View source for Fefferman-Garsia inequality

<!--
f1100401.png
$#A+1 = 26 n = 3
$#C+1 = 26 : ~/encyclopedia/old_files/data/F110/F.1100040 Fefferman\ANDGarsia inequality
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

C. Fefferman [[#References|[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|Functional analysis]]; [[Hardy spaces|Hardy spaces]]; [[Duality|Duality]]; [[BMO-space| $  { \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|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 [[#References|[a4]]] proved an analogous inequality for  $  H _ {p} $
with  $  1 < p \leq  1 $.

S. Ishak and J. Mogyorodi [[#References|[a5]]] extended the validity of the Fefferman–Garsia inequality to all  $  p \geq  1 $. 
In 1983, [[#References|[a6]]], [[#References|[a7]]], [[#References|[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|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 [[#References|[a1]]], [[#References|[a2]]] that the generalized Fefferman–Garsia inequality holds if and only if the right-hand side of the corresponding [[Burkholder–Davis–Gundy inequality|Burkholder–Davis–Gundy inequality]] holds.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.L. Bassily,   "Approximation theory" , ''Proc. Conf. Kecksemet, Hungary, 1990'' , ''Colloq. Math. Soc. Janos Bolyai'' , '''58'''  (1991)  pp. 85–96</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.L. Bassily,   "Probability theory and applications. Essays in memory of J. Mogyorodi"  ''Math. Appl.'' , '''80'''  (1992)  pp. 33–45</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C. Fefferman,   "Characterisation of bounded mean oscillation"  ''Amer. Math. Soc.'' , '''77'''  (1971)  pp. 587–588</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.M. Garsia,   "Martingale inequalities. Seminar notes on recent progress" , ''Mathematics Lecture Notes'' , Benjamin  (1973)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Ishak,   J. Mogyorodi,   "On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110040/f11004027.png" />-spaces and the generalization of Herz's and Fefferman inequalities I"  ''Studia Math. Hung.'' , '''17'''  (1982)  pp. 229–234</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S. Ishak,   J. Mogyorodi,   "On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110040/f11004028.png" />-spaces and the generalization of Herz's and Fefferman inequalities II"  ''Studia Math. Hung.'' , '''18'''  (1983)  pp. 205–210</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  S. Ishak,   J. Mogyorodi,   "On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110040/f11004029.png" />-spaces and the generalization of Herz's and Fefferman inequalities III"  ''Studia Math. Hung.'' , '''18'''  (1983)  pp. 211–219</TD></TR></table>