Lipschitz algebras and derivations II: exterior differentiation

Thick Families of Geodesics and Differentiation

The differentiation theory of Lipschitz functions taking values in a Banach space with the Radon-Nikodým property (RNP), originally developed by Cheeger-Kleiner, has proven to be a powerful tool to prove non-biLipschitz embeddability of metric spaces into these Banach spaces. Important examples of metric spaces to which this theory applies include nonabelian Carnot groups and Laakso spaces. In search of a metric characterization of the RNP, Ostrovskii found another class of spaces that do not biLipschitz embed into RNP spaces, namely spaces containing thick families of geodesics. Our first res…

某些 Banach 空间中的 Lipschitz 数值指数与线性数值指数

Lipschitz vs Linear Numerical Index in certain Banach spaces

针对实 Banach 空间中 Lipschitz 与线性数值指数是否一致的问题，本文利用 Gaussian 卷积微分和不变均值两种线性化技术，证明在可分或对偶空间中二者相等。

Heat flow and quantitative differentiation

For every Banach space $(Y,\|\cdot\|_Y)$ that admits an equivalent uniformly convex norm we prove that there exists $c=c(Y)\in (0,\infty)$ with the following property. Suppose that $n\in \mathbb{N}$ and that $X$ is an $n$-dimensional normed space with unit ball $B_X$. Then for every $1$-Lipschitz function $f:B_X\to Y$ and for every $\varepsilon\in (0,1/2]$ there exists a radius $r\ge\exp(-1/\varepsilon^{cn})$, a point $x\in B_X$ with $x+rB_X\subset B_X$, and an affine mapping $Λ:X\to Y$ such that $\|f(y)-Λ(y)\|_Y\le \varepsilon r$ for every $y\in x+rB_X$.

On Linearity of Nonclassical Differentiation

We introduce a real vector space composed of set-valued maps on an open set X and note it by S. It is a complete metric space and a complete lattice. The set of continuous functions on X is dense in S as in a metric space and as in a lattice. Thus the constructed space plays the same role for the space of continuous functions with uniform convergence as the field of reals plays for the field of rationals. The classical gradient may be extended in the space S as a close operator. If a function f belongs to its domain then f is locally lipschitzian and the values of our gradient coincide with th…

Potential maps, Hardy spaces, and tent spaces on special Lipschitz domains

Suppose that $Ω$ is the open region in $\mathbb{R}^n$ above a Lipschitz graph and let $d$ denote the exterior derivative on $\mathbb{R}^n$. We construct a convolution operator $T $ which preserves support in $\bar{Ω$}, is smoothing of order 1 on the homogeneous function spaces, and is a potential map in the sense that $dT$ is the identity on spaces of exact forms with support in $\barΩ$. Thus if $f$ is exact and supported in $\barΩ$, then there is a potential $u$, given by $u=Tf$, of optimal regularity and supported in $\barΩ$, such that $du=f$. This has implications for the regularity in homo…

Differentiating maps into L^1 and the geometry of BV functions

This is one of a series of papers examining the interplay between differentiation theory for Lipschitz maps, X-->V, and bi-Lipschitz nonembeddability, where X is a metric measure space and V is a Banach space. Here, we consider the case V=L^1 where differentiability fails. We establish another kind of differentiability for certain X, including R^n and H, the Heisenberg group with its Carnot-Cartheodory metric. It follows that H does not bi-Lipschitz embed into L^1, as conjectured by J. Lee and A. Naor. When combined with their work, this provides a natural counter example to the Goemans-Linial…

逐点 Lipschitz 映射的链式法则

Chain rule for pointwise Lipschitz mappings

本文提出逐点 Lipschitz 映射的几乎处处链式法则，去除全局可微假设，并推广至无限维空间，同时加强 Rademacher-Stepanov 定理。

The Frolicher--Kriegl differentiabilities as a particular case of the Bertram--Glockner--Neeb construction

We prove that the order $k$ differentiability classes for $k=0,1,...\infty$ in the "arc-generated" interpretation of the Lipschitz theory of differentiation by Frolicher and Kriegl can be obtained as particular cases of the general construction by Bertram, Glockner and Neeb leading to $C^k$ differentiabilities from a given $C^0$ concept.

Hodge-Dirac, Hodge-Laplacian and Hodge-Stokes operators in L^p spaces on Lipschitz domains

This paper concerns Hodge-Dirac operators D = d + $δ$ acting in L p ($Ω$, λ) where $Ω$ is a bounded open subset of R n satisfying some kind of Lipschitz condition, λ is the exterior algebra of R n , d is the exterior derivative acting on the de Rham complex of differential forms on $Ω$, and $δ$ is the interior derivative with tangential boundary conditions. In L 2 ($Ω$, λ), $δ$ = d * and D is self-adjoint, thus having bounded resolvents (I + itD) --1 t$\in$R as well as a bounded functional calculus in L 2 ($Ω$, λ). We investigate the range of values p H \textless{} p \textless{} p H about p = …

Seip's differentiability concepts as a particular case of the Bertram--Gloeckner--Neeb construction

From the point of view of unification of differentiation theory, it is of interest to note that the general construction principle of Bertram, Gloeckner and Neeb leading to a C^k differentiability concept from a given C^0 one, besides subsuming the Keller--Bastiani C_c^k differentiabilities on real Hausdorff locally convex spaces, also does the same to the "arc-generated" interpretation of the Lipschitz theory of differentiation by Frolicher and Kriegl, and likewise to the "compactly generated" theory of Seip's continuous differentiabilities. In this article, we give the details of the proof f…