Projection Operators and Cesàro Means in L ^P Spaces

Abstract

In analogy with the Bochner–Riesz means on ${\mathbb{R}}^{d}$, the Cesàro (C, δ) means of the spherical harmonic expansions on ${\mathbb{S}}^{d-1}$ can be bounded in L ^p space for δ below the critical index $\frac{d-2} {2}$, and furthermore, they are bounded under the same condition as that of the Bochner–Riesz means. In this chapter, we establish such results for h-harmonic expansions with respect to the product $h_{k}^{2}(x) = \prod\nolimits_{i=1}^{d}{|{x_i}|^{2ki}}$, which cover results for ordinary spherical harmonic expansions. The proof of such results depends on the boundedness of proj ection operators, which will be established in the first section, assuming a critical estimate.