Spectral multipliers for sub-Laplacians on amenable Lie groups with exponential volume growth
Let \(G\) be a Lie group, \(\mathcal G\) be its Lie algebra realized by left invariant vector fields on \(G\). Suppose that \(\eta_1,\dots,\eta_m\in\mathcal G\) generate \(\mathcal G\) as a Lie algebra and set \(\Delta=-\sum_{k=1}^m\eta^2_k\); this is the left invariant sub-Laplacian on \(G\). It extends to a selfadjoint operator in \(L^2(G,\sigma)\), where \(\sigma\) is the right invariant Haar measure. Hence \(f(\Delta)\) is well defined in \(L^2(G,\sigma)\) for any bounded Borel \(f\). If \(f(\Delta)\) is bounded in \(L^p(G,\sigma)\), \(p\neq2\), then \(f\) is called an \(L^p\)-multiplier for \(\Delta\). There are examples of \(\Delta\) and solvable \(G\) such that any \(L^p\)-multiplier extends holomorphically to a neighbourhood of a point of the \(L^2\)-spectrum. In this paper, the author shows that some differentiability conditions are sufficient for \(f\) with compact support to be an \(L^p\)-multiplier. The group \(G\) is assumed to be the semidirect product of a nilpotent normal subgroup \(N\) with one parameter group \(R\) and a compact group \(K\), where the action of \(R\) in \(N\) is generated by a derivation \(D\) whose eigenvalues lie in the open right halfplane. Let \(Q\) be the trace of \(D\). If \(f\) belongs to the Sobolev class \(H^\kappa({R})\), where \(\kappa>{{Q\over{2p}}+{5\over2}}\), \(1\leq p\leq\infty\), then \(f\) is an \(L^p\)-multiplier. The operator \(f(\Delta)\) is a convolution by right with some function in \(L^1(G,\sigma)\). The problem is reduced to the estimation of the heat kernel \(\phi_z\) corresponding to \(\Delta\) which is defined by \(e^{-z\Delta}f=f*\phi_z\). If \(N\) is the 3-dimensional Heisenberg group, \(K\) is trivial and \(D\) has nonnegative eigenvalues \(\alpha,\beta,\alpha+\beta\), then the assertion holds for \(f\in H^{s+{1\over2}+\varepsilon}\), where \(s={{\alpha+\beta}\over{\min\{\alpha,\beta\}}}\) if \(\alpha,\beta\in[{1\over3},3]\) and \(s=6\) otherwise, \(\varepsilon>0\) (thus \(s\) is bounded).
scientific article; zbMATH DE number 2078042