# The Poincaré-Hardy Inequality on the Complement of a Cantor Set

• Cristian S. Calude
• Boris Pavlov
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 134)

## Abstract

The Poincare-Hardy inequality on the complement of the Cantor set E
$$\int {\frac{{\left| u \right|^2 }}{{dist^2 \left( {x,E} \right)}}dm \leqslant 4K^2 \cdot \int {\left| {\nabla u} \right|^2 dm} }$$
holds for every u 2 1 (R 3). Corresponding higher-order inequalities will be also derived. 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. Three applications will be briefly discussed. In the second one, the constant 1/2K -1 ≈ 0.0002 plays the role of an estimate for the dimensionless Planck constant in the corresponding uncertainty principle.

## Keywords

Uncertainty Principle Binary String Quasiregular Mapping Minkowski Dimension Special Tiling
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
A. Ancona. On strong barriers and an inequality of Hardy for domains in R„J. London Math. Soc.(2) 34 (1986), 274–290.
2. 2.
M.S. Birman. On spectrum of boundary value problemsMat.Sb. (N.S.)55, (1961), 125–174. (in Russian)
3. 3.
F. Bruhat and J. Tits. Groupes réductifs sur un corps localInst. Hautes Études Sci. Publ. Math.41 (1972), 5–251. (in French)
4. 4.
C. Calude.Information and RandomnessAn Algorithmic PerspectiveSpringer-Verlag, Berlin, (1994).Google Scholar
5. 5.
L. Carleson.Selected Problems on Exceptional SetsVan-Nostrand, Princeton (1967).Google Scholar
6. 6.
E.B. Davies. A review of Hardy inequalities, inThe Maz’ya Anniversary Collec­tionVol.2 (Rostock, 1998), 55–67,Oper. Theory Adv. Appl.110, Birkhäuser, Basel, (1999).Google Scholar
7. 7.
Gerald A. Edgar.Measure Topology and Fractal GeometrySpringer-Verlag, New York (1990).Google Scholar
8. 8.
W.N. Everitt (ed.).Inequalities: Fifty Years on from Hardy Littlewood and PolyaProceedings of the International Conference held at the University of Birmingham, Birmingham, July 13–17, 1987, Lecture Notes in Pure and Applied Mathematics, 129, Marcel Dekker, Inc., New York, (1991).Google Scholar
9. 9.
L. Carding. Dirichlet problem for linear elliptic differential equationsMath. Scand. 1(1953), 55–72.
10. 10.
G.H. Hardy, J.E. Littlewood, J.E. Polya.InequalitiesCambridge Univ. Press, Cam­bridge, (1934).Google Scholar
11. 11.
D.J.L. Herrmann, T. Janssen. On spectral properties of Harper-like modelsJ. Math. Physics40 3 (1999), 1197–1214.
12. 12.
P. Järvi, M. Vuorinen. Uniformly perfect sets and quasiregular mappingsJ. London Math. Soc.(2) 54 (1966), 515–529.Google Scholar
13. 13.
P. Järvi, M. Vuorinen. Self-similar Cantor sets and quasiregular mappings.J. Reine Angew. Math.424 (1992), 31–45.
14. 14.
P. Kurka. Simplicity criteria for dynamical systems, inAnalysis of Dynamical and Cognitive Systems (Stockholm 1993)189–225, Lecture Notes in Comput. Sci., 888, Springer-Verlag, Berlin, (1995).Google Scholar
15. 15.
O.A. Ladyzhenskaja.Problems in the Dynamics of Viscous Incompressible FlowGordon & Breach, New York, (1963).Google Scholar
16. 16.
D. Lind, B. Marcus. AnIntroduction to Symbolic Dynamics and CodingCambridge University Press, Cambridge, (1995).
17. 17.
V.M. Maz’ja.Sobolev SpacesSpringer-Verlag, Berlin, (1985).
18. 18.
V.L. Oleinik. Carleson measures and uniformly perfect setsZap. Nauchn. Sem. S.­Peterburg. Otdel. Mat. Inst. Steklov. (POMI)255 (1998)Issled. po Linein. Oper. i Teor. Funkts.26, 92–103, 251 (in Russian).Google Scholar
19. 19.
B. Pavlov. Boundary conditions on thin manifolds and the semi-boundedness of the three-particle Schrödinger operator with pointwise potentialMath. USSR Sbornik64, 1 (1989), 161–175.
20. 20.
C.A. Pickover.Mazes for the Mind: Computers and the UnexpectedSt. Martin’s Press, New York, (1992).Google Scholar
21. 21.
Ch. Pommerenke. Uniformly perfect sets and the Poincaré metricArch. Math.32 (1979), 192–199.
22. 22.
L.I. Schiff.Quantum Mechanics3rd edition McGraw-Hill, New York, (1968).Google Scholar
23. 23.
H.G. Schuster.Deterministic ChaosPhysik-Verlag, Weiheim, (1984).Google Scholar
24. 24.
M. Reed, B. Simon.Methods of Modern Mathematical PhysicsVol. 3, Academic Press, 3rd edition, New York, (1987).Google Scholar