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.     

Weak uncertainty principle for fractals, graphs and metric measure spaces

We develop a new approach to formulate and prove the weak uncertainty inequality, which was recently introduced by Okoudjou and Strichartz. We assume either an appropriate measure growth condition with respect to the effective resistance metric, or, in the absence of such a metric, we assume the Poincaré inequality and reverse volume doubling property. We also consider the weak uncertainty inequality in the context of Nash-type inequalities. Our results can be applied to a wide variety of metric measure spaces, including graphs, fractals and manifolds.

     

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     

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.

     

Analysis on products of fractals

For a class of post-critically finite (p.c.f.) fractals, which includes the Sierpinski gasket (SG), there is a satisfactory theory of analysis due to Kigami, including energy, harmonic functions and Laplacians. In particular, the Laplacian coincides with the generator of a stochastic process constructed independently by probabilistic methods. The probabilistic method is also available for non-p.c.f. fractals such as the Sierpinski carpet. In this paper we show how to extend Kigami's construction to products of p.c.f. fractals. Since the products are not themselves p.c.f., this gives the first glimpse of what the analytic theory could accomplish in the non-p.c.f. setting. There are some important differences that arise in this setting. It is no longer true that points have positive capacity, so functions of finite energy are not necessarily continuous. Also the boundary of the fractal is no longer finite, so boundary conditions need to be dealt with in a more involved manner. All in all, the theory resembles PDE theory while in the p.c.f. case it is much closer to ODE theory.

     

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.     

Multivalued fractals in <Emphasis Type=Italic>b</Emphasis>-metric spaces

Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems theory in several topics of applied sciences. It is known that examples of fractals and multivalued fractals are coming from fixed point theory for single-valued and multivalued operators, via the so-called fractal and multi-fractal operators. On the other hand, the most common setting for the study of fractals and multi-fractals is the case of operators on complete or compact metric spaces. The purpose of this paper is to extend the study of fractal operator theory for multivalued operators on complete b-metric spaces.     

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.