## Vector-valued Fourier multipliers on symmetric spaces of the noncompact type.(English)Zbl 0828.43009

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, $$1 < p < \infty$$.
Reviewer: Zhu Fuliu (Hubei)

### MSC:

 43A85 Harmonic analysis on homogeneous spaces 22E20 General properties and structure of other Lie groups
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### References:

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