Spectral Multipliers on 2-Step Stratified Groups, II
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Spectral Multipliers on 2-Step Stratified Groups, II Mattia Calzi1 Received: 19 November 2018 © Mathematica Josephina, Inc. 2019
Abstract Given a graded group G and commuting, formally self-adjoint, left-invariant, homogeneous differential operators L1 , . . . , Ln on G, one of which is Rockland, we study the convolution operators m(L1 , . . . , Ln ) and their convolution kernels, with particular reference to the case in which G is abelian and n = 1, and the case in which G is a 2-step stratified group which satisfies a slight strengthening of the Moore–Wolf condition and L1 , . . . , Ln are either sub-Laplacians or central elements of the Lie algebra of G. Under suitable conditions, we prove that (i) if the convolution kernel of the operator m(L1 , . . . , Ln ) belongs to L 1 , then m equals almost everywhere a continuous function vanishing at ∞ (‘Riemann–Lebesgue lemma’); (ii) if the convolution kernel of the operator m(L1 , . . . , Ln ) is a Schwartz function, then m equals almost everywhere a Schwartz function. Keywords Spectral multipliers · Stratified groups · H-type groups · Rockland operators Mathematics Subject Classification Primary 22E30 · 43A32 · Secondary 22E25 · 43A20
1 Introduction Given a Rockland family1 (L1 , . . . , Ln ) on a homogeneous group G, following [32,39] (see also [15]) we define a ‘kernel transform’ K which to every measurable function m : Rn → C such that m(L1 , . . . , Ln ) is defined on D(G) associates a unique distribution K(m) such that m(L1 , . . . , Ln ) ϕ = ϕ ∗ K(m) 1 See Sect. 2 for precise definitions
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Mattia Calzi [email protected] Scuola Normale Superiore Pisa, Pisa, Italy
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M. Calzi
for every ϕ ∈ D(G). The so-defined kernel transform K enjoys some relevant properties, which we list below; see [32,39] for their proofs and further information. • there is a unique positive Radon measure β on Rn such that K(m) ∈ L 2 (G) if and only if m ∈ L 2 (β), and K induces an isometry of L 2 (β) into L 2 (G); • there is a unique χ ∈ L ∞ (Rn × G, β ⊗ ν), where ν denotes a Haar measure on G, such that for every m ∈ L 1 (β) K(m)(g) = m(λ)χ (λ, g) dβ(λ) Rn
for almost every g ∈ G; • K maps S(Rn ) into S(G). We consider also some additional properties of particular interest, such as (R L) if K(m) ∈ L 1 (G), then we can take m so as to belong to C0 (Rn ); (S) if K(m) ∈ S(G), then we can take m so as to belong to S(Rn ). In this paper, we shall investigate the validity of properties (R L) and (S) in two particular cases: that of a Rockland operator on an abelian group, and that of homogeneous sub-Laplacians and elements of the centre on an M W + group (cf. Definition 4.1). Here is a plan of the following sections. In Sect. 2, we recall the basic definitions and notation, as well as some relevant results proved in [15]. In Sect. 3, we then consider abelian groups, and characterize the Rockland operators which satisfy property (S) thereon. In Sect. 4, we prepare the machinery for the study of homogeneous subLaplacians and elements of the centre on M W + groups, referring to [15] f
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