Paley's inequality for discrete groups
type: Item
instance of: scholarly article
title: Paley's inequality for discrete groups
author: Chian Yeong Chuah
author: Yazhou Han
author: Zhen-Chuan Liu
author: Tao Mei
published in: The Journal of Fourier Analysis and Applications
publication date: 2022-10-07T00:00:00Z
review text: The article under the review studies the Paley's theory of lacunary Fourier series for von Neumann algebra of discrete groups. Let \(H^1(\mathbb{T})\) be the Hardy space on the unit circle \(\mathbb{T}\) defined by \[ H^1(\mathbb{T})=\{f\in L^1(\mathbb{T}):\|f\|_{H^1(\mathbb{T})}=\|f\|_{L^1(\mathbb(T))}+\|H(f)\|_{L^1(\mathbb{T})}<\infty\}, \] where \(H^1(f)\) denotes the Hilbert transform of \(f\). Consider a sequence \((j_k)_{k\in\mathbb{N}}\) of integers which is lacunary à la Hadamard, i.e. there exists \(\delta>0\) such that for all \(n\in\mathbb{N}\), \(\vert j_{k+1}\vert>(1+\delta)\vert j_k\vert\). Paley's theory says that for any \((c_k)_{k\in\mathbb{N}}\in\ell^2\), \[ \left(\sum\vert c_k\vert^2\right)^{1/2}\simeq_\delta\inf \{\|f\|_{H^1(\mathbb{T})}:f\in H^1(\mathbb(T)),\hat{f}(j_k)=c_k\},\tag{1} \] where \(\hat{f}\) denotes the Fourier transform of \(f\). By the Fefferman-Stein's \(H^1\)-BMO duality theory (1) has the following equivalent formulation \[ \left(\sum\vert c_k\vert^2\right)^{1/2}\simeq_\delta\|\sum c_kz^{j_k}\|_{BMO(\mathbb{T})}, \] where \(\|\cdot\|_{BMO(T)}\) denotes the bounded mean oscillation norm. To extend Paley's theory to the setting of nonabelian discrete groups the authors make use of an interpretation of Paley's theory in the semigroup language. For each \(t>0\), let \(P_t\) denote the Poisson integral operator that sends \(e^{ik\theta}\) to \(e^{-\vert k\vert t}e^{ik\theta}\). For \(f\in L^1(\mathbb{T})\), \[ \|f\|_{BMO(T)}\simeq\sup_{t>0}\|P_t\big[\vert f-P_t(f)\vert^2\big]\|_{L^\infty(\mathbb{T})}^{1/2},\tag{2} \] \[ \|f\|_{H^1(\mathbb{T})}\simeq\|\left(\int_0^\infty\lvert\frac{\partial}{\partial t}P_t(f)\rvert^2tdt\right)^{1/2}\|_{L^1(\mathbb{T})}.\tag{3} \] Consider a discrete group \(G\). A function \(\psi:G\to\mathbb{R}^+\) is called a conditionally negative definite length on \(G\) if \begin{itemize} \item[(i)] \(\psi(g)=0\) if and only if \(g=e\). \item[(ii)] \(\psi(g)=\psi(g^{-1})\), for all \(g\in G\). \item[(iii)] \(\sum…
reviewed by: Safoura Zadeh
Mathematics Subject Classification ID: 46L52
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 42A55
zbMATH DE Number: 7597565
zbMATH Keywords: Paley's inequality
zbMATH Keywords: semigroup of operators
zbMATH Keywords: group von Neumann algebra
zbMATH Keywords: noncommutative \(L^p\) spaces
zbMATH Keywords: free group
MaRDI profile type: Publication
full work available at URL: s00041-022-09971-1
OpenAlex ID: W4297094260
cites work: Analyticity in Operator Algebras
cites work: Q3755956
cites work: Complex interpolation of noncommutative Hardy spaces associated with semifinite von Neumann algebras
cites work: Theorems of Hardy and Paley for Vector-Valued Analytic Functions and Related Classes of Banach Spaces
cites work: Q5447311
cites work: Q3719238
cites work: Noncommutative Khintchine inequalities in interpolation spaces of \(L_p\)-spaces
cites work: Harmonic analysis and BMO-spaces of free Araki--Woods factors
cites work: BMO-estimates for non-commutative vector valued Lipschitz functions
cites work: Harmonic analysis on quantum tori
cites work: Orders on Trees and Free Products of Left-ordered Groups
cites work: \(H^{\infty}\)-calculus for semigroup generators on BMO
cites work: An example of a non nuclear C*-algebra, which has the metric approximation property
cites work: On the best constants in noncommutative Khintchine-type inequalities
cites work: Bounded linear operators between \(C^*\)-algebras
cites work: Fourier analysis, Schur multipliers on $S^p$ and non-commutative Λ(p)-sets
cites work: No positive cone in a free product is regular
cites work: Sidon Sets are Proportionally Sidon with Small Sidon Constants
cites work: Q5492306
cites work: BMO spaces associated with semigroups of operators
cites work: Smooth Fourier multipliers on group von Neumann algebras
cites work: Algebraic Calderón-Zygmund theory
cites work: BMO estimates for lacunary series
cites work: Espaces BMO, inégalités de Paley et multiplicateurs idempotents
cites work: Non commutative Khintchine and Paley inequalities
cites work: Operator valued Hardy spaces
cites work: Tent spaces associated with semigroups of operators
cites work: Noncommutative 𝐻² spaces
cites work: Q4529324
cites work: The dual of noncommutative H^1
cites work: Sharp estimates in vector Carleson imbedding theorem and for vector paraproducts
cites work: On the lacunary coefficients of power series
cites work: Twisted Hilbert transforms vs Kakeya sets of directions
cites work: Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol
cites work: Interpolation between \(H^ p\) spaces and non-commutative generalizations. I
cites work: Multipliers and Lacunary Sets in Non-Amenable Groups
cites work: Q4215796
cites work: Real Interpolation between Row and Column Spaces
cites work: Grothendieck’s Theorem, past and present
cites work: Completely Sidon sets in discrete groups
cites work: Completely Sidon sets in \(C^*\)-algebras
cites work: Spectral Gap Properties of the Unitary Groups: Around Rider’s Results on Non-commutative Sidon Sets
cites work: Q4440813
cites work: THE NON-COMMUTATIVE KHINTCHINE INEQUALITIES FOR
cites work: A Markov dilation for self-adjoint Schur multipliers
cites work: Q3265794
cites work: Q5890171
cites work: Q5797063
cites work: Lacunary Fourier series for compact quantum groups
cites work: Q5490299
cites work: On Fourier coefficients and transforms of functions of two variables
DOI: 10.1007/S00041-022-09971-1
description: scientific article; zbMATH DE number 7597565