Spectral Multipliers on 2-step Stratified Groups, I

Abstract

Given a $2$-step stratified group which does not satisfy a slight strengthening of the Moore-Wolf condition, a sub-Laplacian $\mathcal{L}$ and a family $\mathcal{T}$ of elements of the derived algebra, we study the convolution kernels associated with the operators of the form $m(\mathcal{L},-i\mathcal{T})$. Under suitable conditions, we prove that: i) if the convolution kernel of the operator $m(\mathcal{L},-i\mathcal{T})$ belongs to $L^1$, then $m$ equals almost everywhere a continuous function vanishing at $\infty$ (`Riemann-Lebesgue lemma'); ii) if the convolution kernel of the operator $m(\mathcal{L},-i\mathcal{T})$ is a Schwartz function, then $m$ equals almost everywhere a Schwartz function.

References in corpus (7)

• Spectral theory for commutative algebras of differential operators on Lie groups
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• Spectral multiplier theorems of Euclidean type on new classes of 2-step stratified groups
• $L^p$ spectral multipliers on the free group $N_{3,2}$
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Cited by in corpus (3)

• Besov Spaces of Analytic Type: Interpolation, Convolution, Fourier Multipliers, Inclusions
• Functional Calculus on Non-Homogeneous Operators on Nilpotent Groups
• A sharp multiplier theorem for solvable extensions of Heisenberg and related groups