New Sobolev spaces via generalized Poincaré inequalities on metric measure spaces

Keywords

Sobolev space / Hajłasz-Sobolev space / space of homogeneous type / approximation of the identity / sharp maximal function / Poincaré inequality / embedding theorem / 42B35 / 46E35 / 43A99

References

[1] Gilbarg D., Trudinger N. S. Elliptic Partial Differential Equations of Second Order, 1983, Berlin: Springer-Verlag.

[2] Maz’ya V. G. Sobolev Spaces, 1985, Berlin: Springer-Verlag.

[3] Triebel H. Theory of Function Spaces II, 1992, Basel: Birkhäuser Verlag.

[4] Hajłasz P. Sobolev spaces on an arbitrary metric space. Potential Anal, 1996, 5: 403-415.

[5] Hajłasz P., Koskela P. Sobolev met Poincaré. Mem Amer Math Soc, 2000, 145(688): 1-101.

[6] Hajłasz P., Koskela P. Sobolev meets Poincaré. C R Acad Sci Paris Ser I Math, 1995, 320: 1211-1215.

[7] Franchi B., Hajłasz P., Koskela P. Definitions of Sobolev classes on metric spaces. Ann Inst Fourier (Grenoble), 1999, 49: 1903-1924.

[8] Heinonen J. Lectures on Analysis on Metric Spaces, 2001, Berlin: Springer-Verlag.

[9] Grigor’yan A., Hu J., Lau K. Heat kernels on metric measure spaces and an application to semilinear elliptic equations. Trans Amer Math Soc, 2003, 355: 2065-2095.

[10] Shanmugalingam N. Newtonian spaces: An extension of Sobolev spaces to metric measure spaces. Rev Mat Iberoam, 2000, 16: 243-279.

[11] Cheeger J. Differentiability of Lipschitz functions on metric measure spaces. Geom Funct Anal, 1999, 9: 428-517.

[12] Korevaar N. J., Schoen R. M. Sobolev spaces and harmonic maps for metric space targets. Comm Analysis Geom, 1993, 1: 561-659.

[13] Liu Y., Lu G., Wheeden R. L. Some equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces. Math Ann, 2002, 323: 157-174.

[14] Triebel H. Fractals and Spectra, 1997, Basel: Birkhäuser Verlag.

[15] Triebel H. Theory of Function Spaces III, 2006, Basel: Birkhäuser Verlag.

[16] Jonsson A., Wallin H. Function Spaces on Subsets of ℝⁿ. Math Rep 2, 1984, London: Harwood Academic Publ.

[17] Strichartz R. S. Function spaces on fractals. J Funct Anal, 2003, 198: 43-83.

[18] Hu J. A note on Hajłasz-Sobolev spaces on fractals. J Math Anal Appl, 2003, 280: 91-101.

[19] Yang D. New characterizations of Hajłasz-Sobolev spaces on metric spaces. Sci in China, Ser A, 2003, 46: 675-689.

[20] Yang D. Some function spaces relative to Morrey-Campanato spaces on metric spaces. Nagoya Math J, 2005, 177: 1-29.

[21] Coifman R., Weiss G. Analyse Harmonique Non-commutative sur Certains Espaces Homogènes, 1971, Berlin: Springer-Verlag.

[22] Duong X. T., McIntosh A. Singular integral operators with non-smooth kernels on irregular domains. Rev Mat Iberoam, 1999, 15: 233-265.

[23] Martell J. M. Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications. Studia Math, 2004, 161: 113-145.

[24] Auscher P., Coulhon T., Duong X. T. Riesz transform on manifolds and heat kernel regularity. Ann Sci École Norm Sup (4), 2004, 37: 911-957.

[25] Hofmann S., Martell J. M. L^p bounds for Riesz transforms and square roots associated to second order elliptic operators. Publ Mat, 2003, 47: 497-515.

[26] Duong X. T., Yan L. New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications. Comm Pure Appl Math, 2005, 58: 1375-1420.

[27] Deng D., Duong X. T., Yan L. A characterization of the Morrey-Campanato spaces. Math Z, 2005, 250: 641-655.

[28] Duong X. T., Yan L. Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J Amer Math Soc, 2005, 18: 943-973.