrequencies of fractal drums

A method for the computation of eigenfrequencies and eigenmodes of fractal drums is presented. The approach involves first conformally mapping the unit disk to a polygon approximating the fractal and then solving a weighted eigenvalue problem on the unit disk by a spectral collocation method. The numerical computation of the complicated conformal mapping was made feasible by the use of the fast multipole method as described in [L. Banjai, L.N. Trefethen, A multipole method for Schwarz–Christoffel mapping of polygons with thousands of sides, SIAM J. Sci. Comput. 25(3) (2003) 1042–1065]. The linear system arising from the spectral discretization is large and dense. To circumvent this problem we devise a fast method for the inversion of such a system. Consequently, the eigenvalue problem is solved iteratively. We obtain eight digits for the first eigenvalue of the Koch snowflake and at least five digits for eigenvalues up to the 20th. Numerical results for two more fractals are shown.     

A uniformly convex hereditarily indecomposable banach space

We construct a uniformly convex hereditarily indecomposable Banach space, using a method similar to the one of Gowers and Maurey in [GM], and the theory of complex interpolation for a family of Banach spaces of Coifman, Cwikel, Rochberg, Sagher, and Weiss ([CCRSW1]).     

Estimation of fractal dimension and fractal curvatures from digital images

Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several geometric characteristics, namely all the intrinsic volumes (i.e. volume, surface area, Euler characteristic, etc.) of the parallel sets of a fractal. Motivated by recent results on their limiting behavior, we use these functionals to estimate the fractal dimension of sets from digital images. Simultaneously, we also obtain estimates of the fractal curvatures of these sets, some fractal counterpart of intrinsic volumes, allowing a finer classification of fractal sets than by means of fractal dimension only. We show the consistency of our estimators and test them on some digital images of self-similar sets.     

The Fibonacci fractal is a new fractal type

We propose a uniform method for estimating fractal characteristics of systems satisfying some type of scaling principle. This method is based on representing such systems as generating Bethe-Cayley tree graphs. These graphs appear from the formalism of the group bundle of Fibonacci-Penrose inverse semigroups. We consistently consider the standard schemes of Cantor and Koch in the new method. We prove the fractal property of the proper Fibonacci system, which has neither a negative nor a positive redundancy type. We illustrate the Fibonacci fractal by an original procedure and in the coordinate representation. The golden ratio and specific inversion property intrinsic to the Fibonacci system underlie the Fibonacci fractal. This property is reflected in the structure of the Fibonacci generator.     

On G-locally primitive graphs of locally Twisted Wreath type and a conjecture of Weiss

Let Γ be a connected G-vertex-transitive graph and let v be a vertex of Γ. The graph Γ is said to be G-locally primitive if the action of the vertex-stabiliser Gv on the neighbourhood Γ(v) of v is primitive. Furthermore, Γ is said to be of locally Twisted Wreath type if Gv is a primitive group of Twisted Wreath type in its action on Γ(v).Richard Weiss conjectured in 1978 that, there exists a function f:N→N such that if Γ is a connected G-vertex-transitive locally primitive graph of valency d and v is a vertex of Γ, then |Gv|?f(d). In this paper we prove this conjecture when Γ is of locally Twisted Wreath type.     

A new fractal dimension: The topological Hausdorff dimension

We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension.