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Some Function Spaces Relative to Morrey-Campanato Spaces on Metric Spaces

Published online by Cambridge University Press:  11 January 2016

Dachun Yang
Dachun Yang*
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s, Republic of China, dcyang@bnu.edu.cn
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Abstract

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In this paper, the author introduces the Morrey-Campanato spaces Lsp(X) and the spaces Cps(X) on spaces of homogeneous type including metric spaces and some fractals, and establishes some embedding theorems between these spaces under some restrictions and the Besov spaces and the Triebel-Lizorkin spaces. In particular, the author proves that Lsp(X) = Bs∞,∞(X) if 0 < s < ∞ and µ(X) < ∞. The author also introduces some new function spaces Asp(X) and Bsp(X) and proves that these new spaces when 0 < s < 1 and 1 < p < ∞ are just the Triebel-Lizorkin space Fsp,∞(X) if X is a metric space, and the spaces A1p(X) and B1p(X) when 1 < p < ∞ are just the Hajłasz-Sobolev spaces W1p(X). Finally, as an application, the author gives a new characterization of the Hajłasz-Sobolev spaces by making use of the sharp maximal function.

Keywords

42B3546E3543-99
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 177 , 2005 , pp. 1 - 29
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2005

References

[1] Barlow, M. T., Diffusions on fractals, Lectures on probability theory and statistics (Saint-Flour, 1995) (P. Bernard, ed.), Lecture Notes in Math. 1690, Springer, Berlin (1998), pp. 1–121.Google Scholar
[2] Barlow, M. T. and Bass, R. F., Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math., 51 (1999), 673–744.Google Scholar
[3] Christ, M., The extension problem for certain function spaces involving fractional orders of differentiability, Ark. Mat., 22 (1984), 63–81.Google Scholar
[4] Coifman, R. and Weiss, G., Analyse Harmonique Non-commutative sur Certains Espaces Homogènes, Lecture Notes in Math. 242, Springer-Verlag, Berlin, 1971.Google Scholar
[5] David, G., Journé, J. L. and Semmes, S., Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana, 1 (1985), 1–56.Google Scholar
[6] DeVore, R. A. and Sharpley, R. C., Maximal functions measuring smoothness, Memoirs Amer. Math. Soc., 47 (1984), no. 293,1–115.CrossRefGoogle Scholar
[7] Gatto, A. E., Product rule and chain rule estimates for fractional derivatives on spaces that satisfy the doubling condition, J. Funct. Anal., 188 (2002), 27–37.Google Scholar
[8] Grigor’yan, A., Hu, J. and Lau, K., Heat kernels on metric-measure spaces and an application to semi-linear elliptic equations, Trans. Amer. Math. Soc., 355 (2003), 2065–2095.Google Scholar
[9] Hajłasz, P., Sobolev spaces on an arbitrary metric spaces, Potential Anal., 5 (1996), 403–415.Google Scholar
[10] Hajłasz, P. and Koskela, P., Sobolev met Poincaré, Memoirs Amer. Math. Soc., 145 (2000), no. 688,1–101.Google Scholar
[11] Han, Y., Inhomogeneous Calderón reproducing formula on spaces of homogeneous type, J. Geometric Anal., 7 (1997), 259–284.Google Scholar
[12] Han, Y., Lu, S. and Yang, D., Inhomogeneous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type, Approx. Th. and its Appl., 15(3) (1999), 37–65.Google Scholar
[13] Han, Y., Lu, S. and Yang, D., Inhomogeneous Triebel-Lizorkin spaces on spaces of homogeneous type, Math. Sci. Res. Hot-Line, 3(9) (1999), 1–29.Google Scholar
[14] Han, Y., Lu, S. and Yang, D., Inhomogeneous discrete Calderón reproducing formulas for spaces of homogeneous type, J. Fourier Anal. Appl., 7 (2001), 571–600.CrossRefGoogle Scholar
[15] Han, Y. and Sawyer, E. T., Littlewood-Paley theory on spaces of homogeneous type and classical function spaces, Memoirs Amer. Math. Soc., 110 (1994), no. 530,1–126.Google Scholar
[16] Han, Y. and Yang, D., New characterizations and applications of inhomogeneous Besov and Triebel-Lizorkin spaces on homogeneous type spaces and fractals, Dis-sertationes Math. (Rozprawy Mat.), 403 (2002), 1–102.Google Scholar
[17] Han, Y. and Yang, D., Some new spaces of Besov and Triebel-Lizorkin type on homogeneous spaces, Studia Math., 156 (2003), 67–97.Google Scholar
[18] Heinonen, J., Lectures on Analysis on Metric Spaces, Springer-Verlag, Berlin, 2001.Google Scholar
[19] Jonsson, A. and Wallin, H., Function Spaces on Subsets of Rn , Math. Reports, Vol. 2, Harwood Academic Publ., London, 1984.Google Scholar
[20] Kigami, J., Analysis on fractals, Cambridge Tracts in Math. 143, Cambridge University Press, Cambridge, 2001.Google Scholar
[21] Liu, Y., Lu, G. and Wheeden, R. L., Some equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces, Math. Ann., 323 (2002), 157–174.Google Scholar
[22] Macias, R. A. and Segovia, C., Lipschitz functions on spaces of homogeneous type, Adv. in Math., 33 (1979), 257–270.Google Scholar
[23] Mattila, P., Geometry of sets and measures in Euclidean spaces, Cambridge University Press, Cambridge, 1995.Google Scholar
[24] Miyachi, A., Multiplication and factorization of functions in Sobolev spaces and in Cp α spaces on general domains, Math. Nachr., 176 (1995), 209–242.Google Scholar
[25] Miyachi, A., Atomic decomposition for Sobolev spaces and for the Cpa spaces on general domains, Tsukuba J. Math., 21 (1997), 59–96.Google Scholar
[26] Strichartz, R. S., Function spaces on fractals, J. Funct. Anal., 198 (2003), 43–83.Google Scholar
[27] Triebel, H., Theory of Function Spaces, Birkhäuser Verlag, Basel, 1983.Google Scholar
[28] Triebel, H., Theory of Function Spaces II, Birkhäuser Verlag, Basel, 1992.CrossRefGoogle Scholar
[29] Triebel, H., Fractals and Spectra, Birkhäuser Verlag, Basel, 1997.Google Scholar
[30] Triebel, H., The Structure of Functions, Birkhäuser Verlag, Basel, 2001.Google Scholar
[31] Triebel, H., Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers, Revista Mat. Complutense, 15 (2002), 1–50.Google Scholar
[32] Yang, D., Besov spaces on spaces of homogeneous type and fractals, Studia Math., 156 (2003), 15–30.Google Scholar