Explicit algebras with the Leibniz–Mellin translation product

A sub‐calculus of the calculus (“algebra”) of all complete conormal symbols arising in the edge pseudodifferential calculus is constructed. This calculus of complete conormal symbols is suitable for constructing sub‐calculi of the general edge pseudodifferential calculus, for which the edge‐degenerate pseudodifferential operators involved map conormal asymptotics of distributions near the edges in a prescribed manner. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Curvature invariants, differential operators and local homogeneity

We first prove that a Riemannian manifold with globally constant additive Weyl invariants is locally homogeneous. Then we use this result to show that a manifold whose Laplacian commutes with all invariant differential operators is a locally homogeneous space.

     

Partial differential equations with matricial coefficients and generalized translation operators

Let be the Bessel operator with matricial coefficients defined on by

where is a diagonal matrix and let be an matrix-valued function. In this work, we prove that there exists an isomorphism on the space of even , -valued functions which transmutes and . This allows us to define generalized translation operators and to develop harmonic analysis associated with . By use of the Riemann method, we provide an integral representation and we deduce more precise information on these operators.

     

Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions

We consider transfer operators acting on spaces of holomorphic functions, and provide explicit bounds for their eigenvalues. More precisely, if Ω is any open set in Cd, and L is a suitable transfer operator acting on Bergman space A²(Ω), its eigenvalue sequence {λn(L)} is bounded by |λn(L)|?Aexp(−an1/d), where a,A>0 are explicitly given.     

Hermitizable,isospectral complex second-order differential operators

The first aim of the paper is to study the Hermitizability of secondorder differential operators, and then the corresponding isospectral operators. The explicit criteria for the Hermitizable or isospectral properties are presented. The second aim of the paper is to study a non-Hermitian model, which is now well known. In a regular sense, the model does not belong to the class of Hermitizable operators studied in this paper, but we will use the theory developed in the past years, to present an alternative and illustrated proof of the discreteness of its spectrum. The harmonic function plays a critical role in the study of spectrum. Two constructions of the function are presented. The required conclusion for the discrete spectrum is proved by some comparison technique.     

On a class of holomorphic functions representable by Carleman formulas in the interior of an equilateral cone from their values on its rigid base

Let Δ be an equilateral cone in C with vertices at the complex numbers and rigid base M (Section 1). Assume that the positive real semi-axis is the bisectrix of the angle at the origin. For the base M of the cone Δ we derive a Carleman formula representing all those holomorphic functions from their boundary values (if they exist) on M which belong to the class . The class is the class of holomorphic functions in Δ which belong to the Hardy class near the base M (Section 2). As an application of the above characterization, an important result is an extension theorem for a function fL¹(M) to a function .     

Support properties and Holmgren's uniqueness theorem for differential operators with hyperplane singularities

Let W be a finite Coxeter group acting linearly on Rn. In this article we study the support properties of a W-invariant partial differential operator D on Rn with real analytic coefficients. Our assumption is that the principal symbol of D has a special form, related to the root system corresponding to W. In particular the zeros of the principal symbol are supposed to be located on hyperplanes fixed by reflections in W. We show that conv(suppDf)=conv(suppf) holds for all compactly supported smooth functions f so that conv(suppf) is W-invariant. The main tools in the proof are Holmgren's uniqueness theorem and some elementary convex geometry. Several examples and applications linked to the theory of special functions associated with root systems are presented.     

Existence of positive solutions for higher-order functional differential equations

Under suitable conditions on f(t,y(t+θ)), the boundary value problem of higher-order functional differential equation (FDE) of the form

     

Separable anisotropic differential operators and applications

The nonlocal boundary value problems for anisotropic partial differential-operator equations with a dependent coefficients are studied. The principal parts of the appropriate generated differential operators are nonself-adjoint. Several conditions for the maximal regularity and the fredholmness in Banach-valued Lp-spaces of these problems are given. These results permit us to establish that the inverse of corresponding differential operators belongs to Schatten q-class. Some spectral properties of the operators are investigated. In applications, the nonlocal BVP's for quasielliptic partial differential equations and for systems of quasielliptic equations on cylindrical domain are studied.     

Elastic potentials at corners in Sobolev spaces with asymptotics

This paper is aimed at studying the single and double layer potentials related to the boundary value problems of elasticity theory for anisotropic case for the plane, corner domains. We start from the systems of second order elliptic differential equations with constant coefficients, write the fundamental solution and form the single and double layer (elastic) potentials. Applying the pseudo‐differential calculus we obtain the continuity results of the elastic potentials at corners in cone Sobolev spaces without and with asymptotics and characterize asymptotics of solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Asymptotic formulas for sequence factorial of arithmetic progression

This note provides asymptotic formulas for approximating the sequence factorial of members of a finite arithmetic progression by using Stirling, Burnside and other more accurate asymptotic formulas for large factorials that have appeared in the literature.     

Hyperbolic equations, function spaces with exponential weights and Nemytskij operators

Different problems in the theory of hyperbolic equations bases on function spaces of Gevrey type are studied. Beside the original Gevrey classes, spaces defined by the behaviour of the Fourier transform were also used to prove basic results about the well-posedness of Cauchy problems for non-linear hyperbolic systems. In these approaches only the algebra property of the function spaces was used to include analytic non-linearities. Here we will generalize this dependence. First we investigate superposition operators in spaces with exponential weights. Then we show in concrete situations how a priori estimates of strictly hyperbolic type lead to the well-posedness of certain semi-linear hyperbolic Cauchy problems in suitable function spaces with exponential weights of Gevrey type. Mathematics Subject Classification (2000) 46E35, 35L80, 35L15, 47H30     

Bilinear embedding for real elliptic differential operators in divergence form with potentials

We present a simple Bellman function proof of a bilinear estimate for elliptic operators in divergence form with real coefficients and with nonnegative potentials. The constants are dimension-free. The p-range of applicability of this estimate is (1,∞) for any real accretive (nonsymmetric) matrix A of coefficients.     

Nonlinear boundary value problems for second order differential equations with causal operators

In this paper we deal with second order differential equations with causal operators. To obtain sufficient conditions for existence of solutions we use a monotone iterative method. We investigate both differential equations and differential inequalities. An example illustrates the results obtained.     

Pseudo‐differential operators in a Gelfand–Shilov setting

We introduce some general classes of pseudodifferential operators with symbols admitting exponential type growth at infinity and we prove mapping properties for these operators on Gelfand–Shilov spaces. Moreover, we deduce composition and certain invariance properties of these classes.     

Green's function for second-order periodic boundary value problems with piecewise constant arguments

We obtain, under suitable conditions, the Green's function to express the unique solution for a second-order functional differential equation with periodic boundary conditions and functional dependence given by a piecewise constant function. This expression is given in terms of the solutions for certain associated problems. The sign of the solution is determined taking into account the sign of that Green's function.