VII: 14, 136-145, LNM 321 (1973)
MEYER, Paul-André
Le dual de $H^1$ est $BMO$ (cas continu) (
Martingale theory)
The basic results of Fefferman and Fefferman-Stein on functions of bounded mean oscillation in $
R$ and $
R^n$ and the duality between $BMO$ and $H^1$ were almost immediately translated into discrete martingale theory by Herz and Garsia. The next step, due to Getoor-Sharpe ({\sl Invent. Math.}
16, 1972), delt with continuous martingales. The extension to right continuous martingales, a good exercise in martingale theory, is given here
Comment: See
907 for a correction. This material has been published in book form, see for instance Dellacherie-Meyer,
Probabilités et Potentiel, Vol. B, Chapter VII
Keywords: $BMO$,
Hardy spaces,
Fefferman inequalityNature: Original
Retrieve article from Numdam
XI: 12, 132-195, LNM 581 (1977)
MEYER, Paul-André
Le dual de $H^1({\bf R}^\nu)$~: démonstrations probabilistes (
Potential theory,
Applications of martingale theory)
This is a self-contained exposition and proof of the celebrated (Fefferman-Stein) result that the dual of $H^1(
R^n)$ is $BMO$, using methods adapted from the probabilistic Littlewood-Paley theory (of which this is a kind of limiting case). Some details of the proof are interesting in their own right
Comment: Though the proof is complete, it misses an essential point in the Fefferman-Stein theorem, namely, it depends on the Cauchy (Poisson) semigroup while the original result the convolution with quite general smooth functions in its definition of $H^1$. Similar methods were used by Bakry in the case of spheres, see
1818. The reasoning around (3.1) p.178 needs to be corrected
Keywords: Harmonic functions,
Hardy spaces,
Poisson kernel,
Carleson measures,
$BMO$,
Riesz transformsNature: Exposition,
Original additions Retrieve article from Numdam
XI: 32, 482-489, LNM 581 (1977)
MEYER, Paul-André
Sur un théorème de C. Stricker (
Martingale theory)
Some emphasis is put on a technical lemma used by Stricker to prove the well-known result that semimartingales remain so under restriction of filtrations (provided they are still adapted). The result is that a semimartingale up to infinity can be sent into the Hardy space $H^1$ by a suitable choice of an equivalent measure. This leads also to a simple proof and an extension of Jacod's theorem that the set of semimartingale laws is convex
Comment: A gap in a proof is filled in
1251Keywords: Hardy spaces,
Changes of measureNature: Original Retrieve article from Numdam
XII: 12, 98-113, LNM 649 (1978)
DELLACHERIE, Claude;
MEYER, Paul-André;
YOR, Marc
Sur certaines propriétés des espaces de Banach $H^1$ et $BMO$ (
Martingale theory,
Functional analysis)
The general subject is the weak topology $\sigma(H^1,BMO)$ on the space $H^1$. Its relatively compact sets are characterized by a uniform integrability property of the maximal functions. A sequential completeness result (a Vitali-Hahn-Saks like theorem) is proved. Finally, a separate section is devoted to the denseness of $L^\infty$ in $BMO$, a subject which has greatly progressed since (the Garnett-Jones theorem, see
1519; see also
3021 and
3316)
Keywords: Hardy spaces,
$BMO$Nature: Original Retrieve article from Numdam
XII: 56, 757-762, LNM 649 (1978)
MEYER, Paul-André
Inégalités de normes pour les intégrales stochastiques (
Stochastic calculus)
Inequalities of the following kind were introduced by Émery: $$\|X.M\|_{H^p}\le c_p \| X\|_{S^p}\,\| M\|$$ where the left hand side is a stochastic integral of the previsible process $X$ w.r.t. the semimartingale $M$, $S^p$ is a supremum norm, and the norm $H^p$ for semimartingales takes into account the Hardy space norm for the martingale part and the $L^p$ norm of the total variation for the finite variation part. On the right hand side, Émery had used a norm called $H^{\infty}$. Here a weaker $BMO$-like norm for semimartingales is suggested
Keywords: Stochastic integrals,
Hardy spacesNature: Original Retrieve article from Numdam
XII: 59, 775-803, LNM 649 (1978)
MEYER, Paul-André
Martingales locales fonctionnelles additives (two talks) (
Markov processes)
The purpose of the paper is to specialize the standard theory of Hardy spaces of martingales to the subspaces of additive martingales of a Markov process. The theory is not complete: the dual of (additive) $H^1$ seems to be different from (additive) $BMO$
Keywords: Hardy spaces,
Additive functionalsNature: Original Retrieve article from Numdam
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
1518Keywords: $BMO$,
$H^1$ space,
Hardy spacesNature: Original Retrieve article from Numdam
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 Retrieve article from Numdam
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 spacesNature: Original Retrieve article from Numdam
XXXIII: 16, 342-348, LNM 1709 (1999)
GRANDITS, Peter
Some remarks on L$^\infty $, H$^\infty $, and $BMO$ (
Martingale theory)
It is known from
1212 that neither $L^\infty$ nor $H^\infty$ is dense in $BMO$. This article answers a question raised by Durrett (
Brownian Motion and Martingales in Analysis, Wadworth 1984): Does there exist a $BMO$-martingale which has a best approximation in $L^\infty$? The answer is negative, but becomes positive if $L^\infty$ is replaced with $H^\infty$
Keywords: $BMO$,
Hardy spacesNature: Original Retrieve article from Numdam
