Premium
Spherical Maximal Operators on Radial Functions
Author(s) -
Seeger Andreas,
Wainger Stephen,
Wright James
Publication year - 1997
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19971870112
Subject(s) - mathematics , combinatorics , radial function , radius , center (category theory) , function (biology) , maximal operator , lorentz space , exponent , mathematical analysis , space (punctuation) , lorentz transformation , physics , bounded function , crystallography , linguistics , chemistry , computer security , philosophy , evolutionary biology , computer science , biology , classical mechanics
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 .