Article
| Title: | Complex interpolation of function spaces with general weights (English) |
| Author: | Drihem, Douadi |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 64 |
| Issue: | 3 |
| Year: | 2023 |
| Pages: | 289-320 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | 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 $(\omega _{0},\omega _{1})$. (English) |
| Keyword: | Besov space |
| Keyword: | Triebel--Lizorkin space |
| Keyword: | complex interpolation |
| Keyword: | Muckenhoupt class |
| MSC: | 26B35 |
| MSC: | 42B25 |
| MSC: | 42B35 |
| MSC: | 46E35 |
| idZBL: | Zbl 07830510 |
| idMR: | MR4717503 |
| DOI: | 10.14712/1213-7243.2024.003 |
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| Date available: | 2024-03-18T10:40:59Z |
| Last updated: | 2024-08-02 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152300 |
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| Reference: | [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. MR 0604351 |
| Reference: | [2] Bergh J., Löfström J.: Interpolation Spaces. An Introduction.Grundlehren der Mathematischen Wissenschaften, 223, Springer, Berlin, 1976. MR 0482275, 10.1007/978-3-642-66451-9 |
| Reference: | [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. MR 2049464 |
| Reference: | [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. MR 2165915 |
| Reference: | [5] Bownik M.: Anisotropic Triebel–Lizorkin spaces with doubling measures.J. Geom. Anal. 17 (2007), no. 3, 387–424. MR 2358763, 10.1007/BF02922089 |
| Reference: | [6] Bownik M.: Duality and interpolation of anisotropic Triebel–Lizorkin spaces.Math. Z. 259 (2008), no. 1, 131–169. MR 2375620, 10.1007/s00209-007-0216-2 |
| Reference: | [7] Bui H. Q.: Weighted Besov and Triebel spaces: interpolation by the real method.Hiroshima Math. J. 12 (1982), no. 3, 581–605. MR 0676560 |
| Reference: | [8] Calderón A. P.: Intermediate spaces and interpolation, the complex method.Studia Math. 24 (1964), 113–190. MR 0167830, 10.4064/sm-24-2-113-190 |
| Reference: | [9] Cobos F., Fernandez D. L.: Hardy–Sobolev spaces and Besov spaces with a function parameter.Function Spaces and Applications, Lund, 1986, Lecture Notes in Math., 1302, Springer, Berlin, 1988, pages 158–170. MR 0942266, 10.1007/BFb0078872 |
| Reference: | [10] Domínguez O., Tikhonov S.: Function spaces of logarithmic smoothness: embeddings and characterizations.Mem. Amer. Math. Soc. 282 (2023), no. 1393, vii+166 pages. MR 4539365 |
| Reference: | [11] Drihem D.: Besov spaces with general weights.J. Math. Study. 56 (2023), no. 1, 18–92. MR 4560234, 10.4208/jms.v56n1.23.02 |
| Reference: | [12] Drihem D.: Triebel–Lizorkin spaces with general weights.Adv. Oper. Theory 8 (2023) no. 1, Paper No. 5, 69 pages. MR 4510628, 10.1007/s43036-022-00230-0 |
| Reference: | [13] Drihem D.: Duality of Triebel–Lizorkin spaces of general weights.available at ArXiv: 2402.04635v1 [math.FA] (2024), 22 pages. MR 4510628 |
| Reference: | [14] Edmunds D., Triebel H.: Spectral theory for isotropic fractal drums.C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 11, 1269–1274. MR 1649135, 10.1016/S0764-4442(98)80177-8 |
| Reference: | [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. MR 1747890 |
| Reference: | [16] Farkas W., Leopold H.-G.: Characterisations of function spaces of generalised smoothness.Ann. Mat. Pura Appl. (4) 185 (2006), no. 1, 1–62. MR 2179581, 10.1007/s10231-004-0110-z |
| Reference: | [17] Fefferman C., Stein E. M.: Some maximal inequalities.Amer. J. Math. 93 (1971), 107–115. MR 0284802, 10.2307/2373450 |
| Reference: | [18] Frazier M., Jawerth B.: Decomposition of Besov spaces.Indiana Univ. Math. J. 34 (1985), no. 4, 777–799. MR 0808825, 10.1512/iumj.1985.34.34041 |
| Reference: | [19] Frazier M., Jawerth B.: A discrete transform and decomposition of distribution spaces.J. Funct. Anal. 93 (1990), no. 1, 34–170. MR 1070037, 10.1016/0022-1236(90)90137-A |
| Reference: | [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. MR 1107300 |
| Reference: | [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. MR 0807149 |
| Reference: | [22] Goldman M. L.: Description of traces for certain function spaces.Trudy Mat. Inst. Steklov. 150 (1979), 99–127, 322 (Russian). MR 0544006 |
| Reference: | [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). MR 0622227 |
| Reference: | [24] Grafakos L.: Classical Fourier Analysis.Graduate Texts in Mathematics, 249, Springer, New York, 2014. Zbl 1336.00075, MR 3243734 |
| Reference: | [25] Kaljabin G. A.: Descriptions of functions from classes of Besov–Lizorkin–Triebel type.Trudy Mat. Inst. Steklov. 156 (1980), 82–109, 262 (Russian). MR 0622228 |
| Reference: | [26] Kaljabin G. A., Lizorkin P. I.: Spaces of functions of generalized smoothness.Math. Nachr. 133 (1987), 7–32. MR 0912417, 10.1002/mana.19871330102 |
| Reference: | [27] Kalton N., Mayboroda S., Mitrea M.: Interpolation of Hardy–Sobolev–Besov–Triebel–Lizorkin spaces and applications to problems in partial differential equations.Interpolation Theory and Applications, Amer. Math. Soc., Providence, Contemp. Math. 445 (2007), 121–177. MR 2381891, 10.1090/conm/445/08598 |
| Reference: | [28] Kempka H., Vybíral J.: Spaces of variable smoothness and integrability: characterizations by local means and ball means of differences.J. Fourier Anal. Appl. 18 (2012), no. 4, 852–891. MR 2984372, 10.1007/s00041-012-9224-7 |
| Reference: | [29] Kokilashvili V. M.: Maximum inequalities and multipliers in weighted Lizorkin–Triebel spaces.Dokl. Akad. Nauk SSSR 239 (1978), no. 1, 42–45 (Russian). MR 0470592 |
| Reference: | [30] Moura S. D.: Function Spaces of Generalised Smoothness.Dissertationes Math. (Rozprawy Mat.), 398, 2001, 88 pages. MR 1876765 |
| Reference: | [31] Muckenhoupt B.: Weighted norm inequalities for the Hardy maximal function.Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 0293384, 10.1090/S0002-9947-1972-0293384-6 |
| Reference: | [32] Rychkov V. S.: Littlewood–Paley theory and function spaces with $A_{p}^{\mathrm{loc}}$-weights.Math. Nachr. 224 (2001), no. 1, 145–180. MR 1821243, 10.1002/1522-2616(200104)224:1<145::AID-MANA145>3.0.CO;2-2 |
| Reference: | [33] Schott T.: Function spaces with exponential weights. II..Math. Nachr. 196 (1998), 231–250. MR 1658014, 10.1002/mana.19981960110 |
| Reference: | [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. MR 3229143 |
| Reference: | [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. MR 3036711 |
| Reference: | [36] Triebel H.: Interpolation Theory, Function Spaces, Differential Operators.Deutscher Verlag der Wissenschaften, Berlin, 1978. Zbl 0830.46028, MR 0500580 |
| Reference: | [37] Triebel H.: Theory of Function Spaces.Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. Zbl 1104.46001, MR 0781540 |
| Reference: | [38] Triebel H.: Theory of Function Spaces. II..Monographs in Mathematics, 84, Birkhäuser Verlag, Basel, 1992. MR 1163193 |
| Reference: | [39] Tyulenev A. I: Description of traces of functions in the Sobolev space with a Muckenhoupt weight.Tr. Mat. Inst. Steklova 284 (2014), 288–303; translation in Proc. Steklov Inst. Math. 284 (2014), no. 1, 280–295. MR 3479981, 10.1134/S0081543814010209 |
| Reference: | [40] Tyulenev A. I.: Some new function spaces of variable smoothness.Mat. Sb. 206 (2015), no. 6, 85–128; translation in Sb. Math. 206 (2015), no. 5–6, 849–891. MR 3438581, 10.1070/SM2015v206n06ABEH004481 |
| Reference: | [41] Tyulenev A. I.: Besov-type spaces of variable smoothness on rough domains.Nonlinear Anal. 145 (2016), 176–198. MR 3547680 |
| Reference: | [42] Tyulenev A. I.: On various approaches to Besov-type spaces of variable smoothness.J. Math. Anal. Appl. 451 (2017), no. 1, 371–392. MR 3619242, 10.1016/j.jmaa.2017.02.006 |
| Reference: | [43] Wojciechowska A.: Multidimensional Wavelet Bases in Besov and Lizorkin–Triebel Spaces.PhD. Thesis, Adam Mickiewicz University Poznań, Poznań, 2012. |
| Reference: | [44] Yang D., Yuan W., Zhuo C.: Complex interpolation on Besov-type and Triebel–Lizorkin-type spaces.Anal. Appl. (Singap). 11 (2013), no. 5, 1350021, 45 pages. MR 3104106, 10.1142/S0219530513500218 |
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