Acta Mathematica Sinica, English Series
Sep., 2018, Vol. 34, No. 9, pp. 1445–1459
Published online: May 23, 2018
https://doi.org/10.1007/s10114-018-7431-0
http://www.ActaMath.com
Acta Mathematica Sinica,
English Series
Springer-Verlag GmbH Germany &
The Editorial Office of AMS 2018
Non-homogeneous Tb Theorem for Bi-parameter g-function
Ming Ming CAO Qing Ying XUE
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex
Systems, Ministry of Education, Beijing 100875,P.R.China
E-mail : m.cao@mail.bnu.edu.cn qyxue@bnu.edu.cn
Abstract The main result of this paper is a bi-parameter Tbtheorem for Littlewood–Paley g-function,
where b is a tensor product of two pseudo-accretive function. Instead of the doubling measure, we work
with a product measure µ = µ
n
× µ
m
, where the measures µ
n
and µ
m
are only assumed to be upper
doubling. The main techniques of the proof include a bi-parameter b-adapted Haar function decompo-
sition and an averaging identity over good double Whitney regions. Moreover, the non-homogeneous
analysis and probabilistic methods are used again.
Keywords Bi-parameter, square function, probabilistic method, b-adapted Haar functions
MR(2010) Subject Classification 42B25, 42B20
1 Introduction
It is well-known that the multi-parameter harmonic analysis originated in the work of Fef-
ferman and Stein [3], where the bi-parameter singular integral operators of convolution type
are carefully considered. Before long, Journ´e [8] proved the first multi-parameter T 1theorem
for product spaces by treating the singular integral operator as a vector-valued one-parameter
operator. Recently, a new type of T 1 theorem on product spaces was formulated by Pott
and Villarroya [18]. The authors avoided the vector-valued assumptions by the mixed type
conditions including kernel estimates, BMO, and weak boundedness property. Along this way,
Martikainen [10] gave a bi-parameter representation of singular integrals by dyadic shifts, which
extended the famous one-parameter case of Hyt¨onen [5]. Moreover, by means of probabilistic
methods and the techniques of dyadic analysis, Hyt¨onen and Martikainen [6] showed a bi-
parameter T 1 theorem in spaces of non-homogeneous type. Inspired by this, Ou [15] obtained
abi-parameterTb theorem on product Lebesgue spaces, where b isatensorproductoftwo
pseudo-accretive functions. Still more recently, a bi-parameter T 1theoremforbi-parameter
g-function was established by Martikainen [11], although the assumptions imposed on the non-
convolution kernels seem to be somewhat complicated. The proof was based on modern dyadic
probabilistic techniques adapted to the bi-parameter situation.
This paper is devoted to studying the non-homogeneous Tb theorem for bi-parameter
Littlewood–Paley g-function, which is defined by
g(f)(x):=
∞
0
∞
0
|Θ
t
1
,t
2
f(x
1
,x
2
)|
2
dt
1
t
1
dt
2
t
2
1/2
,x=(x
1
,x
2
) ∈ R
n+m
,
Received September 20, 2017, accepted March 12, 2018
Supported by NSFC (Grant No. 11471041)
@Phil_Robichaud
@deepthiw
@JoseServera