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The nontangential maximal function associated to the Poisson semigroup for a compact Riemannian manifold is shown to be weak type (1,1) and LP bounded (p > 1). Let M be a compact Riemannian manifold with Riemannian measure dV and Laplace-Beltrami operator A. Let pt(x,y) be the fundamental solution to L = d 2/dt2 + A = 0; thus, for f continuous on M, Ptf(y) = fM pt(x, y)f(x) dV(x) satisfies LPtf = 0 and limt_0 Ptf(y) = f(y) for each y in M. It is a standard fact that when the class of boundary functions is suitably enlarged, almost everywhere convergence still holds. Indeed, a stronger theorem, proved by C. S. Herz in a more general context [3], is that the radial maximal function P. f = supt>o PtI ff is bounded on LP(M) for p > 1 and is of weak type (1,1) (cf. [4, p. 48] for the latter case). Let d(x, y) be the distance function on M associated to the Riemannian metric, let L.,(y) = {(y', t) E M x (0, oo) I d(y', y) 1) and of weak type (1,1). Standard technique then allows the conclusion of nontangential boundary convergence lim Ptf (y) = f (yo) dV-a.e. for f in LP(M) (p > 1), t-4+o (Y, t) Er (yo) although this result is otherwise easily obtainable by P.D.E. methods. Our method is to prove a two-sided inequality for the Poisson kernel pt (x, y) which immediately reduces our theorem to Herz's. This inequality is obtained by utilizing known behavior of the heat kernel in conjunction with the subordination of the Poisson semigroup to the heat diffusion semigroup, a special case of Bochner's principle of subordination. The author used these ideas in an earlier paper where M was either a compact semisimple Lie group (superseded by the current paper) or a symmetric space of noncompact type [1]; in both cases exact formulae for the heat kernel were used whereas in the present paper we use the Minakshisundaram-Pleijel asymptotic expansion. Received by the editors July 10, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 58G20, 43A32; Secondary 42B25, 31C12. Research supported in part by NSF grant DMS 8402202. ?1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page

Nontangential maximal functions over compact Riemannian manifolds