Sharp Fourier type and cotype with respect to compact semisimple Lie groups

Abstract

Abstract. Sharp Fourier type and cotype of Lebesgue spaces and Schatten classes with respect to an arbitrary compact semisimple Lie group are investigated. In the process, a local variant of the Hausdorff-Young inequality on such groups is given.

The norm of sums of independent noncommutative random variables in Lp(ℓ1)

“…degrees of the irreducible representations of G. That inequality is a particular case of the noncommutative version of the contraction principle given in [15]. This solves the problem posed in [2] not only for compact semisimple Lie groups, but for any non-finite topological compact group.…”

“…We may therefore consider this notion of cotype for arbitrary collections of random unitaries (U ) indexed by ∈ and where d represents the dimension of U . Examples for 's coming from groups are the commutative set of parameters ( 0 = N and d k = 1 for all k 1) which arises from any nonfinite abelian compact group and the set 1 = N with d k = k for k 1, which comes from the classical Lie group SU (2). As we shall see in this paper, these two sets of parameters are the most relevant ones in the theory.…”

“…The main topic of [2] is the sharp Fourier type and cotype of L p spaces. Given 1 p 2, it is showed that L p has sharp Fourier type p with respect to any compact semisimple Lie group.…”

“…Namely, the key point is a Hausdorff-Young type inequality for functions defined on a compact semisimple Lie group with arbitrary small support. However, the sharp Fourier cotype of L p for 1 p 2 is left open in [2]. Now we can solve it by using Corollary 6.5 and the following inequality:…”

Behaviour of the Brascamp–Lieb constant

“…We clarify that one of the two implicit inequalities in (6) is a consequence of the elementary lower semicontinuity of the Brascamp-Lieb constant. One possible application of Conjecture 1 would be in establishing best constants for local versions of Young's inequality for convolution on noncommutative Lie groups, which is intimately related with best constants for the Hausdorff-Young inequality; this topic has been studied by several authors (see, for example, [2,14,18] and a forthcoming paper by M. Cowling, A. Martini, D. Müller and J. Parcet). The Baker-Campbell-Hausdorff formula suggests that, for functions supported on small sets, convolution resembles convolution in R n .…”