Invent. math. (2017) 209:665–748
-estimates for discrete operators of Radon
type: variational estimates
· Elias M. Stein
Received: 20 October 2016 / Accepted: 16 January 2017 / Published online: 31 January 2017
© Springer-Verlag Berlin Heidelberg 2017
Abstract We prove
bounds for p ∈ (1,∞),ofr-variations r ∈
(2,∞), for discrete averaging operators and truncated singular integrals of
Radon type. We shall present a new powerful method which allows us to deal
with these operators in a uniﬁed way and obtain the range of parameters of p
and r which coincide with the ranges of their continuous counterparts.
In this paper we will be concerned with estimates of r -variations for dis-
crete operators of averaging and singular Radon type, and their application
Mariusz Mirek was partially supported by NCN Grant DEC-2015/19/B/ST1/01149. Elias M.
Stein was partially supported by NSF Grant DMS-1265524.
Elias M. Stein
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
Instytut Matematyczny, Uniwersytet Wrocławski, Pl. Grunwaldzki 2/4, 50-384
Department of Mathematics, Princeton University, Princeton, NJ 08544-100, USA
Wrocław University of Technology, Wybrze˙ze Wyspia´nskiego 27, 50-370 Wrocław,