Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPaleys inequality for nonabelian groups
82
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This article studies Paleys theory for lacunary Fourier series on (nonabelian) discrete groups. The results unify and generalize the work of Rudin for abelian discrete groups and the work of Lust-Piquard and Pisier for operator valued functions, and provide new examples of Paley sequences and $Lambda(p)$ sets on free groups.
rate research
Read More
The main result is a Paleys theory for lacunary Fourier series using semigroup-BMO and $H^1$ spaces. This interpretation allows an extension of Paleys theory to general discrete groups, complementing the work of Rudin for abelian groups with a total order, and Lust-Piquard and Pisiers work for lacunary Fourier series with operator-valued coefficients.
Henstock and Macbeath asked in 1953 whether the Brunn-Minkowski inequality can be generalized to nonabelian locally compact groups; questions in the same line were also asked by Hrushovski, McCrudden, and Tao. We obtain here such an inequality and prove that it is sharp for helix-free locally compact groups, which includes real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc. The proof follows an induction on dimension strategy; new ingredients include an understanding of the role played by maximal compact subgroups of Lie groups, a necessary modified form of the inequality which is also applicable to nonunimodular locally compact groups, and a proportionated averaging trick.
The present note surveys my research related to generalizing notions of abelian group theory to non-commutative case and applying them particularly to investigate fundamental groups.
Given a Lie group $G$ with finitely many components and a compact Lie group A which acts on $G$ by automorphisms, we prove that there always exists an A-invariant maximal compact subgroup K of G, and that for every such K, the natural map $H^1(A,K)to H^1(A,G)$ is bijective. This generalizes a classical result of Serre [6] and a recent result in [1].
We prove that for $alpha in (d-1,d]$, one has the trace inequality begin{align*} int_{mathbb{R}^d} |I_alpha F| ;d u leq C |F|(mathbb{R}^d)| u|_{mathcal{M}^{d-alpha}(mathbb{R}^d)} end{align*} for all solenoidal vector measures $F$, i.e., $Fin M_b(mathbb{R}^d,mathbb{R}^d)$ and $operatorname{div}F=0$. Here $I_alpha$ denotes the Riesz potential of order $alpha$ and $mathcal M^{d-alpha}(mathbb{R}^d)$ the Morrey space of $(d-alpha)$-dimensional measures on $mathbb{R}^d$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
