Vector-valued Fourier multipliers on symmetric spaces of the noncompact type
Let \(X = G/K\) be a noncompact Riemannian symmetric space and \({\mathfrak g} = {\mathfrak k} + {\mathfrak p}\) be the Cartan decomposition of the Lie algebra \({\mathfrak g}\) of \(G\), where \({\mathfrak k}\) is the Lie algebra of \(K\). Fix a maximal Abelian subspace \({\mathfrak a}\) in \({\mathfrak p}\), denote by \({\mathfrak a}^*\) (resp. \({\mathfrak a}^*_\mathbb{C})\) the real (resp. complex) dual of \({\mathfrak a}\). Let \(W\) be the Weyl group of \({\mathfrak g}\) and \(\rho\) be the demi-sum of positive roots of \(({\mathfrak g}, {\mathfrak a})\). Denote by \(H(x)\) the Iwasawa projection of \(x \in G\) in \({\mathfrak a}\) and \(a = \dim {\mathfrak a}\); let \({\mathcal I}^\nu = {\mathfrak a}^* + \sqrt {- 1} Co (W \cdot 2 \nu \rho)\) be a tube in \({\mathfrak a}^*_\mathbb{C}\). The author introduces some function spaces \(H^{\sigma, \tau}_{2,r} ({\mathcal I}^{1/2}, \ell_2)\), which are analogous and related to J. Ph. Anker's weighted Sobolev space \(H^{\sigma, \tau}_{q,r}\). Suppose \(K(x) = \{k_{i,j} (x) \}^\infty_{i,j = 1}\) is an infinite matrix with complex coefficients which are locally integrable Schwartz functions on \(X\), and \(f = \{f_i\}^\infty_{i = 1}\) with \(f_i \in C^\infty_c (X)\). Then the author considers the operator \({\mathcal K} f(x) = \int_G K(y^{- 1} x) f(y) dy\) and uses methods analogous to those of Anker to prove the following multiplier theorem of Hörmander-Michlin type for \(p\)-integrable functions which take values in sequence spaces \(l_q\): Let \(m_{i,j} (\lambda) = {\mathcal H} (k_{i,j}) (\lambda) = \int_G k_{i,j} (x)e^{- (\sqrt{-1} \lambda + \rho) H(x^{- 1} k)} dx\) be the Helgason- Fourier transform of \(k_{i,j}\), and let \(\sigma > n = \dim X\) such that \(M = \{m_{i,j} \}^\infty_{i,j = 1} \in H^{\sigma/2, - a/2}_{2, \infty} ({\mathcal I}^{1/2}, l_2)\), (i) then \({\mathcal K}\) is a continuous linear operator in the space \(L_p (X, l_2)\) for all \(p\) with \(1 < p < \infty\) and \((*)\) \(|{\mathcal K} |\leq c |M \mid H_{2, \infty}^{\sigma/2, - a/2} ({\mathcal I}^{1/2}, l_2) |\); (ii) if additionally \(k_{i,j} = 0\) for \(i \neq j\), then \({\mathcal K}\) is a continuous linear operator in the space \(L_p (X, l_q)\) for all \(p,q\) with \(1 < p\), \(q < \infty\), and \({\mathcal K}\) satisfies inequality \((*)\). -- As an application of the above theorem, the author proves an inequality of Littlewood-Paley type for \(L_p\)-functions, \(1 < p < \infty\).
scientific article; zbMATH DE number 733134
scientific article