Weighted Composition Operators on Bergman and Dirichlet Spaces

Let H() denote a functional Hilbert space of analytic functions on a domain . Let w : C and : be such that w f is in H() for every f in H(). The operator wC given by f w f is called a weighted composition operator on H(). In this paper we characterize such operators and those for which (wC )* is a composition operator. Compact weighted composition operators on some functional Hilbert spaces are also characterized. We give sufficient conditions for the compactness of such operators on weighted Dirichlet spaces.     

A variable dimension algorithm with the Dantzig-Wolfe decomposition for structured stationary point problems

Given a set ofR ⁿ and a functionf from intoR ⁿ we consider a problem of finding a pointx ^* in such that(x–x ^*) ^t f(x ^*)0 holds for every pointx in. This problem is called the stationary point problem and the pointx ^* is called a stationary point. We present a variable dimension algorithm for solving the stationary point problem with an affine functionf on a polytope defined by constraints of linear equations and inequalities. We propose a system of equations whose solution set contains a piecewise linear path connecting a trivial starting point in with a stationary point. The path can be followed by solving a series of linear programs which inherit the structure of constraints of. The linear programs are solved efficiently with the Dantzig-Wolfe decomposition method by exploiting fully the structure.Part of this research was carried out when the first author was supported by the Center for Economic Research, Tilburg University, The Netherlands and the third author was supported by the Alexander von Humboldt-Foundation, Federal Republic of Germany.     

Projections on Lp-spaces of polyanalytic functions

The fundamental result: for an arbitrary bounded, simply connected domain in , the subspace L_n,m ^p() of the space L^p(, ) ( is the plane Lebesgue measure, p 1), consisting of the (m, n)-analytic functions in , is complemented in LP(, ) (a function f is said to be (m, n)-analytic if (^m+n/¯Z^mZⁿ)f=0 in ). Consequently, by virtue of a theorem of J. Lindenstrauss and A. Pelczyski, the space L_n,m ^P() is linearly homeomorphic to lP. In particular, for m=n=1 we obtain that the space of all harmonic L^P-functions in is complemented in L^P(, ). This result has been known earlier only for smooth domains.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 15–33, 1991.     

An estimate of the derivatives of the solutions of variational inequalities on the boundary of the domain

For the equation Au=f(x) in the domain Rⁿ, where A is a linear second-order elliptic operator, under conditions of Signorini type on the boundary of the domain , one proves the boundedness of the Hölder continuity of the first derivatives of the solution under the assumption that f L_q(), q>n. The results are applied to the investigation of the regularity of the solutions of variational inequalities in the case of quasilinear operators.Translated from Problemy Matematicheskogo Analiza, No. 10, pp. 92–105, 1986.     

On monotone extensions of boundary data

Summary A functionf C (), is called monotone on if for anyx, y the relation x – y ₊ ^s impliesf(x)f(y). Given a domain with a continuous boundary and given any monotone functionf on we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of and agree withf on . In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when is the unit square [0, 1]² in ², strictly monotone analytic boundary data admit a monotone analytic extension.Research supported by NSF Grant 8922154Research supported by DARPA: AFOSR #90-0323     

Projection-iteration methods for solving constrained minimization problems

The problem of minimization of the functionalf(x) on the set in a Hilbert space H is solved by methods that approximatef(x) by a sequence of functionalsf _n(x_n) defined on the sets _n= H _n (H _n H) and then minimize eachf _n(x_n) on _n by gradient projection methods. Several approximations x_n ⁽ⁱ⁾(i=1,2,...,k_n) are constructed for each functional, and the last approximation is accepted as the starting approximation for the next functional. Convergence theorems are proved and error bounds are obtained.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 3–8, 1988.     

Cofibrations of algebras

Summary The chain complex of a twisted free product A_*t, FK, is chain homotopy equivalent to a differential graded algebra, which is identified to be a confibration of algebras as defined by Quillen. Under certain connectivity conditions we obtain a long exact sequence connecting the homologies of A, K, and A_*t FK. In particular we derive a long exact sequence connecting the homologies of Y, X and (Y U_g CX) (, C, are the loop, the cone and the suspension constructions respectively). A chain complex equivalent to the chain complex of the Milnor free group FX is recognized, from which results a theorem of Bott and Samelson that H(X) is freely generated as a graded algebra by H(X).     

An embedding theorem for a limiting exponent

We consider the function space B _p ^l () of functionsf(x), defined on the domain of a certain class and characterized by specific differential-difference properties in L_p(). We prove a theorem on the embedding B _p,q ^l () L_q in the case whenl=n/p –n/q >0 and its generalization for vectorl, p, q.Translated from Matematicheski Zametki, Vol. 6, No. 2, pp. 129–138, August, 1969.     

Convexes hyperboliques et fonctions quasisymétriques

Hyperbolic convex sets and quasisymmetric functions Every bounded convex open set of R ^m is endowed with its Hilbert metric d . We give a necessary and sufficient condition, called quasisymmetric convexity, for this metric space to be hyperbolic. As a corollary, when the boundary is real analytic, is always hyperbolic.In dimension 2, this condition is: in affine coordinates, the boundary is locally the graph of a C¹ strictly convex function whose derivative is quasisymmetric.      

Generalized Padé-Type Approximation and Integral Representations

In several complex variables, the multivariate Padé-type approximation theory is based on the polynomial interpolation of the multidimensional Cauchy kernel and leads to complicated computations. In this paper, we replace the multidimensional Cauchy kernel by the Bergman kernel function K (z,x) into an open bounded subset of C ⁿ and, by using interpolating generalized polynomials for K (z,x), we define generalized Padé-type approximants to any f in the space OL ²() of all analytic functions on which are of class L ². The characteristic property of such an approximant is that its Fourier series representation with respect to an orthonormal basis for OL ²() matches the Fourier series expansion of f as far as possible. After studying the error formula and the convergence problem, we show that the generalized Padé-type approximants have integral representations which give rise to the consideration of an integral operator – the so-called generalized Padé-type operator – which maps every f OL ²() to a generalized Padé-type approximant to f. By the continuity of this operator, we obtain some convergence results about series of analytic functions of class L ². Our study concludes with the extension of these ideas into every functional Hilbert space H and also with the definition and properties of the generalized Padé-type approximants to a linear operator of H into itself. As an application we prove a Painlevé-type theorem in C ⁿ and we give two examples making use of generalized Padé-type approximants.     

Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements

Summary A generalized Stokes problem is addressed in the framework of a domain decomposition method, in which the physical computational domain is partitioned into two subdomains ₁ and ₂.Three different situations are covered. In the former, the viscous terms are kept in both subdomains. Then we consider the case in which viscosity is dropped out everywhere in . Finally, a hybrid situation in which viscosity is dropped out only in ₁ is addressed. The latter is motivated by physical applications.In all cases, correct transmission conditions across the interface between ₁ and ₂ are devised, and an iterative procedure involving the successive resolution of two subproblems is proposed.The numerical discretization is based upon appropriate finite elements, and stability and convergence analysis is carried out.We also prove that the iteration-by-subdomain algorithms which are associated with the various domain decomposition approaches converge with a rate independent of the finite element mesh size.This work was partially supported by CIRA S.p.A. under the contract Coupling of Euler and Navier-Stokes equations in hypersonic flowsDeceased     

Generalized Cranston–McConnell Inequalities for Discontinuous Superharmonic Functions

Let be an open set in R² with Green function G(x,y) for the Laplace equation. We give a generalization of the Cranston-McConnell inequality concerning the integrability of positive harmonic functions on .     

Existence of bounded solutions for some degenerated quasilinear elliptic equations

Summary We prove the existence of bounded solutions in L () of degenerate elliptic boundary value problems of second order in divergence form with natural growth in the gradient. For the Dirichlet problem our results cover also unbounded domains .Work performed under the auspicies of G.N.A.F.A. of the C.N.R., partially supported by M.P.I. of Italy (40%).     

Groups of Congruences and Restriction Matrices

Let be a domain of the Euclidean space R ^m sent onto itself by a finite group G of congruences. In this paper we first define M elementary restriction matrices related to the group G and to a system of irreducible matrix representations of G. We then describe a general procedure to generate M restriction matrices for any finite-dimensional space V() of real functions defined on , when V() is invariant with respect to G. The number M depends only on the group G. Restriction matrices for the space V() have a block structure and all blocks can be obtained as from an elementary restriction matrix. Restriction matrices related to V() define a decomposition of V() as the sum of M subspaces. Finally, owing to restriction matrices, we propose a result of decomposition for linear systems. Several examples are presented.This revised version was published online in October 2005 with corrections to the Cover Date.     

Gelfand theory of certainC *-algebras on nonasymptotically flat manifolds

Let be a Riemannian manifold with finitely many conical ends. Under certain conditions which do not require to be asymptotically flat, we study aC ^*-algebra containing pseudodifferential operators on . Results on compact commutators and the Gelfand space are presented. Criteria for differential operators or systems within reach to be Fredholm are just simple consequences of Atkinson's theorem and our results.     

Extremal feedback perturbations for families of closed operators

Let T- S, be a family of not necessarily bounded semi-Fredholm operators, where T and S are operators acting between Banach spaces X and Y, and where S is bounded with D(S) D(T). For compact sets , as well as for certain open sets , we investigate existence and minimal rank of bounded feedback perturbations of the form F=BE such that min.ind (T-S+F)=0 for all . Here B is a given operator from a linear space Z to Y and E is some operator from X to Z.We give a simple characterization of that situation, when such regularizing feedback perturbations exist and show that for compact sets the minimal rank never exceeds max { min.ind (T-S) }+1. Moreover, an example shows that the minimal rank, in fact, may increase from max {...} to max {...}+1, if the given B enforces a certain structure of the feedbachk perturbation F.However, the minimal rank is equal to max { min.ind (T-S) }, if is an open set such that min.ind (T-S) already vanishes for all but finitely many points . We illustrate this result by applying it to the stabilization of certain infinite-dimensional dynamical systems in Hilbert space.     

Spectrum of multidimensional periodic operators

Let be a lattice in the n-dimensional Euclidean space Rⁿ and let F be the fundamental domain of the lattice . We denote by H the Schrödinger operator generated in L₂(Rⁿ) by the expression –u + q(x)u(1), and by H^t the operator generated in L₂(F) by the expression (1) and by quasiperiodic boundary conditions, where q(x) is a periodic (with respect to the lattice ) function. Asymptotic formulas for the eigenvalues of the operator H_t are obtained and with the aid of these formulas it is proved that there exists a number (q) such that the interval [(q), ] belongs to the spectrum of the operator H [for n3 in the case of sufficiently smooth potentials q(x), while for n=2 for any potential q(x) from L₂(F)], i.e., the Bethe-Sommerfeld conjecture is proved for arbitrary lattices.Translated from Teoriya Funktsii, Funktsionali'nyi Analiz i Ikh Prilozheniya, No. 49, pp. 17–34, 1988.     

Central orderings in fields of real meromorphic function germs

Let X_oR ⁿ be an irreducible analytic germ and the order space of its field of meromorphicfunetion germs. A formal half-branch in X_o is a kind of C-map germ c[0,)X_o; an ordering is centered at c if it contains the functions which are positive on c. We obtain a partition ¹,...,^d, d=dim X_o, of the set ^* of central (i.e.: centered at some half-branch) orderings, according to the dimension of half-branches. Then we show that all ^e, e= 1,.,d, as well as the set \^* of noncentral orderings, are dense in . Finally, we solve the 17th Hubert Problem for analytic germs.     

Certain problems of vector analysis

In the paper one gives an explicit method of constructing in the domain ^m m2 a vector field, having a prescribed divergencef in and taking prescribed values on the boundary . The differential properties of the field are faithfully determined by the smoothness off, and Simultaneously, one constructs the solutions of a series of other problems of vector analysis, which present an independent interest.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 65–85, 1984.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.