Revistes Catalanes amb Accés Obert (RACO)

Riesz transforms on generalized Heisenberg groups and Riesz transforms

F. Lust-Piquard


Let $1 < q < \infty$. We prove that the Riesz transforms $R_{k}=X_{k} L^{-\frac{1}{2}}$ on a generalized Heisenberg group $G$ satisfy $\left\|\left(\sum_{k=1}^{K}\left| R_{k}(f)\right| ^{2}\right)^{\frac{1}{2}}\right\| _{L^{q}(G)}\leq C(q,J)\left\| f\right\| _{L^{q}(G)}$ where $K$, $J$ are respectively the dimensions of the first and second layer of the Lie algebra of $G$. We prove similar inequalities on Schatten spaces $S^{q}(H)$, with dimension free constants, for Riesz transforms associated to commuting inner $*$-derivations $D_{k}$ and a suitable substitute of the square function. An example is given by the derivations associated to $n$ commuting pairs of operators $(P_{j},Q_{j})$ on a Hilbert space $H$ satisfying the canonical commutation relations [P$_{j},Q_{j}]=iI_{H}$.

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