Note on duality of weighted multi-parameter Triebel-Lizorkin spaces

Wei Ding; Jiao Chen; Yaoming Niu

Czechoslovak Mathematical Journal (2019)

  • Volume: 69, Issue: 3, page 763-779
  • ISSN: 0011-4642

Abstract

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We study the duality theory of the weighted multi-parameter Triebel-Lizorkin spaces F ˙ p α , q ( ω ; ℝ n 1 × ℝ n 2 ) . This space has been introduced and the result ( F ˙ p α , q ( ω ; ℝ n 1 × ℝ n 2 ) ) * = CMO p - α , q ' ( ω ; ℝ n 1 × ℝ n 2 ) for 0 < p ≤ 1 has been proved in Ding, Zhu (2017). In this paper, for 1 < p < ∞ , 0 < q < ∞ we establish its dual space H ˙ p α , q ( ω ; ℝ n 1 × ℝ n 2 ) .

How to cite

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Ding, Wei, Chen, Jiao, and Niu, Yaoming. "Note on duality of weighted multi-parameter Triebel-Lizorkin spaces." Czechoslovak Mathematical Journal 69.3 (2019): 763-779. <http://eudml.org/doc/294263>.

@article{Ding2019,
abstract = {We study the duality theory of the weighted multi-parameter Triebel-Lizorkin spaces $\dot\{F\}^\{\alpha ,q\}_\{p\}(\omega ;\mathbb \{R\}^\{n_\{1\}\}\times \mathbb \{R\}^\{n_\{2\}\})$. This space has been introduced and the result \[(\dot\{F\}^\{\alpha ,q\}\_\{p\}(\omega ;\mathbb \{R\}^\{n\_\{1\}\}\times \mathbb \{R\}^\{n\_\{2\}\}))^\{\ast \}= \{\rm CMO\}^\{-\alpha ,q^\{\prime \}\}\_\{p\}(\omega ;\mathbb \{R\}^\{n\_\{1\}\}\times \mathbb \{R\}^\{n\_\{2\}\})\] for $0<p\le 1$ has been proved in Ding, Zhu (2017). In this paper, for $1<p<\infty $, $0<q<\infty $ we establish its dual space $\dot\{H\}^\{\alpha ,q\}_\{p\}(\omega ;\mathbb \{R\}^\{n_\{1\}\}\times \mathbb \{R\}^\{n_\{2\}\})$.},
author = {Ding, Wei, Chen, Jiao, Niu, Yaoming},
journal = {Czechoslovak Mathematical Journal},
keywords = {Triebel-Lizorkin space; duality; weighted multi-parameter},
language = {eng},
number = {3},
pages = {763-779},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Note on duality of weighted multi-parameter Triebel-Lizorkin spaces},
url = {http://eudml.org/doc/294263},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Ding, Wei
AU - Chen, Jiao
AU - Niu, Yaoming
TI - Note on duality of weighted multi-parameter Triebel-Lizorkin spaces
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 3
SP - 763
EP - 779
AB - We study the duality theory of the weighted multi-parameter Triebel-Lizorkin spaces $\dot{F}^{\alpha ,q}_{p}(\omega ;\mathbb {R}^{n_{1}}\times \mathbb {R}^{n_{2}})$. This space has been introduced and the result \[(\dot{F}^{\alpha ,q}_{p}(\omega ;\mathbb {R}^{n_{1}}\times \mathbb {R}^{n_{2}}))^{\ast }= {\rm CMO}^{-\alpha ,q^{\prime }}_{p}(\omega ;\mathbb {R}^{n_{1}}\times \mathbb {R}^{n_{2}})\] for $0<p\le 1$ has been proved in Ding, Zhu (2017). In this paper, for $1<p<\infty $, $0<q<\infty $ we establish its dual space $\dot{H}^{\alpha ,q}_{p}(\omega ;\mathbb {R}^{n_{1}}\times \mathbb {R}^{n_{2}})$.
LA - eng
KW - Triebel-Lizorkin space; duality; weighted multi-parameter
UR - http://eudml.org/doc/294263
ER -

References

top
  1. Bownik, M., 10.1007/s00209-007-0216-2, Math. Z. 259 (2008), 131-169. (2008) Zbl1213.42062MR2375620DOI10.1007/s00209-007-0216-2
  2. Carleson, L., A counterexample for measures bounded on H p for the bidisc, Mittag-Leffler Report. No. 7 (1974). (1974) MR1555104
  3. Chang, S.-Y. A., Fefferman, R., 10.2307/1971324, Ann. Math. (2) 112 (1980), 179-201. (1980) Zbl0451.42014MR0584078DOI10.2307/1971324
  4. Chang, S.-Y. A., Fefferman, R., 10.2307/2374150, Am. J. Math. 104 (1982), 455-468. (1982) Zbl0513.42019MR0658542DOI10.2307/2374150
  5. Chang, S.-Y. A., Fefferman, R., 10.1090/S0273-0979-1985-15291-7, Bull. Am. Math. Soc., New Ser. 12 (1985), 1-43. (1985) Zbl0557.42007MR0766959DOI10.1090/S0273-0979-1985-15291-7
  6. Cruz-Uribe, D., Martell, J. M., Pérez, C., 10.1016/j.aim.2011.08.013, Adv. Math. 229 (2012), 408-441. (2012) Zbl1236.42010MR2854179DOI10.1016/j.aim.2011.08.013
  7. Ding, W., Lu, G., 10.1090/tran/6576, Trans. Am. Math. Soc. 368 (2016), 7119-7152. (2016) Zbl1338.42025MR3471087DOI10.1090/tran/6576
  8. Ding, W., Zhu, Y., 10.1016/S0252-9602(17)30059-0, Acta Math. Sci., Ser. B, Engl. Ed. 37 (2017), 1083-1104. (2017) Zbl06873879MR3657209DOI10.1016/S0252-9602(17)30059-0
  9. Fan, X., He, J., Li, B., Yang, D., 10.1007/s11425-016-9024-2, Sci. China, Math. 60 (2017), 2093-2154. (2017) Zbl1395.42058MR3714569DOI10.1007/s11425-016-9024-2
  10. Fefferman, R., 10.2307/2374188, Am. J. Math. 103 (1981), 33-40. (1981) Zbl0475.42019MR0601461DOI10.2307/2374188
  11. Fefferman, R., 10.1073/pnas.83.4.840, Proc. Natl. Acad. Sci. USA 83 (1986), 840-843. (1986) Zbl0602.42023MR0828217DOI10.1073/pnas.83.4.840
  12. Fefferman, R., 10.2307/1971346, Ann. Math. (2) 126 (1987), 109-130. (1987) Zbl0644.42017MR0898053DOI10.2307/1971346
  13. Fefferman, R., Stein, E. M., 10.1016/S0001-8708(82)80001-7, Adv. Math. 45 (1982), 117-143. (1982) Zbl0517.42024MR0664621DOI10.1016/S0001-8708(82)80001-7
  14. Ferguson, S. H., Lacey, M. T., 10.1007/BF02392840, Acta Math. 189 (2002), 143-160. (2002) Zbl1039.47022MR1961195DOI10.1007/BF02392840
  15. Frazier, M., Jawerth, B., 10.1016/0022-1236(90)90137-A, J. Funct. Anal. 93 (1990), 34-170. (1990) Zbl0716.46031MR1070037DOI10.1016/0022-1236(90)90137-A
  16. Grafakos, L., Classical and Modern Fourier Analysis, Pearson/Prentice Hall, Upper Saddle River (2004). (2004) Zbl1148.42001MR2449250
  17. Gundy, R. F., Stein, E. M., 10.1073/pnas.76.3.1026, Proc. Natl. Acad. Sci. USA 76 (1979), 1026-1029. (1979) Zbl0405.32002MR0524328DOI10.1073/pnas.76.3.1026
  18. Han, Y., Lee, M.-Y., Lin, C.-C., Lin, Y.-C., 10.1016/j.jfa.2009.10.022, J. Funct. Anal. 258 (2010), 2834-2861. (2010) Zbl1197.42006MR2593346DOI10.1016/j.jfa.2009.10.022
  19. Han, Y., Li, J., Lu, G., 10.2422/2036-2145.2010.4.01, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 9 (2010), 645-685. (2010) Zbl1213.42073MR2789471DOI10.2422/2036-2145.2010.4.01
  20. Han, Y., Li, J., Lu, G., 10.1090/S0002-9947-2012-05638-8, Trans. Am. Math. Soc. 365 (2013), 319-360. (2013) Zbl1275.42035MR2984061DOI10.1090/S0002-9947-2012-05638-8
  21. Han, Y., Lu, G., Ruan, Z., 10.1016/j.jfa.2012.12.006, J. Funct. Anal. 264 (2013), 1238-1268. (2013) Zbl1268.42024MR3010020DOI10.1016/j.jfa.2012.12.006
  22. Han, Y., Lu, G., Ruan, Z., 10.1007/s12220-013-9421-x, J. Geom. Anal. 24 (2014), 2186-2228. (2014) Zbl1302.42024MR3261735DOI10.1007/s12220-013-9421-x
  23. Han, Y., Lin, C., Lu, G., Ruan, Z., Sawyer, E. T., 10.4171/RMI/751, Rev. Mat. Iberoam. 29 (2013), 1127-1157. (2013) Zbl1291.42018MR3148598DOI10.4171/RMI/751
  24. Journé, J.-L., 10.4171/RMI/15, Rev. Mat. Iberoam. 1 (1985), 55-91. (1985) Zbl0634.42015MR0836284DOI10.4171/RMI/15
  25. Journé, J.-L., 10.5802/aif.1125, Ann. Inst. Fourier 38 (1988), 111-132. (1988) Zbl0638.47026MR0949001DOI10.5802/aif.1125
  26. Li, B., Bownik, M., Yang, D., Yuan, W., 10.1007/s11117-011-0119-7, Positivity 16 (2012), 213-244. (2012) Zbl1260.46025MR2929088DOI10.1007/s11117-011-0119-7
  27. Li, B. D., Fan, X., Fu, Z. W., Yang, D., 10.1007/s10114-016-4741-y, Acta Math. Sin., Engl. Ser. 32 (2016), 1391-1414. (2016) Zbl1359.42010MR3557405DOI10.1007/s10114-016-4741-y
  28. Liu, J., Yang, D., Yuan, W., 10.1007/s11425-016-5157-y, Sci. China, Math. 59 (2016), 1669-1720. (2016) Zbl1352.42028MR3536030DOI10.1007/s11425-016-5157-y
  29. Liu, J., Yang, D., Yuan, W., 10.1016/j.jmaa.2017.07.003, J. Math. Anal. Appl. 456 (2017), 356-393. (2017) Zbl1373.42028MR3680972DOI10.1016/j.jmaa.2017.07.003
  30. Liu, J., Yang, D., Yuan, W., 10.1016/S0252-9602(17)30115-7, Acta Math. Sci., Ser. B, Engl. Ed. 38 (2018), 1-33. (2018) Zbl06881863MR3733274DOI10.1016/S0252-9602(17)30115-7
  31. Lu, G. Z., Zhu, Y. P., 10.1007/s10114-012-1402-7, Acta Math. Sin., Engl. Ser. 29 (2013), 39-52. (2013) Zbl1261.42030MR3001008DOI10.1007/s10114-012-1402-7
  32. Pipher, J., 10.1215/S0012-7094-86-05337-8, Duke Math. J. 53 (1986), 683-690. (1986) Zbl0645.42018MR0860666DOI10.1215/S0012-7094-86-05337-8
  33. Pisier, G., 10.1007/BF01450929, Math. Ann. 276 (1986), 105-136. (1986) Zbl0619.47016MR0863711DOI10.1007/BF01450929
  34. Ruan, Z., 10.1016/j.jmaa.2010.02.010, J. Math. Anal. Appl. 367 (2010), 625-639. (2010) Zbl1198.42015MR2607286DOI10.1016/j.jmaa.2010.02.010
  35. Triebel, H., 10.1007/978-3-0346-0416-1, Monographs in Mathematics 78, Birkhäuser, Basel (1983). (1983) Zbl0546.46027MR0781540DOI10.1007/978-3-0346-0416-1
  36. Verbitsky, I. E., 10.2140/pjm.1996.176.529, Pac. J. Math. 176 (1996), 529-556. (1996) Zbl0865.42009MR1435004DOI10.2140/pjm.1996.176.529
  37. Yuan, W., Sickel, W., Yang, D., 10.1007/978-3-642-14606-0, Lecture Notes in Mathematics 2005, Springer, Berlin (2010). (2010) Zbl1207.46002MR2683024DOI10.1007/978-3-642-14606-0

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