Spectral multipliers on Lie groups of polynomial growth

Abstract

Let L L be a left invariant sub-Laplacian on a connected Lie group G G of polynomial volume growth, and let { E λ , λ ⩾ 0 } \{ {E_\lambda },\lambda \geqslant 0\} be the spectral resolution of L L and m m a bounded Borel measurable function on [ 0 , ∞ ) [0,\infty ) . In this article we give a sufficient condition on m m for the operator m ( L ) = ∫ 0 ∞ m ( λ ) d E λ m(L) = \smallint _0^\infty m(\lambda )d{E_\lambda } to extend to an operator bounded on L p ( G ) , 1 > p > ∞ {L^p}(G),\;1 > p > \infty , and also from L 1 ( G ) {L^1}(G) to weak- L 1 ( G ) {L^1}(G) .