Duality and interpolation of anisotropic Triebel–Lizorkin spaces

We study properties of anisotropic Triebel–Lizorkin spaces associated with general expansive dilations and doubling measures on using wavelet transforms. This paper is a continuation of (Bownik in J Geom Anal 2007, to appear, Trans Am Math Soc 358:1469–1510, 2006), where we generalized the isotropic methods of dyadic -transforms of Frazier and Jawerth (J Funct Anal 93:34–170, 1990) to non-isotropic settings. By working at the level of sequence spaces, we identify the duals of anisotropic Triebel–Lizorkin spaces. We also obtain several real and complex interpolation results for these spaces. The author was partially supported by the NSF grants DMS-0441817 and DMS-0653881. The author wishes to thank Michael Frazier and Dachun Yang for valuable comments and discussions on this work.     

Generically there is but one self homeomorphism of the Cantor set

We describe a self homeomorphism of the Cantor set and then show that its conjugacy class in the Polish group of all homeomorphisms of forms a dense subset of . We also provide an example of a locally compact, second countable topological group which has a dense conjugacy class.

     

Minimax inequalities and fixed points of expansive set-valued mappings with noncompact and nonconvex domains and ranges in topological spaces

Some minimax inequalities involving two bifunctions with noncompact and nonconvex domains are first proved in finite continuous topological spaces (in short, FC$FC$-spaces) without convexity structure. As applications some new Fan–Browder type fixed point theorems for expansive set-valued maps with noncompact and nonconvex domains and ranges are obtained in general topological spaces. These results generalize some known results in the recent literature.     

Higher order Schwarzian derivatives and quasisymmetric homeomorphisms北大核心CSCD

We characterize two subclasses of quasisymmetric homeomorphisms of the unit circle in terms of higher order Schwarzian derivatives, which are the strongly symmetric homeomorphisms and the Weil-Petersson class. ©, 2015, Chinese Academy of Sciences. All right reserved.     

Corrigendum to “Extending homeomorphisms from punctured surfaces to handlebodies” [Topology Appl. 155 (6) (2008) 610–621]

We correct the statements of Theorems 9 and 10 of [A. Cattabriga, M. Mulazzani, Extending homeomorphisms from punctured surfaces to handlebodies, Topology Appl. 155 (2008) 610–621], by adding missing generators, and improve the statement of Theorem 10, by removing some redundant generators.     

Equilibria and fixed points of set-valued maps with nonconvex and noncompact domains and ranges

Let C$C$ be a nonempty subset of a topological vector space E$E$. We state and prove new various fixed point theorems of Fan–Browder type for set-valued maps F:C→2^E$F:C\to 2^E$ such that C⊂F(C)$C\subset F\left(C\right)$ (called expansive), without assuming that the sets C$C$ and F(C)$F\left(C\right)$ are convex or compact or equal, and E$E$ is Hausdorff. Let K$K$ be a convex subset of E$E$ and let C$C$ be a nonempty subset of K$K$. Our proofs use a technique based on the investigations of the images of maps and restated for maps f:C×K→R∪{−∞,+∞}$f:C\times K\to \mathbb{R}\cup \{-\infty ,+\infty \}$ of G.X.-Z. Yuan’s results concerning the existence of equilibrium points and minimax inequalities for maps f:K×K→R∪{−∞,+∞}$f:K\times K\to \mathbb{R}\cup \{-\infty ,+\infty \}$. Examples are provided.     

A Survey on Some Anisotropic Hardy-Type Function Spaces

Let $A$ be a general expansive matrix on $\mathbb{R}^n$. The aims of this article are twofold. The first one is to give a survey on the recent developments of anisotropic Hardy-type function spaces on $\mathbb{R}^n$, including anisotropic Hardy–Lorentz spaces, anisotropic variable Hardy spaces and anisotropic variable Hardy–Lorentz spaces as well as anisotropic Musielak–Orlicz Hardy spaces. The second one is to correct some errors and seal some gaps existing in the known articles. Some unsolved problems are also presented.     

RELATIONS BETWEEN SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY SEMIMARTINGALE WITH NON-LIPSCHITZ COEFFICIENTS

In this paper, a class of stochastic differential equations (SDEs) driven by semi-martingale with non-Lipschitz coefficients is studied. We investigate the dependence of solutions to SDEs on the initial value. To obtain a continuous version, we impose the conditions on the local characteristic of semimartingale. In this case, it gives rise to a flow of homeomorphisms if the local characteristic is compactly supported.     

Minimal Periodic Orbit Structure of 2-Dimensional Homeomorphisms

We present a method for estimating the minimal periodic orbit structure, the topological entropy, and a fat representative of the homeomorphism associated with the existence of a finite collection of periodic orbits of an orientation-preserving homeomorphism of the disk D². The method focuses on the concept of fold and recurrent bogus transition and is more direct than existing techniques. In particular, we introduce the notion of complexity to monitor the modification process used to obtain the desired goals. An algorithm implementing the procedure is described and some examples are presented at the end.