The strong ${\ell }^r$ spherical maximal function

Abstract

Let $d\ge 3$ and $2\le r<\infty$. We consider the strong maximal operator associated with Euclidean surface averages over the ${\ell }^r$ unit sphere and independent coordinate dilations. We prove that this operator is bounded on $L^p({\mathbb{R}}^d)$ if and only if \[ p>\max\left\{\frac{d+1}{d-1},\frac{r}{d-1}\right\}. \] We also obtain $L^p\to L^q$ bounds for the corresponding local maximal operator. For each fixed $p$ in certain ranges, the resulting range of $q$ is optimal up to endpoints. The proof uses a dyadic surface decomposition and frequency-localized $L^2$ estimates. We establish these estimates by combining exact multiplier factorization via iterated stationary phase with scalar Fourier inversion and $T{T}^{\ast }$ arguments. In dimension three, these estimates are combined with quantitative multiparameter local smoothing.