Spherical Maximal Operators on Radial Functions

Abstract

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 .

Locations

• arXiv (Cornell University) - View - PDF
• CiteSeer X (The Pennsylvania State University) - View - PDF
• Mathematische Nachrichten - View