Abstract
We characterize functions of finite energy in the plane in terms of their traces on the lines that make up “graph paper” with squares of side length m n for all n and certain 1/2-order Sobolev norms on the graph paper lines. We also obtain analogous results for functions of finite energy on two classical fractals: the Sierpinski gasket and the Sierpinski carpet.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Aougab, S. Dong, and R. Strichartz, Laplacians on a family of quadratic Julia sets II, Comm. Pure Appl. Anal. 12 (2013), 1–58.
M. Barlow, Diffusions on fractals, in Lectures on Probability Theory and Statistics, LNM1690, Springer, Berlin, 1998, pp. 1–121.
M. Barlow, R. Bass, T. Kumagai, and A. Teplyaev, Uniqueness of Brownian motion on Sierpinski carpets, J. Eur.Math. Soc. (JEMS) 12 (2010), 655–701.
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Amer.Math. Soc., Providence, RI, 2013.
T. Flock and R. Strichartz, Laplacians on a family of quadratic Julia sets I, Trans. Amer. Math. Soc. 364 (2012), 3915–3965.
M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter & Co, Berlin, 1994.
M. Hino and T. Kumagai, A trace theorem for Dirichlet forms on fractals, J. Funct. Anal. 238 (2006), 578–611.
A. Jonsson, A trace theorem for the Dirichlet form on the Sierpinski gasket, Math. Z. 250 (2005), 599–609.
A. Jonsson, A Dirichlet form on the Sierpinski gasket, related function spaces, and traces, in Fractal Geometry and Stochastics III, Birkhaüser, Basel, 2004, pp. 235–244.
J. Kigami, Analysis on Fractals, Cambridge Univ. Press, Cambridge, 2001.
V. Mazýa, Sobolev Spaces, Springer-Verlag, Berlin, 1985.
A. Poirier, Critical portraits for postcritically finite polynomials, Fund. Math. 203 (2009), 107–163.
L. Rogers and A. Teplyaev, Laplacians on the basilica Julia set, Comm. Pure Appl. Anal. 9 (2010), 201–231.
C. Spicer, R. Strichartz, and E. Totari, Laplacians on Julia sets III: Cubic Julia sets and formal matings, in Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I: Fractals in Pure Mathematics, Contemporary Math. 600 (2013), 327–348.
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 1970.
R. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031–1060.
R. Strichartz, Differential Equations on Fractals, a Tutorial, Princeton Univ. Press, Princeton, NJ, 2006.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by the National Science Foundation, grant DMS-1162045.
Rights and permissions
About this article
Cite this article
Strichartz, R.S. “Graph paper” trace characterizations of functions of finite energy. JAMA 128, 239–260 (2016). https://doi.org/10.1007/s11854-016-0008-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-016-0008-x