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I: 04, 52-53, LNM 39 (1967)

**DELLACHERIE, Claude**

Un complément au théorème de Weierstrass-Stone (Functional analysis)

An easy but useful remark on the relation between the ``lattice'' and ``algebra'' forms of Stone's theorem, which apparently belongs to the folklore

Keywords: Stone-Weierstrass theorem

Nature: Original

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II: 04, 43-74, LNM 51 (1968)

**DOLÉANS-DADE, Catherine**

Espaces $H^m$ sur les variétés, et applications aux équations aux dérivées partielles sur une variété compacte (Functional analysis)

An attempt to teach to the members of the seminar the basic facts of the analytic theory of diffusion processes

Keywords: Sobolev spaces, Second order elliptic equations

Nature: Exposition

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III: 01, 1-23, LNM 88 (1969)

**ARTZNER, Philippe**

Extension du théorème de Sazonov-Minlos d'après L.~Schwartz (Measure theory, Functional analysis)

Exposition of three notes by L.~Schwartz (*CRAS* **265**, 1967 and **266**, 1968) showing that some classes of maps between spaces $\ell^p$ and $\ell^q$ transform Gaussian cylindrical measures into Radon measures. The result turns out to be an extension of Minlos' theorem

Comment: Self-contained and detailed exposition, possibly still useful

Keywords: Radonifying maps

Nature: Exposition

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VII: 19, 198-204, LNM 321 (1973)

**MEYER, Paul-André**

Limites médiales d'après Mokobodzki (Measure theory, Functional analysis)

Given a sequence of (classes of) random variables on a probability space which converges in some of the standard ways of measure theory, the problem is to find some universal method (independent from the underlying probability) to identify its limit. For convergence in probability, and thus for all strong $L^p$ topologies, Mokobodzki had discovered the procedure of rapid ultrafilters (see 304). The same problem is now solved for weak convergences, using a special kind of Banach limits

Comment: The paper contains a few annoying misprints, in particular p.199 line 9*s.c;s.* should be deleted and line 17 *atomique * should be *absolument continu.* For a misprint-free version see Dellacherie-Meyer, *Probabiliés et Potentiel,* Volume C, Chapter X, **55**--57

Keywords: Continuum axiom, Weak convergence of r.v.'s, Medial limit

Nature: Exposition

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XII: 01, 1-19, LNM 649 (1978)

**PRATELLI, Maurizio**

Une version probabiliste d'un théorème d'interpolation de G. Stampacchia (Martingale theory, Functional analysis)

This theorem is similar to the Marcinkievicz interpolation theorem, in the sense that at one endpoint a weak $L^p$ inequality is involved, but at the other endpoint the spaces involved are some $L^p$ and $BMO$. It concerns linear operators only, not sublinear ones like the Marcinkiewicz theorem. A closely related result, concerning the discrete-time case, had been proved earlier by Stroock,*Comm Pure Appl. Math.*, **26**, 1973

Keywords: Interpolation, $BMO$

Nature: Original

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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

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XIV: 25, 220-222, LNM 784 (1980)

**YAN, Jia-An**

Caractérisation d'ensembles convexes de $L^1$ ou $H^1$ (Stochastic calculus, Functional analysis)

This is a new and simpler approach to the crucial functional analytic lemma in 1354 (the proof that semimartingales are the stochastic integrators in $L^0$). That is, given a convex set $K\subset L^1$ containing $0$, find a condition for the existence of $Z>0$ in $L^\infty$ such that $\sup_{X\in K}E[ZX]<\infty$. A similar result is discussed for $H^1$ instead of $L^1$

Comment: This lemma, instead of the original one, has proved very useful in mathematical finance

Keywords: Semimartingales, Stochastic integrals, Convex functions

Nature: Original

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XVI: 11, 153-158, LNM 920 (1982)

**MEYER, Paul-André**

Interpolation entre espaces d'Orlicz (Functional analysis)

This is an exposition of Calderon's complex interpolation method, in the case of moderate Orlicz spaces, aiming at its application in 1609

Keywords: Interpolation, Orlicz spaces, Moderate convex functions

Nature: Exposition

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Un complément au théorème de Weierstrass-Stone (Functional analysis)

An easy but useful remark on the relation between the ``lattice'' and ``algebra'' forms of Stone's theorem, which apparently belongs to the folklore

Keywords: Stone-Weierstrass theorem

Nature: Original

Retrieve article from Numdam

II: 04, 43-74, LNM 51 (1968)

Espaces $H^m$ sur les variétés, et applications aux équations aux dérivées partielles sur une variété compacte (Functional analysis)

An attempt to teach to the members of the seminar the basic facts of the analytic theory of diffusion processes

Keywords: Sobolev spaces, Second order elliptic equations

Nature: Exposition

Retrieve article from Numdam

III: 01, 1-23, LNM 88 (1969)

Extension du théorème de Sazonov-Minlos d'après L.~Schwartz (Measure theory, Functional analysis)

Exposition of three notes by L.~Schwartz (

Comment: Self-contained and detailed exposition, possibly still useful

Keywords: Radonifying maps

Nature: Exposition

Retrieve article from Numdam

VII: 19, 198-204, LNM 321 (1973)

Limites médiales d'après Mokobodzki (Measure theory, Functional analysis)

Given a sequence of (classes of) random variables on a probability space which converges in some of the standard ways of measure theory, the problem is to find some universal method (independent from the underlying probability) to identify its limit. For convergence in probability, and thus for all strong $L^p$ topologies, Mokobodzki had discovered the procedure of rapid ultrafilters (see 304). The same problem is now solved for weak convergences, using a special kind of Banach limits

Comment: The paper contains a few annoying misprints, in particular p.199 line 9

Keywords: Continuum axiom, Weak convergence of r.v.'s, Medial limit

Nature: Exposition

Retrieve article from Numdam

XII: 01, 1-19, LNM 649 (1978)

Une version probabiliste d'un théorème d'interpolation de G. Stampacchia (Martingale theory, Functional analysis)

This theorem is similar to the Marcinkievicz interpolation theorem, in the sense that at one endpoint a weak $L^p$ inequality is involved, but at the other endpoint the spaces involved are some $L^p$ and $BMO$. It concerns linear operators only, not sublinear ones like the Marcinkiewicz theorem. A closely related result, concerning the discrete-time case, had been proved earlier by Stroock,

Keywords: Interpolation, $BMO$

Nature: Original

Retrieve article from Numdam

XII: 12, 98-113, LNM 649 (1978)

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

XIV: 25, 220-222, LNM 784 (1980)

Caractérisation d'ensembles convexes de $L^1$ ou $H^1$ (Stochastic calculus, Functional analysis)

This is a new and simpler approach to the crucial functional analytic lemma in 1354 (the proof that semimartingales are the stochastic integrators in $L^0$). That is, given a convex set $K\subset L^1$ containing $0$, find a condition for the existence of $Z>0$ in $L^\infty$ such that $\sup_{X\in K}E[ZX]<\infty$. A similar result is discussed for $H^1$ instead of $L^1$

Comment: This lemma, instead of the original one, has proved very useful in mathematical finance

Keywords: Semimartingales, Stochastic integrals, Convex functions

Nature: Original

Retrieve article from Numdam

XVI: 11, 153-158, LNM 920 (1982)

Interpolation entre espaces d'Orlicz (Functional analysis)

This is an exposition of Calderon's complex interpolation method, in the case of moderate Orlicz spaces, aiming at its application in 1609

Keywords: Interpolation, Orlicz spaces, Moderate convex functions

Nature: Exposition

Retrieve article from Numdam