DOI QR코드

DOI QR Code

BOUNDEDNESS OF THE COMMUTATOR OF THE INTRINSIC SQUARE FUNCTION IN VARIABLE EXPONENT SPACES

  • Wang, Liwei (School of Mathematics and Physics Anhui Polytechnic University)
  • 투고 : 2017.08.14
  • 심사 : 2018.02.21
  • 발행 : 2018.07.01

초록

In this paper, we show that the commutator of the intrinsic square function with BMO symbols is bounded on the variable exponent Lebesgue spaces $L^{p({\cdot})}({\mathbb{R}}^n)$ applying a generalization of the classical Rubio de Francia extrapolation. As a consequence we further establish its boundedness on the variable exponent Morrey spaces $\mathcal{M_{p({\cdot}),u}$, Morrey-Herz spaces $M{\dot{K}}^{{\alpha}({\cdot}),{\lambda}}_{q,p({\cdot})}({\mathbb{R}}^n)$ and Herz type Hardy spaces $H{\dot{K}}^{{\alpha}({\cdot}),q}_{p({\cdot})}({\mathbb{R}}^n)$, where the exponents ${\alpha}({\cdot})$ and $p({\cdot})$ are variable. Observe that, even when ${\alpha}({\cdot}){\equiv}{\alpha}$ is constant, the corresponding main results are completely new.

키워드

과제정보

연구 과제 주관 기관 : National Natural Science Foundation of China

참고문헌

  1. A. Almeida and D. Drihem, Maximal, potential and singular type operators on Herz spaces with variable exponents, J. Math. Anal. Appl. 394 (2012), no. 2, 781-795. https://doi.org/10.1016/j.jmaa.2012.04.043
  2. Y. Chen, S. Levine, and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), no. 4, 1383-1406. https://doi.org/10.1137/050624522
  3. D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces, Applied and Numerical Harmonic Analysis, Birkhauser, Springer, Heidelberg, 2013.
  4. D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, The maximal function on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 1, 223-238.
  5. D. Cruz-Uribe, SFO, A. Fiorenza, J. Martell, and C. Perez, The boundedness of classical operators on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 239-264.
  6. L. Diening, Maximal function on generalized Lebesgue spaces $L^{p({\cdot})}$, Math. Inequal. Appl. 7 (2004), no. 2, 245-253.
  7. L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math. 129 (2005), no. 8, 657-700. https://doi.org/10.1016/j.bulsci.2003.10.003
  8. L. Diening, P. Harjulehto, P. Hasto, and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, 2017, Springer, Heidelberg, 2011.
  9. B. Dong and J. Xu, Variable exponent Herz type Hardy spaces and their applications, Anal. Theory Appl. 31 (2015), no. 4, 321-353.
  10. D. Drihem and F. Seghiri, Notes on the Herz-type Hardy spaces of variable smoothness and integrability, Math. Inequal. Appl. 19 (2016), no. 1, 145-165.
  11. C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. https://doi.org/10.2307/2373450
  12. C. Fefferman and E. M. Stein, $Hp$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137-193. https://doi.org/10.1007/BF02392215
  13. V. Guliyev, M. Omarova, and Y. Sawano, Boundedness of intrinsic square functions and their commutators on generalized weighted Orlicz-Morrey spaces, Banach J. Math. Anal. 9 (2015), no. 2, 44-62. https://doi.org/10.15352/bjma/09-2-5
  14. Y. Han, M.-Y. Lee, and C.-C. Lin, Atomic decomposition and boundedness of operators on weighted Hardy spaces, Canad. Math. Bull. 55 (2012), no. 2, 303-314. https://doi.org/10.4153/CMB-2011-072-7
  15. P. Harjulehto, P. Hasto, U. V. Le, and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010), no. 12, 4551-4574. https://doi.org/10.1016/j.na.2010.02.033
  16. K.-P. Ho, The fractional integral operators on Morrey spaces with variable exponent on unbounded domains, Math. Inequal. Appl. 16 (2013), no. 2, 363-373.
  17. K.-P. Ho, Atomic decompositions of weighted Hardy-Morrey spaces, Hokkaido Math. J. 42 (2013), no. 1, 131-157. https://doi.org/10.14492/hokmj/1362406643
  18. K.-P. Ho, Intrinsic square functions on Morrey and block spaces with variable exponents, Bull. Malays. Math. Sci. Soc. 40 (2017), no. 3, 995-1010. https://doi.org/10.1007/s40840-016-0330-6
  19. Y. Hu and Y. Wang, The commutators of intrinsic square functions on weighted Herz spaces, Bull. Malays. Math. Sci. Soc. 39 (2016), no. 4, 1421-1437. https://doi.org/10.1007/s40840-015-0223-0
  20. M. Izuki, Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent, Rend. Circ. Mat. Palermo (2) 59 (2010), no. 3, 461-472. https://doi.org/10.1007/s12215-010-0034-y
  21. M. Izuki, Fractional integrals on Herz-Morrey spaces with variable exponent, Hiroshima Math. J. 40 (2010), no. 3, 343-355.
  22. O. Kovacik and J. Rakosnik, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J. 41 (1991), 592-618.
  23. A. K. Lerner, Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals, Adv. Math. 226 (2011), no. 5, 3912-3926. https://doi.org/10.1016/j.aim.2010.11.009
  24. Y. Liang and D. Yang, Intrinsic square function characterizations of Musielak-Orlicz Hardy spaces, Trans. Amer. Math. Soc. 367 (2015), no. 5, 3225-3256. https://doi.org/10.1090/S0002-9947-2014-06180-1
  25. S. Lu and L. Xu, Boundedness of rough singular integral operators on the homogeneous Morrey-Herz spaces, Hokkaido Math. J. 34 (2005), no. 2, 299-314. https://doi.org/10.14492/hokmj/1285766224
  26. S. Lu, D. Yang, and G. Hu, Herz Type Spaces and Their Applications, Science Press, Beijing, 2008.
  27. Y. Lu and Y. P. Zhu, Boundedness of some sublinear operators and commutators on Morrey-Herz spaces with variable exponents, Czechoslovak Math. J. 64(139) (2014), no. 4, 969-987. https://doi.org/10.1007/s10587-014-0147-0
  28. E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665-3748. https://doi.org/10.1016/j.jfa.2012.01.004
  29. A. Nekvinda, Hardy-Littlewood maximal operator on $L^{p(x)}$ (${\mathbb{R}}$), Math. Inequal. Appl. 7 (2004), no. 2, 255-265.
  30. M. Ruzicka, Electrorheological Fluids: modeling and mathematical theory, Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin, 2000.
  31. J. Tan, Z. Liu, and J. Zhao, On some multilinear commutators in variable Lebesgue spaces, J. Math. Inequal. 11 (2017), no. 3, 715-734.
  32. A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Pure and Applied Mathematics, 123, Academic Press, Inc., Orlando, FL, 1986.
  33. H. Wang, Intrinsic square functions on the weighted Morrey spaces, J. Math. Anal. Appl. 396 (2012), no. 1, 302-314. https://doi.org/10.1016/j.jmaa.2012.06.021
  34. H. Wang, Endpoint estimates for commutators of intrinsic square functions in Morrey type spaces, Math. Inequal. Appl. 18 (2015), no. 3, 801-826.
  35. H. Wang, Commutators of singular integral operator on Herz-type Hardy spaces with variable exponent, J. Korean Math. Soc. 54 (2017), no. 3, 713-732. https://doi.org/10.4134/JKMS.j150771
  36. H. Wang and H. Liu, The intrinsic square function characterizations of weighted Hardy spaces, Illinois J. Math. 56 (2012), no. 2, 367-381.
  37. H. Wang and Z. Liu, The Herz-type Hardy spaces with variable exponent and their applications, Taiwanese J. Math. 16 (2012), no. 4, 1363-1389. https://doi.org/10.11650/twjm/1500406739
  38. M. Wilson, The intrinsic square function, Rev. Mat. Iberoam. 23 (2007), no. 3, 771-791.
  39. M. Wilson, Weighted Littlewood-Paley Theory and Exponential-Square integrability, Lecture Notes in Mathematics, 1924, Springer, Berlin, 2008.
  40. X. Yan, D. Yang, W. Yuan, and C. Zhuo, Variable weak Hardy spaces and their applications, J. Funct. Anal. 271 (2016), no. 10, 2822-2887. https://doi.org/10.1016/j.jfa.2016.07.006
  41. D. Yang, C. Zhuo, and E. Nakai, Characterizations of variable exponent Hardy spaces via Riesz transforms, Rev. Mat. Complut. 29 (2016), no. 2, 245-270. https://doi.org/10.1007/s13163-016-0188-z
  42. D. Yang, C. Zhuo, and W. Yuan, Triebel-Lizorkin type spaces with variable exponents, Banach J. Math. Anal. 9 (2015), no. 4, 146-202. https://doi.org/10.15352/bjma/09-4-9
  43. D. Yang, C. Zhuo, and W. Yuan, Besov-type spaces with variable smoothness and integrability, J. Funct. Anal. 269 (2015), no. 6, 1840-1898. https://doi.org/10.1016/j.jfa.2015.05.016
  44. C. Zhuo, Y. Sawano, and D. Yang, Hardy spaces with variable exponents on RD-spaces and applications, Dissertationes Math. (Rozprawy Mat.) 520 (2016), 74 pp.
  45. C. Zhuo, D. Yang, and Y. Liang, Intrinsic square function characterizations of Hardy spaces with variable exponents, Bull. Malays. Math. Sci. Soc. 39 (2016), no. 4, 1541-1577. https://doi.org/10.1007/s40840-015-0266-2