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Some new Besov and Triebel-Lizorkin spaces associated with para-accretive functions on spaces of homogeneous type

Part of: Harmonic analysis in several variables Special classes of linear operators Linear function spaces and their duals Abstract harmonic analysis General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

Dongguo Deng  and
Dachun Yang
Dongguo Deng
Affiliation:
Department of Mathematics, Zhongshan University, Guangzhou 510275, People's Republic of, China, e-mail: stsdd@zsu.edu.cn
Dachun Yang
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People's Republic of, China, e-mail: dcyang@bnu.edu.cn
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Abstract

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Let (X, ρ, μ)d, θ be a space of homogeneous type with d < 0 and θ ∈ (0, 1], b be a para-accretive function, ε ∈ (0, θ], ∣s∣ > ∈ and a0 ∈ (0, 1) be some constant depending on d, ∈ and s. The authors introduce the Besov space bBspq (X) with a0 > p ≧ ∞, and the Triebel-Lizorkin space bFspq (X) with a0 > p > ∞ and a0 > q ≧∞ by first establishing a Plancherel-Pôlya-type inequality. Moreover, the authors establish the frame and the Littlewood-Paley function characterizations of these spaces. Furthermore, the authors introduce the new Besov space b−1 Bs (X) and the Triebel-Lizorkin space b−1 Fspq (X). The relations among these spaces and the known Hardy-type spaces are presented. As applications, the authors also establish some real interpolation theorems, embedding theorems, T b theorems, and the lifting property by introducing some new Riesz operators of these spaces.

Keywords

space of homogeneous typepara-accretive functionPlancherel-Pôlya inequaalityBesov spaceTriebel-Lizorkin spaceCalderón reproducing formulainterpolationembedding theoremTb theoremRiesz potentiallifting porperty.

MSC classification

Secondary: 42B35: Function spaces arising in harmonic analysis 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 42B25: Maximal functions, Littlewood-Paley theory 43A99: None of the above, but in this section 47B06: Riesz operators; eigenvalue distributions; approximation numbers, $s$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 47A30: Norms (inequalities, more than one norm, etc.) 47B38: Operators on function spaces (general)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 2 , April 2006 , pp. 229 - 262
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Bergh, J. and Löfström, J., Interpolation spaces (Springer, Berlin, 1976).CrossRefGoogle Scholar
[2]Christ, M., ‘A T(b) theorem with remarks on analytic capacity and the Cauchy integral’, Colloq. Math. 40/41 (1990), 601–628.CrossRefGoogle Scholar
[3]Coifman, R. R. and Weiss, G., Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math. 242 (Springer, Berlin, 1971).CrossRefGoogle Scholar
[4]Coifman, R. R. and Weiss, G., ‘Extensions of Hardy spaces and their use in analysis’, Bull. Amer. Math. Soc. 83 (1977), 569–645.CrossRefGoogle Scholar
[5]David, G. and Journé, J. L., ‘A boundedness criterion for generalized Calderón-Zygmund operators’, Ann. of Math. (2) 120 (1984), 371–397.CrossRefGoogle Scholar
[6]David, G., Journé, J. L. and Semmes, S., ‘Opérateurs de Calderón-Zygmund, fonctions paraaccrétives et interpolation’, Rev. Mat. Iberoam. 1 (1985), 1–56.CrossRefGoogle Scholar
[7]Fefferman, C. and Stein, E. M., ‘Some maximal inequalities’, Amer. J. Math. 93 (1971), 107–116.CrossRefGoogle Scholar
[8]Frazier, M. and Jawerth, B., ‘A discrete transform and decompositions of distribution spaces’, J. Funct. Anal. 93 (1990), 34–170.CrossRefGoogle Scholar
[9]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.CrossRefGoogle Scholar
[10]Gatto, A. E., Segovia, C. and Vági, S., ‘On fractional differentiation and integration on spaces of homogeneous type’, Rev. Mat. Iberoam. 12 (1996), 111–145.CrossRefGoogle Scholar
[11]Gatto, A. E. and Vági, S., ‘On Sobolev spaces of fractional order and ε-families of operators on spaces of homogeneous type’, Studia Math. 133 (1999), 19–27.Google Scholar
[12]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.CrossRefGoogle Scholar
[13]Hajlasz, P. and Koskela, P., ‘Sobolev met Poincaré’, Mem. Amer. Math. Soc. 145 (2000), 1–101.Google Scholar
[14]Han, Y., ‘Calderón-type reproducing formula and the T b theorem’, Rev. Mat. Iberoam. 10 (1994), 51–91.CrossRefGoogle Scholar
[15]Han, Y., ‘The embedding theorem for the Besov and Triebel-Lizorkin spaces on spaces of homogeneous type’, Proc. Amer. Math. Soc. 123 (1995), 2181–2189.CrossRefGoogle Scholar
[16]Han, Y., ‘Plancherel-Pôlya type inequality on spaces of homogeneous type and its applications’, Proc. Amer. Math. Soc. 126 (1998), 3315–3327.CrossRefGoogle Scholar
[17]Han, Y., Lee, M. and Lin, C., ‘Hardy spaces and the T b theorem’, J. Geom. Anal. 14 (2004), 291–318.CrossRefGoogle Scholar
[18]Han, Y. and Lin, C., ‘Embedding theorem on spaces of homogeneous type’, J. Fourier Anal. Appl. 8 (2002), 291–307.CrossRefGoogle Scholar
[19]Han, Y., Lu, S. and Yang, D., ‘Inhomogeneous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type’, Approx. Theory Appl. 15 (3) (1999), 37–65.CrossRefGoogle Scholar
[20]Han, Y., Lu, S. and Yang, D., ‘Inhomogeneous Triebel-Lizorkin spaces on spaces of homogeneous type’, Math. Sci. Res. Hot-Line 3 (1999), 1–29.Google Scholar
[21]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
[22]Han, Y. and Sawyer, E. T., ‘Littlewood-Paley theory on spaces of homogeneous type and classical function spaces’, Mem. Amer. Math. Soc. 110 (1994), 1–126.Google Scholar
[23]Han, Y. and Yang, D., ‘New characterizations and applications of inhomogeneous Besov and Triebel-Lizorkin spaces on homogeneous type spaces and fractals’, Dissertationes Math. (Rozprawy Mat.) 403 (2002), 1–102.CrossRefGoogle Scholar
[24]Han, Y. and Yang, D., ‘Some new spaces of Besov and Triebel-Lizorkin type on homogeneous spaces’, Studia Math. 156 (2003), 67–97.CrossRefGoogle Scholar
[25]Heinonen, J., Lectures on analysis on metric spaces (Springer, Berlin, 2001).CrossRefGoogle Scholar
[26]Kigami, J., Analysis on fractals, Cambridge Tracts in Math. 143 (Cambridge University Press, Cambridge, 2001).CrossRefGoogle Scholar
[27]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.CrossRefGoogle Scholar
[28]Macías, R. A. and Segovia, C., ‘A decomposition into atoms of distributions on spaces of homogeneous type’, Adv. Math. 33 (1979), 271–309.CrossRefGoogle Scholar
[29]Macías, R. A. and Segovia, C., ‘Lipschitz functions on spaces of homogeneous type’, Adv. in Math. 33 (1979), 257–270.CrossRefGoogle Scholar
[30]McIntosh, A. and Meyer, Y., ‘Algèbres d'opèrateurs dèfinis par des intègrales singulières’, C. R. Acad. Sci. Paris Sèr. l Math. 301 (1985), 395–397.Google Scholar
[31]Meyer, Y. and Coifman, R. R., Wavelets. Calderón-Zygmund and multilinear operators, Cambridge studies in advanced mathematics 48 (Cambridge Univ. Press, Cambridge, 1997).Google Scholar
[32]Nahmod, A. R., ‘Geometry of operators and spectral analysis on spaces of homogeneous type’, C. R. Acad. Sci. Paris Sér. l Math. 313 (1991), 721–725.Google Scholar
[33]Nahmod, A. R., ‘Generalized uncertainty principles on spaces of homogeneous type’, J. Funct. Anal. 119 (1994), 171–209.CrossRefGoogle Scholar
[34]Semmes, S., ‘An introduction to analysis on metric spaces’, Notice Amer. Math. Soc. 50 (2003), 438–443.Google Scholar
[35]Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals (Princeton Univ. Press, Princeton, N.J., 1993).Google Scholar
[36]Strichartz, R. S., ‘Function spaces on fractals’, J. Funct. Anal. 198 (2003), 43–83.CrossRefGoogle Scholar
[37]Triebel, H., Theory of function spaces (Birkhäuser, Basel, 1983).CrossRefGoogle Scholar
[38]Triebel, H., Interpolation theory, function spaces, differential operators, 2nd edition (Johann Ambrosius Barth, Leipzig, 1995).Google Scholar
[39]Triebel, H., Fractals and spectra (Birkhäuser, Basel, 1997).CrossRefGoogle Scholar
[40]Triebel, H., The structure of functions (Birkhäuser, Basel, 2001).Google Scholar
[41]Triebel, H., ‘Function spaces in Lipschitz domains and inLipschitz manifolds. Characteristic functions as pointwise multipliers’, Rev. Mat. Complut. 15 (2002), 1–50.CrossRefGoogle Scholar
[42]Triebel, H. and Yang, D., ‘Spectral theory of Riesz potentials on quasi-metric spaces’, Math. Nachr. 238 (2002), 160–184.3.0.CO;2-5>CrossRefGoogle Scholar
[43]Yang, D., ‘Frame characterizations of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type and their applications’, Georgian Math. J. 9 (2002), 567–590.CrossRefGoogle Scholar
[44]Yang, D., ‘Embedding theorems of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type’, Sci. China (Ser. A) 46 (2003), 187–199.CrossRefGoogle Scholar
[45]Yang, D., ‘Riesz potentials in Besov and Triebel-Lizorkin spaces over spaces of homogeneous type’, Potential Anal. 19 (2003), 193–210.CrossRefGoogle Scholar
[46]Yang, D., ‘T I theorems on Besov and Triebel-Lizorkin spaces on spaces of homogeneous type and their applications’, Z. Anal. Anwendungen 22 (2003), 53–72.CrossRefGoogle Scholar
[47]Yang, D., ‘Real interpolations for Besov and Triebel-Lizorkin spaces on spaces of homogeneous type’, Math. Nachr. 273 (2004), 96–113.CrossRefGoogle Scholar
[48]Yang, D., ‘Some new Triebel-Lizorkin spaces on spaces of homogeneous type and their frame characterizations’, Sci. China (Ser. A) 48 (2005), 12–39.CrossRefGoogle Scholar