Spectral multipliers for sub-Laplacians on solvable extensions of stratified groups

Abstract

Let $G=N⋊A$, where $N$ is a stratified group and $A=\mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous sub-Laplacians on $N$ and $A$ can be lifted to left-invariant operators on $G$ and their sum is a sub-Laplacian $\Delta$ on $G$. We prove a theorem of Mihlin-Hörmander type for spectral multipliers of $\Delta$. The proof of the theorem hinges on a Calderón-Zygmund theory adapted to a sub-Riemannian structure of $G$ and on $L^1$-estimates of the gradient of the heat kernel associated to the sub-Laplacian $\Delta$.

31 pages

References in corpus (2)

• Analysis on harmonic extensions of H-type groups
• Calderon Zygmund decompositions on amenable groups

Cited by in corpus (5)

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• A family of Hardy type spaces on nondoubling manifolds
• Riesz transforms on solvable extensions of stratified groups
• Riesz transforms on $ax+b$ groups
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