Nikodym sets and maximal functions associated with spheres

2022/10/15 by Chang, Alan, Dosidis, Georgios, Kim, Jongchon · 5 citations
#28A75 #42B25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2210.08320

Abstract

We study spherical analogues of Nikodym sets and related maximal functions. In particular, we prove sharp L^p-estimates for Nikodym maximal functions associated with spheres. As a corollary, any Nikodym set for spheres must have full Hausdorff dimension. In addition, we consider a class of maximal functions which contains the spherical maximal function as a special case. We show that L^p-estimates for these maximal functions can be deduced from local smoothing estimates for the wave equation relative to fractal measures.

Cited by

• Problems on spherical maximal functions
• Weighted wave envelope estimates for the parabola
• Maximal estimates for orthonormal systems of wave equations
• Local smoothing and maximal estimates for average over surfaces of codimension 2 in \mathbb R⁴
• Lacunary δ-Discretised Spherical Maximal Operators

Related

• Problems on spherical maximal functions
• Concerning Nikodym-type sets in 3-dimensional curved spaces
• Restricted weak-type endpoint estimates for discrete k-spherical maximal functions
• Endpoint estimates for the fractal circular maximal function and related local smoothing
• Improved L^p bounds for the strong spherical maximal operator