Multiplicateurs de Mikhlin pour une classe particulière de groupes non-unimodulaires. (Mikhlin multipliers for a particular class of non-unimodular groups.)
Let \(G\) be a connected real Lie group, and let \(X_1, \dots, X_n\) be left invariant vector fields on \(G\) satisfying a certain Hörmander's condition. Let \(\Delta=-\sum^n_{ j=1} X^2_j\) be the associated sub-Laplacian and \(\Delta= \int^\infty_0 \lambda dE_\lambda\) its spectral decomposition. Then, for any bounded measurable function \(m\) on \([0,\infty)\), the operator \(m(\Delta) =\int^\infty_0 m(\lambda) dE_\lambda\) is a bounded linear operator on \(L^2 (G)\). The question of which conditions on \(m\) guarantee that \(m(\Delta)\) is bounded on \(L^p (G)\) for \(1\leq p\leq\infty\), has long been a matter of interest and has been studied by several authors for various types of groups \(G\). When \(G\) is of polynomial growth, in particular, when \(G\) is a stratified nilpotent Lie group, the results are quite satisfying and generalize the classical theorems for \(\mathbb{R}^d\) due to Hörmander and Mikhlin. However, the above problem is considerably more difficult when \(G\) is of exponential growth. Some special results have recently been obtained, e.g. for extensions of the Heisenberg groups by \(\mathbb{R}\). Let \({\mathfrak n}\) be a stratified real nilpotent Lie algebra of rank \(r\), that is, \({\mathfrak n}\) decomposes into a direct sum \({\mathfrak n}= V_1\oplus \cdots \oplus V_r\) of linear subspaces such that \(V_1\) generates the Lie algebra \({\mathfrak n}\), \([V_1,V_r]=0\) and \([V_1,V_j] \subseteq V_{j+1}\), \(1\leq j\leq r-1\). In the paper under review the author studies the semi-direct product group \(G=\mathbb{R} \ltimes N\), where \(N=\exp{\mathfrak n}\) and \(\mathbb{R}\) acts on \({\mathfrak n}\) by \(t\cdot (v_1 \oplus \cdots \oplus v_r) =e^{t_{v_1}} \oplus \cdots \oplus e^{rt} v_r\). Let \(X_1, \dots, X_m\) denote the system of left invariant vector fields associated to a basis of \(V_1\), and consider the sub-Laplacian \(\Delta= -(X^2_1 +\cdots +X^2_m+ {\partial^2 \over \partial t^2})\) on \(G\). It is shown that if \(m\) has compact support on \([0,\infty)\) and belongs to the Sobolev space \(H^{2+\varepsilon} (\mathbb{R})\), \(\varepsilon>0\), then the operator \(m(\Delta)\) is bounded on \(L^p(G)\) for \(1\leq p\leq \infty\). The main tool to establish this result is an \(L^1\)-norm estimate for the heat kernel \(\varphi_{t+is}\), \(t>0\), \(s\in\mathbb{R}\), associated to the semigroup \(e^{-t\Delta}\), \(t>0\). In particular, for some constant \(C>0\), \(\|\varphi_{1+is} \|_{L^1} \leq C(1+ | s |)^{3/2}\) for all \(s\in\mathbb{R}\).
scientific article; zbMATH DE number 1220348
scientific article