Polar actions on Damek-Ricci spaces

A proper isometric Lie group action on a Riemannian manifold is called polar if there exists a closed connected submanifold which meets all orbits orthogonally. In this article we study polar actions on Damek-Ricci spaces. We prove criteria for isometric actions on Damek-Ricci spaces to be polar, find examples and give some partial classifications of polar actions on Damek-Ricci spaces. In particular, we show that non-trivial polar actions exist on all Damek-Ricci spaces.     

Some properties of horocycles on Damek-Ricci spaces

We prove that a Damek-Ricci space is symmetric if and only if the geodesic inversion preserves the set of horocycles.     

Chaotic dynamics of the heat semigroup on the Damek-Ricci spaces

We consider the rank one Riemannian symmetric spaces of noncompact type and their non-symmetric generalization, namely the Damek-Ricci spaces. We show that the heat semigroup generated by a certain perturbation of the Laplace-Beltrami operator of these spaces is chaotic on their L _p -spaces when p > 2. The range of p and the corresponding perturbation are sharp. A precursor to this result is due to Ji and Weber [19] where it was shown that under identical conditions the heat operator is subspace-chaotic on the Riemannian symmetric spaces, which is weaker than it being chaotic. We also extend the results to the Lorentz spaces L _p,q , which are generalizations of the Lebesgue spaces. This enables us to point out that the chaoticity degenerates to subspace-chaoticity only when q = ∞.     

Continuous-time Kreiss resolvent condition on infinite-dimensional spaces

Given the infinitesimal generator of a -semigroup on the Banach space which satisfies the Kreiss resolvent condition, i.e., there exists an such that for all complex with positive real part, we show that for general Banach spaces this condition does not give any information on the growth of the associated -semigroup. For Hilbert spaces the situation is less dramatic. In particular, we show that the semigroup can grow at most like . Furthermore, we show that for every there exists an infinitesimal generator satisfying the Kreiss resolvent condition, but whose semigroup grows at least like . As a consequence, we find that for with the standard Euclidian norm the estimate cannot be replaced by a lower power of or .

     

Analytic continuation of resolvent kernels on noncompact symmetric spaces

For certain real hypersurfaces in the projective space, of signature (1,n), we study the filling problem for small deformations of the CR structure (the other signatures being well understood). We characterize the deformations which are fillable, and prove that they have infinite codimension in the set of all CR structures. This result generalizes the cases of the 3-sphere and of signature (1,1) to higher dimension.The author is a member of EDGE, Research Training Network, HPRN-CT-2000-00101, supported by the European Human Potential Programme.     

The level sets of the resolvent norm and convexity properties of Banach spaces

We construct a bounded linear operator on a separable, reflexive and strictly convex Banach space with the resolvent norm that is constant in a neighbourhood of zero.      

Generalized resolvent matrices and spaces of analytic functions

We introduce triplet spaces for symmetric relations with defect index (1, 1) in a Pontryagin space. Representations of Pontryagin spaces by spaces of vector-valued analytic functions are investigated. These concepts are used to study 2×2-matrix valued analytic functions which satisfy a certain kernel condition.     

Analytic continuation of the resolvent of the Laplacian on symmetric spaces of noncompact type

Let (M,g) be a globally symmetric space of noncompact type, of arbitrary rank, and Δ its Laplacian. We introduce a new method to analyze Δ and the resolvent (Δ-σ)^-1; this has origins in quantum N-body scattering, but is independent of the ‘classical’ theory of spherical functions, and is analytically much more robust. We expect that, suitably modified, it will generalize to locally symmetric spaces of arbitrary rank. As an illustration of this method, we prove the existence of a meromorphic continuation of the resolvent across the continuous spectrum to a Riemann surface multiply covering the plane. We also show how this continuation may be deduced using the theory of spherical functions. In summary, this paper establishes a long-suspected connection between the analysis on symmetric spaces and N-body scattering.     

Uniform stability of a family of resolvent operators in Hilbert spaces

Semigroup Forum - In this paper, two main results on the uniform stability of a resolvent family $${R_{h}(t)}_{tge 0}$$ , depending on a parameter h are presented. First, we discuss a GGP type...     

An algorithm based on resolvent operators for solving variational inequalities in Hilbert spaces

In this paper, a new monotonicity, M$M$-monotonicity, is introduced, and the resolvent operator of an M$M$-monotone operator is proved to be single valued and Lipschitz continuous. With the help of the resolvent operator, an equivalence between the variational inequality VI(C,F+G)$(C,F+G)$ and the fixed point problem of a nonexpansive mapping is established. A proximal point algorithm is constructed to solve the fixed point problem, which is proved to have a global convergence under the condition that F$F$ in the VI problem is strongly monotone and Lipschitz continuous. Furthermore, a convergent path Newton method, which is based on the assumption that the projection mapping _∏C(⋅)${\prod }_C(\cdot )$ is semismooth, is given for calculating ε$\epsilon$-solutions to the sequence of fixed point problems, enabling the proximal point algorithm to be implementable.     

Asymptotic resolvent and theorems on generation of semigroups of operators in locally convex spaces

The concept of the asymptotic resolvent of an operator is introduced and used to establish theorems on the generation of semigroups of distributions and operator semigroups of class (ℒ _loc ^p , ℬ) in a locally convex space. Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 433–442, September, 1977.     

High energy resolvent estimates for Schrödinger operators in Besov spaces

Dedicated to Professor Shmuel Agmon     

Remarks on the resolvent method

The resolvent method for the approximate solution of Fredholm integral equations, proposed by the author, is extended to systems of Fredholm equations and to equations with a weak singularity. The case of a definite irregularity of the integration manifold is also considered.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 5–14, 1976.     

Very weak solutions to the stationary Stokes and Stokes resolvent problem in weighted function spaces

We investigate very weak solutions to the stationary Stokes and Stokes resolvent problem in function spaces with Muckenhoupt weights. The notion used here is similar but even more general than the one used in Amann (Nonhomogeneous Navier–Stokes equations with integrable low-regularity data. Int. Math. Ser., pp. 1–26. Kluwer Academic/Plenum Publishing, New York, 2002) or Galdi et al. (Math. Ann. 331, 41–74, 2005). Consequently the class of solutions is enlarged. To describe boundary conditions we restrict ourselves to more regular data. We introduce a Banach space that admits a restriction operator and that contains the solutions according to such data.