# Fefferman-Garsia inequality

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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 -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
How to Cite This Entry:
Fefferman-Garsia inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fefferman-Garsia_inequality&oldid=46910
This article was adapted from an original article by N.L. Bassily (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article