Spherical averages on nilpotent groups
Volume 282 / 2025
Abstract
We study the spherical maximal operator $\mathcal M_{S^{d-1}(A)}$, which is associated with the sphere $S^{d-1}$ embedded in the horizontal plane of some nilpotent group $\mathbb G_A$. Here, $\mathbb {G}_A$ is the group whose underlying set is identified with $\mathbb R^{d+1}$, and its group law is determined by the $d\times d$ real matrix $A$. Specifically, when $A$ is a skew-symmetric matrix $J$, which is the case of the Heisenberg group, it was known that the spherical maximal operator $\mathcal M_{S^{d-1}(J)}$ is bounded on $L^p(\mathbb G_J)$ for $p$ in the range $(\frac{d}{d-1},\infty ]$. We extend the results for the skew-symmetric matrix $J$ to a more general class of invertible matrices $A$. We aim to classify the range of $p$ for which the operator $\mathcal M_{S^{d-1}(A)}$ is bounded on $L^p(\mathbb G_A)$ based on the eigenvalues of $A$.
