Norm inequalities for higher-order commutators of one-sided oscillatory singular integrals
Shi and Zhang Journal of Inequalities and Applications (2016) 2016:88
DOI 10.1186/s13660-016-1025-0
RESEARCH
Open Access
Norm inequalities for higher-order
commutators of one-sided oscillatory
singular integrals
Shaoguang Shi* and Lei Zhang
*
Correspondence:
Department of Mathematics, Linyi
University, Linyi, 276005, P.R. China
Abstract
In the present paper, we study the weighted norm inequalities for higher-order
commutators formed by a class of one-sided oscillatory singular integrals and BMO
functions. We obtain that the boundedness of these commutators can be deduced
by that of one-sided Calderón-Zygmund singular integral operators.
MSC: Primary 42B20; secondary 42B25
Keywords: commutator; one-sided oscillatory integral; one-sided weight
1 Introduction
The aim of this paper is to further study the one-sided version of the following oscillatory
singular integral, which was first introduced and studied by Ricci and Stein []:
Tf (x) = p.v.
Rn
eiP(x,y) K(x – y)f (y) dy,
where P(x, y) is a nontrivial real-valued polynomial defined on Rn × Rn , and K is a
Calderón-Zygmund kernel. We say that a function in Lloc (Rn \ {}) is a Calderón-Zygmund
kernel if the following properties are satisfied []:
() there exists a finite constant C such that
K(x – y) – K(x) ≤ C |y|
|x|
for all |x| > |y|;
() there exists a finite constant C such that
ε<|x|<N
K(x) dx ≤ C
for all ε and N such that < ε < N;
() there exists a finite constant C such that
K(x) ≤ C
|x|
for all x = .
© 2016 Shi and Zhang. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and
indicate if changes were made.
Shi and Zhang Journal of Inequalities and Applications (2016) 2016:88
Page 2 of 12
Ricci and Stein [] studied the norm inequalities for T on Lp (Rn ) spaces with < p < ∞.
Weighted inequalities arise naturally in Fourier analysis, but their use is best justified by
the variety of applications in which they appear. For example, the theory of weights plays
an important role in the study of boundary value problems inherent in Laplace equations
on Lipschitz domains. Many people are interested in the study of the events that occur
when the weight function belongs to the Muckenhoupt classes Ap :
|B|
w(x) dx
B
|B|
w(x)
–p
p–
≤ C.
dx
B
Here < p < ∞, and B denotes any ball in Rn . The classes A are defined as Mw ≤ Cw,
where M is the classical Hardy-Littlewood maximal operator. Lu and Zhang [, ] gave
the boundedness of T on Lp (w) ( < p < ∞) spaces with weight functions w ∈ Ap . For other
classical works on T, see, for example, [–] and the references therein. In what follows,
we restrict our attention on n = in order to introduce the one-sided operators defined
on R.
Many operators in harmonic analysis have one-sided versions. It is well known that the
one-sided Hardy-Littlewood maximal operators are required in ergodic theory. Sawyer
[] introduced the integral version of these operators as
h> h
x+h
f (y) dy and
M+ f (x) = sup
h> h
x
M– f (x) = sup
x
f (y) dy.
x–h
The good weights for M+ and M– are the one-sided version of Ap classes. Sawyer []
studied these one-sided weights in depth for the first time. We recall their definitions:
A+p : sup
(c
–
a)p
a<b<c
A–p : sup
(c
–
a)p
a<b<c
c
b
w(x) dx
a
w(x)
–p
p–
dx
b
b
c
w(x) dx
w(x)
–p
b
a
and
A– : M+ w ≤ Cw.
≤C
for < p < ∞,
≤C
for < p < ∞,
p–
dx
and
A+ : M– w ≤ Cw
In [], the classes A+∞ and A–∞ were introduced as
A+∞ =
≤p<∞
A+p
and
A–∞ =
A–p .
≤p<∞
The important point to note here is that the one-sided Muckenhoupt classes are larger
than the classical Muckenhoupt classes. For instance, the function ex ∈ A+ , but ex ∈/ A . In
fact, it is easy to see that for ≤ p ≤ ∞, Ap A+p , Ap A–p , and Ap = A+p ∩ A–p . Furthermore,
both the reverse Hölder inequality and the doubling condition are not true for the onesided Muckenhoupt classes. Therefore, some different ideas are needed here to deal with
the weighted norm inequalities for one-sided operators. The classes w ∈ A+p are of interest,
Shi and Zhang Journal of Inequalities and Applications (2016) 2016:88
Page 3 of 12
not only because they control the boundedness of the one-sided Hardy-Littlewood maximal operators, but also they are the right classes for the weighted estimates of one-sided
Calderón-Zygmund singular integral operators [] defined by
T + f (x) = lim+
ε→
∞
K(x – y)f (y) dy
x+ε
and
T – f (x) = lim+
ε→
x–ε
K(x – y)f (y) dy,
–∞
where K is the Calderón-Zygmund kernel with support in R– = (–∞, ) and R+ = (, +∞),
respectively (also called the one-sided Calderón-Zygmund kernel). An example of such a
kernel is
K(x) =
sin(log |x|)
χ(–∞,) (x),
(x log |x|)
where χE denotes the characteristic function of a set E.
Highly inspired by these statements for the oscillatory singular integral operators and
the one-sided operator theory, Fu, Lu, Shi, and their coauthors introduced the one-sided
oscillatory singular integral operators and studied some weighted norm inequalities for
these operators with one-sided weights, including the strong weighted Lp ( < p < ∞)
boundedness [], the weighted weak (, ) type norm inequalities [] and the weighted
norm estimates on one-sided Hardy spaces []. We recall the definition of one-sided oscillatory integral operators:
∞
+
T f (x) = lim+
ε→
eiP(x,y) K(x – y)f (y) dy
x+ε
and
x–ε
–
T f (x) = lim+
ε→
eiP(x,y) K(x – y)f (y) dy,
–∞
where P(x, y) are real-valued polynomials defined on R × R, and K are the one-sided
Calderón-Zygmund kernels.
Let b be a locally integrable function on Rn , and let T be an integral operator. Then we
define the commutator operator for a proper function f by
Tb (f ) := b(T f ) – T (bf ).
The function b is also called the symbol function of Tb . The investigation of the operator
Tb begins with Coifman-Rochberg-Weiss pioneering study of the operator T []. There
are two major reasons for considering the problem of commutators. The first one is that
the boundedness of commutators can produce some characterizations of function spaces
[, ]. The other one is that the theory of commutators plays an important role in the
study of the regularity of solutions to elliptic and parabolic partial differential equations
(PDEs) of the second order [, ]. It is well known that many people are interested in the
study of commutators for which the symbol functions belong to BMO spaces. In harmonic
analysis, a function b of bounded mean oscillation, also known as a BMO function, is a
real-valued function whose mean oscillation is bounded, that is,
(...truncated)
