Inequality on the Complement of a Cantor Set
نویسندگان
چکیده
The Poincaré–Hardy inequality ∫ |u| dist(x,E) dm ≤ K · ∫ | u |dm is derived in R3 on the complement of a Cantor set E. We use a special self-similar tiling and a natural metric on the space of trajectories generated by a Mauldin–Williams graph which is homeomorphic with the space of tiles endowed with the Euclidean distance. A crude estimation of the constant K is 2,100. Two applications will be briefly discussed. In the last one, the constant K−1 ≈ 0.021819 plays the role of an estimate for the dimensionless Plank constant in the corresponding uncertainty principle.
منابع مشابه
A Cantor Set with Hyperbolic Complement
We construct a Cantor set in S3 whose complement admits a complete hyperbolic metric.
متن کاملSOME RESULTS ON THE COMPLEMENT OF THE INTERSECTION GRAPH OF SUBGROUPS OF A FINITE GROUP
Let G be a group. Recall that the intersection graph of subgroups of G is an undirected graph whose vertex set is the set of all nontrivial subgroups of G and distinct vertices H,K are joined by an edge in this graph if and only if the intersection of H and K is nontrivial. The aim of this article is to investigate the interplay between the group-theoretic properties of a finite group G and the...
متن کاملLocalization for a Continuum Cantor-anderson Hamiltonian
We prove localization at the bottom of the spectrum for a random Schrödinger operator in the continuum with a single-site potential probability distribution supported by a Cantor set of zero Lebesgue measure. This distribution is too singular to be treated by the usual methods. In particular, an “a priori” Wegner estimate is not available. To prove the result we perform a multiscale analysis fo...
متن کاملCompleteness in Probabilistic Metric Spaces
The idea of probabilistic metric space was introduced by Menger and he showed that probabilistic metric spaces are generalizations of metric spaces. Thus, in this paper, we prove some of the important features and theorems and conclusions that are found in metric spaces. At the beginning of this paper, the distance distribution functions are proposed. These functions are essential in defining p...
متن کاملOn Smoothness of the Green Function for the Complement of a Rarefied Cantor-Type Set
Smoothness of the Green functions for the complement of rarefied Cantortype sets is described in terms of the function φ(δ) = (1/ log 1 δ ) that gives the logarithmic measure of sets. Markov’s constants of the corresponding sets are evaluated.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2000