Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
${\rm CMO}(\mathbb {R}^{n})$; Fefferman-Stein; Riesz transform
${\rm CMO}(\mathbb {R}^{n})$; Fefferman-Stein; Riesz transform
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.
References:
[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. DOI 10.5802/aif.1915 | MR 1927078 | Zbl 1061.46025
[2] Coifman, R. R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83 (1977), 569-645. DOI 10.1090/S0002-9904-1977-14325-5 | MR 0447954 | Zbl 0358.30023
[3] Conway, J. B.: A Course in Functional Analysis. Graduate Texts in Mathematics 96. Springer, New York (1990). DOI 10.1007/978-1-4757-4383-8 | MR 1070713 | Zbl 0706.46003
[4] Dafni, G.: Local VMO and weak convergence in $h^1$. Can. Math. Bull. 45 (2002), 46-59. DOI 10.4153/CMB-2002-005-2 | MR 1884133 | Zbl 1004.42020
[5] Ding, Y., Mei, T.: Some characterizations of $VMO(\Bbb R^n)$. Anal. Theory Appl. 30 (2014), 387-398. DOI 10.4208/ata.2014.v30.n4.6 | MR 3303365 | Zbl 1340.42061
[6] Fefferman, C. L., Stein, E. M.: $H^p$ spaces of several variables. Acta Math. 129 (1972), 137-193. DOI 10.1007/BF02392215 | MR 0447953 | Zbl 0257.46078
[7] Grafakos, L.: Classical Fourier Analysis. Graduate Texts in Mathematics 249. Springer, New York (2014). DOI 10.1007/978-1-4939-1194-3 | MR 3243734 | Zbl 1304.42001
[8] John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14 (1961), 415-426. DOI 10.1002/cpa.3160140317 | MR 0131498 | Zbl 0102.04302
[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. DOI 10.1007/s10114-011-9118-7 | MR 2853801 | Zbl 1266.42061
[10] Neri, U.: Fractional integration on the space $H^1$ and its dual. Stud. Math. 53 (1975), 175-189. DOI 10.4064/sm-53-2-175-189 | MR 0388074 | Zbl 0269.44012
[11] Sarason, D.: Functions of vanishing mean oscillation. Trans. Am. Math. Soc. 207 (1975), 391-405. DOI 10.1090/S0002-9947-1975-0377518-3 | MR 0377518 | Zbl 0319.42006
[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. DOI 10.1007/BF02546524 | MR 0121579 | Zbl 0097.28501
[13] Uchiyama, A.: On the compactness of operators of Hankel type. Tohoku Math. J., II. Ser. 30 (1978), 163-171. DOI 10.2748/tmj/1178230105 | MR 0467384 | Zbl 0384.47023
Search
Partner of
