Geometric invariants of locally symmetric spaces and multiplier theorems (Q1000596)

An analogous of the Hormander-Mikhlin multiplier theorem is proven, first in an abstract framework and then for the locally symmetric spaces. One considers a Lie group \(G\) which is semi-simple, non-compact, connected, with finite center, and \(K\) its maximal compact subgroup. Then, the symmetric space \(G/K\) is introduced and the convolution operator \(T_{m}\) is defined. In the first part of the paper it is supposed that \(G\) is simple and one considers a space \(Y\) over that \(G\) acts transitively. As an example, one chooses \(Y=G/H\), where \(H\) is a closed subgroup of \(G\) and, in particular, the symmetric space \(G/K\). It is proven that the operator \(T_{m}\) is bounded in \(L^{p}(Y)\), using the property that for a simple group the matrix coefficients of the spherical representations belong to \(L^{p}(G)\). In the second part, it is supposed that G is semi-simple, non-compact, connected, with finite center, and one considers a discrete subgroup \(\Gamma\) without torsion and the locally symmetric space \(M=\Gamma/X\). Then, the multiplier theorem is proven in this case for the convolution operator \(S_{m}\) over \(L^{p}(M)\), considering separately the local and infinite parts of the operator.