# Sparse Bounds for Spherical Maximal Functions

## TL;DR

This paper establishes sharp sparse bounds for lacunary and full spherical maximal functions, providing precise variants of known $L^p$ bounds and deriving new weighted inequalities for specific weight classes.

## Contribution

It introduces the first sharp sparse bounds for both lacunary and full spherical maximal functions, enhancing understanding of their weighted inequalities.

## Key findings

- Sharp sparse bounds for lacunary spherical maximal function.
- Sharp sparse bounds for full spherical maximal function.
- New weighted inequalities for specific weight classes.

## Abstract

We consider the averages of a function $ f$ on $ \mathbb R ^{n}$ over spheres of radius $ 0< r< \infty $ given by $ A_{r} f (x) = \int_{\mathbb S ^{n-1}} f (x-r y) \; d \sigma (y)$, where $ \sigma $ is the normalized rotation invariant measure on $ \mathbb S ^{n-1}$. We prove a sharp range of sparse bounds for two maximal functions, the first the lacunary spherical maximal function, and the second the full maximal function. $$ M_{{lac}} f = \sup_{j\in \mathbb Z } A_{2^j} f , \qquad M_{{full}} f = \sup_{ r>0 } A_{r} f . $$ The sparse bounds are very precise variants of the known $L^p$ bounds for these maximal functions. They are derived from known $ L ^{p}$-improving estimates for the localized versions of these maximal functions, and the indices in our sparse bound are sharp. We derive novel weighted inequalities for weights in the intersection of certain Muckenhoupt and reverse H\"older classes.