Fefferman-Garsia inequality

C. Fefferman [a3] discovered the remarkable fact that the space $ { \mathop{\rm BMO} } $ is none other than the "dual" of the Hardy space $ H _ {1} $ in the sense of function analysis (cf. also Functional analysis; Hardy spaces; Duality; $ { \mathop{\rm BMO} } $- space). In establishing the above duality, Fefferman discovered the following "formal" inequality: if $ X \in H _ {1} $ and $ Y \in { \mathop{\rm BMO} } $, then

$$ \left | { {\mathsf E} ( XY ) } \right | \leq c \left \| X \right \| _ {H _ {1} } \left \| Y \right \| _ { { \mathop{\rm BMO} } } . $$

The word "formal" is used here since $ XY $ does not necessarily have a finite Lebesgue integral. However, one can define $ {\mathsf E} ( XY ) $ by setting $ {\mathsf E} ( XY ) = {\lim\limits } _ {n \rightarrow \infty } {\mathsf E} ( X _ {n} Y _ {n} ) $, since it has been proved that in this case $ {\lim\limits } _ {n \rightarrow \infty } {\mathsf E} ( X _ {n} Y _ {n} ) $ exists. Here, $ X _ {n} = {\mathsf E} ( X \mid {\mathcal F} _ {n} ) $ and $ Y _ {n} = {\mathsf E} ( Y \mid {\mathcal F} _ {n} ) $, $ n \geq 0 $, $ X _ {0} = Y _ {0} = 0 $ a.s., are regular martingales. Later, A.M. Garsia [a4] proved an analogous inequality for $ H _ {p} $ with $ 1 < p \leq 1 $.

S. Ishak and J. Mogyorodi [a5] extended the validity of the Fefferman–Garsia inequality to all $ p \geq 1 $. In 1983, [a6], [a7], [a8], they also proved the following generalization: If $ X \in H _ \Phi $ and $ Y \in K _ \Psi $, where $ ( \Phi, \Psi ) $ is a pair of conjugate Young functions (cf. also Dual functions) such that $ \Phi $ has a finite power, then

$$ \left | { {\mathsf E} ( XY ) } \right | \leq C _ \Phi \left \| X \right \| _ {H _ \Phi } \left \| Y \right \| _ {K _ \Psi } , $$

where $ C _ \Phi > 0 $ is a constant depending only on $ \Phi $ and $ {\mathsf E} ( XY ) $ stands for $ {\lim\limits } _ {n \rightarrow \infty } {\mathsf E} ( X _ {n} Y _ {n} ) $, 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 $\mathcal{P}\Phi$-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 $\mathcal{P}\Phi$-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 $\mathcal{P}\Phi$-spaces and the generalization of Herz's and Fefferman inequalities III" Studia Math. Hung. , 18 (1983) pp. 211–219