Exponential decay estimates for singular integral operators

• Published in: Mathematische annalen Vol. 357; no. 4; pp. 1217 - 1243
• Main Authors: Ortiz-Caraballo, Carmen, Pérez, Carlos, Rela, Ezequiel
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-12-2013
• Subjects: Mathematics; Mathematics and Statistics; Primary 42B20; 47B38; 42B25; Secondary 46B70
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-013-0940-3

The following subexponential estimate for commutators is proved
where
and
are absolute constants,
is a Calderón–Zygmund operator,
is the Hardy Littlewood maximal function and
is any function supported on the cube
. We also obtain that
where
is the median value of
on the cube
and
is Strömberg’s local sharp maximal function with
. As a consequence we derive Karagulyan’s estimate:
from [
21
] improving Buckley’s theorem [
3
]. A completely different approach is used based on a combination of “Lerner’s formula” with some special weighted estimates of Coifman–Fefferman type obtained via Rubio de Francia’s algorithm. The method is flexible enough to derive similar estimates for other operators such as multilinear Calderón–Zygmund operators, dyadic and continuous square functions and vector valued extensions of both maximal functions and Calderón–Zygmund operators. In each case,
will be replaced by a suitable maximal operator.