Spherical Maximal Operators on Radial Functions

✍ Scribed by Andreas Seeger; Stephen Wainger; James Wright

• Publisher: John Wiley and Sons
• Year: 1997
• Tongue: English
• Weight: 818 KB
• Volume: 187
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19971870112

✦ Synopsis

Abstract

Let A~t~f(x) denote the mean of f over a sphere of radius t and center x. We prove sharp estimates for the maximal function M~E~ f(X) = sup~t~∈~E~ |A__tf__(x)| where E is a fixed set in IR^+^ and f is a radial function ∈ L^p^(IR^d^). Let P~d~ = d/(__d−__1) (the critical exponent for Stein's maximal function). For the cases (i) p < p~d~, d ⩾ 2, and (ii) p = p~d~, d ⩽ 3, and for p ⩽ q ⩽ ∞ we prove necessary and sufficient conditions on E for M~E~ to map radial functions in L^p^ to the Lorentz space L^P,q^.