Variation Inequalities for the Commutators of One-Sided Calderón-Zygmund Singular Integrals with Lipschitz Functions ()
1. Introduction
Let
be a family of operators such that the limit
exists. It is nature to study the speed of convergence of the family
. A classic method
is to consider square function of the type
, more generally, define the following oscillation operator:
where
is a fixed sequence decreasing to zero. Let
. The
-variation operator is defined by
where the sup is taken over all sequence
of positive numbers decreasing to zero. The variation inequalities play important roles in probability, ergodic theory, and harmonic analysis. We refer the readers to [1] - [6] and the references therein for more background information. Gillespie and Torrea [7] show that oscillation and variation of Hilbert transforms are bounded on
for
. The weighted oscillation and variation boundedness of differential operators and Calderón-Zygmund singular integral are established in [8] [9]. This paper is devoted to studying weighted boundedness of
-variational operator for the families of commutator generated by one-sided Calderón-Zygmund singular integral with Lipschitz functions. Recently, Liu and Wu [10] presented a criterion on the weighted estimate of the oscillation and variation operators for the commutators of Calderón-Zygmund singular integrals with BMO functions in one dimension. Variation inequalities for the commutators one-sided singular integrals with BMO functions were established in [11]. Zhang and Wu [12] gave the oscillation and variation inequalities for the commutators of singular integrals with Lipschitz functions.
Before stating our main results, we firstly recall some notations and definitions. In [13], Aimar, Forzani and Martín-Reyes introduced the one-sided Calderón-Zygmund singular integrals defined by:
(1.1)
and
where the kernel K is called the one-sided Calderón-Zygmund kernel (OCZK) which satisfies
(1.2)
(1.3)
(1.4)
with support in
or
, where (1.4) is named Hörmander’s condition. Equation (1.3) is also called the size condition for K. An interesting example is
where
denotes the characteristic function of a set E, for more details one can refer to [13].
For
, a function
, if it satisfies
Let K be one-sided Calderón-Zygmund kernel (OCZK) with support in
.
, we define the following one-sided operator
where
(1.5)
In this paper, we study variational inequalities for the commutators of one-sided singular integrals with Lipschitz functions. Our result can be formulated as follows:
Theorem 1.1. Let K be one-sided Calderón-Zygmund kernel (OCZK) with support in
. Let
and
be given as in (1.5) and (1.1), respectively. The operator
is bounded in
for some
. Let
,
. Then, for all
,
,
, we have
The rest of this paper is organized as follows. In Section 2, we introduce and recall some basic facts and auxiliary lemmas. The proof of main theorem will be given in Sections 3.
Throughout this paper, the letter C, sometimes with additional parameters, will stand for positive constants, not necessarily the same one at each occurrence, but independent of the essential variables. We also denote
if
.
2. Preliminaries
In 1986, Sawyer [14] first introduced the one-sided Muckenhoupt weights
and
to treat the one-sided Hardy-Littlewood maximal operators
A positive function
is said to belong to
or
if it satisfies
or
when
; also, for
,
for some constant C. If
, then
and
. Notice that the function
mentioned above is in
but not in
.
Similarly, the double weight classes
and
are denoted by
for all
,
,
and
; also for
,
,
The one-sided
classes not only control the boundedness of one-sided Hardy-Littlewood maximal operators, but also serve as the right weight classes for one-sided singular integral operators. Set
and let K be an OCZK with support in
. Then
is bounded on
if
, see [13].
Lemma 2.1. [15] Suppose that
, then there exists
such that, for all
,
.
Lemma 2.2. [16] Suppose
, then
and
for all
.
Lemma 2.3. [17] Let
and let T be sublinear operator defined in
satisfying
for every
and
; then, for every
,
, and
, the inequality
Lemma 2.4. [18] For every p with
, assume that
, and that
for some
with
. Then
where
and
.
3. The Proof of Theorem 1.1
According to [7], we denote by
the mixed norm Banach space of two variable functions h defined on
such that
where
. Given a family of operators
defined on
, we consider the
-valued operator
on
defined by
where
is an abbreviation for the element of
given by
This implies
This section is devoted to proving Theorem 1.1. To do this, we need establish the following lemma.
Lemma 3.1. Let K be one-sided Calderón-Zygmund kernel (OCZK) with support in
. Let
and
be given as in (1.5) and (1.1), respectively. The operator
is bounded in
for some
. Let
,
. Then, for all
,
,
, we have
(3.1)
Proof. Let
and
. Suppose
. Notice that
It is easy to check
Consider the following three sublinear operators defined on
:
For
, let
, then
. By Lemma 2.1, there exists
such that
. Take
and
. Using Hölder’s inequality and the
boundedness for
, we obtain
By Lemma 2.3, we have that for
and
For
, by Hölder’s inequality and the
boundedness for
, we get
where
for all
. Then
For
and
, it follows from Lemma 2.3 that
It remains to deal with
. For
, we get
(3.2)
In view of (3.2), we have
Since
and
, we get
. Using (1.4), we have
. Note that
Then
where
for all
.
For
, let
and
. Then
Now we estimate
. It is easy to see
For
, using Hölder’s inequality and (1.3), we have
where
for all
. By a similar estimate of
, we have
Now we estimate
. It is easy to see
By noting that
, we have
with
. Using Hölder’s inequality and (1.3), we have
where
for all
. By similar arguments, we have
Following from the estimates of
,
,
and
, we get
This together with the estimate of
implies
where
for all
. Then
For
and
, it follows from Lemma 2.3 that
This completes the proof.
Now, we turn to the proof of theorem 1.1.
Proof. For
, by Lemma 2.2, we have
. Using Lemma 2.4 and Lemma 3.1, we obtain
It remains to prove
. By the similar arguments in the proof of Theorem 1.3 in [10], we can get
. Then Theorem 1.1 is proved.
Supported
Supported by the Natural Science Foundation of Shandong Province (No. ZR2020QA006).