Bochner–Riesz Means and K-Functional on Triebel–Lizorkin Spaces

Abstract

Based on n-dimensional Euclidean spaces and n-dimensional torus as underlying spaces, we investigate the rate for the norm convergence of the generalized Bochner–Riesz means on homogeneous Triebel–Lizorkin spaces, and establish the equivalence between the rate and the K-functional. Particularly, we show that such equivalence is closely related to the Bochner–Riesz conjecture on homogeneous Triebel–Lizorkin spaces. Thus, we obtain natural extensions of some well-known theorems by Fefferman, Tomas and Stein and by Carleson, Söjlin and Hörmander.