XI: 16, 303-323, LNM 581 (1977)
BERNARD, Alain;
MAISONNEUVE, Bernard
Décomposition atomique de martingales de la classe $H^1$ (
Martingale theory)
Atomic decompositions have been used with great success in the analytical theory of Hardy spaces, in particular by Coifman (
Studia Math. 51, 1974). An atomic decomposition of a Banach space consists in finding simple elements (called atoms) in its unit ball, such that every element is a linear combination of atoms $\sum_n \lambda_n a_n$ with $\sum_n \|\lambda_n\|<\infty$, the infimum of this sum defining the norm or an equivalent one. Here an atomic decomposition is given for $H^1$ spaces of martingales in continuous time (defined by their maximal function). Atoms are of two kinds: the first kind consists of martingales bounded uniformly by a constant $c$ and supported by an interval $[T,\infty[$ such that $P\{T<\infty\}\le 1/c$. These atoms do not generate the whole space $H^1$ in general, though they do in a few interesting cases (if all martingales are continuous, or in the discrete dyadic case). To generate the whole space it is sufficient to add martingales of integrable variation (those whose total variation has an $L^1$ norm smaller than $1$ constitute the second kind of atoms). This approach leads to a proof of the $H^1$-$BMO$ duality and the Davis inequality
Comment: See also
1117Keywords: Atomic decompositions,
$H^1$ space,
$BMO$Nature: Original
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XI: 17, 324-326, LNM 581 (1977)
BERNARD, Alain
Complément à l'exposé précédent (
Martingale theory)
This paper is a sequel to
1116, which it completes in two ways: it makes it independent of a previous proof of the Fefferman inequality, which is now proved directly, and it exhibits atoms of the first kind appropriate to the quadratic norm of $H^1$
Keywords: Atomic decompositions,
$H^1$ space,
$BMO$Nature: Original Retrieve article from Numdam
