Paley's inequality for discrete groups
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)] \(\sum\overline{a_g}a_h\psi(g^{-1}h)\leq0\), for any finite collection of coefficients \(a_g\in\mathbb{C}\) with \(\sum_g a_g=0\). \end{itemize} A sequence \((h_k)_{k\in\mathbb{N}}\) of elements of \(G\) is called \(\psi\)-lacunary if there exists a constant \(\delta>0\) such that for any \(k>j\), \(\psi(h_k)\geq(1+\delta)\psi(h_j)\) and \(\psi(h_j^{-1}h_k)\geq\delta\psi(h_k)\). Let \(\lambda\) be the left regular representation of \(G\). Assume \((h_k)\) is a \(\psi\)-lacunary sequence. Given a sequence \((c_k)\) in \(\ell^2(\mathbb{C})\), we may view \(f=\sum c_k\lambda_{h_k}\) as a lacunary ``Fourier Series''. In the article under the review the authors study the Paley's theory of such \(f\)'s. For this end, they made use of noncommutative analogs of (2) and (3). Let \(\tau\) be the cannonical trace on the group von Neumann algebra of \(\hat{G}\). Let \(T_t:\lambda_g\mapsto e^{-t\psi(g)}\lambda_g\) be the semigroup of operators on the group von Neumann algebra \(\hat{G}\) associated with \(\psi\). For \(f=\sum_g c_g\lambda_g\in L^1(\hat{G})\), let \[ \|f\|_{BMO_c(\psi)}=\sup_{s>0}\|T_s\left[\lvert f-T_s(f)\rvert^2\right]\|^{1/2}. \] \[ \|f\|_{H^1_c(\psi)}=\tau\left[\left(\int_0^\infty\lvert\frac{\partial}{\partial s}T_s(f)\rvert^2 sda\right)^{1/2}\right], \] The authors show that Theorem 2.3. Suppose that \((h_k)_{k\in\mathbb{N}}\subseteq G\) is a \(\psi\)-lacunary sequence. Then, for any \(N\in\mathbb{N}\) and \(f=\sum_{k=1}^N c_k \lambda_{h_k}\) with \(c_k\in B(H)\), we have \[ \|f\|_{BMO_c(\psi)}^2\simeq_\delta\left\|\sum_{k,h_k\neq e}\vert c_k\vert^2\right\|. \] For any \((c_k)_{k\in\mathbb{N}}\) in \(S^1(\ell^2_c)\), \[ tr\left[\left(\sum_k\vert c_k\vert^2\right)^{1/2}\right]\simeq_\delta\inf\left\{(tr\otimes\tau)\left[\left(\int_0^\infty\vert\frac{\partial}{\partial s}T_sf\vert^2s ds\right)^{1/2}\right]:\tau(f\lambda_{h_k}^*)=c_k\right\}, \] where the infimum runs over all \(f\in L^1(B(H)\bar{\otimes}\hat{G})\). In particular, Theorem 0.1. Assume \((h_k)\) is a \(\psi\)-lacunary sequence. Then, for any sequence \((c_k)_{k\in\mathbb{N}}\) in \( \ell^2(\mathbb{C})\), the series \(\sum_{k\in\mathbb{N}} c_k\lambda_{h_k}\) converges in \(BMO_c(\psi)\) and \[ \left(\sum_{k\in\mathbb{N}}\lvert c_k\rvert^2\right)^{1/2}\simeq_\delta \|\sum_{k=1}^\infty c_k\lambda_{h_k}\|_{BMO_c(\psi)}. \] Moreover, \[ \left(\sum_k\lvert c_k\rvert^2\right)^{1/2}\simeq_\delta \inf\left\{\|f\|_{H^1_c(\psi)}:\hat{f}(h_k)=c_k,\text{ for all } k\in\mathbb{N}\right\}. \] Assume that \((G,\leq)\) is a countable discrete group with a bi-invariant total order, that is a total order which is invariant under the left and right multiplication. This is equivalent to say that \(G\) contains a normal subsemigroup \(G_+\) such that for \(G_-:=(G_+)^{-1}\), \[ G_+\cup G_-=G,\quad G_+\cap G_-=\{e\}. \] In this case, \(G_+=\{g\in G:e\leq g\}.\) As it is explained in Section 4 of the article, free groups have bi-invariant total orders. A subset \(E\subseteq G_+\) is called lacunary if there exists a constant \(K\) such that \[ N(E):=\sup_{g\in G_+}\#\{h\in E: g\leq h\leq g^2\}\leq K. \] Consider \(\mathbb{F}_2\), the nonabelian free group with two generators \(a, b\). As it is shown in Section 4, among other examples, the subset \(\left\{a^{2^k}b^{2^k}a^{2^k}b^{-2^k}:k\in\mathbb{N}\right\}\) is a lacunary subset of \(\mathbb{F}_2\). In Section 3, the authors generalise and unify the result of \textit{W. Rudin} [Fourier analysis on groups. Paperback edition. New York etc.: John Wiley \&| Sons (1990; Zbl 0698.43001), Section 8.6] and \textit{F. Lust-Piquard} and \textit{G. Pisier} [Ark. Mat. 29, No. 2, 241--260 (1991; Zbl 0755.47029)]. Theorem 3.1. Let \((\mathcal{M},tr)\) be a semifinite von Neumann algebra. Assume that \(E\) is a lacunary subset of \(G_+\). Then, for any sequence \((c_k)_{k\in\mathbb{N}}\subseteq L^1(\mathcal{M})\) and any sequence \((g_k)_{k\in\mathbb{N}}\subseteq E\), we have \[ \|(c_k)_{k\in\mathbb{N}}\|_{L^1(\mathbb{M},\ell^2_{cr})}\simeq\inf \left\{(tr\otimes\tau)(\vert f\vert):f\in L^1(\mathcal{M}\bar{\otimes}\hat{G}), \hat{f}(g_k)=c_k, \hat(f)(g)=0, \forall g<e \right\}. \] In particular, when \(\mathcal{M}=\mathbb{C}\) we have Theorem 0.2. For any sequence \((c_k)_{k\in\mathbb{N}}\) in \(\mathbb{C}\), and any sequence \((h_k)_{k\in\mathbb{N}}\) in a lacunary subset \(E\subseteq G_+\), we have \[ \inf\left\{\tau(\lvert f\rvert):\hat{f}(h_k)=c_k, supp(\hat{f})\subseteq G_+\right\}\simeq\left(\sum\lvert c_k\rvert^2\right)^{1/2}. \] In Remark 3.4, the authors ask if, in view of \textit{G. Pisier} and \textit{ Ricard}'s argument [J. Inst. Math. Jussieu 16, No. 5, 1103--1123 (2017; Zbl 1385.46045)], Theorem 3.1 can be pushed to include \(p<1\).
scientific article; zbMATH DE number 7597565
scientific article