Fourier multipliers on non-Riemannian symmetric spaces (Q1775424)

Let \(G\) be a non-compact semisimple connected Lie group with finite center and an involution \(\sigma\). Let \(H\) be an open subgroup of \(G^{\sigma}\), the fixed point group of \(\sigma\). So \(G/H\) is a semisimple symmetric space. Now assume that there is a Cartan involution \(\theta\) commuting with \(\sigma\) and denote by \(K\) the fixed point set for \(\theta\). Corresponding to the involutions we have the decompositions \(\mathfrak g = \mathfrak h\oplus \mathfrak q=\mathfrak k \oplus \mathfrak p\). Since the involutions commute we also have \(\mathfrak g = \mathfrak h\cap\mathfrak k\oplus\mathfrak h\cap\mathfrak p\oplus\mathfrak q\cap\mathfrak k\oplus\mathfrak q\cap\mathfrak p\). The author proceeds to construct a non-compact Riemannian form, \(G^d/ K^d\), of \(G/H\). Take a maximal split \(\theta\)-invariant Cartan subspace \(\mathfrak b\) of \(\mathfrak q\), i.e. such that \(\mathfrak a=\mathfrak b\cap\mathfrak p\) is maximal Abelian in \(\mathfrak q\cap\mathfrak p\). Set \(\mathfrak b^r = \mathfrak b\cap\mathfrak p+ i\mathfrak b\cap\mathfrak k\). Now \(\mathfrak b^r\) is a maximal split for \(\mathfrak g^d\). Let \(W\) be the Weyl group for the root system \(\sum (\mathfrak a,\mathfrak g)\). Let \(L^p(K\backslash G/H)\) be the space of \(K\)-invariant \(L^p\)-functions on \(G/H\). Let \(\psi\) be a \(W(\mathfrak b^r)\)-invariant function on the support of the Plancherel measure. Using the isomorphism between \(L^p(K \backslash G/H)\) and \(L^p(H^d \backslash G^d/K^d)\) the author defines an operator \(M_{\psi}\) from \(L^p(K \backslash G/H)\) to itself. It is then shown that \(\widehat{M_{\psi} f}(\pi) v = \psi(\lambda) \hat{f}(\pi)v\), where \(v\) is an \(H\)-fixed spherical distribution vector of type \(\lambda\) for the representation \(\pi\), and \(f \in L^p(K \backslash G/H)\). Let \({\mathcal T}\) be the strip \((\mathfrak b^r)^{\ast} + i \text{Conv}(W(\mathfrak b^r)\rho)\), where Conv denotes the convex hull and \(\rho\) represents the half sum of the positive roots. The main result proven in this paper is: Let \(\psi\) be a holomorphic function in the strip \({\mathcal T}\) and continuous up to the boundary satisfying the estimate \[ | \nabla^i \psi(\lambda) | \leq C(1 + | \lambda| )^{-i}, \lambda \in \bar{{\mathcal T}}, i < \left[{ n \over 2}\right] + 1, \] where \(\nabla\) is the gradient, then \(M_{\psi}\) is a bounded operator on \(L^p(K \backslash G/H)\) for \(1 < p < \infty\). An example is worked out for the case \(G = SL(2, {\mathbb R})\) that illustrates the theorem along with the ideas behind it. In the last section of the paper a certain set \(\text{Ũ}_p\) is defined and it is proven that if \(M_{\psi}\) is an \(L^p\)-multiplier then \(\psi(\lambda)\) has a partial holomorphic extension to the set \(\text{Ũ}_p\). The introduction of the paper contains an excellent explanation for the motivation behind the paper.