Multilinear Spectral Multipliers on Besov and Triebel–Lizorkin Spaces on Lie Groups of Polynomial Growth

Abstract

In this paper, on Lie groups of polynomial growth G, we prove the boundedness of multilinear spectral multipliers from the product of Besov spaces $B_{p_1,q_1}^{s_1}(G)\times B_{p_2,q_2}^{s_2}(G) \times \cdots \times B_{p_N,q_N}^{s_N}(G)$ to Lebesgue spaces $L^p(G)$ with $p_1, \ldots ,p_N,q_1, \ldots ,q_N,p\geqslant 1$ and $s_1, \ldots ,s_N\in {\mathbb {R}}$. Then we prove the boundedness from the product of Triebel–Lizorkin spaces $T_{p_1,q_1}^{s_1}(G)\times T_{p_2,q_2}^{s_2}(G) \times \cdots \times T_{p_N,q_N}^{s_N}(G)$ to Lebesgue spaces $L^p(G)$ with $p_1, \ldots ,p_N,q_1, \ldots ,q_N>1$, $p\geqslant 1$, $s_1, \ldots ,s_N\in {\mathbb {R}}$.