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II: 09, 166-170, LNM 51 (1968)

**MEYER, Paul-André**

Une majoration du processus croissant associé à une surmartingale (Martingale theory)

Let $(X_t)$ be the potential generated by a previsible increasing process $(A_t)$. Then a norm equivalence in $L^p,\ 1<p<\infty$ is given between the random variables $X^\ast$ and $A_\infty$

Comment: This paper became obsolete after the $H^1$-$BMO$ theory

Keywords: Inequalities, Potential of an increasing process

Nature: Original

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III: 13, 163-174, LNM 88 (1969)

**MEYER, Paul-André**

Les inégalités de Burkholder en théorie des martingales, d'après Gundy (Martingale theory)

A proof of the famous Burkholder inequalities in discrete time, from Gundy,*Ann. Math. Stat.*, **39**, 1968

Keywords: Burkholder inequalities

Nature: Exposition

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IV: 03, 37-46, LNM 124 (1970)

**CHERSI, Franco**

Martingales et intégrabilité de $X\log^+X$ d'après Gundy (Martingale theory)

Gundy's result (*Studia Math.*, **33**, 1968) is a converse to Doob's inequality: for a positive martingale such that $X_n\leq cX_{n-1}$, the integrability of $\sup_n X_n$ implies boundedness in $L\log^+L$. All martingales satisfy this condition on regular filtrations

Comment: The integrability of $\sup_n |\,X_n\,|$ has become now the $H^1$ theory of martingales

Keywords: Inequalities, Regular martingales

Nature: Exposition, Original additions

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IV: 14, 162-169, LNM 124 (1970)

**MEYER, Paul-André**

Quelques inégalités sur les martingales, d'après Dubins et Freedman (Martingale theory)

The original paper appeared in*Ann. Math. Stat.*, **36**, 1965, and the inequalities are extensions to martingales of the Borel-Cantelli lemma and the strong law of large numbers. For martingales with bounded jumps, exponential bounds are given (Neveu, *Martingales à temps discret,* gives a better one)

Comment: Though the proofs are very clever, so much work has been devoted to martingale inequalities since the paper was written that it is probably obsolete

Keywords: Inequalities

Nature: Exposition

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VI: 13, 151-158, LNM 258 (1972)

**MEYER, Paul-André**

Les résultats récents de Burkholder, Davis et Gundy (Martingale theory)

The well-known norm equivalence between the maximum and the square-function of a martingale in moderate Orlicz spaces is presented following the celebrated papers of Burkholder-Gundy (*Acta Math.,* **124**, 1970), Burkholder-Davis-Gundy (*Proc. 6-th Berkeley Symposium,* **3**, 1972). The technique of proof is now obsolete

Keywords: Burkholder inequalities, Moderate convex functions

Nature: Exposition

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VI: 19, 198-201, LNM 258 (1972)

**RAO, Murali**

Doob's decomposition and Burkholder's inequalities (Martingale theory)

The ``Burkholder inequalities'' referred here are the weak-$L^1$ estimates for the supremum of a martingale transform and for the square function proved by Burkholder (*Ann. Math. Stat.,* **37**, 1966) for $L^1$-bounded discrete time martingales. The original proof was quite sophisticated, while here these inequalities are deduced from an estimate on the (elementary) Doob decomposition of a discrete supermartingale

Comment: This little-known paper would probably deserve a modern translation in continuous time

Keywords: Burkholder inequalities, Decomposition of supermartingales

Nature: Original

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IX: 04, 206-212, LNM 465 (1975)

**CHOU, Ching Sung**

Les inégalités des surmartingales d'après A.M. Garsia (Martingale theory)

A proof is given of a simple and important inequality in discrete martingale theory, controlling a previsible increasing process whose potential is dominated by a positive martingale. It is strong enough to imply the Burkholder-Davis-Gundy inequalities

Keywords: Inequalities, Burkholder inequalities

Nature: Exposition

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IX: 05, 213-225, LNM 465 (1975)

**CHOU, Ching Sung**

Les méthodes d'A. Garsia en théorie des martingales. Extension au cas continu (Martingale theory)

The methods developed in discrete time by Garsia*Martingale Inequalities: Seminar Notes on Recent Progress,* Benjamin, 1973, are extended to continuous time

Comment: See Lenglart-Lépingle-Pratelli 1404. These methods have now become standard, and can be found in a number of books

Keywords: Inequalities, Burkholder inequalities

Nature: Original

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X: 04, 40-43, LNM 511 (1976)

**KAZAMAKI, Norihiko**

A simple remark on the conditioned square functions for martingale transforms (Martingale theory)

This is a problem of discrete martingale theory, giving inequalities between the conditioned square funtions (discrete angle brackets) of martingale transforms of two martingales related through a change of time

Comment: The author has published a paper on a related subject in*Tôhoku Math. J.*, **28**, 1976

Keywords: Angle bracket, Inequalities

Nature: Original

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X: 10, 125-183, LNM 511 (1976)

**MEYER, Paul-André**

Démonstration probabiliste de certaines inégalités de Littlewood-Paley (4 talks) (Applications of martingale theory, Markov processes)

This long paper consists of four talks, suggested by E.M.~Stein's book*Topics in Harmonic Analysis related to the Littlewood-Paley theory,* Princeton 1970. The classical Littlewood-Paley theory shows that the $L^p$ norm ($1<p<\infty$) of a function $f$ on $**R**^n$ is equivalent to that of several kinds of non-linear functionals of $f$ called Littlewood-Paley functions, which are square roots of quadratic expressions involving the harmonic extension of $f$ to the half-space $**R**^n\times **R**_+$, and its derivatives. Using these equivalences, it is easy to prove that the Riesz transforms are bounded in~$L^p$. The classical theory is given a probabilistic interpretation, the L-P functions appearing as conditional expectations of functionals of a Brownian motion on the half-space, given its final position on the limit hyperplane, and then the L-P inequalities follow from the Burkholder inequalities of martingale theory. The original L-P theory concerned the unit disk; Stein had extended it to $**R**^n$ and had started extending it to symmetric semigroups. Here a new tool is introduced, the squared-field operator (carré du champ) introduced by J.P.~Roth (*CRAS Paris,* **278A**, 1974, p.1103) in potential theory and by Kunita (*Nagoya M. J.*, **36**, 1969) in probability. This paper consists of 4 talks, and in the last one theorems 1' and 3 are false

Comment: This paper was rediscovered by Varopoulos (*J. Funct. Anal.*, **38**, 1980), and was then rewritten by Meyer in 1510 in a simpler form. Its main application has been to the Ornstein-Uhlenbeck semigroup in 1816. It has been superseded by the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912, and Meyer 1908 reporting on Cowling's extension of Stein's work. An erratum is given in 1253

Keywords: Littlewood-Paley theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ, Infinitesimal generators, Semigroup theory

Nature: Original

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X: 17, 245-400, LNM 511 (1976)

**MEYER, Paul-André**

Un cours sur les intégrales stochastiques (6 chapters) (Stochastic calculus, Martingale theory, General theory of processes)

This is a systematic exposition of the theory of stochastic integration with respect to semimartingales, with the exception of stochastic differential equations. Chapter I is devoted to a quick exposition of the general theory of processes, and of the trivial stochastic integral with respect to a process of finite variation. Chapter II is the Kunita-Watanabe theory of square integrables martingales, angle and square bracket, stable subspaces, compensated sums of jumps, and the corresponding $L^2$ theory of stochastic integration. Chapter III studies a restricted class of semimartingales and introduces the Ito formula, with its celebrated applications due to Watanabe, to Brownian motion and the Poisson process. Chapter IV localizes the theory and gives the general definitions of semimartingales and special semimartingales, and studies the stochastic exponential, multiplicative decomposition. It also sketches a theory of multiple stochastic integrals. Chapter V deals with the application of the spaces $H^1$ and $BMO$ to the theory of stochastic integration, and to martingales inequalities (it contains the extension to continuous time of Garsia's ``Fefferman implies Davis implies Burkholder'' approach). Chapter VI contains more special topics: Stratonovich integrals, Girsanov's theorem, local times, representation of elements of $BMO$

Comment: This set of lectures was well circulated in its time, an intermediate stage between a research paper and a polished book form. See also 1131. Now the material can be found in many books

Keywords: Increasing processes, Stable subpaces, Angle bracket, Square bracket, Stochastic integrals, Optional stochastic integrals, Previsible representation, Change of variable formula, Semimartingales, Stochastic exponentials, Multiplicative decomposition, Fefferman inequality, Davis inequality, Stratonovich integrals, Burkholder inequalities, $BMO$, Multiple stochastic integrals, Girsanov's theorem

Nature: Exposition, Original additions

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X: 18, 401-413, LNM 511 (1976)

**PRATELLI, Maurizio**

Sur certains espaces de martingales de carré intégrable (Martingale theory)

The main purpose of this paper is to define spaces similar to the $H^p$ and $BMO$ spaces (which we may call here $h^p$ and $bmo$) using the angle bracket of a local martingale instead of the square bracket (this concerns only locally square integrable martingales). It is shown that for $1<p<\infty$ $h^p$ is reflexive with dual the natural $h^q$, and that the conjugate (dual) space of $h^1$ is $bmo$

Comment: This paper contains some interesting martingale inequalities, which are developed in Lenglart-Lépingle-Pratelli, 1404. An error is corrected in 1250

Keywords: Inequalities, Angle bracket, $BMO$

Nature: Original

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XI: 25, 383-389, LNM 581 (1977)

**DOLÉANS-DADE, Catherine**; **MEYER, Paul-André**

Une caractérisation de $BMO$ (Martingale theory)

Kazamaki gave in 1027 a criterion for a continuous martingale to belong to $BMO$, involving its stochastic exponential. This criterion is extended, though in a different form, to non-continuous local martingales: $M$ belongs to $BMO$ if and only if for $|\lambda|$ small enough, its stochastic exponential ${\cal E}(\lambda M)$ is a (positive) multiplicatively bounded process---a class of processes, which looked promising but did not attract attention

Comment: Related subjects occur in 1328. The reference to ``note VI'' on p.384 probably refers to an earlier preprint, and is no longer intelligible

Keywords: $BMO$, Stochastic exponentials, Martingale inequalities

Nature: Original

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XII: 15, 134-137, LNM 649 (1978)

**LÉPINGLE, Dominique**

Une inégalité de martingales (Martingale theory)

The following inequality for a discrete time adapted process $(a_n)$ and its conditional expectations $b_n=E[a_n\,|\,{\cal F}_{n-1}]$ is proved: $$\|(\sum_n b_n^2)^{1/2}\|_1\le 2\|(\sum_n a_n^2)^{1/2}\|_1\ .$$ A similar inequality in $L^p$, $1\!<\!p\!<\!\infty$, does not require adaptedness, and is due to Stein

Keywords: Inequalities, Quadratic variation

Nature: Original

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XII: 19, 170-179, LNM 649 (1978)

**MÉTRAUX, C.**

Quelques inégalités pour martingales à paramètre bidimensionnel (Several parameter processes)

This paper extends to two-parameter discrete martingales the classical Burkholder inequalities ($1<p<\infty$) and a few more inequalities

Keywords: Burkholder inequalities

Nature: Original

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XII: 50, 739-739, LNM 649 (1978)

**LÉPINGLE, Dominique**

Correction au Séminaire X (Martingale theory)

Corrects a detail in 1018

Keywords: Inequalities, Angle bracket, $BMO$

Nature: Correction

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XIII: 26, 294-306, LNM 721 (1979)

**BONAMI, Aline**; **LÉPINGLE, Dominique**

Fonction maximale et variation quadratique des martingales en présence d'un poids (Martingale theory)

Weighted norm inequalities in martingale theory assert that a martingale inequality---relating under the law $**P**$ two functionals of a $**P**$-martingale---remains true, possibly with new constants, when $**P**$ is replaced by an equivalent law $Z.**P**$. To this order, the ``weight'' $Z$ must satisfy special conditions, among which a probabilistic version of Muckenhoupt's (1972) $(A_p)$ condition and a condition of multiplicative boundedness on the jumps of the martingale $E[Z\,|\,{\cal F}_t]$. This volume contains three papers on weighted norms inequalities, 1326, 1327, 1328, with considerable overlap. Here the main topic is the weighted-norm extension of the Burkholder-Gundy inequalities

Comment: Recently (1997) weighted norm inequalities have proved useful in mathematical finance

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

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XIII: 27, 307-312, LNM 721 (1979)

**IZUMISAWA, Masataka**; **SEKIGUCHI, Takesi**

Weighted norm inequalities for martingales (Martingale theory)

See the review of 1326. The topic is the same, though the proof is different

Comment: See the paper by Kazamaki-Izumisawa in*Tôhoku Math. J.* **29**, 1977. For a modern reference see also Kazamaki, *Continuous Exponential Martingales and $\,BMO$,* LNM. 1579, 1994

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

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XIII: 28, 313-331, LNM 721 (1979)

**DOLÉANS-DADE, Catherine**; **MEYER, Paul-André**

Inégalités de normes avec poids (Martingale theory)

See the review of 1326. This is a rather systematic exposition of the subject in the frame of martingale theory

Comment: An exponent $1/\lambda$ is missing in formula (4), p.315

Keywords: Weighted norm inequalities

Nature: Exposition, Original additions

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XIII: 31, 371-377, LNM 721 (1979)

**DELLACHERIE, Claude**

Inégalités de convexité pour les processus croissants et les sousmartingales (Martingale theory)

Several inequalities concerning general convex functions are classical in martingale theory (e.g. generalizations of Doob's inequality) and the general theory of processes (e.g. estimates on dual projections of increasing processes). The proof of such inequalities given here is slightly more natural than those in Dellacherie-Meyer,*Probabilités et Potentiels B,* Chapter VI

Keywords: Martingale inequalities, Convex functions

Nature: Exposition, Original proofs

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XIV: 04, 26-48, LNM 784 (1980)

**LENGLART, Érik**; **LÉPINGLE, Dominique**; **PRATELLI, Maurizio**

Présentation unifiée de certaines inégalités de la théorie des martingales (Martingale theory)

This paper is a synthesis of many years of work on martingale inequalities, and certainly one of the most influential among the papers which appeared in these volumes. It is shown how all main inequalities can be reduced to simple principles: 1) Basic distribution inequalities between pairs of random variables (``Doob'', ``domination'', ``good lambda'' and ``Garsia-Neveu''), and 2) Simple lemmas from the general theory of processes

Comment: This paper has been rewritten as Chapter XXIII of Dellacherie-Meyer,*Probabilités et Potentiel E *; see also 1621. A striking example of the power of these methods is Barlow-Yor, {\sl Jour. Funct. Anal.} **49**,1982

Keywords: Moderate convex functions, Inequalities, Martingale inequalities, Burkholder inequalities, Good lambda inequalities, Domination inequalities

Nature: Original

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XIV: 05, 49-52, LNM 784 (1980)

**LENGLART, Érik**

Appendice à l'exposé précédent~: inégalités de semimartingales (Martingale theory, Stochastic calculus)

This paper contains several applications of the methods of 1404 to the case of semimartingales instead of martingales

Keywords: Inequalities, Semimartingales

Nature: Original

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XV: 10, 151-166, LNM 850 (1981)

**MEYER, Paul-André**

Retour sur la théorie de Littlewood-Paley (Applications of martingale theory, Markov processes)

The word ``original'' may be considered misleading, since this paper is essentially a re-issue of 1010 (see the corresponding review), with a slightly better pedagogy, and the correction of a mistake. Meanwhile, Varopoulos had independently rediscovered the subject (*J. Funct. Anal.*, 38, 1980)

Comment: See an application to the Ornstein-Uhlenbeck semigroup 1816, see 1818 for a related topic, and the report 1908 on Cowling's extension of Stein's work. Bouleau-Lamberton 2013 replace the auxiliary Brownian motion by a stable process to obtain further inequalities. In another direction, the subject is developed in the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912; a general account of this point of view in semigroup theory is given by Bakry in his 1992 Saint-Flour lectures (LN 1581)

Keywords: Littlewood-Paley theory, Semigroup theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ

Nature: Original

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XV: 20, 285-289, LNM 850 (1981)

**CHOU, Ching Sung**

Une inégalité de martingales avec poids (Martingale theory)

Chevalier has strengthened the Burkholder inequalities into an equivalence of $L^p$ norms between $M^{\ast}\lor Q(M)$ and $M^{\ast}\land Q(M)$, where $M$ is a martingale, $M^{\ast}$ is its maximal function and $Q(M)$ its quadratic variation. This has been extended to all moderate Orlicz spaces in 1404. The present paper further extends the result to the Orlicz spaces of a law $\widehat P$ equivalent to $P$, provided the density is an $(A_p)$ weight (see 1326)

Keywords: Weighted norm inequalities, Burkholder inequalities, Moderate convex functions

Nature: Original

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XVI: 09, 138-150, LNM 920 (1982)

**BAKRY, Dominique**; **MEYER, Paul-André**

Sur les inégalités de Sobolev logarithmiques (two parts) (Applications of martingale theory)

These two papers are variations on a paper of G.F. Feissner (*Trans. Amer Math. Soc.*, **210**, 1965). Let $\mu$ be a Gaussian measure, $P_t$ be the corresponding Ornstein-Uhlenbeck semigroup. Nelson's hypercontractivity theorem states (roughly) that $P_t$ is bounded from $L^p(\mu)$ to some $L^q(\mu)$ with $q\ge p$. In another celebrated paper, Gross showed this to be equivalent to a logarithmic Sobolev inequality, meaning that if a function $f$ is in $L^2$ as well as $Af$, where $A$ is the Ornstein-Uhlenbeck generator, then $f$ belongs to the Orlicz space $L^2Log_+L$. The starting point of Feissner was to translate this again as a result on the ``Riesz potentials'' of the semi-group (defined whenever $f\in L^2$ has integral $0$) $$R^{\alpha}={1\over \Gamma(\alpha)}\int_0^\infty t^{\alpha-1}P_t\,dt\;.$$ Note that $R^{\alpha}R^{\beta}=R^{\alpha+\beta}$. Then the theorem of Gross implies that $R^{1/2}$ is bounded from $L^2$ to $L^2Log_+L$. This suggests the following question: which are in general the smoothing properties of $R^\alpha$? (Feissner in fact considers a slightly different family of potentials).\par The complete result then is the following : for $\alpha$ complex, with real part $\ge0$, $R^\alpha$ is bounded from $L^pLog^r_+L$ to $L^pLog^{r+p\alpha}_+L$. The method uses complex interpolation between two cases: a generalization to Orlicz spaces of a result of Stein, when $\alpha$ is purely imaginary, and the case already known where $\alpha$ has real part $1/2$. The first of these two results, proved by martingale theory, is of a quite general nature

Keywords: Logarithmic Sobolev inequalities, Hypercontractivity, Gaussian measures, Riesz potentials

Nature: Original

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XVI: 10, 151-152, LNM 920 (1982)

**MEYER, Paul-André**

Sur une inégalité de Stein (Applications of martingale theory)

In his book*Topics in harmonic analysis related to the Littlewood-Paley theory * (1970) Stein uses interpolation between two results, one of which is a discrete martingale inequality deduced from the Burkholder inequalities, whose precise statement we omit. This note states and proves directly the continuous time analogue of this inequality---a mere exercise in translation

Keywords: Littlewood-Paley theory, Martingale inequalities

Nature: Exposition, Original additions

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XVI: 19, 221-233, LNM 920 (1982)

**YOR, Marc**

Application de la relation de domination à certains renforcements des inégalités de martingales (Martingale theory)

The domination relation (Lenglart 1977) between a positive, right-continuous process $X$ and a previsible increasing process $A$ holds whenever $E[X_T]\le E[A_T]$ at stopping times. It plays an important role in the paper 1404 of Lenglart-Lepingle-Pratelli on martingale inequalities. Here it is shown to imply a general inequality involving $X^\ast_{\infty}$ and $1/A_{\infty}$, from which follow a number of inequalities for a continuous local martingale $M$. Among them, estimates on the ratios of the three quantities $M^\ast_{\infty}$, $<M>_{\infty}$, $\sup_{a,t} L^a_t$. One can recover also the stronger version of Doob's inequality, proved by Pitman 1517

Comment: See an earlier paper of the author on this subject,*Stochastics,* **3**, 1979. The author mentions that part of the results were discovered slightly earlier by R.~Gundy

Keywords: Martingale inequalities, Domination inequalities

Nature: Original

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XX: 13, 162-185, LNM 1204 (1986)

**BOULEAU, Nicolas**; **LAMBERTON, Damien**

Théorie de Littlewood-Paley et processus stables (Applications of martingale theory, Markov processes)

Meyer' probabilistic approach to Littlewood-Paley inequalities (1010, 1510) is extended by replacing the underlying Brownian motion with a stable process. The following spectral multiplicator theorem is obtained: If $(P_t)_{t\geq 0}$ is a symmetric Markov semigroup with spectral representation $P_t=\int_{[0,\infty)}e^{-t\lambda} dE_{\lambda}$, and if $M$ is a function on $**R**_+$ defined by $M(\lambda)=\lambda\int_0^\infty r(y)e^{-y\lambda}dy,$ where $r(y)$ is bounded and Borel on $**R**_+$, then the operator $T_M=\int_{[0,\infty)}M(\lambda)dE_{\lambda},$ which is obviously bounded on $L^2$, is actually bounded on all $L^p$ spaces of the invariant measure, $1<p<\infty$. The method also leads to new Littlewood-Paley inequalities for semigroups admitting a carré du champ operator

Keywords: Littlewood-Paley theory, Semigroup theory, Riesz transforms, Stable processes, Inequalities, Singular integrals, Carré du champ

Nature: Original

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XXI: 09, 173-175, LNM 1247 (1987)

**ÉMERY, Michel**; **YUKICH, Joseph E.**

A simple proof of the logarithmic Sobolev inequality on the circle (Real analysis)

The same kind of semi-group argument as in Bakry-Émery 1912 gives an elementary proof of the logarithmic Sobolev inequality on the circle

Keywords: Logarithmic Sobolev inequalities

Nature: New proof of known results

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Une majoration du processus croissant associé à une surmartingale (Martingale theory)

Let $(X_t)$ be the potential generated by a previsible increasing process $(A_t)$. Then a norm equivalence in $L^p,\ 1<p<\infty$ is given between the random variables $X^\ast$ and $A_\infty$

Comment: This paper became obsolete after the $H^1$-$BMO$ theory

Keywords: Inequalities, Potential of an increasing process

Nature: Original

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III: 13, 163-174, LNM 88 (1969)

Les inégalités de Burkholder en théorie des martingales, d'après Gundy (Martingale theory)

A proof of the famous Burkholder inequalities in discrete time, from Gundy,

Keywords: Burkholder inequalities

Nature: Exposition

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IV: 03, 37-46, LNM 124 (1970)

Martingales et intégrabilité de $X\log^+X$ d'après Gundy (Martingale theory)

Gundy's result (

Comment: The integrability of $\sup_n |\,X_n\,|$ has become now the $H^1$ theory of martingales

Keywords: Inequalities, Regular martingales

Nature: Exposition, Original additions

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IV: 14, 162-169, LNM 124 (1970)

Quelques inégalités sur les martingales, d'après Dubins et Freedman (Martingale theory)

The original paper appeared in

Comment: Though the proofs are very clever, so much work has been devoted to martingale inequalities since the paper was written that it is probably obsolete

Keywords: Inequalities

Nature: Exposition

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VI: 13, 151-158, LNM 258 (1972)

Les résultats récents de Burkholder, Davis et Gundy (Martingale theory)

The well-known norm equivalence between the maximum and the square-function of a martingale in moderate Orlicz spaces is presented following the celebrated papers of Burkholder-Gundy (

Keywords: Burkholder inequalities, Moderate convex functions

Nature: Exposition

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VI: 19, 198-201, LNM 258 (1972)

Doob's decomposition and Burkholder's inequalities (Martingale theory)

The ``Burkholder inequalities'' referred here are the weak-$L^1$ estimates for the supremum of a martingale transform and for the square function proved by Burkholder (

Comment: This little-known paper would probably deserve a modern translation in continuous time

Keywords: Burkholder inequalities, Decomposition of supermartingales

Nature: Original

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IX: 04, 206-212, LNM 465 (1975)

Les inégalités des surmartingales d'après A.M. Garsia (Martingale theory)

A proof is given of a simple and important inequality in discrete martingale theory, controlling a previsible increasing process whose potential is dominated by a positive martingale. It is strong enough to imply the Burkholder-Davis-Gundy inequalities

Keywords: Inequalities, Burkholder inequalities

Nature: Exposition

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IX: 05, 213-225, LNM 465 (1975)

Les méthodes d'A. Garsia en théorie des martingales. Extension au cas continu (Martingale theory)

The methods developed in discrete time by Garsia

Comment: See Lenglart-Lépingle-Pratelli 1404. These methods have now become standard, and can be found in a number of books

Keywords: Inequalities, Burkholder inequalities

Nature: Original

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X: 04, 40-43, LNM 511 (1976)

A simple remark on the conditioned square functions for martingale transforms (Martingale theory)

This is a problem of discrete martingale theory, giving inequalities between the conditioned square funtions (discrete angle brackets) of martingale transforms of two martingales related through a change of time

Comment: The author has published a paper on a related subject in

Keywords: Angle bracket, Inequalities

Nature: Original

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X: 10, 125-183, LNM 511 (1976)

Démonstration probabiliste de certaines inégalités de Littlewood-Paley (4 talks) (Applications of martingale theory, Markov processes)

This long paper consists of four talks, suggested by E.M.~Stein's book

Comment: This paper was rediscovered by Varopoulos (

Keywords: Littlewood-Paley theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ, Infinitesimal generators, Semigroup theory

Nature: Original

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X: 17, 245-400, LNM 511 (1976)

Un cours sur les intégrales stochastiques (6 chapters) (Stochastic calculus, Martingale theory, General theory of processes)

This is a systematic exposition of the theory of stochastic integration with respect to semimartingales, with the exception of stochastic differential equations. Chapter I is devoted to a quick exposition of the general theory of processes, and of the trivial stochastic integral with respect to a process of finite variation. Chapter II is the Kunita-Watanabe theory of square integrables martingales, angle and square bracket, stable subspaces, compensated sums of jumps, and the corresponding $L^2$ theory of stochastic integration. Chapter III studies a restricted class of semimartingales and introduces the Ito formula, with its celebrated applications due to Watanabe, to Brownian motion and the Poisson process. Chapter IV localizes the theory and gives the general definitions of semimartingales and special semimartingales, and studies the stochastic exponential, multiplicative decomposition. It also sketches a theory of multiple stochastic integrals. Chapter V deals with the application of the spaces $H^1$ and $BMO$ to the theory of stochastic integration, and to martingales inequalities (it contains the extension to continuous time of Garsia's ``Fefferman implies Davis implies Burkholder'' approach). Chapter VI contains more special topics: Stratonovich integrals, Girsanov's theorem, local times, representation of elements of $BMO$

Comment: This set of lectures was well circulated in its time, an intermediate stage between a research paper and a polished book form. See also 1131. Now the material can be found in many books

Keywords: Increasing processes, Stable subpaces, Angle bracket, Square bracket, Stochastic integrals, Optional stochastic integrals, Previsible representation, Change of variable formula, Semimartingales, Stochastic exponentials, Multiplicative decomposition, Fefferman inequality, Davis inequality, Stratonovich integrals, Burkholder inequalities, $BMO$, Multiple stochastic integrals, Girsanov's theorem

Nature: Exposition, Original additions

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X: 18, 401-413, LNM 511 (1976)

Sur certains espaces de martingales de carré intégrable (Martingale theory)

The main purpose of this paper is to define spaces similar to the $H^p$ and $BMO$ spaces (which we may call here $h^p$ and $bmo$) using the angle bracket of a local martingale instead of the square bracket (this concerns only locally square integrable martingales). It is shown that for $1<p<\infty$ $h^p$ is reflexive with dual the natural $h^q$, and that the conjugate (dual) space of $h^1$ is $bmo$

Comment: This paper contains some interesting martingale inequalities, which are developed in Lenglart-Lépingle-Pratelli, 1404. An error is corrected in 1250

Keywords: Inequalities, Angle bracket, $BMO$

Nature: Original

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XI: 25, 383-389, LNM 581 (1977)

Une caractérisation de $BMO$ (Martingale theory)

Kazamaki gave in 1027 a criterion for a continuous martingale to belong to $BMO$, involving its stochastic exponential. This criterion is extended, though in a different form, to non-continuous local martingales: $M$ belongs to $BMO$ if and only if for $|\lambda|$ small enough, its stochastic exponential ${\cal E}(\lambda M)$ is a (positive) multiplicatively bounded process---a class of processes, which looked promising but did not attract attention

Comment: Related subjects occur in 1328. The reference to ``note VI'' on p.384 probably refers to an earlier preprint, and is no longer intelligible

Keywords: $BMO$, Stochastic exponentials, Martingale inequalities

Nature: Original

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XII: 15, 134-137, LNM 649 (1978)

Une inégalité de martingales (Martingale theory)

The following inequality for a discrete time adapted process $(a_n)$ and its conditional expectations $b_n=E[a_n\,|\,{\cal F}_{n-1}]$ is proved: $$\|(\sum_n b_n^2)^{1/2}\|_1\le 2\|(\sum_n a_n^2)^{1/2}\|_1\ .$$ A similar inequality in $L^p$, $1\!<\!p\!<\!\infty$, does not require adaptedness, and is due to Stein

Keywords: Inequalities, Quadratic variation

Nature: Original

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XII: 19, 170-179, LNM 649 (1978)

Quelques inégalités pour martingales à paramètre bidimensionnel (Several parameter processes)

This paper extends to two-parameter discrete martingales the classical Burkholder inequalities ($1<p<\infty$) and a few more inequalities

Keywords: Burkholder inequalities

Nature: Original

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XII: 50, 739-739, LNM 649 (1978)

Correction au Séminaire X (Martingale theory)

Corrects a detail in 1018

Keywords: Inequalities, Angle bracket, $BMO$

Nature: Correction

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XIII: 26, 294-306, LNM 721 (1979)

Fonction maximale et variation quadratique des martingales en présence d'un poids (Martingale theory)

Weighted norm inequalities in martingale theory assert that a martingale inequality---relating under the law $

Comment: Recently (1997) weighted norm inequalities have proved useful in mathematical finance

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

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XIII: 27, 307-312, LNM 721 (1979)

Weighted norm inequalities for martingales (Martingale theory)

See the review of 1326. The topic is the same, though the proof is different

Comment: See the paper by Kazamaki-Izumisawa in

Keywords: Weighted norm inequalities, Burkholder inequalities

Nature: Original

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XIII: 28, 313-331, LNM 721 (1979)

Inégalités de normes avec poids (Martingale theory)

See the review of 1326. This is a rather systematic exposition of the subject in the frame of martingale theory

Comment: An exponent $1/\lambda$ is missing in formula (4), p.315

Keywords: Weighted norm inequalities

Nature: Exposition, Original additions

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XIII: 31, 371-377, LNM 721 (1979)

Inégalités de convexité pour les processus croissants et les sousmartingales (Martingale theory)

Several inequalities concerning general convex functions are classical in martingale theory (e.g. generalizations of Doob's inequality) and the general theory of processes (e.g. estimates on dual projections of increasing processes). The proof of such inequalities given here is slightly more natural than those in Dellacherie-Meyer,

Keywords: Martingale inequalities, Convex functions

Nature: Exposition, Original proofs

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XIV: 04, 26-48, LNM 784 (1980)

Présentation unifiée de certaines inégalités de la théorie des martingales (Martingale theory)

This paper is a synthesis of many years of work on martingale inequalities, and certainly one of the most influential among the papers which appeared in these volumes. It is shown how all main inequalities can be reduced to simple principles: 1) Basic distribution inequalities between pairs of random variables (``Doob'', ``domination'', ``good lambda'' and ``Garsia-Neveu''), and 2) Simple lemmas from the general theory of processes

Comment: This paper has been rewritten as Chapter XXIII of Dellacherie-Meyer,

Keywords: Moderate convex functions, Inequalities, Martingale inequalities, Burkholder inequalities, Good lambda inequalities, Domination inequalities

Nature: Original

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XIV: 05, 49-52, LNM 784 (1980)

Appendice à l'exposé précédent~: inégalités de semimartingales (Martingale theory, Stochastic calculus)

This paper contains several applications of the methods of 1404 to the case of semimartingales instead of martingales

Keywords: Inequalities, Semimartingales

Nature: Original

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XV: 10, 151-166, LNM 850 (1981)

Retour sur la théorie de Littlewood-Paley (Applications of martingale theory, Markov processes)

The word ``original'' may be considered misleading, since this paper is essentially a re-issue of 1010 (see the corresponding review), with a slightly better pedagogy, and the correction of a mistake. Meanwhile, Varopoulos had independently rediscovered the subject (

Comment: See an application to the Ornstein-Uhlenbeck semigroup 1816, see 1818 for a related topic, and the report 1908 on Cowling's extension of Stein's work. Bouleau-Lamberton 2013 replace the auxiliary Brownian motion by a stable process to obtain further inequalities. In another direction, the subject is developed in the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912; a general account of this point of view in semigroup theory is given by Bakry in his 1992 Saint-Flour lectures (LN 1581)

Keywords: Littlewood-Paley theory, Semigroup theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ

Nature: Original

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XV: 20, 285-289, LNM 850 (1981)

Une inégalité de martingales avec poids (Martingale theory)

Chevalier has strengthened the Burkholder inequalities into an equivalence of $L^p$ norms between $M^{\ast}\lor Q(M)$ and $M^{\ast}\land Q(M)$, where $M$ is a martingale, $M^{\ast}$ is its maximal function and $Q(M)$ its quadratic variation. This has been extended to all moderate Orlicz spaces in 1404. The present paper further extends the result to the Orlicz spaces of a law $\widehat P$ equivalent to $P$, provided the density is an $(A_p)$ weight (see 1326)

Keywords: Weighted norm inequalities, Burkholder inequalities, Moderate convex functions

Nature: Original

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XVI: 09, 138-150, LNM 920 (1982)

Sur les inégalités de Sobolev logarithmiques (two parts) (Applications of martingale theory)

These two papers are variations on a paper of G.F. Feissner (

Keywords: Logarithmic Sobolev inequalities, Hypercontractivity, Gaussian measures, Riesz potentials

Nature: Original

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XVI: 10, 151-152, LNM 920 (1982)

Sur une inégalité de Stein (Applications of martingale theory)

In his book

Keywords: Littlewood-Paley theory, Martingale inequalities

Nature: Exposition, Original additions

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XVI: 19, 221-233, LNM 920 (1982)

Application de la relation de domination à certains renforcements des inégalités de martingales (Martingale theory)

The domination relation (Lenglart 1977) between a positive, right-continuous process $X$ and a previsible increasing process $A$ holds whenever $E[X_T]\le E[A_T]$ at stopping times. It plays an important role in the paper 1404 of Lenglart-Lepingle-Pratelli on martingale inequalities. Here it is shown to imply a general inequality involving $X^\ast_{\infty}$ and $1/A_{\infty}$, from which follow a number of inequalities for a continuous local martingale $M$. Among them, estimates on the ratios of the three quantities $M^\ast_{\infty}$, $<M>_{\infty}$, $\sup_{a,t} L^a_t$. One can recover also the stronger version of Doob's inequality, proved by Pitman 1517

Comment: See an earlier paper of the author on this subject,

Keywords: Martingale inequalities, Domination inequalities

Nature: Original

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XX: 13, 162-185, LNM 1204 (1986)

Théorie de Littlewood-Paley et processus stables (Applications of martingale theory, Markov processes)

Meyer' probabilistic approach to Littlewood-Paley inequalities (1010, 1510) is extended by replacing the underlying Brownian motion with a stable process. The following spectral multiplicator theorem is obtained: If $(P_t)_{t\geq 0}$ is a symmetric Markov semigroup with spectral representation $P_t=\int_{[0,\infty)}e^{-t\lambda} dE_{\lambda}$, and if $M$ is a function on $

Keywords: Littlewood-Paley theory, Semigroup theory, Riesz transforms, Stable processes, Inequalities, Singular integrals, Carré du champ

Nature: Original

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XXI: 09, 173-175, LNM 1247 (1987)

A simple proof of the logarithmic Sobolev inequality on the circle (Real analysis)

The same kind of semi-group argument as in Bakry-Émery 1912 gives an elementary proof of the logarithmic Sobolev inequality on the circle

Keywords: Logarithmic Sobolev inequalities

Nature: New proof of known results

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