Michael Kunzinger
Based on Colombeau's theory of algebras of generalized functions we introduce the concepts of generalized functions taking values in differentiable manifolds as well as of generalized vector bundle homomorphisms. We study their basic properties, in particular with respect to some new point value concepts for generalized functions and indicate applications of the resulting theory in general relativity.
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We discuss multiscale representations of discrete manifold-valued data. As it turns out that we cannot expect general manifold-analogues of biorthogonal wavelets to possess perfect reconstruction, we focus our attention on those constructions which are based on upscaling operators which are either interpolating or midpoint-interpolating. For definable multiscale decompositions we obtain a stability result.
We discuss the nature of structure-preserving maps of varies function algebras. In particular, we identify isomorphisms between special Colombeau algebras on manifolds with invertible manifold-valued generalized functions in the case of smooth parametrization. As a consequence, and to underline the consistency and validity of this approach, we see that this generalized version on algebra isomorphisms in turn implies the classical result on algebras of smooth functions.
This paper is part of an ongoing program to develop a theory of generalized differential geometry. We consider the space $\mathcal{G}[X,Y]$ of Colombeau generalized functions defined on a manifold $X$ and taking values in a manifold $Y$. This space is essential in order to study concepts such as flows of generalized vector fields or geodesics of generalized metrics. We introduce an embedding of the space of continuous mappings $\mathcal{C}(X,Y)$ into $\mathcal{G}[X,Y]$ and study the sheaf properties of $\mathcal{G}[X,Y]$. Similar results are obtained for spaces of generalized vector bundle hom…
We introduce the notion of generalized function taking values in a smooth manifold into the setting of full Colombeau algebras. After deriving a number of characterization results we also introduce a corresponding concept of generalized vector bundle homomorphisms and, based on this, provide a definition of tangent map for such generalized functions.
Stokes theorem holds for Lipschitz forms on a smooth manifold.
We derive sufficient conditions for the surjectivity of the Cauchy-Riemann operator $\overline{\partial}$ between spaces of weighted smooth Fréchet-valued functions. This is done by establishing an analog of Hörmander's theorem on the solvability of the inhomogeneous Cauchy-Riemann equation in a space of smooth $\mathbb{C}$-valued functions whose topologyis given by a whole family of weights. Our proof relies on a weakened variant of weak reducibility of the corresponding subspace of holomorphic functions in combination with the Mittag-Leffler procedure. Using tensor products, we deduce the co…
Corrigendum to "On Smooth extensions of vector-valued functions defined on closed subsets of Banach spaces" [Math. Ann. (2013) 355: 1201--1219]
本文是对2013年发表于Math. Ann.关于Banach空间闭子集上向量值函数光滑延拓论文的勘误说明。
We prove that a $BV$ map with values into the projective space $\mathbb{RP}^{d-1}$ has a $BV$ lifting with values into the unit sphere $\mathbb S^{d-1}$ that satisfies an optimal $BV$-estimate. As an application to liquid crystals, this result is also stated for $BV$ maps with values into the set of uniaxial $Q$-tensors. In order to quantify $BV$ liftings, we prove an explicit formula for an intrinsic $BV$-energy of maps with values into any compact smooth manifold.
The concept of generalized functions taking values in a differentiable manifold is extended to a functorial theory. We establish several characterization results which allow a global intrinsic formulation both of the theory of manifold-valued generalized functions and of generalized vector bundle homomorphisms. As a consequence, a characterization of equivalence that does not resort to derivatives (already known for the scalar-valued cases of Colombeau's construction) is achieved. These results are employed to derive a point value description of all types of generalized functions valued in man…
Let $\mathcal{N}$ be a smooth, compact, connected Riemannian manifold without boundary. Let $\mathcal{E}\to\mathcal{N}$ be the Riemannian universal covering of $\mathcal{N}$. For any bounded, smooth domain $Ω\subseteq\mathbb{R}^d$ and any $u\in\mathrm{BV}(Ω, \, \mathcal{N})$, we show that $u$ has a lifting $v\in\mathrm{BV}(Ω, \, \mathcal{E})$. Our result proves a conjecture by Bethuel and Chiron.
Extending the construction of the (intrinsically defined) full algebra of scalar valued Colombeau functions on a smooth manifold M (Grosser et al., Adv. Math. 166 (2002), 179-206) we present a suitable basic space for eventually obtaining tensor valued generalized functions on M, via the usual quotient construction. This basic space canonically contains the tensor valued distributions and permits a natural extension of the classical Lie derivative. Its members are smooth functions depending - via a third slot - on so-called transport operators, in addition to slots one (smooth n-forms on M) an…
We introduce an operator $\mathbf{S}$ on vector-valued maps $u$ which has the ability to capture the relevant topological information carried by $u$. In particular, this operator is defined on maps that take values in a closed submanifold $N$ of the Euclidean space $\mathbb{R}^m$, and coincides with the distributional Jacobian in case $N$ is a sphere. The range of $\mathbf{S}$ is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. In this paper, we use $\mathbf{S}$ to characterise strong limits of smooth, $N$-valued maps with respect to Sobolev norm…
Let $X$ and $Z$ be Banach spaces, $A$ a closed subset of $X$ and a mapping $f:A \to Z$. We give necessary and sufficient conditions to obtain a $C^1$ smooth mapping $F:X \to Z$ such that $F_{\mid_A}=f$, when either (i) $X$ and $Z$ are Hilbert spaces and $X$ is separable, or (ii) $X^*$ is separable and $Z$ is an absolute Lipschitz retract, or (iii) $X=L_2$ and $Z=L_p$ with $1<p<2$, or (iv) $X=L_p$ and $Z=L_2$ with $2<p<\infty$.
For suitable finite-dimensional smooth manifolds M (possibly with various kinds of boundary or corners), locally convex topological vector spaces F and non-negative integers k, we construct continuous linear operators S_n from the space of F-valued k times continuously differentiable functions on M to the corresponding space of smooth functions such that S_n(f) converges to f in C^k(M,F) as n tends to infinity, uniformly for f in compact subsets of C^k(M,F). We also study the existence of continuous linear right inverses for restriction maps from C^k(M,F) to C^k(L,F) if L is a closed subset of…
Local symmetry and smoothness in the space of vector-valued continuous functions
本文研究了向量值连续函数空间的局部对称性与光滑性，刻画了 $C(K,X)$ 的左右对称点条件，并识别了 $C_0(K,X)$ 中的光滑点。
Function spaces on formal manifolds
作为形式流形相关系列研究的一部分，本文在已有形式流形概念与基础框架的基础上，研究形式流形上的各类函数空间，实现光滑流形上向量值广义函数与分布的推广。
In this paper we use the notion of operator-valued symbol in order to compute the index of Toeplitz operators on compact Lie groups. Our approach combines the Connes index theorem and the infinite-dimensional operator-valued symbolic calculus of Ruzhansky-Turunen. We also give applications to the index of Toeplitz operators on simply connected closed $3$-manifolds $\mathbb{M}\simeq \mathbb{S}^3\simeq \textnormal{SU}(2) ,$ by using, as a fundamental tool, the Poincaré theorem (see Perelman [31,32,33,34])
Topological point singularities of manifold-valued maps and approximability by smooth maps
研究从(p+1)维欧氏区域到闭黎曼流形N的W^{1,p}映射，在π_1(N)在π_p(N)上作用平凡条件下，利用π_p(N)系数的平坦链刻画弱序列闭包中映射的点奇异，其消失刻画局部强可光滑逼近性。
In this paper we study $R$-boundedness of operator families $\mathcal{T}\subset \calL(X,Y)$, where $X$ and $Y$ are Banach spaces. Under cotype and type assumptions on $X$ and $Y$ we give sufficient conditions for $R$-boundedness. In the first part we show that certain integral operator are $R$-bounded. This will be used to obtain $R$-boundedness in the case that $\mathcal{T}$ is the range of an operator-valued function $T:\R^d\to \calL(X,Y)$ which is in a certain Besov space $B^{d/r}_{r,1}(\R^d;\calL(X,Y))$. The results will be applied to obtain $R$-boundedness of semigroups and evolution fami…
We present a generalization of the Radon-Riesz property to sequences of continuous functions with values in uniformly convex and uniformly smooth Banach spaces.
This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator on spaces $\mathcal{EV}(Ω,E)$ of $\mathcal{C}^{\infty}$-smooth vector-valued functions whose growth on strips along the real axis with holes $K$ is induced by a family of continuous weights $\mathcal{V}$. Vector-valued means that these functions have values in a locally convex Hausdorff space $E$ over $\mathbb{C}$. We characterise the weights $\mathcal{V}$ which give a counterpart of the Grothendieck-Köthe-Silva duality $\mathcal{O}(\mathbb{C}\setminus K)/\mathcal{O}(\mathbb{C})\cong\mathscr{A}(K)$ with non-…
Let $M$ be a smooth compact manifold and $P$ be either $R^1$ or $S^1$. There is a natural action of the groups $Diff(M)$ and $Diff(M) \times Diff(P)$ on the space of smooth mappings $C^{\infty}(M,P)$. For $f\in C^{\infty}(M,P)$ let $S_f$, $S_{MP}$, $O_f$, and $O_{MP}$ be the stabilizers and orbits of $f$ under these actions. Recently, the author proved that under mild conditions on $f\in C^{\infty}(M,R^1)$ the corresponding stabilizers and orbits are homotopy equivalent: $S_{MR} \sim S_{M}$ and $O_{MR} \sim O_M$. These results are extended here to the actions on $C^{\infty}(M,S^1)$. It is prov…
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