Remarks on the embedding of spaces of distributions into spaces of Colombeau generalized functions

Schwartz 分布在渐近函数代数中的一个嵌入

An Embedding of Schwartz Distributions in the Algebra of Asymptotic Functions

本文通过将分布空间嵌入到一种名为“渐近函数”的微分代数中，解决了 Schwartz 分布的乘法问题，该代数与 Colombeau 代数相似但不同。

Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity

We present new types of regularity for nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of Colombeau's simplified model. This generalizes the notion of G^{\infty }-regularity introduced by M. Oberguggenberger. A key point is that these regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to microanalysis of singularities of generalized functions, with respect to these regularities. We present a complete study of this topic, including proper…

Nonlinear generalised functions on manifolds

This paper lays the foundations for a nonlinear theory of differential geometry that is developed in a subsequent paper which is based on Colombeau algebras of tensor distributions on manifolds. We adopt a new approach and construct a global theory of algebras of generalised functions on manifolds based on the concept of smoothing operators. This produces a generalisation of previous theories in a form which is suitable for applications to differential geometry. The generalised Lie derivative is introduced and shown to commute with the embedding of distributions. It is also shown that the cova…

Rapidly converging approximations and regularity theory

We consider distributions on a closed compact manifold $M$ as maps on smoothing operators. Thus spaces of certain maps between $Ψ^{-\infty}(M)\to \mathcal{C}^{\infty}(M)$ are considered as generalized functions. For any collection of regularizing processes we produce an algebra of generalized functions and a diffeomorphism equivariant embedding of distributions into this algebra. We provide examples invariant under certain group actions. The regularity for such generalized functions is provided in terms of a certain tameness of maps between graded Frechét spaces. This notion of regularity impl…

Non-Standard Analysis, Multiplication of Schwartz Distributions and Delta-Like Solution of Hopf's Equation

We construct an algebra of generalized functions $^*\mathcal{E}(\mathbb{R}^d)$. We also construct an embedding of the space of Schwartz distributions $\mathcal{D}^\prime(\mathbb{R}^d)$ into $^*\mathcal{E}(\mathbb{R}^d)$ and thus present a solution of the problem of multiplication of Schwartz distributions which improves J.F. Colombeau's solution. As an application we prove the existence of a weak delta-like solution in ${^*\mathcal{E}(\mathbb{R}^d)}$ of the Hopf equation. This solution does not have a counterpart in the classical theory of partial differential equations. Our result improves a …

A global theory of algebras of generalized functions

We present a geometric approach to defining an algebra $\hat{\mathcal G}(M)$ (the Colombeau algebra) of generalized functions on a smooth manifold $M$ containing the space ${\mathcal D}'(M)$ of distributions on $M$. Based on differential calculus in convenient vector spaces we achieve an intrinsic construction of $\hat{\mathcal G}(M)$. $\hat{\mathcal G}(M)$ is a{\em differential} algebra, its elements possessing Lie derivatives with respect to arbitrary smooth vector fields. Moreover, we construct a canonical linear embedding of ${\mathcal D}'(M)$ into $\hat{\mathcal G}(M)$ that renders ${\mat…

关于Banach空间的点特征及到$c_0(\Ga)$的非线性嵌入的注记

Remarks on the point character of Banach spaces and non-linear embeddings into~$c_0(\Ga)$

本文综述了Banach空间到$c_0(\Ga)$的粗或一致嵌入及点特征，证明了若干新结果（如确定$L_p(\mu)$的点特征），解决了多个开放问题，并给出了双Lipschitz嵌入的充要条件。

Remarks on Sobolev-Morrey-Campanato spaces defined on $C^{0,γ}$ domains

We discuss a few old results concerning embedding theorems for Campanato and Sobolev-Morrey spaces adapting the formulations to the case of domains of class $C^{0,γ}$, and we present more recent results concerning the extension of functions from Sobolev-Morrey spaces defined on those domains. As a corollary of the extension theorem we obtain an embedding theorem for Sobolev-Morrey spaces on arbitrary $C^{0,γ}$ domains.

Embeddings of vector-valued Bergman spaces

We remark that a dyadic version of the Carleson embedding theorem for the Bergman space extends to vector-valued functions and operator-valued measures. This is in contrast to a result by Nazarov, Treil, Volberg in the context of the Hardy space. We also discuss some embeddings for analytic vector-valued functions.

Sheaves of nonlinear generalized functions and manifold-valued distributions

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…

A remark on embedding of a cylinder on a real commutative Banach algebra

Let $A$ be a real commutative Banach algebra with unity. Let $a_0\in A\setminus\{0\}$. Let $\mathbb Z a_0:=\{na_0\}_{n\in \mathbb Z}$. Then, $\mathbb Z a_0$ is a discrete subgroup of $A$. For any $n\in \mathbb Z$, the Frechet derivative of the mapping $$x \, \in \, A \ \ \ \mapsto \ \ \ x+na_0 \, \in \, A$$ is the identity map on $A$ and, especially, an $A$-linear transformation on $A$. So, the quotient group $A/(\mathbb Z a_0)$ is a $1$-dimensional $A$-manifold and the covering projection $$x \, \in \, A \ \ \ \mapsto \ \ \ x+\mathbb Z a_0 \, \in \, A/(\mathbb Z a_0)$$ is an $A$-map. We call …

A remark on a generalization of a logarithmic Sobolev inequality to the Holder class

In a recent work of the author, a parabolic extension of the elliptic Ogawa type inequality has been established. This inequality is originated from the Brezis-Gallouet-Wainger logarithmic type inequalities revealing Sobolev embeddings in the critical case. In this paper, we improve the parabolic version of Ogawa inequality by allowing it to cover not only the class of functions from Sobolev spaces, but the wider class of Holder continuous functions.

Some remarks on universality properties of $\ell_\infty / c_0$

We prove that if continuum is not a Kunen cardinal, then there is a uniform Eberlein compact space $K$ such that the Banach space $C(K)$ does not embed isometrically into $\ell_\infty/c_0$. We prove a similar result for isomorphic embeddings. We also construct a consistent example of a uniform Eberlein compactum whose space of continuous functions embeds isomorphically into $\ell_\infty/c_0$, but fails to embed isometrically. As far as we know it is the first example of this kind.

Some remarks on $L^1$ embeddings in the subelliptic setting

In this paper we establish an optimal Lorentz estimate for the Riesz potential in the $L^1$ regime in the setting of a stratified group $G$: Let $Q\geq 2$ be the homogeneous dimension of $G$ and $\mathcal{I}_α$ denote the Riesz potential of order $α$ on $G$. Then, for every $α\in (0,Q)$, there exists a constant $C=C(α,Q)>0$ such that \begin{align} \| \mathcal{I}_αf \|_{L^{Q/(Q-α),1}(G)} \leq C\| X \mathcal{I}_1 f \|_{L^1(G)} \end{align} for distributions $f$ such that $X \mathcal{I}_1 f \in L^1(G)$, where $X$ denotes the horizontal gradient.

Remarks on quasi-isometric non-embeddability into uniformly convex Banach spaces

We construct a locally finite graph and a bounded geometry metric space which do not admit a quasi-isometric embedding into any uniformly convex Banach space. Connections with the geometry of $c_0$ and superreflexivity are discussed.

Remarks on Gurarii spaces

We present selected known results and some of their improvements, involving Gurarii spaces. A Banach space is Gurarii if it has certain natural extension property for almost isometric embeddings of finite-dimensional spaces. Deleting the word "almost", we get the notion of a strong Gurarii space. There exists a unique (up to isometry) separable Gurarii space, however strong Gurarii spaces cannot be separable. The structure of the class of non-separable Gurarii spaces seems to be not very well understood. We discuss some of their properties and state some open questions. In particular, we chara…

作为广义全纯函数的狄拉克δ函数

Dirac delta as a generalized holomorphic function

该研究通过复数域的非阿基米德扩张定义广义全纯函数，推广多个经典复分析定理，解决Colombeau理论的局限，为证明含分布PDE等的柯西-柯瓦列夫斯卡娅定理奠基。

Linearly rigid metric spaces and the embedding problem

We consider the problem of isometric embedding of metric spaces to the Banach spaces; and introduce and study the remarkable class of so-called linearly rigid metric spaces: these are the spaces that admit a unique, up to isometry, linearly dense isometric embedding into a Banach space. The first nontrivial example of such a space was given by R. Holmes; he proved that the universal Urysohn space has this property. We give a criterion of linear rigidity of a metric space, which allows us to give a simple proof of the linear rigidity of the Urysohn space and some other metric spaces. The variou…