Wavelets on fractals and besov spaces

Wavelets on self-similar fractals are introduced. It is shown that for certain totally disconnected fractals, function spaces may be characterized by means of the magnitude of the wavelet coefficients of the functions.     

Function spaces on local fields

We study the function spaces on local fields in this paper, such as Triebel B-type and F-type spaces, Holder type spaces, Sobolev type spaces, and so on, moreover, study the relationship between the p-type derivatives and the Holder type spaces. Our obtained results show that there exists quite difference between the functions defined on Euclidean spaces and local fields, respectively. Furthermore, many properties of functions defined on local fields motivate the new idea of solving some important topics on fractal analysis.     

Function spaces and product topologies on powers of spaces

A study is made of two classes of product topologies on powers of spaces: the general box product topologies, and the general uniform product topologies. Some examples are given and some results are shown about the properties of these general product spaces. This is applied to show that certain spaces of continuous functions with the fine topology are homeomorphic to box product spaces, and certain spaces of continuous functions with the uniform topology are homeomorphic to uniform product spaces.     

Function spaces between BMO and critical Sobolev spaces

The function spaces Dk(Rn) are introduced and studied. The definition of these spaces is based on a regularity property for the critical Sobolev spaces Ws,p(Rn), where sp=n, obtained by J. Bourgain, H. Brezis, New estimates for the Laplacian, the div-curl, and related Hodge systems, C. R. Math. Acad. Sci. Paris 338 (7) (2004) 539-543 (see also J. Van Schaftingen, Estimates for L¹-vector fields, C. R. Math. Acad. Sci. Paris 339 (3) (2004) 181-186). The spaces Dk(Rn) contain all the critical Sobolev spaces. They are embedded in BMO(Rn), but not in VMO(Rn). Moreover, they have some extension and trace properties that BMO(Rn) does not have.     

Homogenization on nested fractals

Summary We study the homogenization problem on nested fractals. LetX _t be the continuous time Markov chain on the pre-nested fractal given by puttingi.i.d. random resistors on each cell. It is proved that under some conditions, converges in law to a constant time change of the Brownian motion on the fractal asn, where is the contraction rate andt _E is a time scale constant. As the Brownian motion on fractals is not a semi-martingale, we need a different approach from the well-developed martingale method.Dedicated to Professor Masatoshi Fukushima on his 60th birthdayResearch partially supported by the Yukawa Foundation     

Badly approximable vectors on fractals

For a large class of closed subsetsC of ℝ ⁿ , we show that the intersection ofC with the set of badly approximable vectors has the same Hausdorff dimension asC. The sets are described in terms of measures they support. Examples include (but are not limited to) self-similar sets such as Cantor’s ternary sets or attractors for irreducible systems of similarities satisfying Hutchinson’s open set condition.     

Function spaces from core compact coherent spaces to continuous B-domains

It is proved in this paper that for a continuous B-domain L, the function space [X→L] is continuous for each core compact and coherent space X. Further, applications are given. It is proved that:

(1)

the function space from the unit interval to any bifinite domain which is not an L-domain is not Lawson compact;

(2)

the Isbell and Scott topologies on [X→L] agree for each continuous B-domain L and core compact coherent space X.

     

Function spaces and the aleksandrov-urysohn conjecture

Summary The Aleksandrov-Urysohn conjecture about the cardinality of a first countable compact spare X is here given an equivalent formulation in terms of a generalized Lindel?f condition in the weak topology on X generated by the Baire functions. A number of related conditions are shown to be equivalent (without assuming that X is first countable); these include two sequential properties of the pointwise topology on various spaces of real-valued functions on X, and the condition that X is dispersed (i.e., has no non-void perfect subsets). Added in prof: The Alexandrov-Urysohn conjecture has been generalized and proved by A. V. Archangel' skii (On the cardinality of bicompacta satisfying the first axiom of countability, Dokl. Akad. Nauk SSSR 187 (1969) (Russian); translated as Soviet Math. Dokl. 10 (1969) pp. 951–955). In his more general theorem the compactness hypothesis is weakened to Lindel?f. Dedicated to the sixtieth birthday of Prof. Edgar R. Lorch This research was partially supported by the National Science Foundation.     

Function spaces of Lizorkin-Triebel type on an irregular domain

On an irregular domain G ⊂ ℝ ⁿ of a certain type, we introduce function spaces of fractional smoothness s > 0 that are similar to the Lizorkin-Triebel spaces. We prove embedding theorems that show how these spaces are related to the Sobolev and Lebesgue spaces W _p ^m (G) and L _p (G). Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 260, pp. 32–43.     

Function spaces arising from kernel operators

Given a probability space (Ω, μ) and a rearrangement invariant space X on [0,1], in certain situations inequalities for spaces of ${\mathbb {R}}$ -valued functions on Ω are equivalent to the boundedness of an associated operator T _K : L ^∞ ([0, 1]) → X generated by a kernel K ≥ 0 on the unit square (e.g. Sobolev type inequalities or Riesz potentials on subsets ${\Omega \subset \mathbb {R}^n}$ ). A natural class of spaces for treating such inequalities is given by ${[T_{K}, X](\Omega) := \{u : \Omega\to \mathbb {R} : T_{K} u^* \in X\}}$ together with the functional ${u \mapsto ||T_{K} u^*||_X}$ , where u* is the decreasing rearrangement of u. The investigation of these spaces is our main aim; the nature of the base space X and of K (via its monotonicity/growth properties) play a crucial role.