Norms of Complex Harmonic Projection Operators

In this paper we estimate the (L^p-L^2)-norm of the complex harmonic projectors (pi)llʹ, 1 <= p< 2, uniformly with respect to the indexes l,lʹ. We provide sharp estimates both for the projectors (pi)llʹ, when l,lʹ belong to a proper angular sector in N * N, and for the projectors (pi)l0 and (pi)0l. The proof is based on an extension of a complex interpolation argument by C.Sogge. In the appendix, we prove in a direct way the uniform boundedness of a particular zonal kernel in the L1 norm on the unit sphere of R^(2n).