C. Fefferman [a3] discovered the remarkable fact that the space is none other than the "dual" of the Hardy space in the sense of function analysis (cf. also Functional analysis; Hardy spaces; Duality; -space). In establishing the above duality, Fefferman discovered the following "formal" inequality: if and , then
The word "formal" is used here since does not necessarily have a finite Lebesgue integral. However, one can define by setting , since it has been proved that in this case exists. Here, and , , a.s., are regular martingales. Later, A.M. Garsia [a4] proved an analogous inequality for with .
S. Ishak and J. Mogyorodi [a5] extended the validity of the Fefferman–Garsia inequality to all . In 1983, [a6], [a7], [a8], they also proved the following generalization: If and , where is a pair of conjugate Young functions (cf. also Dual functions) such that has a finite power, then
where is a constant depending only on and stands for , which exists.
|[a1]||N.L. Bassily, "Approximation theory" , Proc. Conf. Kecksemet, Hungary, 1990 , Colloq. Math. Soc. Janos Bolyai , 58 (1991) pp. 85–96|
|[a2]||N.L. Bassily, "Probability theory and applications. Essays in memory of J. Mogyorodi" Math. Appl. , 80 (1992) pp. 33–45|
|[a3]||C. Fefferman, "Characterisation of bounded mean oscillation" Amer. Math. Soc. , 77 (1971) pp. 587–588|
|[a4]||A.M. Garsia, "Martingale inequalities. Seminar notes on recent progress" , Mathematics Lecture Notes , Benjamin (1973)|
|[a5]||S. Ishak, J. Mogyorodi, "On the generalization of the Fefferman–Garsia inequality" , Proc. 3rd IFIP-WG17/1 Working Conf. , Lecture Notes in Control and Information Science , 36 , Springer (1981) pp. 85–97|
|[a6]||S. Ishak, J. Mogyorodi, "On the -spaces and the generalization of Herz's and Fefferman inequalities I" Studia Math. Hung. , 17 (1982) pp. 229–234|
|[a7]||S. Ishak, J. Mogyorodi, "On the -spaces and the generalization of Herz's and Fefferman inequalities II" Studia Math. Hung. , 18 (1983) pp. 205–210|
|[a8]||S. Ishak, J. Mogyorodi, "On the -spaces and the generalization of Herz's and Fefferman inequalities III" Studia Math. Hung. , 18 (1983) pp. 211–219|
Fefferman-Garsia inequality. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Fefferman-Garsia_inequality&oldid=22403