Free Hilbert transforms Article Swipe
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· 2017
· Open Access
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· DOI: https://doi.org/10.1215/00127094-2017-0007
· OA: W2393766649
We study analogues of classical Hilbert transforms as fourier multipliers on\nfree groups. We prove their complete boundedness on non commutative $L^p$\nspaces associated with the free group von Neumann algebras for all\n$1<p<\\infty$. This implies that the decomposition of the free group $\\F_\\infty$\ninto reduced words starting with distinct free generators is completely\nunconditional in $L^p$. We study the case of Voiculescu's amalgamated free\nproducts of von Neumann algebras as well. As by-products, we obtain a positive\nanswer to a compactness-problem posed by Ozawa, a length independent estimate\nfor Junge-Parcet-Xu's free Rosenthal inequality, a Littlewood-Paley-Stein type\ninequality for geodesic paths of free groups, and a length reduction formula\nfor $L^p$-norms of free group von Neumann algebras.\n
