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 It has been superseded by the theory of $\Gamma_2$ due to Bakry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
1253Keywords: Littlewood-Paley theory,
Riesz transforms,
Brownian motion,
Inequalities,
Harmonic functions,
Singular integrals,
Carré du champ,
Infinitesimal generators,
Semigroup theoryNature: 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
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 champNature: Original Retrieve article from Numdam
XVI: 06, 95-132, LNM 920 (1982)
MEYER, Paul-André
Note sur les processus d'Ornstein-Uhlenbeck (
Malliavin's calculus)
With every Gaussian measure $\mu$ one can associate an Ornstein-Uhlenbeck semigroup, for which $\mu$ is a reversible invariant measure. When $\mu$ is Wiener's measure on ${\cal C}(
R)$, this semigroup is a fundamental tool in Malliavin's own approach to the ``Malliavin calculus''. See for instance Stroock's exposition of it in
Math. Systems Theory, 13, 1981. With this semigroup one can associate its generator $L$ which plays the role of the classical Laplacian, and the positive bilinear functional $\Gamma(f,g)= L(fg)-fLg-gLf$---leaving aside domain problems for simplicity---sometimes called ``carré du champ'', which plays the role of the squared classical gradient. As in classical analysis, one can define it as $\sum_i \nabla_i f\nabla i g$, the derivatives being relative to an orthonormal basis of the Cameron-Martin space. We may define Sobolev-like spaces of order one in two ways: either by the fact that $Cf$ belongs to $L^p$, where $C=-\sqrt{-L}$ is the ``Cauchy generator'', or by the fact that $\sqrt{\Gamma(f,f)}$ belongs to $L^p$. A result which greatly simplifies the analytical part of the ``Malliavin calculus'' is the fact that both definitions are equivalent. This is the main topic of the paper, and its proof uses the Littlewood-Paley-Stein theory for semigroups as presented in
1010,
1510Comment: An important problem is the extension to higher order Sobolev-like spaces. For instance, we could define the Sobolev space of order 2 either by the fact that $C^2f=-Lf$ belongs to $L^p$, and on the other hand define $\Gamma_2(f,g)=\sum_{ij} \nabla_i\nabla_j f \nabla_i\nabla_j g$ (derivatives of order 2) and ask that $\sqrt{\Gamma_2(f,f)}\in L^p$. For the equivalence of these two definitions and general higher order ones, see
1816, which anyhow contains many improvements over
1606. Also, proofs of these results have been given which do not involve Littlewood-Paley methods. For instance, Pisier has a proof which only uses the boundedness in $L^p$ of classical Riesz transforms.\par Another trend of research has been the correct definition of ``higher gradients'' within semigroup theory (the preceding definition of $\Gamma_2(f,g)$ makes use of the Gaussian structure). Bakry investigated the fundamental role of ``true'' $\Gamma_2$, the bilinear form $\Gamma_2(f,g)=L\Gamma(f,g)-\Gamma(Lf,g)-\Gamma(Lf,g)$, which is positive in the case of the Ornstein-Uhlenbeck semigroup but is not always so. See
1909,
1910,
1912Keywords: Ornstein-Uhlenbeck process,
Gaussian measures,
Littlewood-Paley theory,
Hypercontractivity,
Hermite polynomials,
Riesz transforms,
Test functionsNature: Exposition,
Original additions Retrieve article from Numdam
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 champNature: Original Retrieve article from Numdam
