Article
| Title: | A simple proof of Fefferman-Stein type characterization of ${\rm CMO}(\mathbb {R}^{n})$ space (English) |
| Author: | Guo, Qingdong |
| Author: | Linli, Zeqiang |
| Author: | Hu, Kang |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 2 |
| Year: | 2025 |
| Pages: | 599-610 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | We give a simple proof of Fefferman-Stein type characterization of the space ${\rm CMO}(\mathbb {R}^{n})$, that is, $f\in {\rm CMO} (\mathbb {R}^{n})$ if and only if $$ f=\phi +\sum _{j=1}^{n}R_{j}\varphi _{j}, $$ where $\phi ,\varphi _{j}\in {C_{0}(\mathbb {R}^{n})}$ and $R_{j}$, $j=1,2,\ldots ,n$, are the Riesz transforms. Notice that this result was established by G. Bourdaud (2002), but his proof depends on the Fefferman-Stein type decomposition of the space ${\rm VMO}(\mathbb {R}^{n})$ obtained by D. Sarason (1975). We will provide a direct method to prove this conclusion. (English) |
| Keyword: | ${\rm CMO}(\mathbb {R}^{n})$ |
| Keyword: | Fefferman-Stein |
| Keyword: | Riesz transform |
| MSC: | 42B20 |
| MSC: | 42B35 |
| MSC: | 42B99 |
| DOI: | 10.21136/CMJ.2025.0331-24 |
| . | |
| Date available: | 2025-05-20T11:48:13Z |
| Last updated: | 2025-05-26 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152960 |
| . | |
| Reference: | [1] Bourdaud, G.: Remarques sur certains sous-espaces de BMO$(\Bbb R^n)$ et de bmo$(\Bbb R^n)$.Ann. Inst. Fourier 52 (2002), 1187-1218 French. Zbl 1061.46025, MR 1927078, 10.5802/aif.1915 |
| Reference: | [2] Coifman, R. R., Weiss, G.: Extensions of Hardy spaces and their use in analysis.Bull. Am. Math. Soc. 83 (1977), 569-645. Zbl 0358.30023, MR 0447954, 10.1090/S0002-9904-1977-14325-5 |
| Reference: | [3] Conway, J. B.: A Course in Functional Analysis.Graduate Texts in Mathematics 96. Springer, New York (1990). Zbl 0706.46003, MR 1070713, 10.1007/978-1-4757-4383-8 |
| Reference: | [4] Dafni, G.: Local VMO and weak convergence in $h^1$.Can. Math. Bull. 45 (2002), 46-59. Zbl 1004.42020, MR 1884133, 10.4153/CMB-2002-005-2 |
| Reference: | [5] Ding, Y., Mei, T.: Some characterizations of $VMO(\Bbb R^n)$.Anal. Theory Appl. 30 (2014), 387-398. Zbl 1340.42061, MR 3303365, 10.4208/ata.2014.v30.n4.6 |
| Reference: | [6] Fefferman, C. L., Stein, E. M.: $H^p$ spaces of several variables.Acta Math. 129 (1972), 137-193. Zbl 0257.46078, MR 0447953, 10.1007/BF02392215 |
| Reference: | [7] Grafakos, L.: Classical Fourier Analysis.Graduate Texts in Mathematics 249. Springer, New York (2014). Zbl 1304.42001, MR 3243734, 10.1007/978-1-4939-1194-3 |
| Reference: | [8] John, F., Nirenberg, L.: On functions of bounded mean oscillation.Commun. Pure Appl. Math. 14 (1961), 415-426. Zbl 0102.04302, MR 0131498, 10.1002/cpa.3160140317 |
| Reference: | [9] Liu, L. G., Yang, D. C., Yang, D. Y.: Atomic Hardy-type spaces between $H^1$ and $L^1$ on metric spaces with non-doubling measures.Acta Math. Sin., Engl. Ser. 27 (2011), 2445-2468. Zbl 1266.42061, MR 2853801, 10.1007/s10114-011-9118-7 |
| Reference: | [10] Neri, U.: Fractional integration on the space $H^1$ and its dual.Stud. Math. 53 (1975), 175-189. Zbl 0269.44012, MR 0388074, 10.4064/sm-53-2-175-189 |
| Reference: | [11] Sarason, D.: Functions of vanishing mean oscillation.Trans. Am. Math. Soc. 207 (1975), 391-405. Zbl 0319.42006, MR 0377518, 10.1090/S0002-9947-1975-0377518-3 |
| Reference: | [12] Stein, E. M., Weiss, G.: On the theory of harmonic functions of several variables. I. The theory of $H^p$-spaces.Acta Math. 103 (1960), 25-62. Zbl 0097.28501, MR 0121579, 10.1007/BF02546524 |
| Reference: | [13] Uchiyama, A.: On the compactness of operators of Hankel type.Tohoku Math. J., II. Ser. 30 (1978), 163-171. Zbl 0384.47023, MR 0467384, 10.2748/tmj/1178230105 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
