Global elliptic estimates on symmetric spaces

The domination properties of elliptic invariant diierential operators on symmetric spaces of noncompact type are investigated. Using the relation between parametrices and fundamental solutions on symmetric space we will show that the invariant diierential operator applied to a function can be uniformly estimated by function and an elliptic operator of higher order applied to the function in L p spaces for all 1 p 1. As a consequence, by algebraic methods we will give a simple unifying proof that derivatives of a function can be uniformly estimated by function and its Laplacian. 0 Introduction In this paper we will investigate the question of the domination of the (invariant) diierential operator by the elliptic operator of higher order on symmetric spaces of noncompact type. We are also interested in the estimates of the derivative of a function by the function and its Laplacian in L p spaces for 1 p 1. Similar estimates are known for p = 1 for manifolds with bounded curvature ((1]). On the other hand estimates for 1 < p < 1 are well known for Euclidean spaces ((6]) and are closely related to the question of the continuity of pseudo-diierential operators of order zero, which hold only locally on general manifolds. We will give a unifying algebraic proof for global estimates in L p-space for all 1 p 1 on symmetric spaces of the noncompact type. In Section 2 we discuss local integrability properties of distributional kernels of pseudo diierential operators on manifolds and x the notation. It will be applied to establish a relation between parametrices and fundamental solutions of elliptic operators on symmetric space in Section 3 (Lemma 3). Based on this lemma, we will prove the domination property for invariant diierential operators (Theorem 1). Then, reformulating the problem in Lie group terms, we will show how the invariance condition can be dropped for the rst order diierential operators reducing it to a problem on Lie group (Theorem 2). We will give a simple proof of it in the last section. The general theory of the second order diierential operators on Lie groups can be found in 5], 7]. I would like to thank E.G.F.Thomas for drawing my attention to the subject and many valuable discussions. I would also like to thank J.J.Duistermaat for broadening my understanding of the topic and J.A.C.Kolk for his remarks about contents and presentation of the material.