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 It has been superseded by the theory of $\Gamma_2$ due to Bakry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
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
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
