# Boundedness of iterated spherical average on modulation spaces

## TL;DR

This paper establishes the precise conditions under which the iterated spherical average operator, combined with the Laplacian, is bounded between different modulation spaces, advancing understanding in harmonic analysis and approximation theory.

## Contribution

It provides necessary and sufficient conditions for the boundedness of the operator $ riangle (A_1)^N$ on modulation spaces, a novel result in harmonic analysis.

## Key findings

- Derived conditions for boundedness between modulation spaces.
- Extended understanding of spherical averages in harmonic analysis.
- Connected spherical averages with approximation theory via Laplacian.

## 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\leq p_{1},p_{2},q_{1},q_{2}\leq \infty $ and $s_{1},s_{2}\in \mathbb{R}$.