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
