Waldemar Hebisch

Spectral multipliers on exponential growth solvable Lie groups

Let M be a measure space and let L be a positive definite operator on L2(M). By the spectral theorem, for any bounded Borel measurable function F : [0, ∞) 7→ C the operator F (L)f = ∫ ∞ 0 F (λ)dE(λ)f is bounded on L2(M). We are interested in sufficient conditions on F for F (L) to be bounded on Lp(M), p 6= 2. We direct the reader to [1], [3], [4], [8], [9], [10], [12] and [13] for more background on various multiplier theorems. In this paper we assume F is compactly supported and have some smoothness (finite number of derivatives) and we consider only the case p = 1. Our measure space G is semidirect product of stratified nilpotent Lie group N and the real line. The operator L is (minus) sublaplacian on G. Our group has exponential volume growth. The earlier theory suggested that one needs holomorphic F for F (L) to be bounded on L1, however the recent results [5], [6], [7] showed that estimates on only a finite number of derivatives of F imply boundedness of F (L) on L1 on some solvable G of exponential growth. In this case we say that G (more precisely L) has Ckfunctional calculus. On the other hand, Christ and Muller give an example of a solvable Lie group on which F must be holomorphic. The problem is to find the condition on G (and possibly L) which decides whether G has a Ck-functional calculus or not. Here, our condition is in terms of roots of adjoint representation of the Lie algebra of G. Our groups are of “rank one”,

Spectral multipliers on metabelian groups

Let G be a Lie group, Xj right invariant vector fields on G, which generate (as a Lie algebra) the Lie algebra of G, L = -S Xj2. (...) In this paper we consider L1(G) boundedness of F(L) for (some) metabelian G and a distinguished L on G. Of the main interest is that the group is of exponential growth, and possibly higher rank. Previously positive results about higher rank groups were only about Iwasawa type groups. Also, our groups may be unimodular, so it is the second positive result (after [13]) about unimodular groups, and the first giving a family of examples.

Scattering of small solutions to generalized Benjamin-Bona-Mahony equation in several space dimensions

We show that the supremum norm of solutions with small initial data of the generalized Benjamin-Bona-Mahony equation ut-△ut=(b,▽u)+up(a,▽u)in x∊Rn,n≥2, with integer p≥3 , decays to zero like t-2/3 if n=2 and like t-1+6, for any δ0, if n≥3, when t tends to infinity. The proofs of these results are based on an analysis of the linear equation ut-△=(b,▽u)) and the associated oscillatory integral which may have nonisolated stationary points of the phase function.

A remark on discontinuous translation invariant functionals onL p (G) for certain compact groupsG

We show some criterion for the existence of discontinuous translation invariant linear functionals onC (G) andLp(G) for a compact groupG.

Transference for amenable actions

Suppose G acts amenably on a measure space X with quasi-invariant cr-finite measure m. Let a be an isometric representation of G on L p (X,dm) and μ a finite Radon measure on G. We show that the operator σ μ f(x) = ∫ G (σ(g)f)(x)dμ(g) has L p (X,dm)-operator norm not exceeding the L p (G)-operator norm of the convolution operator defined by μ. We shall also prove an analogous result for the maximal function M associated to a countable family of Radon measures μ n .