Boundedness of iterated spherical average on modulation spaces

Abstract

The spherical average $A_1(f)$ and its iteration $(A_1{)}^N$ are important operators in harmonic analysis and probability theory. Also $\Delta (A_1{)}^N$ is used to study the $K$ functional in approximation theory, where $\Delta$ is the Laplace operator. In this paper, we obtain the sufficient and necessary conditions to ensure the boundedness of $\Delta (A_1{)}^N$ from the modulation space $M_{p_1,q_1}^{s_1}$ to the modulation space $M_{p_2,q_2}^{s_2}$ for $1\le p_1,p_2,q_1,q_2\le \infty$ and $s_1,s_2\in \mathbb{R}$.