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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 1117

Keywords: 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

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XIII: 30, 360-370, LNM 721 (1979)

**JEULIN, Thierry**; **YOR, Marc**

Sur l'expression de la dualité entre $H^1$ et $BMO$ (Martingale theory)

The problem is to find pairs of martingales $X,Y$ belonging to $H^1$ and $BMO$ such that the duality functional can be expressed as $E[X_{\infty}Y_{\infty}]$

Comment: On the same topic see 1518

Keywords: $BMO$, $H^1$ space, Hardy spaces

Nature: Original

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XIII: 34, 400-406, LNM 721 (1979)

**YOR, Marc**

Quelques épilogues (General theory of processes, Martingale theory, Stochastic calculus)

This is an account of current folklore, i.e., small remarks which settle natural questions, possibly published elsewhere but difficult to locate. Among the quotable results, one may mention that if a sequence of martingales converges in $L^1$, one can stop them at arbitrary large stopping times so that the stopped processes converge in $H^1$

Keywords: Local time, Enlargement of filtrations, $H^1$ space, Hardy spaces, $BMO$

Nature: Original

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XV: 18, 259-277, LNM 850 (1981)

**BRU, Bernard**; **HEINICH, Henri**; **LOOTGIETER, Jean-Claude**

Autour de la dualité $(H^1,BMO)$ (Martingale theory)

This is a sequel to 1330. Given two martingales $(X,Y)$ in $H^1$ and $BMO$, it is investigated whether their duality functional can be safely estimated as $E[X_{\infty}Y_{\infty}]$. The simple result is that if $X_{\infty}Y_{\infty}$ belongs to $L^1$, or merely is bounded upwards by an element of $L^1$, then the answer is positive. The second (and longer) part of the paper searches for subspaces of $H^1$ and $BMO$ such that the property would hold between their elements, and here the results are fragmentary (a question of 1330 is answered). An appendix discusses a result of Talagrand

Keywords: $BMO$, $H^1$ space, Hardy spaces

Nature: Original

Retrieve article from Numdam

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 (

Comment: See also 1117

Keywords: Atomic decompositions, $H^1$ space, $BMO$

Nature: Original

Retrieve article from Numdam

XI: 17, 324-326, LNM 581 (1977)

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

XIII: 30, 360-370, LNM 721 (1979)

Sur l'expression de la dualité entre $H^1$ et $BMO$ (Martingale theory)

The problem is to find pairs of martingales $X,Y$ belonging to $H^1$ and $BMO$ such that the duality functional can be expressed as $E[X_{\infty}Y_{\infty}]$

Comment: On the same topic see 1518

Keywords: $BMO$, $H^1$ space, Hardy spaces

Nature: Original

Retrieve article from Numdam

XIII: 34, 400-406, LNM 721 (1979)

Quelques épilogues (General theory of processes, Martingale theory, Stochastic calculus)

This is an account of current folklore, i.e., small remarks which settle natural questions, possibly published elsewhere but difficult to locate. Among the quotable results, one may mention that if a sequence of martingales converges in $L^1$, one can stop them at arbitrary large stopping times so that the stopped processes converge in $H^1$

Keywords: Local time, Enlargement of filtrations, $H^1$ space, Hardy spaces, $BMO$

Nature: Original

Retrieve article from Numdam

XV: 18, 259-277, LNM 850 (1981)

Autour de la dualité $(H^1,BMO)$ (Martingale theory)

This is a sequel to 1330. Given two martingales $(X,Y)$ in $H^1$ and $BMO$, it is investigated whether their duality functional can be safely estimated as $E[X_{\infty}Y_{\infty}]$. The simple result is that if $X_{\infty}Y_{\infty}$ belongs to $L^1$, or merely is bounded upwards by an element of $L^1$, then the answer is positive. The second (and longer) part of the paper searches for subspaces of $H^1$ and $BMO$ such that the property would hold between their elements, and here the results are fragmentary (a question of 1330 is answered). An appendix discusses a result of Talagrand

Keywords: $BMO$, $H^1$ space, Hardy spaces

Nature: Original

Retrieve article from Numdam