The Berezin transform and Laplace-Beltrami operator

We study the Berezin transform of bounded operators on the Bergman space on a bounded symmetric domain Ω in Cn. The invariance of range of the Berezin transform with respect to G=Aut(Ω), the automorphism group of biholomorphic maps on Ω, is derived based on the general framework on invariant symbolic calculi on symmetric domains established by Arazy and Upmeier. Moreover we show that as a smooth bounded function, the Berezin transform of any bounded operator is also bounded under the action of the algebra of invariant differential operators generated by the Laplace-Beltrami operator on the unit disk and even on the unit ball of higher dimensions.     

Shape sensitivity for the Laplace-Beltrami operator with singularities

This paper considers shape sensitivity analysis for the Laplace-Beltrami operator formulated on a two-dimensional manifold with a fracture. We characterize the shape gradient of a functional as a bounded measure on the manifold and decompose it into a “distributed gradient” supported on the manifold, plus a singular part that we derive as the limit of a “jump” through the crack and Dirac measures at the crack extremities. The important point is that we introduce a technique that is not dimension dependent, and makes no use of classical arguments such as the maximum principle or continuation uniqueness. The technique makes use of a family of envelopes surrounding the fracture which enable us to relax certain terms and to overcome the lack of regularity resulting from the presence of the fracture. We use the min-max differentiation in order to avoid taking the derivative of the state equation and to manage the crack's singularities. Therefore, we write the functional in a min-max formulation on a space which takes into account the hidden boundary regularity established by the tangential extractor method.     

Applications of a Laplace-Beltrami operator for Jack polynomials

An explicit computation is made for a Laplace-Beltrami type operator for Jack polynomials. As applications we obtain: combinatorial formula, determinantal formula and raising operator formula for Jack polynomials, as well as an iterative formula for the Littlewood-Richardson coefficients. One special case of our results implies Mimachi-Yamada’s result on Jack polynomials of rectangular shapes.     

Inequalities and separation for the Laplace-Beltrami differential operator in Hilbert spaces

In this paper we have studied the separation for the Laplace-Beltrami differential operator of the form

     

Estimates of sums of zero multiplicities for eigenfunctions of the Laplace-Beltrami operator

We obtain an upper estimate N?_χ(M) for the sum Q _N of singular zero multiplicities of the Nth eigenfunction of the Laplace-Beltrami operator on the two-dimensional, compact, connected Riemann manifold M, where _χ M is the Euler characteristic ofM. Stronger estimates, but equivalent asymptotically (N å ∞), are given for the cases of the sphere S ² and the projective plane ?². Asymptotically sharper estimates are shown for the case of a domain on the plane.     

Convexity properties for interior operator games

Interior operator games arose by abstracting some properties of several types of cooperative games (for instance: peer group games, big boss games, clan games and information market games). This reason allow us to focus on different problems in the same way. We introduced these games in Bilbao et al. (Ann. Oper. Res. 137:141–160, 2005) by a set system with structure of antimatroid, that determines the feasible coalitions, and a non-negative vector, that represents a payoff distribution over the players. These games, in general, are not convex games. The main goal of this paper is to study under which conditions an interior operator game verifies other convexity properties: 1-convexity, k-convexity (k≥2 ) or semiconvexity. But, we will study these properties over structures more general than antimatroids: the interior operator structures. In every case, several characterizations in terms of the gap function and the initial vector are obtained. We also find the family of interior operator structures (particularly antimatroids) where every interior operator game satisfies one of these properties.     

On the Hencl's notion of absolute continuity

We prove that a slight modification of the notion of α-absolute continuity introduced in [D. Bongiorno, Absolutely continuous functions in , J. Math. Anal. Appl. 303 (2005) 119–134] is equivalent to the notion of n, λ-absolute continuity given by S. Hencl in [S. Hencl, On the notions of absolute continuity for functions of several variables, Fund. Math. 173 (2002) 175–189].     

Real polynomial chaos and absolute continuity

Summary Under very general conditions, we prove that the members of a so-called real polynomial chaos have absolutely continuous probability distributions.Partly supported by the Swedish Natural Science Research Council contract F-FU 1443-101, KTO:511443101-7     

Local Lipschitz continuity of the stop operator

On a closed convex set Z in ^N with sufficiently smooth (W ^2,) boundary, the stop operator is locally Lipschitz continuous from W ^1,1([0,T]^N) × Z into W ^1,1([0,T],^N). The smoothness of the boundary is essential: A counterexample shows that C ¹-smoothness is not sufficient.     

Focusing problem and the asymptotics of the spectral function of the Laplace-Beltrami operator. II

In the paper a formal high-frequency solution of the problem of a point source of oscillations near a reflecting boundary is constructed. The boundary is geodesically concave which makes it possible to use methods developed in diffraction theory by V. A. Fock and J. B. Keller. On the basis of the formal solution of the problem of a point source of oscillations, it is possible to construct the asymptotics of the spectral functions of the Laplace-Beltrami operator.Translated fromZapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 14–53, 1979.     

Regularized trace of the perturbed Laplace-Beltrami operator on two-dimensional manifolds with closed geodesics

The main result of the paper is the determination of the regularized trace of the Laplace-Beltrami operator with potential on the manifold given by a function family of smooth almost Liouville metrics on the sphere (besides, all the geodesics of these metrics are closed and have equal length).     

Formula for the first regularized trace for the Laplace-Beltrami operator with a potential

The first regularized trace of the Laplace-Beltrami operator with an odd potential on two-dimensional sphere is computed. Bibliography: 4 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 18, pp. 79–105. 1995.     

Problem of focusing and the asymptotics of the spectral function of the Laplace-Beltrami operator. I

The connection between the asymptotics of the spectral function and the formal shortwave expansion of the solution of the problem of the asymptotics of the Green function near a geodesically concave boundary of a two-dimensional surface is considered in the paper.     

Modulus of continuity of the matrix absolute value

Lipschitz continuity of the matrix absolute value |A| = (A*A)^1/2 is studied. Let A and B be invertible, and let M ₁ = max(‖A‖, ‖B‖), M ₂ = max(‖A ⁻¹‖, ‖B ⁻¹‖). Then it is shown that

Estimates of operator moduli of continuity

In Aleksandrov and Peller (2010) [2] we obtained general estimates of the operator moduli of continuity of functions on the real line. In this paper we improve the estimates obtained in Aleksandrov and Peller (2010) [2] for certain special classes of functions. In particular, we improve estimates of Kato (1973) [18] and show that