Complex interpolation of function spaces with general weights

Douadi Drihem

Commentationes Mathematicae Universitatis Carolinae (2023)

  • Volume: 64, Issue: 3, page 289-320
  • ISSN: 0010-2628

Abstract

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We present the complex interpolation of Besov and Triebel–Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel–Lizorkin spaces. As a corollary of our results, we obtain the complex interpolation between the weighted Triebel–Lizorkin spaces F ˙ p 0 , q 0 s 0 ( ω 0 ) and F ˙ ∞ , q 1 s 1 ( ω 1 ) with suitable assumptions on the parameters s 0 , s 1 , p 0 , q 0 and q 1 , and the pair of weights ( ω 0 , ω 1 ) .

How to cite

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Drihem, Douadi. "Complex interpolation of function spaces with general weights." Commentationes Mathematicae Universitatis Carolinae 64.3 (2023): 289-320. <http://eudml.org/doc/299210>.

@article{Drihem2023,
abstract = {We present the complex interpolation of Besov and Triebel–Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel–Lizorkin spaces. As a corollary of our results, we obtain the complex interpolation between the weighted Triebel–Lizorkin spaces $\dot\{F\}_\{p_\{0\},q_\{0\}\}^\{s_\{0\}\} (\omega _\{0\})$ and $\dot\{F\}_\{\infty ,q_\{1\}\}^\{s_\{1\}\}(\omega _\{1\}) $ with suitable assumptions on the parameters $ s_\{0\},s_\{1\},p_\{0\}, q_\{0\}$ and $q_\{1\}$, and the pair of weights $(\omega _\{0\},\omega _\{1\})$.},
author = {Drihem, Douadi},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Besov space; Triebel--Lizorkin space; complex interpolation; Muckenhoupt class},
language = {eng},
number = {3},
pages = {289-320},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Complex interpolation of function spaces with general weights},
url = {http://eudml.org/doc/299210},
volume = {64},
year = {2023},
}

TY - JOUR
AU - Drihem, Douadi
TI - Complex interpolation of function spaces with general weights
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2023
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 64
IS - 3
SP - 289
EP - 320
AB - We present the complex interpolation of Besov and Triebel–Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel–Lizorkin spaces. As a corollary of our results, we obtain the complex interpolation between the weighted Triebel–Lizorkin spaces $\dot{F}_{p_{0},q_{0}}^{s_{0}} (\omega _{0})$ and $\dot{F}_{\infty ,q_{1}}^{s_{1}}(\omega _{1}) $ with suitable assumptions on the parameters $ s_{0},s_{1},p_{0}, q_{0}$ and $q_{1}$, and the pair of weights $(\omega _{0},\omega _{1})$.
LA - eng
KW - Besov space; Triebel--Lizorkin space; complex interpolation; Muckenhoupt class
UR - http://eudml.org/doc/299210
ER -

References

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  1. Andersen K. F., John R. T., Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math. 69 (1980/81), no. 1, 19–31. MR0604351
  2. Bergh J., Löfström J., 10.1007/978-3-642-66451-9, Grundlehren der Mathematischen Wissenschaften, 223, Springer, Berlin, 1976. MR0482275DOI10.1007/978-3-642-66451-9
  3. Besov O. V., Equivalent normings of spaces of functions of variable smoothness, Funkts. Prostran., Priblizh., Differ. Uravn., Tr. Mat. Inst. Steklova 243 (2003), 87–95 (Russian); translation in Proc. Steklov Inst. Math. 243 (2003), no. 4, 80–88. MR2049464
  4. Besov O. V., Interpolation, embedding, and extension of spaces of functions of variable smoothness, Issled. po Teor. Funkts. i Differ. Uravn., Tr. Mat. Inst. Steklova 248 (2005), 52–63 (Russian); translation in Proc. Steklov Inst. Math. 248 (2005), no. 1, 47–58. MR2165915
  5. Bownik M., 10.1007/BF02922089, J. Geom. Anal. 17 (2007), no. 3, 387–424. MR2358763DOI10.1007/BF02922089
  6. Bownik M., 10.1007/s00209-007-0216-2, Math. Z. 259 (2008), no. 1, 131–169. MR2375620DOI10.1007/s00209-007-0216-2
  7. Bui H. Q., Weighted Besov and Triebel spaces: interpolation by the real method, Hiroshima Math. J. 12 (1982), no. 3, 581–605. MR0676560
  8. Calderón A. P., 10.4064/sm-24-2-113-190, Studia Math. 24 (1964), 113–190. MR0167830DOI10.4064/sm-24-2-113-190
  9. Cobos F., Fernandez D. L., 10.1007/BFb0078872, Function Spaces and Applications, Lund, 1986, Lecture Notes in Math., 1302, Springer, Berlin, 1988, pages 158–170. MR0942266DOI10.1007/BFb0078872
  10. Domínguez O., Tikhonov S., Function spaces of logarithmic smoothness: embeddings and characterizations, Mem. Amer. Math. Soc. 282 (2023), no. 1393, vii+166 pages. MR4539365
  11. Drihem D., 10.4208/jms.v56n1.23.02, J. Math. Study. 56 (2023), no. 1, 18–92. MR4560234DOI10.4208/jms.v56n1.23.02
  12. Drihem D., 10.1007/s43036-022-00230-0, Adv. Oper. Theory 8 (2023) no. 1, Paper No. 5, 69 pages. MR4510628DOI10.1007/s43036-022-00230-0
  13. Drihem D., Duality of Triebel–Lizorkin spaces of general weights, available at ArXiv: 2402.04635v1 [math.FA] (2024), 22 pages. MR4510628
  14. Edmunds D., Triebel H., 10.1016/S0764-4442(98)80177-8, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 11, 1269–1274. MR1649135DOI10.1016/S0764-4442(98)80177-8
  15. Edmunds D., Triebel H., Eigenfrequencies of isotropic fractal drums, The Maz'ya anniversary collection, 2, Rostock, 1998, Birkhäuser Verlag, Basel, Oper. Theory Adv. Appl. 110 (1999), pages 81–102. MR1747890
  16. Farkas W., Leopold H.-G., 10.1007/s10231-004-0110-z, Ann. Mat. Pura Appl. (4) 185 (2006), no. 1, 1–62. MR2179581DOI10.1007/s10231-004-0110-z
  17. Fefferman C., Stein E. M., 10.2307/2373450, Amer. J. Math. 93 (1971), 107–115. MR0284802DOI10.2307/2373450
  18. Frazier M., Jawerth B., 10.1512/iumj.1985.34.34041, Indiana Univ. Math. J. 34 (1985), no. 4, 777–799. MR0808825DOI10.1512/iumj.1985.34.34041
  19. Frazier M., Jawerth B., 10.1016/0022-1236(90)90137-A, J. Funct. Anal. 93 (1990), no. 1, 34–170. MR1070037DOI10.1016/0022-1236(90)90137-A
  20. Frazier M., Jawerth B., Weiss G., 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., American Mathematical Society, Providence, 1991. MR1107300
  21. García-Cuerva J., Rubio de Francia J. L., Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies, 116, Notas de Matemática, 104, North-Holland Publishing Co., Amsterdam, 1985. MR0807149
  22. Goldman M. L., Description of traces for certain function spaces, Trudy Mat. Inst. Steklov. 150 (1979), 99–127, 322 (Russian). MR0544006
  23. Goldman M. L., The method of coverings for describing general spaces of Besov type, Trudy Mat. Inst. Steklov. 156 (1980), 47–81, 262 (Russian). MR0622227
  24. Grafakos L., Classical Fourier Analysis, Graduate Texts in Mathematics, 249, Springer, New York, 2014. Zbl1336.00075MR3243734
  25. Kaljabin G. A., Descriptions of functions from classes of Besov–Lizorkin–Triebel type, Trudy Mat. Inst. Steklov. 156 (1980), 82–109, 262 (Russian). MR0622228
  26. Kaljabin G. A., Lizorkin P. I., 10.1002/mana.19871330102, Math. Nachr. 133 (1987), 7–32. MR0912417DOI10.1002/mana.19871330102
  27. Kalton N., Mayboroda S., Mitrea M., 10.1090/conm/445/08598, Interpolation Theory and Applications, Amer. Math. Soc., Providence, Contemp. Math. 445 (2007), 121–177. MR2381891DOI10.1090/conm/445/08598
  28. Kempka H., Vybíral J., 10.1007/s00041-012-9224-7, J. Fourier Anal. Appl. 18 (2012), no. 4, 852–891. MR2984372DOI10.1007/s00041-012-9224-7
  29. Kokilashvili V. M., Maximum inequalities and multipliers in weighted Lizorkin–Triebel spaces, Dokl. Akad. Nauk SSSR 239 (1978), no. 1, 42–45 (Russian). MR0470592
  30. Moura S. D., Function Spaces of Generalised Smoothness, Dissertationes Math. (Rozprawy Mat.), 398, 2001, 88 pages. MR1876765
  31. Muckenhoupt B., 10.1090/S0002-9947-1972-0293384-6, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR0293384DOI10.1090/S0002-9947-1972-0293384-6
  32. Rychkov V. S., 10.1002/1522-2616(200104)224:1<145::AID-MANA145>3.0.CO;2-2, Math. Nachr. 224 (2001), no. 1, 145–180. MR1821243DOI10.1002/1522-2616(200104)224:1<145::AID-MANA145>3.0.CO;2-2
  33. Schott T., 10.1002/mana.19981960110, Math. Nachr. 196 (1998), 231–250. MR1658014DOI10.1002/mana.19981960110
  34. Sickel W., Skrzypczak L., Vybíral J., Complex interpolation of weighted Besov and Lizorkin–Triebel spaces, Acta. Math. Sci. (Engl. Ser.) 30 (2014), no. 8, 1297–1323. MR3229143
  35. Tang C., A note on weighted Besov-type and Triebel–Lizorkin-type spaces, J. Funct. Spaces Appl. 2013 (2013), Article ID 865835, 12 pages. MR3036711
  36. Triebel H., Interpolation Theory, Function Spaces, Differential Operators, Deutscher Verlag der Wissenschaften, Berlin, 1978. Zbl0830.46028MR0500580
  37. Triebel H., Theory of Function Spaces, Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. Zbl1104.46001MR0781540
  38. Triebel H., Theory of Function Spaces. II., Monographs in Mathematics, 84, Birkhäuser Verlag, Basel, 1992. MR1163193
  39. Tyulenev A. I, 10.1134/S0081543814010209, Tr. Mat. Inst. Steklova 284 (2014), 288–303; translation in Proc. Steklov Inst. Math. 284 (2014), no. 1, 280–295. MR3479981DOI10.1134/S0081543814010209
  40. Tyulenev A. I., 10.1070/SM2015v206n06ABEH004481, Mat. Sb. 206 (2015), no. 6, 85–128; translation in Sb. Math. 206 (2015), no. 5–6, 849–891. MR3438581DOI10.1070/SM2015v206n06ABEH004481
  41. Tyulenev A. I., Besov-type spaces of variable smoothness on rough domains, Nonlinear Anal. 145 (2016), 176–198. MR3547680
  42. Tyulenev A. I., 10.1016/j.jmaa.2017.02.006, J. Math. Anal. Appl. 451 (2017), no. 1, 371–392. MR3619242DOI10.1016/j.jmaa.2017.02.006
  43. Wojciechowska A., Multidimensional Wavelet Bases in Besov and Lizorkin–Triebel Spaces, PhD. Thesis, Adam Mickiewicz University Poznań, Poznań, 2012. 
  44. Yang D., Yuan W., Zhuo C., 10.1142/S0219530513500218, Anal. Appl. (Singap). 11 (2013), no. 5, 1350021, 45 pages. MR3104106DOI10.1142/S0219530513500218

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