Noncommutative Riesz transforms -- a probabilistic approach

By Marius Junge, Tao Mei. 2008

83 citations recorded by the source corpus.

For $2\le p<\infty$ we show the lower estimates \[ \|A^{\frac 12}x\|_p \kl c(p)\max\{\pl \|\Gamma(x,x)^{{1/2}}\|_p,\pl \|\Gamma(x^*,x^*)^{{1/2}}\|_p\} \] for the Riesz transform associated to a semigroup $(T_t)$ of completely positive maps on a von Neumann algebra with negative generator $T_t=e^{-tA}$, and gradient form \[ 2\Gamma(x,y)\lel Ax^*y+x^*Ay-A(x^*y)\pl .\] As additional hypothesis we assume that $\Gamma^2\gl 0$ and the existence of a Markov dilation for $(T_t)$. We give applications to quantum metric spaces and show the equivalence of semigroup Hardy norms and martingale Hardy norms derived from the Markov dilation. In the limiting case we obtain a viable definition of BMO spaces for general semigroups of completely positive maps which can be used as an endpoint for interpolation. For torsion free ordered groups we construct a connection between Riesz transforms and the Hilbert transform induced by the order.

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Citations in the corpus

Singular integrals in quantum Euclidean spaces (2017, unknown): “Duality was developed by Mei [51, 52] and endpoint estimates for imaginary powers A of infinitesimal generators, noncommutative Riesz transforms or more general Fourier multipliers have been studied in [13, 40, 41, 42, 63].”
Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori (2015, methodology): “[35] M. Junge and T. Mei and J. Parcet.”
Smooth Fourier multipliers on group von Neumann algebras (2010, background): “11 has been essential at this point, see [25].”
Pseudo-Localization of Singular Integrals and Noncommutative Littlewood–Paley Inequalities (2008, methodology): “We refer to [6, 8, 10, 11, 17, 24, 27] and the references therein.”
Noncommutative Riesz transforms -- Dimension free bounds and Fourier multipliers (2014, background): “The main result in [33] provides lower estimates for p ≥ 2 and regular Markov semigroups satisfying Γ(2) ≥ 0.”
BMO spaces associated with semigroups of operators (2011, methodology): “ ∈ A. It is easy to see Γ2 ≥ 0 also implies Γ(Pvf,Pvf) ≤ PvΓ(f,f) for any v > 0. Denote by the gradient form associated with (Pt)t by Γ A 1 2 . We will need the following Lemma proved in [Jun] and [JM10]. We add a short proof for the convenience of the reader. 4 M. JUNGE AND T. MEI Lemma 1.1. (i)”
Noncommutative martingale deviation and Poincaré type inequalities with applications (2012, background): “We refer the readers to [26, 28, 30] for further details about the facts mentioned in this section.”