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November  2014, 13(6): 2155-2175. doi: 10.3934/cpaa.2014.13.2155

Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators

Marius Ionescu 1, and  Luke G. Rogers 2,

1. 

Department of Mathematics, United States Naval Academy, Annapolis, MD, 21402, United States
2. 

Department of Mathematics, University of Connecticut, Storrs CT 06269-3009

Received  June 2011 Revised  July 2012 Published  July 2014

We give the first natural examples of Calderón-Zygmund operators in the theory of analysis on post-critically finite self-similar fractals. This is achieved by showing that the purely imaginary Riesz and Bessel potentials on nested fractals with $3$ or more boundary points are of this type. It follows that these operators are bounded on $L^{p}$, $1 < p < \infty$ and satisfy weak $1$-$1$ bounds. The analysis may be extended to infinite blow-ups of these fractals, and to product spaces based on the fractal or its blow-up.
Keywords: Bessel potentials., Calderón-Zygmund operators, Riesz potentials, Analysis on fractals.
Mathematics Subject Classification: Primary: 28A80, 46F12; Secondary: 42C99, 81Q1.
Citation: Marius Ionescu, Luke G. Rogers. Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators. Communications on Pure &amp; Applied Analysis, 2014, 13 (6) : 2155-2175. doi: 10.3934/cpaa.2014.13.2155
References:
[1]

Martin T. Barlow and Edwin A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields, 79 (1988), 543-623.  Google Scholar

[2]

Oren Ben-Bassat, Robert S. Strichartz and Alexander Teplyaev, What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. Funct. Anal., 166 (1999), 197-217.  Google Scholar

[3]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1990.  Google Scholar

[4]

Pat J. Fitzsimmons, Ben M. Hambly and Takashi Kumagai, Transition density estimates for Brownian motion on affine nested fractals, Comm. Math. Phys., 165 (1994), 595-620.  Google Scholar

[5]

Gerald B. Folland, Real Analysis, Pure and Applied Mathematics, New York, Second edition, Modern techniques and their applications, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1999.  Google Scholar

[6]

M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35. doi: 10.1007/BF00249784.  Google Scholar

[7]

B. M. Hambly and T. Kumagai, Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. London Math. Soc., 78 (1999), 431-458.  Google Scholar

[8]

B. M. Hambly and T. Kumagai, Diffusion processes on fractal fields: heat kernel estimates and large deviations, Probab. Theory Related Fields, 127 (2003), 305-352.  Google Scholar

[9]

Jiaxin Hu and Martina Zähle, Potential spaces on fractals, Studia Math., 170 (2005), 259-281. doi: 10.4064/sm170-3-4.  Google Scholar

[10]

Jiaxin Hu and Martina Zähle, Generalized Bessel and Riesz potentials on metric measure spaces, Potential Anal., 30 (2009), 315-340. doi: 10.1007/s11118-009-9117-9.  Google Scholar

[11]

John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[12]

Jun Kigami, Analysis on Fractals, Cambridge Tracts in Mathematics, 143, Cambridge, 2001.  Google Scholar

[13]

Jun Kigami, Harmonic analysis for resistance forms, J. Funct. Anal., 204 (2003), 399-444. doi: 10.1016/S0022-1236(02)00149-0.  Google Scholar

[14]

Tom Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc., 83 (1990), 420.  Google Scholar

[15]

Jonathan Needleman, Robert S. Strichartz, Alexander Teplyaev and Po-Lam Yung, Calculus on the Sierpinski gasket. I. Polynomials, exponentials and power series, J. Funct. Anal., 215 (2004), 290-340. doi: 10.1016/j.jfa.2003.11.011.  Google Scholar

[16]

Luke G. Rogers, Estimates for the resolvent kernel of the Laplacian on p.c.f. self-similar fractals and blowups, Trans. Amer. Math. Soc., 364 (2012), 1633-1685. doi: 10.1090/S0002-9947-2011-05551-0.  Google Scholar

[17]

Christophe Sabot, Pure point spectrum for the Laplacian on unbounded nested fractals, J. Funct. Anal., 173 (2000), 497-524.  Google Scholar

[18]

R. T. Seeley, Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), 288-307, Amer. Math. Soc., Providence, R.I., 1967.  Google Scholar

[19]

R. T. Seeley, Analytic extension of the trace associated with elliptic boundary problems, Amer. J. Math., 91 (1969), 963-983.  Google Scholar

[20]

Adam Sikora, Multivariable spectral multipliers and analysis of quasielliptic operators on fractals, Indiana Univ. Math. J., 58 (2009), 317-334. doi: 10.1512/iumj.2009.58.3745.  Google Scholar

[21]

Elias M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[22]

Elias M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[23]

Elias M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[24]

Elias M. Stein and Rami Shakarchi, Complex Analysis, Princeton Lectures in Analysis, II, Princeton University Press, Princeton, NJ, 2003.  Google Scholar

[25]

Robert S. Strichartz, Fractals in the large, Canad. J. Math., 50 (1998), 638-657.  Google Scholar

[26]

Robert S. Strichartz, Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc., 355 (2003), 4019-4043.  Google Scholar

[27]

Robert S. Strichartz, Function spaces on fractals, J. Funct. Anal., 198 (2003), 43-83.  Google Scholar

[28]

Robert S. Strichartz, Analysis on products of fractals, Trans. Amer. Math. Soc., 357 (2005), 571-615.  Google Scholar

[29]

Robert S. Strichartz, Differential Equations on Fractals, A tutorial, Princeton University Press, Princeton, NJ, 2006.  Google Scholar

[30]

Robert S. Strichartz, A fractal quantum mechanical model with Coulomb potential, Commun. Pure Appl. Anal., 8 (2009), 743-755. doi: 10.3934/cpaa.2009.8.743.  Google Scholar

[31]

Michael E. Taylor, Pseudodifferential Operators, Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981.  Google Scholar

[32]

Alexander Teplyaev, Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal., 159 (1998), 537-567.  Google Scholar

[33]

Xuan Thinh Duong, El Maati Ouhabaz and Adam Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., 196 (2002), 443-485.  Google Scholar

show all references

References:
[1]

Martin T. Barlow and Edwin A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields, 79 (1988), 543-623.  Google Scholar

[2]

Oren Ben-Bassat, Robert S. Strichartz and Alexander Teplyaev, What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. Funct. Anal., 166 (1999), 197-217.  Google Scholar

[3]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1990.  Google Scholar

[4]

Pat J. Fitzsimmons, Ben M. Hambly and Takashi Kumagai, Transition density estimates for Brownian motion on affine nested fractals, Comm. Math. Phys., 165 (1994), 595-620.  Google Scholar

[5]

Gerald B. Folland, Real Analysis, Pure and Applied Mathematics, New York, Second edition, Modern techniques and their applications, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1999.  Google Scholar

[6]

M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35. doi: 10.1007/BF00249784.  Google Scholar

[7]

B. M. Hambly and T. Kumagai, Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. London Math. Soc., 78 (1999), 431-458.  Google Scholar

[8]

B. M. Hambly and T. Kumagai, Diffusion processes on fractal fields: heat kernel estimates and large deviations, Probab. Theory Related Fields, 127 (2003), 305-352.  Google Scholar

[9]

Jiaxin Hu and Martina Zähle, Potential spaces on fractals, Studia Math., 170 (2005), 259-281. doi: 10.4064/sm170-3-4.  Google Scholar

[10]

Jiaxin Hu and Martina Zähle, Generalized Bessel and Riesz potentials on metric measure spaces, Potential Anal., 30 (2009), 315-340. doi: 10.1007/s11118-009-9117-9.  Google Scholar

[11]

John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[12]

Jun Kigami, Analysis on Fractals, Cambridge Tracts in Mathematics, 143, Cambridge, 2001.  Google Scholar

[13]

Jun Kigami, Harmonic analysis for resistance forms, J. Funct. Anal., 204 (2003), 399-444. doi: 10.1016/S0022-1236(02)00149-0.  Google Scholar

[14]

Tom Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc., 83 (1990), 420.  Google Scholar

[15]

Jonathan Needleman, Robert S. Strichartz, Alexander Teplyaev and Po-Lam Yung, Calculus on the Sierpinski gasket. I. Polynomials, exponentials and power series, J. Funct. Anal., 215 (2004), 290-340. doi: 10.1016/j.jfa.2003.11.011.  Google Scholar

[16]

Luke G. Rogers, Estimates for the resolvent kernel of the Laplacian on p.c.f. self-similar fractals and blowups, Trans. Amer. Math. Soc., 364 (2012), 1633-1685. doi: 10.1090/S0002-9947-2011-05551-0.  Google Scholar

[17]

Christophe Sabot, Pure point spectrum for the Laplacian on unbounded nested fractals, J. Funct. Anal., 173 (2000), 497-524.  Google Scholar

[18]

R. T. Seeley, Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), 288-307, Amer. Math. Soc., Providence, R.I., 1967.  Google Scholar

[19]

R. T. Seeley, Analytic extension of the trace associated with elliptic boundary problems, Amer. J. Math., 91 (1969), 963-983.  Google Scholar

[20]

Adam Sikora, Multivariable spectral multipliers and analysis of quasielliptic operators on fractals, Indiana Univ. Math. J., 58 (2009), 317-334. doi: 10.1512/iumj.2009.58.3745.  Google Scholar

[21]

Elias M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[22]

Elias M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[23]

Elias M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[24]

Elias M. Stein and Rami Shakarchi, Complex Analysis, Princeton Lectures in Analysis, II, Princeton University Press, Princeton, NJ, 2003.  Google Scholar

[25]

Robert S. Strichartz, Fractals in the large, Canad. J. Math., 50 (1998), 638-657.  Google Scholar

[26]

Robert S. Strichartz, Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc., 355 (2003), 4019-4043.  Google Scholar

[27]

Robert S. Strichartz, Function spaces on fractals, J. Funct. Anal., 198 (2003), 43-83.  Google Scholar

[28]

Robert S. Strichartz, Analysis on products of fractals, Trans. Amer. Math. Soc., 357 (2005), 571-615.  Google Scholar

[29]

Robert S. Strichartz, Differential Equations on Fractals, A tutorial, Princeton University Press, Princeton, NJ, 2006.  Google Scholar

[30]

Robert S. Strichartz, A fractal quantum mechanical model with Coulomb potential, Commun. Pure Appl. Anal., 8 (2009), 743-755. doi: 10.3934/cpaa.2009.8.743.  Google Scholar

[31]

Michael E. Taylor, Pseudodifferential Operators, Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981.  Google Scholar

[32]

Alexander Teplyaev, Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal., 159 (1998), 537-567.  Google Scholar

[33]

Xuan Thinh Duong, El Maati Ouhabaz and Adam Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., 196 (2002), 443-485.  Google Scholar

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