American Institute of Mathematical Sciences

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

 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.
Citation: Marius Ionescu, Luke G. Rogers. Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators. Communications on Pure and 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. [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. [3] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1990. [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. [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. [6] M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35. doi: 10.1007/BF00249784. [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. [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. [9] Jiaxin Hu and Martina Zähle, Potential spaces on fractals, Studia Math., 170 (2005), 259-281. doi: 10.4064/sm170-3-4. [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. [11] John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055. [12] Jun Kigami, Analysis on Fractals, Cambridge Tracts in Mathematics, 143, Cambridge, 2001. [13] Jun Kigami, Harmonic analysis for resistance forms, J. Funct. Anal., 204 (2003), 399-444. doi: 10.1016/S0022-1236(02)00149-0. [14] Tom Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc., 83 (1990), 420. [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. [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. [17] Christophe Sabot, Pure point spectrum for the Laplacian on unbounded nested fractals, J. Funct. Anal., 173 (2000), 497-524. [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. [19] R. T. Seeley, Analytic extension of the trace associated with elliptic boundary problems, Amer. J. Math., 91 (1969), 963-983. [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. [21] Elias M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. [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. [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. [24] Elias M. Stein and Rami Shakarchi, Complex Analysis, Princeton Lectures in Analysis, II, Princeton University Press, Princeton, NJ, 2003. [25] Robert S. Strichartz, Fractals in the large, Canad. J. Math., 50 (1998), 638-657. [26] Robert S. Strichartz, Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc., 355 (2003), 4019-4043. [27] Robert S. Strichartz, Function spaces on fractals, J. Funct. Anal., 198 (2003), 43-83. [28] Robert S. Strichartz, Analysis on products of fractals, Trans. Amer. Math. Soc., 357 (2005), 571-615. [29] Robert S. Strichartz, Differential Equations on Fractals, A tutorial, Princeton University Press, Princeton, NJ, 2006. [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. [31] Michael E. Taylor, Pseudodifferential Operators, Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981. [32] Alexander Teplyaev, Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal., 159 (1998), 537-567. [33] Xuan Thinh Duong, El Maati Ouhabaz and Adam Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., 196 (2002), 443-485.

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References:
 [1] Martin T. Barlow and Edwin A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields, 79 (1988), 543-623. [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. [3] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge, 1990. [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. [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. [6] M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35. doi: 10.1007/BF00249784. [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. [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. [9] Jiaxin Hu and Martina Zähle, Potential spaces on fractals, Studia Math., 170 (2005), 259-281. doi: 10.4064/sm170-3-4. [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. [11] John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055. [12] Jun Kigami, Analysis on Fractals, Cambridge Tracts in Mathematics, 143, Cambridge, 2001. [13] Jun Kigami, Harmonic analysis for resistance forms, J. Funct. Anal., 204 (2003), 399-444. doi: 10.1016/S0022-1236(02)00149-0. [14] Tom Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc., 83 (1990), 420. [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. [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. [17] Christophe Sabot, Pure point spectrum for the Laplacian on unbounded nested fractals, J. Funct. Anal., 173 (2000), 497-524. [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. [19] R. T. Seeley, Analytic extension of the trace associated with elliptic boundary problems, Amer. J. Math., 91 (1969), 963-983. [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. [21] Elias M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. [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. [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. [24] Elias M. Stein and Rami Shakarchi, Complex Analysis, Princeton Lectures in Analysis, II, Princeton University Press, Princeton, NJ, 2003. [25] Robert S. Strichartz, Fractals in the large, Canad. J. Math., 50 (1998), 638-657. [26] Robert S. Strichartz, Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc., 355 (2003), 4019-4043. [27] Robert S. Strichartz, Function spaces on fractals, J. Funct. Anal., 198 (2003), 43-83. [28] Robert S. Strichartz, Analysis on products of fractals, Trans. Amer. Math. Soc., 357 (2005), 571-615. [29] Robert S. Strichartz, Differential Equations on Fractals, A tutorial, Princeton University Press, Princeton, NJ, 2006. [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. [31] Michael E. Taylor, Pseudodifferential Operators, Princeton Mathematical Series, 34, Princeton University Press, Princeton, N.J., 1981. [32] Alexander Teplyaev, Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal., 159 (1998), 537-567. [33] Xuan Thinh Duong, El Maati Ouhabaz and Adam Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., 196 (2002), 443-485.
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