Comparaison des champs de vecteurs et des puissances du Laplacien sur une variété Riemannienne à courbure non positive. (French) Zbl 0605.58051

Lohoué, Noël

J. Funct. Anal. 61, 164-201 (1985).

Some classical inequalities of Calderon-Zygmund are generalized to smooth, n-dimensional, complete, simply connected Riemannian manifolds with non-positive sectional curvature. For example, let M be such a manifold whose curvature tensor and covariant derivatives up to order two are bounded, \(\Delta\) the Laplace operator on M, \(1<r<\infty\) and X a unit vector field on M. For \(f\in C_ 0^{\infty}(M)\) \((=\) the space of smooth complex-valued functions with compact support on M), let \(\| f\|^ r_ r=\int_{M}| f(x)|^ rd\sigma (x),\) where \(d\sigma\) denotes the Riemannian measure on M. Suppose that there exists a constant \(C>0\) such that \(\| f\|_ 2\leq C\| \Delta f\|_ 2\) for any \(f\in C_ 0^{\infty}(M)\). Under these assumptions, the author proves the existence of a constant C(r) such that \((I)\quad \| X(-\Delta)^{-}f\|_ r\leq C(r)\| f\|_ r\) for any \(f\in C_ 0^{\infty}(M)\). (I) is then extended to the case in which instead of X one considers \(2\ell\) unit vector fields on M (\(\ell \geq 1)\) satisfying a certain condition. The non-Euclidean symmetric spaces of non-compact type satisfy the hypotheses that provide the validity of (I) and of its generalization.

Reviewer: M.Craioveanu

MSC:

58J99	Partial differential equations on manifolds; differential operators
58J35	Heat and other parabolic equation methods for PDEs on manifolds

Keywords:

inequalities of Calderon-Zygmund; Riemannian manifolds; non-positive sectional curvature

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