Bulletin of the Korean Mathematical Society (대한수학회보)

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A NOTE OF LITTLEWOOD-PALEY FUNCTIONS ON TRIEBEL-LIZORKIN SPACES

  • Liu, Feng (College of Mathematics and Systems Science Shandong University of Science and Technology)
  • Received : 2017.03.10
  • Accepted : 2017.11.03
  • Published : 2018.03.31
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Abstract

In this note we prove that several classes of Littlewood-Paley square operators defined by the kernels without any regularity are bounded on Triebel-Lizorkin spaces $F^{p,q}_{\alpha}({\mathbb{R}}^n)$ and Besov spaces $B^{p,q}_{\alpha}({\mathbb{R}}^n)$ for 0 < ${\alpha}$ < 1 and 1 < p, q < ${\infty}$.

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References

  1. H. Al-Qassem and Y. Pan, On certain estimates for Marcinkiewicz integrals and extrapolation, Collect. Math. 60 (2009), no. 2, 123-145. https://doi.org/10.1007/BF03191206
  2. A. Benedek, A. P. Calderon, and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), no. 3, 356-365. https://doi.org/10.1073/pnas.48.3.356
  3. L. C. Cheng, On Littlewood-Paley functions, Proc. Amer. Math. Soc. 135 (2007), no. 10, 3241-3247. https://doi.org/10.1090/S0002-9939-07-08917-4
  4. Y. Ding, D. Fan, and Y. Pan, On Littlewood-Paley functions and singular integrals, Hokkaido Math. J. 29 (2000), no. 3, 537-552. https://doi.org/10.14492/hokmj/1350912990
  5. Y. Ding and S. Sato, Littlewood-Paley functions on homogeneous groups, Forum Math. 28 (2016), no. 1, 43-55.
  6. J. Duoandikoetxea, Sharp $L_p$ boundedness for a class of square functions, Rev. Mat. Complut. 26 (2013), no. 2, 535-548. https://doi.org/10.1007/s13163-012-0106-y
  7. J. Duoandikoetxea and E. Seijo, Weighted inequalities for rough square functions through extrapolation, Studia Math. 149 (2002), no. 3, 239-252. https://doi.org/10.4064/sm149-3-2
  8. D. Fan and S. Sato, Remarks on Littlewood-Paley functions and singular integrals, J. Math. Soc. Japan 54 (2002), no. 3, 565-585. https://doi.org/10.2969/jmsj/1191593909
  9. M. Frazier, B. Jawerth, and G. Weiss, Littlewood-Paley theory and the study of function spaces, CBMS Regional Conference Series in Mathematics, 79, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1991.
  10. S. Korry, A class of bounded operators on Sobolev spaces, Arch. Math. (Basel) 82 (2004), no. 1, 40-50. https://doi.org/10.1007/s00013-003-0416-x
  11. S. Sato, Remarks on square functions in the Littlewood-Paley theory, Bull. Austral. Math. Soc. 58 (1998), no. 2, 199-211. https://doi.org/10.1017/S0004972700032172
  12. S. Sato, Estimates for Littlewood-Paley functions and extrapolation, Integral Equations Operator Theory 62 (2008), no. 3, 429-440. https://doi.org/10.1007/s00020-008-1631-4
  13. E. M. Stein, On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430-466. https://doi.org/10.1090/S0002-9947-1958-0112932-2
  14. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970.
  15. E. M. Stein, Harmonic Analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993.
  16. H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78, Birkhauser Verlag, Basel, 1983.
  17. H. Xu, J. Chen, and Y. Ying, A note on Marcinkiewicz integrals with $H^1$ kernels, Acta Math. Sci. Ser. B Engl. Ed. 23 (2003), no. 1, 133-138.
  18. K. Yabuta, Triebel-Lizorkin space boundedness of Marcinkiewicz integrals associated to surfaces, Appl. Math. J. Chinese Univ. Ser. B 30 (2015), no. 4, 418-446. https://doi.org/10.1007/s11766-015-3358-8
  19. K. Yabuta, Remarks on $L_2$ boundedness of Littlewood-Paley operators, Analysis (Berlin) 33 (2013), no. 3, 209-218.
  20. C. Zhang and J. Chen, Boundedness of g-functions on Triebel-Lizorkin spaces, Taiwanese J. Math. 13 (2009), no. 3, 973-981. https://doi.org/10.11650/twjm/1500405452