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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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