Algebras of generalized functions with smooth parameter dependence

A note on the trace theorem for Besov-type spaces of generalized smoothness on $d$-sets

The main goal of this paper is to give a complete proof of the trace theorem for Besov-type spaces of generalized smoothness associated with complete Bernstein functions satisfying certain scaling conditions on $d$-sets $D\subset\mathbb R^n$, $d\leq n$. The proof closely follows the classical approach by Jonsson and Wallin and the trace theorem for classical Besov spaces. Here, the trace space is defined by means of differences. When $d=n$, as an application of the trace theorem, we give a condition under which the test functions $C_c^\infty(D)$ are dense in the trace space on $D$.

Group invariants in algebras of generalized functions

We study invariance properties of Colombeau generalized functions under actions of smooth Lie transformation groups. Several characterization results analogous to the smooth setting are derived and applications to generalized rotational invariance are given.

Non-smooth differential geometry and algebras of generalized functions

Algebras of generalized functions offer possibilities beyond the purely distributional approach in modelling singular quantities in non-smooth differential geometry. This article presents an introductory survey of recent developments in this field and highlights some applications in mathematical physics.

Generalized functions valued in a smooth manifold

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.

Categorical frameworks for generalized functions

We tackle the problem of finding a suitable categorical framework for generalized functions used in mathematical physics for linear and non-linear PDEs. We are looking for a Cartesian closed category which contains both Schwartz distributions and Colombeau generalized functions as natural objects. We study Frölicher spaces, diffeological spaces and functionally generated spaces as frameworks for generalized functions. The latter are similar to Frölicher spaces, but starting from locally defined functionals. Functionally generated spaces strictly lie between Frölicher spaces and diffeological s…

A convenient notion of compact set for generalized functions

We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functions on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional analytic foundations of these spaces.

Real Paley-Wiener Theorem for the Generalized Weinstein transform in quantum calculus

We first characterize the image of the compactly supported smooth even functions under the q-Weinstein transform as a subspace of the Schwartz space. We then describe the space of smooth $L_{α, q, a}^{2}$-functions whose q-Weinstein transform has compact support as a subspace of the space of $L_{α, q, a}^{2}$-functions.

广义光滑性的插值空间及其在椭圆方程中的应用

Interpolation spaces of generalized smoothness and their applications to elliptic equations

本文通过插值方法构造以O-正则变化函数为正则性阶数的广义光滑性分布空间，推广经典Sobolev、Nikolskii-Besov和Triebel-Lizorkin空间，并将其应用于光滑流形上的椭圆方程与椭圆问题。

Persistence and Smooth Dependence on Parameters of Periodic Orbits in Functional Differential Equations Close to an ODE or an Evolutionary PDE

We consider functional differential equations(FDEs) which are perturbations of smooth ordinary differential equations(ODEs). The FDE can involve multiple state-dependent delays or distributed delays (forward or backward). We show that, under some mild assumptions, if the ODE has a nondegenerate periodic orbit, then the FDE has a smooth periodic orbit. Moreover, we get smooth dependence of the periodic orbit and its frequency on parameters with high regularity. The result also applies to FDEs which are perturbations of some evolutionary partial differential equations(PDEs). The proof consists i…

Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality

Let $\M$ be a smooth connected manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ which is symmetric with respect to $μ$. We assume that $L$ satisfies a generalized curvature dimension inequality as introduced by Baudoin-Garofalo \cite{BG1}. Our goal is to discuss functional inequalities for $μ$ like the Poincaré inequality, the log-Sobolev inequality or the Gaussian logarithmic isoperimetric inequality.

具有光滑剖面的广义Constantin-Lax-Majda模型自相似有限时间爆破

Self-similar finite-time blowups with smooth profiles of the generalized Constantin-Lax-Majda model

本文通过非线性映射不动点方法，证明了广义Constantin-Lax-Majda模型对所有参数$a\leq 1$存在精确自相似有限时间爆破解，统一了现有离散结果并解释了数值观察。

An algebraic approach to manifold-valued generalized functions

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.

广义Newton-Leibniz公式与带主导混合光滑性的Sobolev函数到Hölder空间的嵌入

Generalized Newton-Leibniz Formula and the Embedding of the Sobolev Functions with Dominating Mixed Smoothness into Hölder Spaces

本文推广Newton-Leibniz公式至n维矩形，结合Sobolev迹嵌入归纳证明，给出与可积指数无关的弱可微条件，刻画Sobolev函数到Hölder空间的嵌入，并证明带主导混合光滑性的Sobolev空间的对应嵌入结论。