Taiwanese Journal of Mathematics

ROUGH SINGULAR INTEGRALS SUPPORTED BY SUBMANIFOLDS IN TRIEBEL-LIZORKIN SPACES AND BESOVE SPACES

Feng Liu and Huoxiong Wu

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Abstract

This paper is devoted to studying the singular integral operators associated to polynomial mappings as well as the corresponding compound submanifolds. By imposing a restrictive condition on the kernels of the operators in the radial direction, the boundedness for such operators on Triebel-Lizorkin spaces and Besov spaces are established, provided that the kernels satisfy a rather weak size condition on the unit sphere, which is distinct from the Hardy space functions. Some previous results are essentially improved and generalized.

Article information

Source
Taiwanese J. Math., Volume 18, Number 1 (2014), 127-146.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706335

Digital Object Identifier
doi:10.11650/tjm.18.2014.3147

Mathematical Reviews number (MathSciNet)
MR3162117

Zentralblatt MATH identifier
1357.42026

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory

Keywords
singular integrals rough kernels Triebel-Lizorkin spaces Besov spaces vector-valued norm inequalities compound submanifolds

Citation

Liu, Feng; Wu, Huoxiong. ROUGH SINGULAR INTEGRALS SUPPORTED BY SUBMANIFOLDS IN TRIEBEL-LIZORKIN SPACES AND BESOVE SPACES. Taiwanese J. Math. 18 (2014), no. 1, 127--146. doi:10.11650/tjm.18.2014.3147. https://projecteuclid.org/euclid.twjm/1499706335


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References

  • H. M. Al-Qassem, On the boundedness of maxiaml operators and singular operators with kernels in $L(\log^+L)^{\alpha}(S^{n-1})$, J. Inequal. Appl., 2006 (2006), 1-16.
  • A. Al-Salman and Y. Pan, Singular integrals with rough kernels in $L\log^+L(S^{n-1})$, J. London Math. Soc., 66(2) (2002), 153-174.
  • A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. Soc., 78 (1956), 289-309.
  • Y. Chen, Y. Ding and H. Liu, Rough singular integrals supported on submanifolds, J. Math. Anal. Appl., 368 (2010), 677-691.
  • L. Colzani, Hardy Spaces on Spheres, Ph.D thesis, Washington University, St. Louis, 1982.
  • J. Duoandikoetxea and J. L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math., 84 (1986), 541-561.
  • Y. Ding, Q. Xue and Y. Yabuta, On singular interal operators with rough kernel along surfaces, Integr. Equ. Oper. Theory, 68 (2010), 151-161.
  • M. Frazier and B. Jawerth, A discrete transform an decompositions of distribution spaces, J. Funct. Anal., 93 (1990), 34-170.
  • M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley theory and the study of Function Spaces, CBMS Reg. Conf. Ser., Vol. 79. Amer. Math. Soc., Providence, RI, 1991.
  • D. Fan and Y. Pan, Singular integral operators with rough kernels supported by subvarieties, Amer. J. Math., 119 (1997), 799-839.
  • D. Fan and S. Sato, A note on the singular integrals associated with a variable surface of revolution, Math. Ineq. Appl., 12(2) (2009), 441-454.
  • D. Fan and H. Wu, Non-isotropic singular integrals and maximal operators along surfaces of revolution, Math. Ineq. Appl., 16(2) (2013), 461-476.
  • L. Grafakos, Classical and Modern Fourier Analysis, Prentice Hall, Upper Saddle River, NJ, 2003.
  • H. Viet Le, Singular integrals with dominating mixed smoothness in Triebel-Lizorkin spaces, Preprint, 2013.
  • F. Ricci and G. Weiss, A Characterization of $H^1(\sum_{n-1})$, Harmonic Analysis in Euclidean Space, Proc. Sympos. Pure Math., WilliamsColl., Williamstown, Mass., 1978, Part 1, pp. 163-165, Proc. Sympos. Pure Math. 35, Part, Amer. Math. Soc., Providence, R.I., 1979.
  • S. Sato, Estimates for singular integrals and extrapolation, Studia Math., 192 (2009), 219-233.
  • E. M. Stein, Problems in Harmonic Analysis Related to Curvature and Oscillatory Integrals, Proc. Intern. Cong. Math., Berkeley, 1986, pp. 196-221.
  • E. M. Stein, Harmonic Analysis: Real-variable methods, orthogonality and oscillatory integral, Princeton University Press, 1993.
  • H. Triebel, Theory of Function Spaces, Monogr. Math., Vol. 78, Birkhäser Verlag, Basel, 1983.