1 Introduction
Recently in the field of the interaction of complex analysis of several variables and harmonic analysis, there has been interest in strongly pseudoconvex domains with boundaries satisfying the minimum regularity condition of class
$C^{2}$
, with new phenomena emerging in the analysis and geometry of domains of this kind. Lanzani and Stein [Reference Lanzani and Stein13] studied the Cauchy–Szegő projection operator in such domains by introducing a family of Cauchy integrals
, and established the
$L^{p}(bD)$
(
$1<p<\infty $
, with respect to the Leray–Levi measure) boundedness of
. Different from the case of smooth strongly pseudoconvex domains, the kernels of these Cauchy integral operators do not satisfy the standard size or smoothness conditions for Calderón–Zygmund operators. It is important to study the harmonic analysis of such Cauchy–Szegő operators and Cauchy integral operators in complex analysis, for example, to establish the theory of holomorphic Hardy spaces over such domains. Duong et al. [Reference Duong, Lanzani, Li and Wick4] proved the characterization of boundedness and compactness for commutators of such Cauchy type integral operators. In an abuse of notation, we omit the subscript
$\epsilon $
.
Theorem A [Reference Duong, Lanzani, Li and Wick4].
Suppose
$D\subset \mathbb C^{n}$
,
$n\geq 2$
, is a bounded domain whose boundary is of class
$C^{2}$
and is strongly pseudoconvex. Suppose
. Then, for
$1<p<\infty $
:
-
(1)
if and only if the commutator
is bounded on
; -
(2)
if and only if the commutator
is compact on
.
Because the characterization of the boundedness and compactness of Calderón–Zygmund operator commutators on certain function spaces has many applications in various areas, such as harmonic analysis, complex analysis, (nonlinear) partial differential equations, researchers have recently been seeking to establish such a characterization for singular integral operators (especially non-Calderón–Zygmund operators) over various function spaces. For example, the characterization of boundedness and compactness was proved for Calderón–Zygmund operator commutators on Morrey spaces over Euclidean space by Di Fazio and Ragusa [Reference Duong, Lacey, Li, Wick and Wu3] and Chen et al. [Reference Chen, Ding and Wang2], and the boundedness of such commutators was proved by Komori and Shirai [Reference Komori and Shirai9] on weighted Morrey spaces. The characterization for Cauchy integrals and Beurling–Ahlfors transformation commutators on
$\mathbb C$
over weighted Morrey spaces was established by Tao et al. [Reference Tao, Yang and Yang16, Reference Tao, Yang and Yang17]. In this paper, we extend the characterization of the boundedness and compactness of commutators of Cauchy integral operators in Theorem A to weighted Morrey spaces.
Let
$p\in (1,\infty )$
. A nonnegative function
$v\in L^{1}(bD)$
is in
$A_{p}(bD)$
if
where the supremum is taken over all balls B in
$bD$
. A nonnegative function
$v\in L^{1}(bD)$
is in
$A_{1}(bD)$
if there exists a constant C such that for all balls
$B\subset bD$
,
For
$p=\infty $
, we define
$$ \begin{align*} A_{\infty }(bD)= \bigcup_{1\leq p<\infty }A_{p}(bD). \end{align*} $$
Let
$p\in (1,\infty )$
,
$\kappa \in (0,1)$
and
$v\in A_{p}(bD)$
. The weighted Morrey space
$L_{v}^{p,\,\kappa }(bD)$
(see [Reference Kokilashvili and Meskhi8]) is defined by
$$ \begin{align*} L_{v}^{p,\,\kappa}(bD):=\{f\in L_{\mathrm{loc}}^{p}(bD):\|f\|_{L_{v}^{p,\,\kappa}(bD)}<\infty\} \end{align*} $$
with
where
For the history of Morrey spaces one can refer to [Reference Adams and Xiao1].
The main result of our paper is to characterize the boundedness and compactness of the commutator of Cauchy type integral
on the weighted Morrey space, following the ideas and approach in [Reference Duong, Lanzani, Li and Wick4].
Theorem 1.1. Suppose
$D\subset \mathbb C^{n}$
,
$n\geq 2$
, is a bounded domain whose boundary is of class
$C^{2}$
and is strongly pseudoconvex. Suppose
$b\in L^{1}(bD),1<p<\infty $
,
$0<\kappa <1$
and
$v\in A_{p}$
. Then
$b\in \textrm{BMO}(bD)$
if and only if the commutator
is bounded on
$L_{v}^{p,\kappa }(bD)$
.
Theorem 1.2. Suppose
$D\subset \mathbb C^{n}$
,
$n\geq 2$
, is a bounded domain whose boundary is of class
$C^{2}$
and is strongly pseudoconvex. Suppose
$b\in L^{1}(bD),1<p<\infty $
,
$0<\kappa <1$
and
$v\in A_{p}$
. Then
$b\in \text{VMO}(bD)$
if and only if the commutator
is compact on
$L_{v}^{p,\kappa }(bD)$
.
We also consider the Cauchy–Leray integral on a bounded domain in
$\mathbb C^{n}$
, which is strongly
$\mathbb C$
-linearly convex and whose boundary
$bD$
satisfies the minimum regularity
$C^{1,1}$
(for the details we refer the reader to Section 3 below); such integral operators are studied by Lanzani and Stein in [Reference Lanzani and Stein12]. They obtained the
$L^{p}(bD)$
boundedness (
$1<p<\infty $
) of the Cauchy–Leray transform
$\mathcal C$
by showing that the kernel
$K(w,z)$
of
$\mathcal C$
satisfies the standard size and smoothness conditions of Calderón–Zygmund operators (for details of these definitions and notation we refer the reader to Section 3), and that
$\mathcal C$
satisfies a suitable version of the
$T(1)$
theorem. In [Reference Duong, Lanzani, Li and Wick4], the authors also obtained the characterization of boundedness and compactness for commutators of such transforms. Following a similar approach to that in the proof of Theorems 1.1 and 1.2, we obtain the following results on the Cauchy–Leray transform and its commutator.
Theorem 1.3. Let D be a bounded domain in
$\mathbb C^{n}$
of class
$C^{1,1}$
that is strongly
$\mathbb C$
-linearly convex. Let
$1<p<\infty $
,
$v\in A_{p}$
. Then there exists a positive constant C such that
$$ \begin{align*} \|\mathcal C(f)\|_{L_{v}^{p}(bD)}\leq C \|f\|_{L_{v}^{p}(bD)} \end{align*} $$
for every function
$f\in L_{v}^{p}(bD).$
Theorem 1.4. Let D be a bounded domain in
$\mathbb C^{n}$
of class
$C^{1,1}$
that is strongly
$\mathbb C$
-linearly convex and let
$b\in L^{1}(bD)$
,
$1<p<\infty $
,
$0<\kappa <1$
and
$v\in A_{p}$
. Let
$\mathcal C$
be the Cauchy–Leray transform
$($
as in [Reference Lanzani and Stein12]
$)$
. Then, for
$1<p<\infty $
:
-
(1)
$b\in \textrm{BMO}(bD)$
if and only if the commutator
$[b, \mathcal C]$
is bounded on
$L_{v}^{p, \kappa }(bD)$
; -
(2)
$b\in \text{VMO}(bD)$
if and only if the commutator
$[b, \mathcal C]$
is compact on
$L_{v}^{p, \kappa }(bD)$
.
This paper is organized as follows. In Section 2 we recall the notation and definitions related to a family of Cauchy integrals for bounded strongly pseudoconvex domains in
$\mathbb C^{n}$
with minimal smoothness, and then we prove Theorems 1.1 and 1.2. In Section 3 we recall the notation and definitions related to the Cauchy–Leray integral for bounded
$\mathbb C$
-linearly convex domains in
$\mathbb C^{n}$
with minimal smoothness and give the proofs of Theorems 1.3 and 1.4.
2 Commutator of Cauchy type integral for bounded strongly pseudoconvex domains with minimal smoothness
Throughout this section we assume that D is a bounded strongly pseudoconvex domain whose boundary is of class
$C^{2}$
.
Let
$\texttt { d}(w,z)$
be the quasidistance on the boundary
$bD$
, which is defined as in [Reference Lanzani and Stein13, Section 2.3] and satisfies the following conditions: there exist constants
$A_{1}>0$
and
$C_{d}>1$
such that for all
$w,z,z^{\prime }\in bD$
,
$$ \begin{align*} \begin{split} & \mathtt{d}(w,z)=0\quad \mbox{if and only if } w=z;\\[3pt] & \mathtt{d}(w,z)\leq C_{d}( \mathtt{d}(w,z^{\prime}) +\mathtt{d}(z^{\prime},z));\\[3pt] &A_{1}^{-1} \mathtt{d}(z,w)\leq \mathtt{d}(w,z) \leq A_{1} \mathtt{d}(z,w). \end{split} \end{align*} $$
Let
be the Leray–Levi measure on
$bD$
(cf. [Reference Lanzani and Stein13, 138]). Then one has
where
$d\sigma $
is the induced Lebesgue measure on
$bD$
and
$\Lambda (w)$
is a continuous function such that
$ c\leq \Lambda (w)\leq \tilde c, w\in bD$
, for some positive constants c and
$\widetilde {c}$
.
We also recall the boundary balls
$ B_{r}(w) $
determined via the quasidistance
$\texttt { d }$
, that is,
$$ \begin{align*} B_{r}(w) :=\{ z\in bD:\ \mathtt{d}(z,w)<r \}, \quad \text{where}\ w\in bD. \end{align*} $$
According to [Reference Lanzani and Stein13, page 139],
for some
. Without loss of generality, by rescaling, we can assume the diameter of the domain to be 1. So we only consider balls of radius less than 1 in this paper
In [Reference Lanzani and Stein13], the authors defined a family of Cauchy integrals
and studied their properties when
$\epsilon $
is kept fixed. For convenience of notation we henceforth drop explicit reference to
$\epsilon $
. To study the Cauchy transform
, which is the restriction of such a Cauchy integral on
$bD$
, one of the key steps in [Reference Lanzani and Stein13] is that they provided a constructive decomposition of
as follows:
with essential part
and remainder
If we write
then
$$ \begin{align*}C(w,z)=C^{\sharp}(w,z)+ R(w,z),\end{align*} $$
where the kernel
$C^{\sharp }(w,z)$
satisfies the standard size and smoothness conditions for Calderón–Zygmund operators, that is, there exists a positive constant
$A_{2}$
such that for every
$w,z\in bD$
with
$w\not =z$
,
$$ \begin{align*} \begin{split} & |C^{\sharp}(w,z)|\leq A_{2} {\displaystyle1\over\displaystyle\mathtt{d}(w,z)^{2n}};\\[3pt] &|C^{\sharp}(w,z) - C^{\sharp}(w,z^{\prime})|\leq A_{2} {\displaystyle \mathtt{d}(z,z^{\prime})\over \displaystyle \mathtt{d}(w,z)^{2n+1} },\quad \textrm{if}\ \mathtt{d}(w,z)\geq c \mathtt{d}(z,z^{\prime});\\[3pt] &|C^{\sharp}(w,z) - C^{\sharp}(w^{\prime},z)|\leq A_{2} {\displaystyle \mathtt{d}(w,w^{\prime})\over \displaystyle \mathtt{d}(w,z)^{2n+1} },\quad \textrm{if}\ \mathtt{d}(w,z)\geq c\mathtt{d}(w,w^{\prime}), \end{split} \end{align*} $$
for an appropriate constant
$c>0$
. However, the kernel
$R(w,z)$
of
satisfies a size condition and a smoothness condition for only one of the variables as follows: there exists a positive constant
$C_{R}$
such that for every
$w,z\in bD$
with
$w\not =z$
,
$$ \begin{align} \begin{split} &|R(w,z)|\leq C_{R} {\displaystyle1\over \displaystyle\mathtt{d}(w,z)^{2n-1}};\\[3pt] & |R(w,z)-R(w,z^{\prime})|\leq C_{R} {\displaystyle\mathtt{d}(z,z^{\prime})\over\displaystyle \mathtt{d}(w,z)^{2n}},\quad \textrm{if\ } \mathtt{d}(w,z)\geq c_{R}\, \mathtt{d}(z,z^{\prime}), \end{split} \end{align} $$
for an appropriate large constant
$c_{R}$
.
We also denote by
$BUC(bD)$
the space of all bounded uniformly continuous functions on
$bD$
. We first point out that the Leray–Levi measure
on
$bD$
is a doubling measure, and satisfies condition (1.1) in [Reference Krantz and Li10].
Lemma 2.1 [Reference Duong, Lanzani, Li and Wick4].
The Leray–Levi measure
on
$bD$
is doubling, that is, there is a positive constant C such that for all
$x\in bD$
and
$0<r\leq 1$
,
Moreover,
satisfies the condition that there exist a constant
$\epsilon _{0}\in (0,1)$
and a positive constant C such that
for all
$x,y\in bD$
and
$\texttt { d}(x,y)\leq r\leq 1$
.
We now recall the bounded mean oscillation (BMO) and vanishing mean oscillation (VMO) spaces on
$bD$
. Consider
as a space of homogeneous type with
$bD$
compact. Then
$\text {BMO}(bD)$
is defined as the set of all
$b\in L^{1}(bD)$
such that
where
and the norm is defined as
$$ \begin{align*}\|b\|_{\text{BMO}(bD)}:=\|b\|_{*}+ \|b\|_{L^{1}(bD)}. \end{align*} $$
The space
$\text {VMO }(bD)$
is the subspace of
$\mathrm {BMO}(bD)$
whose elements satisfy the further requirement that
where
$r_{B}$
is the radius of B.
The maximal function
$Mf$
is defined as
The sharp function
$f^\#$
is defined as
where
$f_{B}$
is defined in (2-3).
Note that from Lemma 2.1,
is a doubling measure and hence we have the following results.
Lemma 2.2 [Reference Kokilashvili and Meskhi8].
Let
$v\in A_{p}(bD)$
,
$p\geq 1$
. Then there exist constants
$C, \sigma>0$
such that for every ball B and measurable subset
$E\subset B$
the inequality
holds.
Lemma 2.3 [Reference Macías and Segovia14].
Let
$v\in A_{p}(bD)$
,
$1< p<\infty $
. There exists a constant C such that for every
$f\in L_{v}^{p}(bD)$
,
$$ \begin{align*} \|Mf\|_{L_{v}^{p}(bD)}\leq C\|f\|_{L_{v}^{p}(bD)}, \end{align*} $$
where
.
Lemma 2.4 [Reference Pradolini and Salinas15].
Let
$v\in A_{p}(bD)$
,
$1< p<\infty $
. There exists a constant C such that if
$\|f\|_{L_{v}^{p}(bD)}<\infty $
, then
$$ \begin{align*} \|f\|_{L_{v}^{p}(bD)}^{p}\leq C(v(bD)(f_{bD})^{p}+\|f^\#\|_{L_{v}^{p}(bD)}^{p}). \end{align*} $$
Lemma 2.5 [Reference Kronz11].
If
$f\in \textrm{BMO}(bD)$
, then there exist positive constants
$C_{1}$
and
$C_{2}$
such that for every ball
$B\subset bD$
and every
$\alpha>0$
,
According to [Reference Hart and Torres7, Theorem 5.5], we have the following result for BMO functions on
$bD$
.
Lemma 2.6. Let
$0<p<\infty $
,
$v\in A_{\infty }(bD)$
,
$f\in \text { BMO}(bD)$
. Then
where
2.1 Characterization of
$\textbf{BMO}(bD)$
via the commutator
Proof (Proof of necessity of Theorem 1.1).
We first prove necessity, namely that
$b\in \textrm{BMO}(bD)$
implies the boundedness of
.
We can write
Since the kernel of
is a standard kernel on
$bD\times bD$
, according to [Reference Gong, Li, Pozzi and Vempati6, Theorem 1.2], we can obtain that
is bounded on
$L_{v}^{p,\kappa }(bD)$
and
Thus, it suffices to show that
Now fix a ball
$B=B_{r}(z_{0})\subset bD$
and decompose
${f=f\chi _{bD\cap 2B}+f\chi _{bD\setminus 2B}=:f_{1}+f_{2}}$
. Then
For term I, by the proof of Theorem 3.2 in [Reference Di Fazio and Ragusa5], we have
For term
$II$
, observe that for
$z\in B$
, by (2-2), we have
where
Then
For
${II}_{1}$
, recall that
is a bounded space of homogeneous type and there exists a positive integer
$J_{0}$
depending on r such that
$2^{J_{0}+1}B=bD$
and
$2^{J_{0}}B\varsubsetneq bD$
. Then, by the Hölder inequality, Lemmas 2.6 and 2.2,
For
${II}_{2}$
, by the Hölder inequality,
For
${II}_{21}$
, since
$v\in A_{p}(bD)$
, we have
$v^{1-p^{\prime }}\!\!\in \!A_{p^{\prime }}(bD)$
, where
$1/p+1/p^{\prime }\!=\!1$
. By Lemma 2.6, we can obtain that
$$ \begin{align*}{II}_{21}\lesssim\|b\|_{\text{{BMO}}(bD)}[v^{1-p^{\prime}}(2^{j+1}B)]^{1/p^{\prime}}.\end{align*} $$
For
${II}_{22}$
, by Lemma 2.6,
Since
$b\in \text {{BMO}}(bD)$
, by Lemmas 2.2 and 2.5, there exist
$\bar C_{1}>0$
and
$\bar C_{2}>0$
such that for any ball B and
$\alpha>0$
,
$$ \begin{align*}v( \{z\in B: |b(z)-b_{B}|>\alpha \})\leq \bar C_{1}v(B)e^{-({\bar C_{2}\alpha\sigma/\|b\|_{\text{{BMO}}(bD)}})}, \end{align*} $$
for some
$\sigma>0$
. Therefore,
Similarly,
Then,
$$ \begin{align*}{II}_{22}\lesssim (\kern2pt j+1)\|b\|_{\text{{BMO}}(bD)}[v^{1-p^{\prime}}(2^{j+1}B)]^{1/p^{\prime}}.\end{align*} $$
Now together with Lemma 2.2,
Consequently,
$$ \begin{align*}II\lesssim \|b\|^{p}_{\text{{BMO}}(bD)}\|f\|^{p}_{L_{v}^{p,\kappa}(bD)}.\end{align*} $$
As a result,
This completes the proof of the necessity part.□
In order to prove the sufficiency of Theorem 1.1, we need the following lemma, which can be obtained by the proof of (3.10) in [Reference Di Fazio and Ragusa5, Theorem 3.2] and of (2.18) in [Reference Duong, Lanzani, Li and Wick4, Theorem 1.1].
Lemma 2.7. Denote by
$ C_{1}(w,z)$
and
$ C_{2}(w,z)$
the real and imaginary parts of
$ C(w,z)$
, respectively. Then at least one of
$C_{1}$
and
$C_{2}$
satisfies the following argument. There exist positive constants
$\gamma _{0}, A$
such that for every ball
$B=B_{r}(z_{0})\subset bD$
with
$r<\gamma _{0}$
, there exists another ball
$\tilde B = B_{r}(w_{0})\subset bD$
with
$Ar \leq \texttt { d}(w_{0},z_{0}) \leq (A+1)r$
such that for every
$z\in B$
and
$w\in \tilde B$
,
$ C_{i}(w,z)$
does not change sign and
$$ \begin{align*}| C_{i}(w,z)|\geq { c\over \mathtt{ d}(w,z)^{2n} }.\end{align*} $$
${\kern-4pt}$
Proof (Proof of sufficiency of Theorem 1.1).
We turn to the proof of the sufficiency condition, namely that if
is bounded on
$L_{v}^{p,\kappa }(bD)$
, then
$b\in \textrm{BMO}(bD)$
. We mainly follow the method and technique in [Reference Duong, Lanzani, Li and Wick4].
Assume that b is in
$L^{1}(bD)$
and that
. Let
$\gamma _{0}$
be the constant in Lemma 2.7. We test the
condition on the case of balls with big radius and small radius.
Case 1: In this case we work with balls with large radius,
$r\geq \gamma _{0}$
.
By (2-1) and by the fact that
,
Case 2: In this case we work with balls with small radius,
$r<\gamma _{0}$
.
We aim to prove that for every fixed ball
$B=B_{r}(z_{0})\subset bD$
with radius
$r<\gamma _{0}$
,
Now let
$\tilde B=B_{r}(w_{0})$
be the ball chosen as in Lemma 2.7. Without loss of generality we may assume that
$ C_{1}(w,z)$
does not change sign and
$$ \begin{align} | C_{1}(w,z)|\geq { c\over \mathtt{d}(w,z)^{2n} }. \end{align} $$
Let
$m_{b}(\tilde B)$
be the median value of b on the ball
$\tilde B$
with respect to the measure
. Following the sufficiency proof of Theorem 1.1(1) in [Reference Duong, Lanzani, Li and Wick4], we define sets
$ F_{1}= \{ w\in \tilde B: b(w)\leq m_{b}(\tilde B) \}$
,
$F_{2}=\{ w\in \tilde B: b(w)\geq m_{b}(\tilde B) \}$
,
$E_{1}=\{ z\in B: b(z)\geq m_{b}(\tilde B) \}$
and
$E_{2}=\{ z\in B: b(z)< m_{b}(\tilde B) \}$
. Then it is immediate that
$\tilde B=F_{1}\cup F_{2}$
,
$B=E_{1}\cup E_{2}$
and
$E_{1}\cap E_{2}=\emptyset $
, and from definition of the median value we can see that
It is clear that
$$ \begin{align*}\begin{cases} b(z)-b(w) \geq 0, \quad (z,w)\in E_{1}\times F_{1},\\ b(z)-b(w) < 0, \quad (z,w)\in E_{2}\times F_{2}. \end{cases}\end{align*} $$
Then, for
$(z,w)\in (E_{1}\times F_{1} )\cup (E_{2}\times F_{2})$
,
$$ \begin{align} |b(z)-b(w)| = |b(z)-m_{b}(\tilde B) +m_{b}(\tilde B) -b(w)| \geq |b(z)-m_{b}(\tilde B)|. \end{align} $$
Together with (2-5), (2-6) and (2-4), we have
Then by Hölder’s inequality and the fact that
,
Similarly,
and hence,
Consequently,
This finishes the proof of sufficiency of Theorem 1.1.□
2.2 Characterization of
via the Commutator
Based on Lemma 2.1, we have the following fundamental lemma from [Reference Krantz and Li10, Lemma 1.2].
Lemma 2.8. Let
. Then, for any
$\xi>0$
, there is a function
$b_{\xi }\in BUC(bD)$
such that
$$ \begin{align*} \|b_{\xi} -b\|_{*}<\xi. \end{align*} $$
Moreover,
$b_{\xi }$
satisfies the following conditions: there is an
$\epsilon \in (0,1)$
such that
$$ \begin{align} |b_{\xi}(w) -b_{\xi}(z)|<C_{\xi} \mathtt{d}(w,z)^{\epsilon},\quad \text{for all } w,z\in bD. \end{align} $$
For each
$0<\eta <<1$
, we let
$R^{\eta }(w,z)$
be a continuous extension of the kernel
$R(w,z)$
of
from
$bD\times bD\backslash \{(w,z): \texttt { d}(w,z)<\eta \}$
to
$bD\times bD$
such that
$$ \begin{align*} R^{\eta}(w,z) =R(w,z),&\quad \text{if}\ \ \mathtt{d}(w,z)\geq\eta;\\ |R^{\eta}(w,z)|\lesssim {1\over \mathtt{d}(w,z)^{2n-1}}, &\quad \text{if}\ \mathtt{d}(w,z)<\eta;\\ R^{\eta}(w,z)=0, \quad \text{if}\ \ \mathtt{d}(w,z)<\eta/c,&\quad {\text {for some}}\ c>1. \end{align*} $$
Let
be the integral operator associated to the kernel
$R^{\eta }(w,z)$
. Then we have the following approximation result.
Lemma 2.9. Let
$b\in BUC(bD)$
satisfy
$$ \begin{align} |b(w) -b(z)|<c~ \mathtt{d}(w,z)^{\epsilon},\quad \mbox{for some } c \geq1,\ \epsilon\in(0,1),\ \text{for all } w,z\in bD. \end{align} $$
Then, for
$1<p<\infty $
,
$0<\kappa <1$
and
$v\in A_{p}$
,
as
$\eta \to 0$
.
Proof. Let
$f\in L_{v}^{p,\kappa }(bD)$
. For any
$z\in bD$
,
Then
which implies that
This concludes the proof.
We can now give the proof of Theorem 1.2.
Proof (Proof of Theorem 1.2).
We begin with sufficiency. Assume that
$v\in A_{p}, 1<p<\infty $
and that
is compact on
$L_{v}^{p,\kappa }(bD)$
. Then
is bounded on
$L_{v}^{p,\kappa }(bD)$
. By Theorem 1.1, we have
$b\in \textrm{BMO}(bD)$
. Without loss of generality, we may assume that
$\|b\|_{ \textrm{BMO}(bD) }=1$
.
To show
$b\in \text{VMO}(bD)$
, we seek a contradiction. In its simplest form, the contradiction is that there is no bounded operator
$T \;:\; \ell ^{p} (\mathbb N) \to \ell ^{p} (\mathbb N)$
with
$Te_{j } = T e_{k} \neq 0$
for all
$j,k\in \mathbb N$
. Here,
$e_{j}$
is the standard basis for
$\ell ^{p} (\mathbb N)$
.
The main step is to construct the approximation to a standard basis in
$\ell ^{p}$
, namely a sequence of functions
$\{g_{j}\}$
such that
$ \lVert g_{j } \rVert _{L_{v}^{p,\kappa }(bD)} \simeq 1$
, and such that for a nonzero
$ \phi $
, we have
.
Suppose that
$b\notin \text{VMO}(bD)$
. Then there exist
$\delta _{0}>0$
and a sequence
$\{B_{j}\}_{j=1}^{\infty }:=\{B_{r_{j}}(z_{j})\}_{j=1}^{\infty }$
of balls such that
Without loss of generality, we assume that
$r_{j}<\gamma _{0}$
for all j, where
$\gamma _{0}$
is as in Lemma 2.7.
Now choose a subsequence
$\{B_{j_{i}}\}$
of
$\{B_{j}\}$
such that
where
is the constant as in (2-1).
For the sake of simplicity we drop the subscript i, that is, we denote
$\{B_{j_{i}}\}$
by
$\{B_{j}\}$
.
Following the sufficiency proof of Theorem 1.1(2) in [Reference Duong, Lanzani, Li and Wick4], for each such
$B_{j}$
, we can choose a corresponding ball
$\tilde B_{j}$
and the corresponding disjoint subsets
$E_{j,1}, E_{j,2}\subset B_{j}$
and
$\widetilde F_{j,1}, \widetilde F_{j,2}\subset \tilde B_{j}$
, such that
$B_{j}=E_{j,1}\cup E_{j,2}$
, and we have
and
Next, by using Hölder’s inequality and
$v\in A_{p}$
,
where in the above inequalities we denote
$$ \begin{align*} f_{j} := v(B_{j})^{{(\kappa-1)/ p}}\chi_{\widetilde F_{j,1}} .\end{align*} $$
Combining the above estimates, we have that
Note that
Since
$v\in A_{p}$
, it follows that there exist positive constants
$C_{1},C_{2}$
and
$\sigma \in (0,1)$
such that, for any measurable set
$E\subset B$
,
Combining this and (2-9),
$$ \begin{align*} \sup_{B}\bigg\{\frac{v(B\cap \widetilde F_{j,1})}{[v(B)]^{\kappa}} \bigg\}^{1/p}&\leq \sup_{B} \{ v(B\cap \widetilde F_{j,1})^{1-\kappa} \}^{1/p}\\[3pt] &\leq v( \widetilde F_{j,1})^{(1-\kappa)/ p}\leq v( \widetilde B_{j })^{(1-\kappa)/ p}\thickapprox v( B_{j })^{(1-\kappa)/ p} \end{align*} $$
and
$$ \begin{align*}\sup_{B} \bigg\{\frac{v(B\cap \widetilde F_{j,1})}{[v(B)]^{\kappa}} \bigg\}^{1/p} \geq \bigg\{\frac{v( \widetilde F_{j,1})}{[v(\widetilde B_{j })]^{\kappa}} \bigg\}^{1/p} \gtrsim v( \widetilde B_{j })^{(1-\kappa)/ p}\thickapprox v( B_{j })^{(1-\kappa)/ p}. \end{align*} $$
This implies that
$ \|f_{j}\|_{L_{v}^{p,\kappa }(bD)} \approx 1. $
Consequently,
$\{f_{j}\}_{j}$
is a bounded sequence in
$L_{v}^{p,\kappa }(bD)$
with a uniform
$L_{v}^{p,\kappa }(bD)$
lower bound away from zero.
Since
is compact, we obtain that the sequence
has a convergent subsequence, denoted by
We denote the limit function by
$g_{0}$
, that is,
Moreover,
$g_{0} \neq 0$
.
After taking a further subsequence, labelled
$g_{j}$
, we may assume that
-
•
$\lVert g_{j} \rVert _{L_{v}^{p,\kappa }(bD)} \simeq 1$
; -
•
$g_{j}$
are disjointly supported; and -
•
.
Take
$a_{j } = j^{ -( ({p+1})/{2p})}$
, so that
$ \{a_{j}\} \in \ell ^{p} \setminus \ell ^{1}$
. It is immediate that
$ \gamma = \sum _{j} a_{j} g_{j} \in L_{v}^{p,\kappa }(bD)$
, and hence,
. But
$ g_{0} \sum _{j} a_{j} \equiv \infty $
, and yet
This contradiction shows that
$b\in \text{VMO}(bD)$
.
We now turn to necessity. Recall that
. Since the kernel
is a standard kernel,
is compact on
$L_{v}^{p,\kappa }(bD)$
. Therefore, we only need to show that
is also compact on
$L_{v}^{p,\kappa }(bD)$
.
From Lemma 2.8, for any
$\xi>0$
, there exists
$b_{\xi }\in BUC(bD)$
such that
$\|b-b_{\xi }\|_{*}<\xi $
. Then, by Theorem 1.1,
Thus, to prove that
is compact on
$L_{v}^{p,\kappa }(bD)$
, it suffices to prove that
is compact on
$L_{v}^{p,\kappa }(bD)$
. By Lemma 2.8 and (2-7), without loss of generality, we may assume that
$b\in BUC(bD)$
and (2-8) holds. By Lemma 2.9, it suffices to prove that, for any fixed
$\eta $
satisfying
$0<\eta \ll 1$
,
is compact on
$L_{v}^{p,\kappa }(bD)$
.
Since
$R(w,z)$
is continuous on
$bD\times bD\backslash \{(z,z): z\in bD\}$
, for any
$f\in L_{v}^{p,\kappa }(bD)$
, we see that
is continuous on
$bD$
. To conclude the proof, we now argue that the image of the unit ball of
$L_{v}^{p,\kappa }(bD)$
under the commutator
is an equicontinuous family. Compactness follows from the Arzelà–Ascoli theorem.
It remains to prove equicontinuity. For any
$z,w\in bD$
with
$\texttt{d}(w,z)<1$
,
For
$I(z,w)$
, by Hölder’s inequality,
where the last inequality is due to the fact that
$v\in A_{p}, R^{\eta }(u,z)\in C(bD\times bD)$
and
$bD$
is bounded.
Since b is bounded, if we let
$\texttt { d}(w,z)<{\eta / c\cdot c_{R}}$
, by a discussion similar to [Reference Krantz and Li10, page 645], we can obtain that
As a consequence,
is an equicontinuous family, where
$\mathcal U$
is the unit ball in
$L_{v}^{p,\kappa }(bD)$
. This finishes the proof of Theorem 1.2.□
3 Commutator of Cauchy–Leray integral for strongly
$\mathbb C$
-linearly convex domains with minimal smoothness
In this section we focus on the bounded domain
$D\subset \mathbb C^{n}$
which is strongly
$\mathbb C$
-linearly convex and whose boundary satisfies the minimal regularity condition of class
$C^{1,1}$
[Reference Lanzani and Stein12].
Suppose D is a bounded domain in
$\mathbb C^{n}$
with defining function
$\rho $
satisfying the following two conditions.
-
(1) D is of class
$C^{1,1}$
, that is, the first derivatives of its defining function
$\rho $
are Lipschitz, and
$|\nabla \rho (w)|>0$
whenever
$w\in \{w:\rho (w)=0\}=bD$
. -
(2) D is strongly
$\mathbb C$
-linearly convex, that is, D is a bounded domain of
$C^{1}$
, and at any boundary point it satisfies either of the two equivalent conditions
$$ \begin{align*} |\Delta(w,z)|&\geq c|w-z|^{2}, \\ d_{E}(z,w+T_{w}^{\mathbb C})&\geq \tilde c|w-z|^{2}, \end{align*} $$
for some
$c,\tilde c>0$
, where
$$ \begin{align*} \Delta(w,z)=\langle\partial \rho(w), w-z\rangle \end{align*} $$
and
$d_{E}(z,w+T_{w}^{\mathbb C})$
denotes the Euclidean distance from z to the affine subspace
$w+T_{w}^{\mathbb C}$
. Note that
$T_{w}^{\mathbb C}:=\{v:\langle \partial \rho (w),v\rangle =0\}$
is the complex tangent space referred to the origin, and
$w+T_{w}^{\mathbb C}$
is its geometric realization as an affine space tangent to
$bD$
at w.
On
$bD$
there is a quasidistance
$\texttt { d}$
, which is defined as
$$ \begin{align*} \mathtt{d}(w,z)= |\Delta(w,z)|^{1/2} = |\langle \partial\rho,w-z \rangle|^{1/2},\quad w,z\in bD. \end{align*} $$
Let
be the Leray–Levi measure
on
$bD$
(see [Reference Lanzani and Stein12]). According to [Reference Lanzani and Stein12, Proposition 3.4],
is also equivalent to the induced Lebesgue measure
$d\sigma $
on
$bD$
in the following sense:
and there are two strictly positive constants
$c_{1}$
and
$c_{2}$
so that
$$ \begin{align*} c_{1}\leq \tilde\Lambda(w)\leq c_{2} \quad \text{for }\sigma\mbox{-almost every } w\in bD.\end{align*} $$
We also denote by
$ B_{r}(w) =\{ z\in bD:\ \texttt { d}(w,z)<r \}$
the boundary balls determined via the quasidistance
$\texttt { d }$
. By [Reference Lanzani and Stein12, Proposition 3.5], we also have
The Cauchy–Leray integral of a suitable function f on
$bD$
, denoted
$\textbf {C}(f)$
, is formally defined by
When restricting z to the boundary
$bD$
, we have the Cauchy–Leray transform
$f\mapsto \mathcal C(f)$
, defined as
where the function f satisfies the Hölder-like condition
$$ \begin{align*} |f(w_{1})-f(w_{2})|\lesssim \mathtt{d}(w_{1},w_{2})^{\alpha},\quad w_{1},w_{2}\in bD, \end{align*} $$
for some
$0<\alpha \leq 1$
.
Take
$K(w,z)$
to be the function defined for
$w,z\in bD$
, with
$w\not = z$
, by
$$ \begin{align*}K(w,z) = {1\over \Delta(w,z)^{n}}.\end{align*} $$
This function is the ‘kernel’ of the operator
$\mathcal C$
, in the sense that
whenever z lies outside of the support of f and f satisfies the Hölder-like condition for some
$\alpha $
. The size and regularity estimates that are relevant for us are:
$$ \begin{align} & |K(w,z)|\lesssim {1\over\mathtt{d}(w,z)^{2n}},\nonumber\\[3pt] & |K(w,z) - K(w^{\prime},z)|\lesssim { \mathtt{d}(w,w^{\prime})\over \mathtt{d}(w,z)^{2n+1} },\quad \text{if}\ \mathtt{d}(w,z)\geq c_{K}\mathtt{d}(w,w^{\prime}),\\[3pt] & |K(w,z) - K(w,z^{\prime})|\lesssim { \mathtt{d}(z,z^{\prime})\over \mathtt{d}(w,z)^{2n+1} },\quad \text{if}\ \mathtt{d}(w,z)\geq c_{K} \mathtt{d}(z,z^{\prime}),\nonumber \end{align} $$
for an appropriate constant
$c_{K}>0$
. Moreover, for the size estimates we actually have
$$ \begin{align} |K(w,z)|= {1\over\mathtt{d}(w,z)^{2n}}. \end{align} $$
We need the
$L^{p}(bD)$
boundedness of the Cauchy–Leray transform in [Reference Lanzani and Stein12, Theorem 5.1].
Lemma 3.1. The Cauchy–Leray transform
$f\mapsto \mathcal C(f)$
, initially defined for functions f that satisfy the Hölder-like condition for some
$\alpha $
, extends to a bounded linear operator on
$L^{p}(bD)$
for
$1<p<\infty $
.
Proof (Proof of Theorems 1.3 and 1.4).
We point out that the proof of Theorem 1.3 follows from the proof of Theorem 1.5 in [Reference Di Fazio and Ragusa5], and the proof of Theorem 1.4 follows from the proofs of Theorems 1.1 and 1.2. In fact, these are simpler, since the operator
$\mathcal C$
is a Calderón–Zygmund operator.□
