Sobolev spaces and harmonic maps between singular spaces

Recently Korevaar and Schoen developed a Sobolev theory for maps from smooth (at least ) manifolds into general metric spaces by proving that the weak limit of appropriate average difference quotients is well behaved. Here we extend this theory to functions defined over Lipschitz manifold. As an application we then prove an existence theorem for harmonic maps from Lipschitz manifolds to NPC metric spaces. Received December 6, 1996 / Accepted March 4, 1997     

Some function spaces on spaces of homogeneous type

We introduce Campanato, Morrey, BMO and Sobolev-type spaces for mappings from a space of homogeneous type into a complete metric space which possess properties comparable to their classical analogues. In particular we show integral characterizations, the validity of the John–Nirenberg theorem, Poincarè and Sobolev inequalities, Sobolev's embedding theorem and estimates on the pointwise behavior of Sobolev-type mappings. Received: 4 December 2000 / Revised version: 5 July 2001     

Local isometries of function spaces

Abstract. We show that for a large class of function spaces any isometry that coincides locally with a surjective isometry must be automatically surjective. This class includes finite-codimensional subspaces of and spaces of E-valued continuous functions for finite-dimensional or uniformly convex and algebraically reflexive E. Received: 5 November 2001 / Published online: 14 February 2003 Thanks: Research of both authors partially supported by a grant \# 1102386 from the NSF and a grant \# DST/INT/US(NSF-RP041)/2000 from the DST     

Berezin transform on¶compact Hermitian symmetric spaces

Let X=G ^* be a compact Hermitian symmetric space. We study the Berezin transform on L ²(X) and calculate its spectrum under the decomposition of L ²(X) into the irreducible representations of G ^*. As applications we find the expansion of powers of the canonical polynomial (Bergman reproducing kernel for the canonical line bundle) in terms of the spherical polynomials on X, and we find the irreducible decomposition of tensor products of Bergman spaces on X. Received: 10 September 1996 / Revised version: 10 September 1997     

Nonexistence of invariant semigroups in affine symmetric spaces

Let be an affine symmetric space with a simple Lie group, an involutive automorphism of and an open subgroup of the -fixed point group . It is proved here that the existence of a proper semigroup with and implies that is of Hermitian type, as conjectured by Hilgert and Neeb [4]. When exists, it turns out that it leaves invariant an open -orbit in a minimal flag manifold of . A byproduct of our approach is an alternate proof of the maximality of the compression semigroup of an open orbit (see Hilgert and Neeb [31]). Received: 10 September 1999 / Published online: 23 July 2001     

Orthogonal decompositions of Sobolev spaces in Clifford analysis

The space L ₂(G;ℂ _m ) of Clifford-algebra-valued functions in bounded domains G of ℝ ^m is decomposed into the orthogonal sum of the subspace of poly-left-monogenic functions of arbitrary order k≥1 and its orthogonal complement and as well into the orthogonal sum of the subspace of polyharmonic functions of arbitrary order k≥1 and its orthogonal complement. The complementary subspaces are given explicitly. In the particular case m=2, complex functions are involved. Although this case has to be treated separately, the results are as before. The proofs are based on proper higher-order Cauchy–Pompeiu formulas and Green functions for powers of the Laplacian. Received: July 4, 2000; in final form: January 7, 2001?Published online: December 19, 2001     

Edge Sobolev spaces and weakly hyperbolic equations

Summary. Edge Sobolev spaces are proposed as a main new tool for the investigation of weakly hyperbolic equations. The well-posedness of the linear and semilinear Cauchy problem in the class of these edge Sobolev spaces is proved. An application to the propagation of singularities for solutions to the semilinear problem is considered. Received: October 3, 2000 Published online: December 19, 2001     

On SS-rigid groups and A. Weil's criterion for local rigidity. I

The fundamental group Γ of a compact complete affine manifold is represented as an affine crystallographic subgroup of . L.S. Auslander conjectured that Γ is virtually solvable. Our purpose is to find the algebraic condition on Γ which leads affirmative answer to the conjecture. Received: 26 May 1997 / Revised version: 17 December 1997     

A new factorization technique of the matrix mask of univariate refinable functions

Summary. A univariate compactly supported refinable function can always be written as the convolution product , with the B-spline of order k,f a compactly supported distribution, and k the approximation orders provided by the underlying shift-invariant space . Factorizations of univariate refinable vectors were also studied and utilized in the literature. One of the by-products of this article is a rigorous analysis of that factorization notion, including, possibly, the first precise definition of that process. The main goal of this article is the introduction of a special factorization algorithm of refinable vectors that generalizes the scalar case as closely (and unexpectedly) as possible: the original vector is shown to be `almost' in the form , with F still compactly supported and refinable, andk the approximation order of . The algorithm guarantees F to retain the possible favorable properties of , such as the stability of the shifts of and/or the polynomiality of the mask symbol. At the same time, the theory and the algorithm are derived under relatively mild conditions and, in particular, apply to whose shifts are not stable, as well as to refinable vectors which are not compactly supported. The usefulness of this specific factorization for the study of the smoothness of FSI wavelets (known also as `multiwavelets' and `multiple wavelets') is explained. The analysis invokes in an essential way the theory of finitely generated shift-invariant (FSI) spaces, and, in particular, the tool of superfunction theory. Received June 10, 1998 / Revised version received June 14, 1999 / Published online August 2, 2000     

Sobolev spaces on an arbitrary metric space

We define Sobolev space W ^1,p for 1<p on an arbitrary metric space with finite diameter and equipped with finite, positive Borel measure. In the Euclidean case it coincides with standard Sobolev space. Several classical imbedding theorems are special cases of general results which hold in the metric case. We apply our results to weighted Sobolev space with Muckenhoupt weight.This work is supported by KBN grant no. 2 1057 91 01     

Computation of high order derivatives in optimal shape design

Summary. In shape optimization problems, each computation of the cost function by the finite element method leads to an expensive analysis. The use of the second order derivative can help to reduce the number of analyses. Fujii ([4], [10]) was the first to study this problem. J. Simon [19] gave the second order derivative for the Navier-Stokes problem, and the authors describe in [8], [11], a method which gives an intrinsic expression of the first and second order derivatives on the boundary of the involved domain. In this paper we study higher order derivatives. But one can ask the following questions: -- are they expensive to calculate? -- are they complicated to use? -- are they imprecise? -- are they useless? \medskip\noindent At first sight, the answer seems to be positive, but classical results of V. Strassen [20] and J. Morgenstern [13] tell us that the higher order derivatives are not expensive to calculate, and can be computed automatically. The purpose of this paper is to give an answer to the third question by proving that the higher order derivatives of a function can be computed with the same precision as the function itself. We prove also that the derivatives so computed are equal to the derivatives of the discrete problem (see Diagram 1). We call the discrete problem the finite dimensional problem processed by the computer. This result allows the use of automatic differentiation ([5], [6]), which works only on discrete problems. Furthermore, the computations of Taylor's expansions which are proposed at the end of this paper, could be a partial answer to the last question. Received January 27, 1993/Revised version received July 20, 1993