Another characterization of BMO

Coifman, R. R.; Rochberg, R.

doi:10.1090/S0002-9939-1980-0565349-8

Another characterization of BMO
HTML articles powered by AMS MathViewer

by R. R. Coifman and R. Rochberg PDF

Proc. Amer. Math. Soc. 79 (1980), 249-254 Request permission

Abstract:

The following characterization of functions of bounded mean oscillation (BMO) is proved. f is in BMO if and only if \[ f = \alpha \log {g^ \ast } - \beta \log {h^ \ast } + b\] where ${g^ \ast },({h^ \ast })$ is the Hardy-Littlewood maximal function of g, (h), respectively, b is bounded and ${\left \| f \right \|_{{\text {BMO}}}} \leqslant c(\alpha + \beta + {\left \| b \right \|_\infty })$.

References

Lennart Carleson, Two remarks on $H^{1}$ and BMO, Advances in Math. 22 (1976), no. 3, 269–277. MR 477058, DOI 10.1016/0001-8708(76)90095-5
R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
John B. Garnett and Peter W. Jones, The distance in BMO to $L^{\infty }$, Ann. of Math. (2) 108 (1978), no. 2, 373–393. MR 506992, DOI 10.2307/1971171
F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. MR 131498, DOI 10.1002/cpa.3160140317
Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, DOI 10.1090/S0002-9947-1972-0293384-6
E. M. Stein, Note on the class $L$ $\textrm {log}$ $L$, Studia Math. 32 (1969), 305–310. MR 247534, DOI 10.4064/sm-32-3-305-310
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972