Fefferman-Garsia inequality

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.

It was proved in [a1], [a2] that the generalized Fefferman–Garsia inequality holds if and only if the right-hand side of the corresponding Burkholder–Davis–Gundy inequality holds.

References

[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