Homogeneous Besov spaces on locally compact Vilenkin groups
We prove that, for a distinguished laplacian on an Iwasawa AN group corresponding to a complex semisimple Lie group, a Hörmander type multiplier theorem holds. Our argument is based on Littlewood-Paley theory.
Let L¹(G)** be the second dual of the group algebra L¹(G) of a locally compact group G. We study the question of involutions on L¹(G)**. A new class of subamenable groups is introduced which is universal for all groups. There is no involution on L¹(G)** for a subamenable group G.
Let L be a homogeneous sublaplacian on the 6-dimensional free 2-step nilpotent Lie group on three generators. We prove a theorem of Mikhlin-Hörmander type for the functional calculus of L, where the order of differentiability s > 6/2 is required on the multiplier.
There are several algebras associated with a locally compact group 𝓖 which determine 𝓖 in the category of topological groups, such as L¹(𝓖), M(𝓖), and their second duals. In this article we add a fairly large family of locally convex algebras to this list. More precisely, we show that for two infinite locally compact groups 𝓖₁ and 𝓖₂, there are infinitely many locally convex topologies τ₁ and τ₂ on the measure algebras M(𝓖₁) and M(𝓖₂), respectively, such that (M(𝓖₁),τ₁)** is isometrically...
We give a Hörmander-type sufficient condition on an operator-valued function M that implies the Lp-boundedness result for the operator TM defined by (TMf)^ = Mf^ on the (2n + 1)-dimensional Heisenberg group Hn. Here ^ denotes the Fourier transform on Hn defined in terms of the Fock representations. We also show the H1-L1 boundedness of TM, ||TMf||L1 ≤ C||f||H1, for Hn under the same hypotheses of Lp-boundedness.
A linear map T from a Banach algebra A into another B preserves zero products if T(a)T(b) = 0 whenever a,b ∈ A are such that ab = 0. This paper is mainly concerned with the question of whether every continuous linear surjective map T: A → B that preserves zero products is a weighted homomorphism. We show that this is indeed the case for a large class of Banach algebras which includes group algebras. Our method involves continuous bilinear maps ϕ: A × A → X (for some Banach space X) with the property...