\(L^p\) spectral multipliers on the free group \(N_{3,2}\) (Q2847826)

Consider the free 2-step nilpotent Lie group \(N_{3,2}\) and a homogeneous sublaplacian \(L\) on it. Since \(L\) is a self-adjoint operator on the Hilbert space \(L^2(N_{3,2})\), the Borel functional calculus for \(L\) can be defined, and \(F(L)\) is a bounded operator on \(L^2(N_{3,2})\) whenever \(F\) is a bounded Borel function \(\mathbb{R}\to\mathbb{C}\). The authors give a sufficient condition for \(F(L)\) to be bounded as an operator on \(L^p(N_{3,2})\) for \(p\in (1,\infty)\) in term of the smoothness of \(F\). A similar result for more general Lie groups has been proved before in [\textit{M. Christ}, Trans. Am. Math. Soc. 328, No. 1, 73--81 (1991; Zbl 0739.42010)] and [\textit{G. Mauceri} and \textit{S. Meda}, Rev. Mat. Iberoam. 6, No. 3--4, 141--154 (1990; Zbl 0763.43005)], but for the specific group \(N_{3,2}\), the authors have been able to weaken the requirement on the order of smoothness from \(s>9/2\) to \(s>6/2\), where \(9\) and \(6\) are, respectively, the homogeneous and topological dimensions of \(N_{3,2}\).