Estimate of area functions on Riemannian manifolds with non-positive curvature (Q1335543)

Let \(M\) be a complete simply connected Riemannian manifold. The curvature of \(M\) is assumed to be negative, bounded, with bounded covariant derivatives up to order two. This paper contains \(L^ p\) estimates on \(M\) of the following form: \[ \| S_ \alpha f\|_ p \leq C \| f\|_ p, \] where \(f\) is a compactly supported \(C^ \infty\) function on \(M\), and the following notations are used: \[ F(x,t) = \int_ M f(y) P_ t (x,y) dv(y) \] (where \(x\), \(y\) are points of \(M\), \(dv\) is the Riemannian volume and \(P_ t\) is the Poisson kernel) and \[ S_ \alpha f(y) = \left ( \int_{d(x,y) \leq \alpha t} | \text{grad}_{x,t} F(x,t) |^ 2 {dv(x) dt\over \text{vol }B(x,t)}\right)^{1\over 2} \] (where \(d\) is the geodesic distance, \(\alpha\) is a positive constant and \(\text{vol }B(x,t)\) is the volume of the geodesic ball with centre \(x\) and radius \(t\)). The above \(L^ p\) estimates are proved under some additional technical assumptions. Improved results are obtained when \(M\) is a symmetric space of noncompact type.