Volume 6, Issue 4

This is essentially a survey on tube domains over irreducible symmetric cones or their bounded realizations. It includes the fact that in rank 2 these inequalities have now been proved for the whole range of possible exponents p , due to recent work of Bourgain and Demeter. We also discuss L p estimates with loss for the bounded domains and conclude with some remarks and questions related to Bloch functions.

So far, the uncertainty principles for solvable non-exponential Lie groups have been treated only in few cases. The first author and Kaniuth produced an analogue of Hardy's theorem for a diamond Lie group, which is a semi-direct product of ℝ d with the Heisenberg group ℍ2d+1${\mathbb {H}_{2d+1}}$ In this setting, we formulate and prove in this paper some other uncertainty principles (Miyachi, Cowling–Price and L p - L q Morgan). This allows us to provide a refined version of Hardy's theorem and to study the sharpness problems.

We prove various versions of uncertainty principles for a certain Fourier transform ℱ A . Here, A is a Chébli function (that is, a Sturm–Liouville function with additional hypotheses). We mainly establish an analogue of Beurling's theorem, and its relatives such as theorems of Gelfand–Shilov type, of Morgan type, of Hardy type, and of Cowling–Price type, for ℱ A and relate them to the characterization of the heat kernel corresponding to ℱ A . Heisenberg's and local uncertainty inequalities are also proved.

It was one of great successes of Kirillov's orbit method to see that the unitary dual of an exponential Lie group is in bijective correspondence with the orbit space associated with the linear dual of the Lie algebra of the group in question. To show that this correspondence is an homeomorphism turned out to be unexpectedly difficult. Only in 1994 H. Leptin and J. Ludwig gave a proof using the notion of variable groups. In this article their proof in the case of completely solvable Lie group is reorganized, some “philosophy” and some new arguments are added. The purpose is to contribute to a better understanding of this proof.

We consider a random matrix X uniformly distributed on an orbit for the action of the orthogonal group on the space of real symmetric matrices or of the unitary group on the space of Hermitian matrices. The problem is to evaluate the distribution of the eigenvalues of a compression of X . We give a survey about this question and present some new results. Baryshnikov's formula and Olshanski's determinantal formula are revisited, and a Markov–Krein type formula is established.