Boundedness of Projection Operators and Cesàro Means

Abstract

The Cesàro (C, δ) means are important tools, and their boundedness in appropriate function spaces often serves as a gauge of our understanding of the underlying structure. In this chapter, we establish the boundedness of the Cesáro means for h-harmonic expansions with respect to the product weights h _κ ²(x) = ∏ _{i = 1} ^d | x _i | ^2κ _i on the sphere. The main results are stated and discussed in the first section. The central piece of the proof is a pointwise estimate of the integral kernel of the means, which involves a multiple beta integral of the Jacobi polynomials. These integrals will be estimated in the second section, and the pointwise estimate of the kernels is given in the third section, from which the upper bound of the norm of (C, δ) means is deduced in the fourth section. Finally, a lower estimate of the norm is given in the fifth section.