On a sharp inequality of L. Fontana for compact Riemannian manifolds

Abstract

In this paper, we refine a sharp inequality established by Fontana (Comment. Math. Helvetici 68:415–454, 1993). Precisely, for any compact Riemannian manifold without boundary, there exists a uniform constant depending on positive lower bound of the injectivity radius, upper bound of the diameter, and bounds of the curvature tensor and its covariant derivatives such that Fontana’s inequality holds. Also we establish an analog on oriented compact Riemannian manifold with smooth boundary. Though our method is based on the argument of L. Fontana and thereby closely on that of Adams (Ann. Math. 128:385–398, 1988), some technical difficulties must be smoothed. For the first problem, we need a uniform elliptic estimate on manifolds; While for the second problem, we use the Green function for a larger manifold with boundary to represent functions supported in the original one.