Nous établissons une inégalité entre les sommes de Cesáro et la fonction maximale associées á une fonction définie sur la sphére, et nous en déduisons divers résultats de convergence en norme L p {L^p} , convergence presque partout, localisation des développements en harmoniques sphériques, ainsi qu’un théorème de multiplicateurs qui généralise le théorème classique de Marcinkiewicz sur les séries trigonométriques. La même étude est faite pour les développements suivant les polynômes ultrasphériques. Nous montrons de plus que les sommes partielles du développement en harmoniques sphériques d’une fonction de L p ( Σ n ) , p ≠ 2 {L^p}({\Sigma _n}),p \ne 2 , ne convergent pas forcément en norme.

Let V be a Hermitian symmetric space of the non-compact type, w its Kaehler form. For A a geodesic triangle in £>, we compute explicitly the integral J a;, generalizing previous results (see [D-T]). As a consequence, if X is a manifold which admits V as universal cover, we calculate the Gromov norm of [u>] E /f 2 (X ) R). The formula for f. u is extended to ideal triangles. Precise estimates are given and triangles for which the bound is achieved are studied. For tube-type domains we show the connection of these integrals with the Maslov index we introduced in a previous paper (see [C-0]). Introduction.Let M be a Hermitian symmetric space of the non-compact type, which for simplicity, we assume to be irreducible. Let G be the neutral component of the group of biholomorphic automorphisms of M. The space M admits a natural (G-invariant) Kaehler form UJ. This real differential form of degree 2 is closed and hence can be integrated along any 2-cycle, in particular geodesic triangles (to mean triangles the sides of which are geodesic segments). When M is of type 1,11 or III (in E. Cartan's classification), the integral / A a; (the symplectic area of the geodesic triangle A) was computed in [D-T]. By using their techniques, we give the result in the general case. It turns out that these quantities have an upper bound, and with the appropriate normalization, the bound depends only on the rank r of M. We extend these computations to ideal triangles, and we prove (new) sharp estimates for the areas. In particular, we determine precisely the triangles for which the upper bound is achieved. This turns out be of great geometric significance, as the summits of such an extremal triangle are contained in the image of a tight holomorphic totally geodesic imbedding of the complex unit disc into M (Theorem 4.7). Generally speaking, our study of the integrals / A u requires the fine structure of Hermitian symmetric spaces : special role played by the tube-type case, behaviour of geodesies at infinity and structure of G-orbits in the boundary, use of partial Cayley transforms.This study is also related to a previous work (see [C-0]) where we extended the notion of Maslov index to the Shilov boundary S of a Hermitian symmetric space of tube-type. The Maslov index is (up to a factor TT) nothing but the symplectic area of ideal triangles with summits in 5, and in some sense the present work can be understood as a continuation of [C-0].The computation of the integrals was used in [D-T] to calculate the Gromov norm of the Kaehler class of a compact Hermitian locally symmetric manifold X = r\M, where M is of type I and F a discrete, torsion-free, co-compact subgroup of the group G. They observed that it has a nice topological corollary. Let S be a Riemann surface of genus g > 1 and / : S -> X a continuous map. Then /.

Let G be a real noncompact semi-simple Lie group with finite center and K a maximal compact sub-group. The symmetric space M = GIK carries a measure invariant under the action of G. The operators which map LP(M) continuously into itself and commute with the action of G, can be easily characterized when p = 2 or p = 1. This note gives some results on "singular integrals" which map LP into itself (1 < p < + co). 3911However, a strong necessary condition is needed in the case of a noncompact symmetric space, due to the "holomorphic extension" of Fourier transforms of L"-functions (1 < p < 2).Denote by e the convex hull of the images of p under the Weyl group WI by e (0 < + < 1) the t-dilation of e = Cl, and by 5gthe tube over the polygonC:: = ga*R + icky THEOREM 1. Let m be a bounded measurable function on a*R and suppose that the associated G-invariant operator T on L2(M) extends as a continuous operator on L"(M), for some p, 1 < p < 2. Then in extends as a holomorphic function in the interior of the tube 5,, where t -/p2 1, and is uniformly bounded there.To prove the theorem, notice that T must preserve the class of K-invariant functions on M which are in L", but also the related L'-class, because of a simple relation between T and its adjoint on these classes. But now elementary spherical functions belong to some Lq, q > 2. In fact, using complex interpolation and the inequalities of Harish-Chandta' for Vo one can prove the following lemma. If*II < A, IfII (1 < p . + co).We may assume p is finite, and write f*(m) < f*1i(m) + (x * If (m), where f* has the same definition as f*, but the sup is taken only over balls of radius less than one, and X is the biinvariant function given by x(n) = inf(c, B(oAn) where o is the origin in M(o = e.K), and c is some adequate I Harish-Chandra (1958) "Spherical functions on a semisimpleLie group, I," Amer. J. Math. 80, 241-310.

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