We derive $L_p$-Poincar\'{e} inequalities for quantum Markov semigroups on finite von Neumann algebras, from a spectral gap (PI) or from a modified logarithmic Sobolev inequality (MLSI). For $p\ge2$, the corresponding coefficients have optimal order $O(p)$ and $O(\sqrt p)$, respectively. Both estimates follow from Beckner inequalities by integration along semigroups. We also establish a non-tracial extension to quantum Markov semigroups satisfying GNS detailed balance. As a direct byproduct, we apply $L_p$-Poincar\'{e} inequalities to Schatten-valued functions on graphs in order to obtain the Schatten distortion of expanders.
We prove a noncommutative maximal ergodic inequality for averages along polynomial sequences. Let $\gamma$ be a trace-preserving automorphism of a semifinite von Neumann algebra $(\mathcal N,\tau)$. We show that the associated polynomial averages \begin{equation*} A_Nf:=\frac1N\sum_{n=1}^N\gamma^{P(n)}(f), \qquad N\in\...
Gui-Xiang Hong, Wen-Bo Li, É. Ricard et al.· 0 citations
For every $1<p<\infty$, with $p^*=\max\{p,p/(p-1)\}$, we determine the sharp Hilbert norm $C_p=\cot(\pi/(2p^*))$ on $L^p(\mathbb{R};\mathcal{S}_p)$ and the sharp uniform bound $D_p=\csc(\pi/p)$ for retained and strict Riesz projections of maximal subdiagonal algebras in finite tracial von Neumann algebras. The Hilbert...
We study the exact $L^p$ norms for a number of maximal operators, which are prominent in harmonic analysis, semigroup theory, and probability. We prove that a strongly continuous positive symmetric sub-Markovian semigroup $(T_t)_{t\ge0}$ on a nonatomic $\sigma$-finite measure space $(X,\mathcal F,\mu)$ has maximal $L^p...
Valentina Ciccone, M. Hormozi, José Madrid et al.· 1 citation· ⚡1
We prove finiteness and Poincar\'e duality for pro-\'etale $\mathbf{Q}_p$-local systems on proper $p$-adic rigid-analytic spaces in the framework of Banach--Colmez spaces and establish their optimal cohomological vanishing bounds. As a consequence, we also obtain ordinary finiteness and duality for their arithmetic pro...
Shi-Zhang Li, Wiesława Nizioł, Emanuel Reinecke et al.· 0 citations
Let $F$ be the unramified quadratic extension of $\mathbb{Q}_p$, and let $P_n(T)$ denote the polynomials arising in the $p$-adic Fourier theory of Schneider and Teitelbaum. We determine explicitly the norms of the functions $P_n(x\Omega)$ on the Banach spaces of locally $F$-analytic functions of fixed order, where $\Om...
We consider a family of non-local and non-convex functionals depending on three real parameters $(\gamma,p,\lambda)$, which was introduced by H. Brezis, A. Seeger, J. Van Schaftingen and P. L. Yung. In the case $\gamma<-p$ we prove a lower bound for the Gamma-liminf of these functionals as $\lambda\to 0^+$ and a Poinca...
Nicola Picenni· 1 citation· ⚡1
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