Nonlinear Energy Forms and Lipschitz Spaces on the Koch Curve

Raffaela Capitanelli; Maria Rosaria Lancia

doi:10.68381/jca09012

Abstract

We consider the nonlinear convex energy forms

\mathcal{E}^{(p)}

on the Koch curve

K

and we prove that the corresponding domains coincide with the spaces

\mathit{Lip}_{\alpha, D_f} (p, \infty, K)

. Then we give a precise interpretation of the smoothness index

\alpha

in terms of the structural constants of the fractal.

Author information

Contact details are reproduced from the original publication and may be historical.

Raffaela Capitanelli

Dip. di Metodi e Modelli Matematici per le Scienze Applicate, Università di Roma I, Via A. Scarpa 16, 00161 Roma, Italy

raffaela.capitanelli@uniroma1.it

Maria Rosaria Lancia

Dip. di Metodi e Modelli Matematici per le Scienze Applicate, Università di Roma I, Via A. Scarpa 16, 00161 Roma, Italy

lancia@dmmm.uniroma1.it

Keywords

Nonlinear convex energy forms
fractals
Lipschitz spaces

References

H. Bauer: Probability Theory and Elements of Measure Theory, Academic Press (1981).Google Scholar
R. Capitanelli: Nonlinear energy forms on certain fractal curves, J. Nonlinear Convex Analysis, to appear.Google Scholar
R. Capitanelli: Functional inequalities for measure-valued Lagrangians on homogeneous spaces, preprint.Google Scholar
J. E. Hutchinson: Fractals and selfsimilarity, Indiana Univ. Math. J. 30 (1981) 713–747.https://doi.org/10.1512/iumj.1981.30.30055 Google Scholar
A. Jonsson: Brownian motion on fractals and function spaces, Math. Z. 6 (1996) 495–504.https://doi.org/10.1007/BF02621879 Google Scholar
A. Jonsson: Haar wavelets of higher order on fractals and regularity of functions, Research Reports 7, Umeȧ Univ. (1998).Google Scholar
A. Jonsson, H. Wallin: Function spaces on subsets of Rn, Math. Rep. 2, Harwood Acad. Publ. (1984).Google Scholar
T. Kumagai: Brownian motion penetrating fractals: an application of the trace theorem of Besov spaces, J. Funct. Anal. 170 (2000) 69–92.https://doi.org/10.1006/jfan.1999.3500 Google Scholar
M. R. Lancia, M. A. Vivaldi: Lipschitz spaces and Besov traces on self-similar fractals, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 117, XXIII (1999) 101–116.Google Scholar
U. Mosco: Variational fractals, Special Volume in Memory of E. De Giorgi, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. 4 (1997) 683–712.Google Scholar
U. Mosco: Variational metrics on self-similar fractals, C. R. Acad. Sci. Paris, Sér. I Math. 321 (1995) 715–720.Google Scholar
U. Mosco: Dirichlet forms and self-similarity, in: New Directions in Dirichlet Forms, J. Jost et al., AMS/IP Stud. Adv. Math. 8, Amer. Math. Soc., Providence R. I. (1998) 117–155.https://doi.org/10.1090/amsip/008/03 Google Scholar
K. P. Paluba: Some function spaces related to the Brownian motion on simple nested fractals, Stoch. and Stoch. Rep. 67 (1999) 267–285.https://doi.org/10.1080/17442509908834214 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2002. Rights now held by Banach Press LLC.

[1] H. Bauer: Probability Theory and Elements of Measure Theory, Academic Press (1981).Google Scholar

[2] R. Capitanelli: Nonlinear energy forms on certain fractal curves, J. Nonlinear Convex Analysis, to appear.Google Scholar

[3] R. Capitanelli: Functional inequalities for measure-valued Lagrangians on homogeneous spaces, preprint.Google Scholar

[4] J. E. Hutchinson: Fractals and selfsimilarity, Indiana Univ. Math. J. 30 (1981) 713–747.https://doi.org/10.1512/iumj.1981.30.30055 Google Scholar

[5] A. Jonsson: Brownian motion on fractals and function spaces, Math. Z. 6 (1996) 495–504.https://doi.org/10.1007/BF02621879 Google Scholar

[6] A. Jonsson: Haar wavelets of higher order on fractals and regularity of functions, Research Reports 7, Umeȧ Univ. (1998).Google Scholar

[7] A. Jonsson, H. Wallin: Function spaces on subsets of Rn, Math. Rep. 2, Harwood Acad. Publ. (1984).Google Scholar

[8] T. Kumagai: Brownian motion penetrating fractals: an application of the trace theorem of Besov spaces, J. Funct. Anal. 170 (2000) 69–92.https://doi.org/10.1006/jfan.1999.3500 Google Scholar

[9] M. R. Lancia, M. A. Vivaldi: Lipschitz spaces and Besov traces on self-similar fractals, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 117, XXIII (1999) 101–116.Google Scholar

[10] U. Mosco: Variational fractals, Special Volume in Memory of E. De Giorgi, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. 4 (1997) 683–712.Google Scholar

[11] U. Mosco: Variational metrics on self-similar fractals, C. R. Acad. Sci. Paris, Sér. I Math. 321 (1995) 715–720.Google Scholar

[12] U. Mosco: Dirichlet forms and self-similarity, in: New Directions in Dirichlet Forms, J. Jost et al., AMS/IP Stud. Adv. Math. 8, Amer. Math. Soc., Providence R. I. (1998) 117–155.https://doi.org/10.1090/amsip/008/03 Google Scholar

[13] K. P. Paluba: Some function spaces related to the Brownian motion on simple nested fractals, Stoch. and Stoch. Rep. 67 (1999) 267–285.https://doi.org/10.1080/17442509908834214 Google Scholar