# Liouville-type results in two dimensions for stationary points of
functionals with linear growth

Michael Bildhauer & Martin Fuchs

###### Abstract

111AMS Subject Classification: 49J40, 35J50

We consider variational integrals of linear growth satisfying the condition of
$\mu$-ellipticity for some exponent $\mu>1$ and prove that stationary points
$u$: $\mathbb{R}^{2}\to\mathbb{R}^{N}$ with the property

$\limsup_{|x|\to\infty}\frac{|u(x)|}{|x|}<\infty$

must be affine functions. The latter condition can be dropped in the scalar
case together with appropriate assumptions on the energy density providing an
extension of Bernstein’s theorem.

##  1 Introduction

In this note we mainly present results of Liouville-type for entire solutions
$u$: $\mathbb{R}^{2}\to\mathbb{R}^{N}$ of the system

$\operatorname{div}\big{[}\nabla F(\nabla u)\big{]}=0\quad\mbox{on
$\mathbb{R}^{2}$}\,,$ (1.1)

concentrating on the case of energy densities $F$:
$\mathbb{R}^{2N}\to\mathbb{R}$ with linear growth.

To be precise we assume that $F$ is of class
$C^{2}\big{(}\mathbb{R}^{2N}\big{)}$ satisfying with constants $M$, $\lambda$,
$\Lambda>0$ and for some exponent $\mu>1$

$\displaystyle|\nabla F(Z)|$ $\displaystyle\leq$ $\displaystyle M\,,$ (1.2)
$\displaystyle\lambda(1+|Z|)^{-\mu}|Y|^{2}$ $\displaystyle\leq$ $\displaystyle
D^{2}F(Z)(Y,Y)\leq\Lambda(1+|Z|)^{-1}|Y|^{2}$ (1.3)

for all $Y$, $Z\in\mathbb{R}^{2N}$, where the first inequality in (1.3)
expresses the fact that $F$ is a $\mu$-elliptic integrand. Note that (1.2) and
(1.3) exactly correspond to the requirements of Assumption 4.1 in [2] and as
outlined in Remark 4.2 of this reference, conditions (1.2) and (1.3) imply
that $F$ is of linear growth in the sense that

$a|Z|-b\leq F(Z)\leq A|Z|+B\,,\quad Z\in\mathbb{R}^{2N}\,,$

holds with constants $a$, $A>0$, $B$, $b$, $\geq 0$.

Note also that the “minimal surface case” is included by letting
$F(Z):=(1+|Z|^{2})^{1/2}$. In this case we have the validity of (1.3) with the
choice $\mu=3$, and two families of densities satisfying (1.2) and (1.3) with
prescribed exponent $\mu>1$ are given by

$F(Z):=\left\\{\begin{array}[]{l}\displaystyle\int_{0}^{|Z|}\int_{0}^{s}(1+r)^{-\mu}\,{\rm
d}r\,{\rm d}s\\\\[17.22217pt]
\displaystyle\int_{0}^{|Z|}\int_{0}^{s}(1+r^{2})^{-\mu/2}\,{\rm d}r\,{\rm
d}s\end{array}\right\\}\,,\quad Z\in\mathbb{R}^{2N}\,.$

Our results on the behaviour of global solutions of the Euler equations (1.1)
with $\mu$-elliptic densities $F$ are as follows.

######  Theorem 1.1.

Let $u\in C^{2}\big{(}\mathbb{R}^{2},\mathbb{R}^{N}\big{)}$ denote a solution
of (1.1) with density $F$ such that (1.2) and (1.3) hold.

  1. a)

Suppose that in addition

$\lim_{|x|\to\infty}\frac{|u(x)|}{|x|}=0\,.$ (1.4)

Then $u$ is a constant function.

  2. b)

If the function $u$ has the property

$\sup_{x\in\mathbb{R}^{2}}|\nabla u(x)|<\infty\,,$ (1.5)

then $u$ is affine.

  3. c)

If we have

$\limsup_{|x|\to\infty}\frac{|u(x)|}{|x|}<\infty\,,$ (1.6)

then the conclusion of $b)$ holds.

######  Remark 1.1.

  1. a)

Clearly (1.4) holds in the case that $u$ is a bounded solution, and evidently
(1.5) implies (1.6).

  2. b)

We do not know if there are versions of Theorem 1.1 for entire solutions $u$:
$\mathbb{R}^{n}\to\mathbb{R}^{N}$ of (1.1) in the case $n\geq 3$.

  3. c)

Our discussion of smooth solutions of the system (1.1) includes the vector
case $N>1$ for densities $F$ of linear growth. The existence of smooth
solutions is known provided that $\mu$ is not too large and provided that
$F(Z)=f(|Z|)$. It is a challenging question whether the smoothness of
solutions remains true (to some extend) if the second hypothesis is dropped.

Before presenting the proof of Theorem 1.1 we wish to mention that there
exists a variety of Liouville-type theorems for entire solutions $u$:
$\mathbb{R}^{n}\to\mathbb{R}^{N}$, $n\geq 2$, $N\geq 1$, of systems of the
form (1.1) (and even for nonhomogeneous systems not generated by a density
$F$) assuming that $F$ is of superlinear growth. The interested reader should
consult the references on this topic quoted for example in the textbooks [12],
[13], [14], [16], [22] and [25]. A nice survey is also presented in [6].

Besides this more general discussion the validity of Liouville theorems for
harmonic maps between Riemannian manifolds turned out to be a useful tool for
the analysis of the geometric properties of the underlying manifolds. Without
being complete we refer to [4], [5], [17], [18], [19], [20], [27] and [28].

Liouville theorems are also of interest in the setting of fluid mechanics,
where in the stationary case (1.1) is replaced by a nonlinear variant of the
Navier-Stokes equation with dissipative potential $F$ of superlinear growth
and the incompressibility condition $\operatorname{div}u=0$ for the velocity
field $u$: $\mathbb{R}^{n}\to\mathbb{R}^{n}$ has to be added. The validity of
Liouville theorems has been established in the $2$-D-case, i.e. for $n=2$, for
instance in the papers [3], [8], [9], [10], [11], [15], [21], [23], [29] and
[30]. We like to mention that the case of potentials $F$ satisfying (1.2) and
(1.3) is treated in [10] assuming $\mu<2$.

As it stands, the conclusions of Theorem 1.1 b) and c) are in the spirit of
Bernstein’s theorem (see [1]) for nonparametric minimal surfaces, where in
this particular setting conditions like (1.5) or (1.6) are seen to be
superfluous. For completeness we specialize the Bernstein result obtained by
Farina, Sciunzi and Valdinoci in Theorem 1.4 of their paper [7] to the case of
linear growth integrands.

######  Theorem 1.2.

Consider a function $g\in C^{2}\big{(}[0,\infty)\big{)}$ such that with
constants $a_{1}$, $a_{3}$, $a_{5}>0$, $a_{2}$, $a_{4}\geq 0$ we have for some
exponent $\mu\geq 1$

$\displaystyle\displaystyle g^{\prime}(0)=0\,,\quad
g^{\prime\prime}(t)>0\quad\mbox{for}\quad t>0\,,$ (1.7)
$\displaystyle\displaystyle a_{1}t-a_{2}\leq g(t)\leq
a_{3}t+a_{4}\quad\mbox{for}\quad t\geq 0\,,$ (1.8) $\displaystyle\displaystyle
g^{\prime\prime}(t)\leq a_{5}(1+t)^{-\mu}\quad\mbox{for}\quad t\geq 0\,.$
(1.9)

Let $F$: $\mathbb{R}^{2}\to\mathbb{R}$, $F(Z):=g\big{(}|Z|\big{)}$, and
consider a solution $u$: $\mathbb{R}^{2}\to\mathbb{R}$ of (1.1) being of class
$C^{2}$. Then $u$ is an affine function provided that $\mu\geq 3$.

######  Remark 1.2.

  1. a)

Note that the minimal surface case is included with the choices
$g(t)=\sqrt{1+t^{2}}$ and $\mu=3$, moreover, we can cover the examples stated
in front of Theorem 1.1 provided that $\mu\geq 3$.

  2. b)

To our knowledge it is an unsolved problem, if Theorem 1.2 remains true for
exponents $\mu\in(1,3)$.

  3. c)

Roughly speaking it follows from the work [26] of J.C.C. and J. Nitsche that
the Bernstein property fails for the equation

$0=\operatorname{div}\Bigg{[}\frac{g^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}\nabla u\Bigg{]}\,,$

if the density of $g$ is elliptic and of superlinear growth including even the
nearly linear case $g(t)=t\ln(1+t)$, i.e. there exist non-affine solutions
$u$: $\mathbb{R}^{2}\to\mathbb{R}$. However, the Nitsche criterion does not
apply to integrands of linear growth as considered in Theorem 1.2 (see Remark
4.1).

  4. d)

From the identity

$\displaystyle D^{2}F(Z)(X,X)$ $\displaystyle=$
$\displaystyle\frac{1}{|Z|}g^{\prime}\big{(}|Z|\big{)}\Big{[}|X|^{2}-\frac{1}{|Z|^{2}}(X\cdot
Z)^{2}\Big{]}$
$\displaystyle+g^{\prime\prime}\big{(}|Z|\big{)}\frac{1}{|Z|^{2}}(X\cdot
Z)^{2}\,,\quad X\,\,Z\in\mathbb{R}^{2}\,,$

it follows that (observing the boundedness of $g^{\prime}$)

$\min\Bigg{\\{}g^{\prime\prime}\big{(}|Z|\big{)},\frac{g^{\prime}\big{(}|Z|\big{)}}{|Z|}\Bigg{\\}}|X|^{2}\leq
D^{2}F(Z)(X,X)\leq\Lambda\big{(}1+|Z|\big{)}^{-1}|X|^{2}\,,$

i.e. the second inequality in (1.3) holds with some constant $\Lambda>0$. For
$t\geq 1$ we have the lower bound $g^{\prime}(t)/t\geq c/t$, which by (1.9)
means that in fact $g^{\prime\prime}\big{(}|Z|\big{)}$ measures the degree of
ellipticity of $D^{2}F(Z)$. This shows that the integrand $F$ in general is
not $\mu$-elliptic in the sense of the first inequality from (1.3): according
to (1.9) the power $t^{-\mu}$ just acts as an upper bound for the values
$g^{\prime\prime}(t)$. Thus we have the “Bernstein property” for any density
$F(Z)=g\big{(}|Z|\big{)}$ of linear growth and for which
$g^{\prime\prime}(t)=O(t^{-3})$ as $t\to\infty$.

##  2 Proof of Theorem 1.1, Part a)

In the weak formulation of (1.1), i.e. in the equation

$\int_{\mathbb{R}^{2}}\nabla F(\nabla u):\nabla\varphi\,{\rm
d}x=0\,,\quad\varphi\in
C^{1}_{0}\big{(}\mathbb{R}^{2},\mathbb{R}^{N}\big{)}\,,$ (2.1)

the function $\varphi$ is replaced by $\partial_{\alpha}\varphi$
($\alpha\in\\{1,2\\}$ fixed), where now $\varphi\in
C^{2}_{0}\big{(}\mathbb{R}^{2},\mathbb{R}^{N})$ is assumed. With an
integration by parts we obtain from (2.1)

$\int_{rz^{2}}D^{2}F(\nabla u)\big{(}\partial_{\alpha}\nabla
u,\nabla\varphi\big{)}\,{\rm d}x=0\,.$ (2.2)

Now we choose $\varphi=\eta^{2}\partial_{\alpha}u\in
C^{1}_{0}\big{(}\mathbb{R}^{2},\mathbb{R}^{N}\big{)}$ in (2.2), where $\eta\in
C^{1}_{0}\big{(}\mathbb{R}^{2}\big{)}$, $\operatorname{spt}\eta\subset
B_{2R}(0)$, $\eta\equiv 1$ on $B_{R}(0)$, $0\leq\eta\leq 1$, $|\nabla\eta|\leq
cR^{-1}$. Then by Cauchy-Schwarz’s and Young’s inequality we have (summation
w.r.t. $\alpha=1$, $2$)

$\displaystyle\int_{B_{2R}(0)}\eta^{2}D^{2}F(\nabla
u)\big{(}\partial_{\alpha}\nabla u,\partial_{\alpha}\nabla u\big{)}\,{\rm d}x$
(2.3) $\displaystyle\leq$ $\displaystyle c\int_{B_{2R}(0)}D^{2}F(\nabla
u)\big{(}\nabla\eta\otimes\partial_{\alpha}u,\nabla\eta\otimes\partial_{\alpha}u\big{)}\,{\rm
d}x\,.$

The hypotheses (1.2) and (1.3) yield

$\displaystyle\int_{B_{R}(0)}\big{(}1+|\nabla
u|\big{)}^{-\mu}|\nabla^{2}u|^{2}\,{\rm d}x$ $\displaystyle\leq$
$\displaystyle cR^{-2}\int_{B_{2R}(0)-B_{R}(0)}\frac{|\nabla
u|^{2}}{\sqrt{1+|\nabla u|^{2}}}\,{\rm d}x$ (2.4) $\displaystyle\leq$
$\displaystyle cR^{-2}\int_{B_{2R}(0)-B_{R}(0)}|\nabla u|\,{\rm d}x$

and using the auxiliary inequality (2.9) of Lemma 2.1 given below we obtain
for any $\varepsilon>0$

$\displaystyle\int_{B_{R}(0)}\big{(}1+|\nabla
u|\big{)}^{-\mu}|\nabla^{2}u|^{2}\,{\rm d}x$ (2.5) $\displaystyle\leq$
$\displaystyle\frac{c}{R^{2}}\int_{B_{2R}(0)-B_{R}(0)}\Big{[}\varepsilon+c(\varepsilon)\big{(}\nabla
F(\nabla u)-\nabla F(0)\big{)}:\nabla u\Big{]}\,{\rm d}x\,.$

With (2.1) we also have

$\int_{\mathbb{R}^{2}}\big{(}\nabla F(\nabla u)-\nabla
F(0)\big{)}:\nabla\varphi\,{\rm d}x=0\,,\quad\varphi\in
C^{1}_{0}\big{(}\mathbb{R}^{2},\mathbb{R}^{N}\big{)}\,,$ (2.6)

where we now choose $\varphi=\tilde{\eta}^{2}u$, $\tilde{\eta}\in
C^{1}_{0}\big{(}\mathbb{R}^{2}\big{)}$, $\tilde{\eta}\equiv 1$ on
$B_{2R}(0)-B_{R}(0)$, $\operatorname{spt}\eta\subset
B_{5R/2}(0)-\overline{B}_{R/2}(0)$, $0\leq\tilde{\eta}\leq 1$,
$|\nabla\tilde{\eta}|\leq c/R$.

With this choice (2.6) gives

$\displaystyle\int_{\mathbb{R}^{2}}\big{(}\nabla F(\nabla u)-\nabla
F(0)\big{)}:\nabla u\tilde{\eta}^{2}\,{\rm d}x$ (2.7) $\displaystyle=$
$\displaystyle-2\int_{\mathbb{R}^{2}}\tilde{\eta}\big{(}\nabla F(\nabla
u)-\nabla F(0)\big{)}:(\nabla\tilde{\eta}\otimes u)\,{\rm d}x$
$\displaystyle\leq$ $\displaystyle
cR^{-1}\int_{B_{5R/2}(0)-B_{R/2}(0)}|u|\,{\rm d}x$ $\displaystyle\leq$
$\displaystyle cR\sup_{B_{5R/2}(0)-\overline{B}_{R/2}(0)}|u|\,,$

where our assumption (1.2) is used.

By the definition of $\tilde{\eta}$ we obtain using (2.7)

$\displaystyle\int_{B_{2R}(0)-B_{R}(0)}\big{(}\nabla F(\nabla u)-\nabla
F(0)\big{)}:\nabla u\,{\rm d}x$ (2.8) $\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{2}}\big{(}\nabla F(\nabla u)-\nabla
F(0)\big{)}:\nabla u\tilde{\eta}^{2}\,{\rm d}x$ $\displaystyle\leq$
$\displaystyle cR\sup_{B_{5R/2}(0)-\overline{B}_{R/2}(0)}|u|\,.$

If we insert (2.8) into inequality (2.5) and pass to the limit $R\to\infty$
recalling (1.4), we obtain for any $\varepsilon>0$

$\int_{\mathbb{R}^{2}}\big{(}1+|\nabla u|\big{)}^{-\mu}|\nabla^{2}u|^{2}\,{\rm
d}x\leq c\varepsilon\,,$

hence $\nabla^{2}u\equiv 0$ and therefore we find $A\in\mathbb{R}^{2N}$,
$a\in\mathbb{R}^{N}$ such that

$u(x)=Ax+a\,.$

Again we apply of the growth condition (1.4) and obtain $A=0$, hence the first
part of Theorem 1.1 is established.

During the proof we made use of the elementary lemma

######  Lemma 2.1.

Let $F\in C^{2}\big{(}\mathbb{R}^{2N}\big{)}$ just satisfy the first
inequality of (1.3) and let

$\theta(r):=\frac{\lambda}{\mu-1}\big{[}1-(1+r)^{1-\mu}\big{]}\,,\quad r\geq
0\,.$

Then it holds for any $\varepsilon>0$ and all $Z\in\mathbb{R}^{nN}$

$|Z|\leq\varepsilon+\theta^{-1}(\varepsilon)\big{[}\nabla F(Z)-\nabla
F(0)\big{]}:Z\,.$ (2.9)

_Proof of Lemma 2.1._ We fix $\varepsilon>0$. If $|Z|\geq\varepsilon$ then

$|Z|\theta\big{(}|Z|\big{)}\geq|Z|\theta(\varepsilon)\,,$

which implies

$|Z|\leq\theta^{-1}(\varepsilon)\,|Z|\,\theta\big{(}|Z|\big{)}\,,$

and if $Z\in\mathbb{R}^{2N}$ is arbitrarily given, we have

$|Z|\leq\varepsilon+\theta^{-1}(\varepsilon)\,|Z|\,\theta\big{(}|Z|\big{)}\,.$

Moreover,

$\theta\big{(}|Z|\big{)}\,|Z|\leq\big{[}\nabla F(Z)-\nabla F(0)\big{]}:Z\ $
(2.10)

easily follows from the first inequality in (1.3) as outlined in [2], formula
(1), p. 98., and (2.10) gives (2.9). $\Box$

######  Remark 2.1.

Clearly Lemma 2.1 is not limited to the case $n=2$ and without condition (1.3)
it would be sufficient to assume (2.10) for an increasing non-negative
function $\theta$: $[0,\infty)\to\mathbb{R}$.

##  3 Proof of Theorem 1.1, Parts b) and c)

For Part b) we remark, that the idea of applying a Liouville argument to the
derivatives of solutions, which are seen to solve an appropriate elliptic
equation, has been successfully used by Moser [24], Theorem 6, with the result
that entire solutions of the minimal surface equation with bounded gradients
in fact must be affine functions in any dimension $n\geq 2$.

In our setting, i.e. for $n=2$ together with $N\geq 1$, one may just follow
the arguments presented in [12], Chapter III, p. 82, for an elementary proof
essentially based on the “hole-filling” technique.

In Theorem 1.1, Part b) turns out to be a corollary of Part c), which we now
prove following some ideas given in [11].

As in the proof of the first part of Theorem 1.1 we obtain from (2.3) the
following variant of inequality (2.4)

$\int_{B_{R}(0)}D^{2}F(\nabla u)\big{(}\partial_{\alpha}\nabla
u,\partial_{\alpha}\nabla u\big{)}\,{\rm d}x\leq
cR^{-2}\int_{B_{2R}(0)-B_{R}(0)}|\nabla u|\,{\rm d}x$ (3.1)

and, as outlined after (2.4), (3.1) gives for all $R>0$ and with the choice
$\varepsilon=1$

$\int_{B_{R}(0)}D^{2}F(\nabla u)\big{(}\partial_{\alpha}\nabla
u,\partial_{\alpha}\nabla u\big{)}\,{\rm d}x\leq
c\Big{[}1+R^{-1}\sup_{B_{5R/2}(0)-B_{R/2}(0)}|u|\Big{]}\,.$ (3.2)

Inequality (3.2) shows, using (1.6),

$\int_{\mathbb{R}^{2}}D^{2}F(\nabla u)\big{(}\partial_{\alpha}\nabla
u,\partial_{\alpha}\nabla u\big{)}\,{\rm d}x<\infty\,.$ (3.3)

We finally claim that

$\int_{\mathbb{R}^{2}}D^{2}F(\nabla u)\big{(}\partial_{\alpha}\nabla
u,\partial_{\alpha}\nabla u\big{)}\,{\rm d}x=0\,,$ (3.4)

which gives $|\nabla^{2}u|=0$, hence the proof will be complete.

To prove (3.4) we again consider (2.2) and choose $\varphi$ as done after this
inequality. We obtain with $T_{R}:=B_{2R}(0)-\overline{B}_{R/2}(0)$ using the
Cauchy-Schwarz inequality

$\displaystyle\int_{\mathbb{R}^{2}}D^{2}F(\nabla
u)\big{(}\partial_{\alpha}\nabla u,\partial_{\alpha}\nabla
u\big{)}\eta^{2}\,{\rm d}x$ $\displaystyle=$
$\displaystyle-2\int_{T_{R}}D^{2}F(\nabla u)\big{(}\eta\partial_{\alpha}\nabla
u,\nabla\eta\otimes\partial_{\alpha}u\big{)}\,{\rm d}x$ $\displaystyle\leq$
$\displaystyle c\Bigg{[}\int_{T_{R}}\eta^{2}D^{2}F(\nabla
u)\big{(}\partial_{\alpha}\nabla u,\partial_{\alpha}\nabla u\big{)}\,{\rm
d}x\Bigg{]}^{\frac{1}{2}}$
$\displaystyle\cdot\Bigg{[}\int_{T_{R}}D^{2}F(\nabla
u)\big{(}\nabla\eta\otimes\partial_{\alpha}u,\nabla\eta\otimes\partial_{\alpha}u\big{)}\,{\rm
d}x\Bigg{]}^{\frac{1}{2}}$ $\displaystyle=:$ $\displaystyle I_{1}(R)\cdot
I_{2}(R)\,.$

We recall (3.3) which gives

$I_{1}(R)\to 0\quad\mbox{as $R\to\infty$.}$

Assumption (1.3) yields the estimate

$I_{2}(R)\leq c\Bigg{[}R^{-2}\int_{T_{R}}|\nabla u|\,{\rm
d}x\Bigg{]}^{\frac{1}{2}}\,.$

thus we obtain (3.4), if we can prove

$\int_{B_{R}(0)}|\nabla u|\,{\rm d}x\leq c\big{(}1+R^{2}\big{)}\,.$ (3.5)

For (3.5) we use (2.9) (recall $\eta\equiv 1$ on $B_{R}(0)$) with the choice
$\varepsilon=1$, hence (compare the derivation of (2.7))

$\displaystyle\int_{B_{R}(0)}|\nabla u|\,{\rm d}x$ $\displaystyle\leq$
$\displaystyle|B_{R}(0)|+c\int_{B_{R}(0)}\big{[}\nabla F(\nabla u)-\nabla
F(0)\big{]}:\nabla u\,{\rm d}x$ $\displaystyle\leq$
$\displaystyle|B_{R}(0)|+c\int_{B_{2}R(0)}\eta^{2}\big{[}\nabla F(\nabla
u)-\nabla F(0)\big{]}:\nabla u\,{\rm d}x$ $\displaystyle\leq$ $\displaystyle
c\Big{[}R^{2}+R\sup_{T_{R}}|u|\Big{]}$ $\displaystyle=$ $\displaystyle
cR^{2}\Bigg{[}1+\frac{1}{R}\sup_{T_{R}}|u|\Bigg{]}\,,$

and our hypothesis (1.6) gives (3.4), hence the proof of Theorem 1.1 is
complete. $\Box$

##  4 Proof of Theorem 1.2

We follow the lines of [7] by checking the hypotheses of Theorem 1.4 in this
reference. We let

$a(t):=\frac{g^{\prime}(t)}{t}\,,\quad t>0\,,$ (4.1)

and observe that on account of (1.7) the function $a$ continuously extends to
$t=0$ by letting $a(0)=g^{\prime\prime}(0)$. Obviously $a$ satisfies (1.2)
from [7] ($g^{\prime}$ is strictly increasing and thereby positive on
$(0,\infty)$ due to (1.7)), and since for any $t>0$ it holds (compare (4.1))

$\lambda_{1}(t):=a(t)+ta^{\prime}(t)=g^{\prime\prime}(t)$ (4.2)

we get (1.3) in [7]. At the same time assumption (A2) from [7] is obvious by
formula (4.1) and our requirements concerning $g$. Moreover, the stability
condition (see (1.11) in [7]) follows from

$\int_{\mathbb{R}^{2}}\big{(}A(\nabla u)\nabla\phi\big{)}\cdot\nabla\phi\,{\rm
d}x\geq 0$ (4.3)

for any $\phi\in C^{1}_{0}(\mathbb{R}^{2})$ with matrix
($Z\in\mathbb{R}^{2}-\\{0\\}$)

$A_{ij}(Z):=|Z|^{-1}a^{\prime}\big{(}|Z|\big{)}Z_{i}Z_{j}+a\big{(}|Z|\big{)}\delta_{ij}$

by observing that (recall (4.1), (1.7))

$\displaystyle\big{(}A(Y)Z\big{)}\cdot Z$ $\displaystyle=$
$\displaystyle|Y|^{-1}\Big{[}g^{\prime\prime}\big{(}|Y|\big{)}|Y|^{-1}-g^{\prime}\big{(}|Y|\big{)}|Y|^{-2}\Big{]}(Y\cdot
Z)^{2}$ $\displaystyle+|Y|^{-1}g^{\prime}\big{(}|Y|\big{)}|Z|^{2}$
$\displaystyle\geq$ $\displaystyle
g^{\prime}\big{(}|Y|\big{)}\Big{[}|Y|^{-1}|Z|^{2}-|Y|^{-3}(Y\cdot
Z)^{2}\Big{]}\geq 0$

on account of $g^{\prime}\big{(}|Y|\big{)}>0$ for $Y\not=0$.

It remains to check (1.17) and (1.18) in [7]: since $g$ is convex (see (1.7))
and of linear growth (compare (1.8)) the boundedness of $g^{\prime}$ follows,
hence we get (1.17) and by monotonicity
$g^{\prime}_{\infty}:=\lim_{t\to\infty}g^{\prime}(t)$ exists in $(0,\infty)$.
By (4.2) the function $\lambda_{1}$ defined in (4.2) (see (1.5) of [7]) is
just $g^{\prime\prime}$ so that (1.18) is a consequence of (1.9) and the
aforementioned limit behaviour of $g^{\prime}$ provided we assume $\mu\geq 3$.
From Theorem 1.4 in [7] it follows that $u(x)=\tilde{u}(\omega\cdot x)$ for
some $\omega\in\mathbb{R}^{2}$, $|\omega|=1$, and a function $\tilde{u}$:
$\mathbb{R}\to\mathbb{R}$. Assuming $\omega=(1,0)$ (w.l.o.g.) we see that
(1.1) implies

$\frac{{\rm d}}{\,{\rm
d}t}\Bigg{[}\frac{1}{|\tilde{u}^{\prime}|}g^{\prime}\big{(}|\tilde{u}^{\prime}|\big{)}\tilde{u}^{\prime}\Bigg{]}=0\,,$

hence

$|\tilde{u}^{\prime}|^{-1}g^{\prime}\big{(}|\tilde{u}^{\prime}|\big{)}\tilde{u}^{\prime}\equiv
c$ (4.4)

for some $c\in\mathbb{R}$.

_Case 1, $c=0$._ Recalling $g^{\prime}>0$ on $(0,\infty)$ equation (4.4)
yields $\tilde{u}^{\prime}\equiv 0$ and we are done.

_Case 2, $c\not=0$._ Then (4.4) shows $\tilde{u}^{\prime}(t)\not=0$ for any
$t\in\mathbb{R}$, thus $g^{\prime}\big{(}|\tilde{u}^{\prime}|\big{)}\equiv|c|$
and in conclusion

$\tilde{u}^{\prime}(t)\in\Big{\\{}-(g^{\prime})^{-1}\big{(}|c|\big{)},(g^{\prime})^{-1}\big{(}|c|\big{)}\Big{\\}}$

for any $t\in\mathbb{R}$. But this immediately implies the constancy of
$\tilde{u}^{\prime}$ and our claim follows. ∎

######  Remark 4.1.

Let us end with a short remark on the failure of the Nitsche criterion
(condition (4) in [26]) for densities $g$ with the properties (1.7) and (1.8)
from Theorem 1.2. The boundedness of $g^{\prime}$ in particular shows that

$\int_{1}^{\infty}g^{\prime\prime}(t)\,{\rm
d}t<\infty\,,\quad\mbox{thus}\quad\int_{1}^{\infty}\frac{1}{\sqrt{s}}g^{\prime\prime}(\sqrt{s})\,{\rm
d}s<\infty\,.$ (4.5)

Let us introduce the functions (compare [26])

$f(t):=g(\sqrt{t})\,,\quad\lambda(t):=\frac{2f^{\prime\prime}(t)}{f^{\prime}(t)}\,.$

Elementary calculations show

$\displaystyle\frac{1+t\lambda(t)}{2+t\lambda(t)}\cdot\frac{1}{t}$
$\displaystyle=$
$\displaystyle\frac{g^{\prime\prime}(\sqrt{t})}{\frac{1}{\sqrt{t}}g^{\prime}(\sqrt{t})+g^{\prime\prime}(\sqrt{t})}\cdot\frac{1}{t}=\frac{1}{t+\frac{\sqrt{t}}{g^{\prime\prime}(\sqrt{t})}g^{\prime}(\sqrt{t})}$

and for $t\geq 1$ we obtain by the monotonicity of $g^{\prime}$

$\frac{1+t\lambda(t)}{2+t\lambda(t)}\frac{1}{t}\leq\frac{1}{g^{\prime}(1)}\frac{g^{\prime\prime}(\sqrt{t})}{\sqrt{t}}\,.$

Recalling (4.5) it follows

$\int_{1}^{\infty}\frac{1+t\lambda(t)}{2+t\lambda(t)}\frac{\,{\rm
d}t}{t}<\infty\,,$

which means that the “Satz” on p. 295 of [26] does not apply to the linear
growth case.

## References

  * [1] Bernstein, S.  Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus.  Math. Z, 26(1):551–558, 1927.
  * [2] Bildhauer, M.  Convex variational problems. Linear, nearly linear and anisotropic growth conditions, volume 1818 of Lecture Notes in Mathematics.  Springer, Berlin, 2003.
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  * [4] Cheng, S.Y.  Liouville theorem for harmonic maps.  Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., pages 147–151, 1980.
  * [5] Eells, J.; Lemaire, L..  Another report on harmonic maps.  Bull. London Math. Soc., 20(5):385–524, 1988.
  * [6] Farina, A.  Liouville-type theorems for elliptic problems.  Handbook of differential equations: stationary partial differential equations. Vol. IV, Elsevier/North-Holland, Amsterdam, pages 61–116, 2007.
  * [7] Farina, A.; Sciunzi, B.; Valdinoci, E.  Bernstein and De Giorgi type problems: new results via a geometric approach.  Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7(4):741–791, 2008.
  * [8] Fuchs, M.  Liouville theorems for stationary flows of shear thickening fluids in the plane.  J. Math. Fluid Mech., 14(3):421–444, 2012.
  * [9] Fuchs, M.  Variations on Liouville’s theorem in the setting of stationary flows of generalized Newtonian fluids in the plane.  Proceedings of the St. Petersburg Mathematical Society. Vol. XV. Advances in mathematical analysis of partial differential equations, 232, Amer. Math. Soc. Transl. Ser. 2, Providence, RI:79–98, 2014.
  * [10] Fuchs, M.; Müller, J.  A Liouville Theorem for Stationary Incompressible Fluids of Von Mises Type.  Acta Math. Sci. Ser. B (Engl. Ed.), 39(1):1–10, 2019.
  * [11] Fuchs, M.; Zhang, G.  Liouville theorems for entire local minimizers of energies defined on the class LlogL and for entire solutions of the stationary Prandtl-Eyring fluid model.  Calc. Var. Partial Differential Equations, 44(1-2):271–295, 2012\.
  * [12] Giaquinta, M.  Multiple intergals in the calculus of variations and nonlinear ellipitc systems, volume 105 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 1983.
  * [13] Giaquinta, M.; Martinazzi, L.  An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, volume 11 of Lecture Notes. Scuola Normale Superiore di Pisa (New Series).  Edizioni della Normale Superiore di Pisa (Nuova Serie), Pisa, second edition, 2012.
  * [14] Gilbarg, D.; Trudinger, N.  Elliptic partial differential equations of second order, volume 224 of Grundlehren der math. Wiss. Springer, Berlin, Revised Third Printing, second edition, 1998.
  * [15] Gilbarg, D.; Weinberger, H.F.  Asymptotic properties of steady plane solutions of the Navier-Stokes equations with bounded Dirichlet integral.  Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5(2):381–404, 1978\.
  * [16] Giusti, E.  Direct methods in the calculus of variations.  World Scientific Publishing Co., River Edge, NJ, 2003.
  * [17] Hildebrandt, S.  Liouville theorems for harmonic mappings, and an approach to Bernstein theorems.  Ann. of Math. Stud., Seminar on Differential Geometry, Princeton University, Princeton, N.J., 102:107–131, 1982.
  * [18] Hildebrandt, S.; Jost, J.; Widman, K.O.  Harmonic mappings and minimal submanifolds.  Invent. Math., 62(2):269–298, 1980.
  * [19] Hildebrandt, S.; Kaul, H.  Two-dimensional variational problems with obstructions, and Plateau’s problem for H-surfaces in a Riemannian manifold.  Comm. Pure Appl. Math., 25:187–223, 1972.
  * [20] Hildebrandt, S.; Kaul, H.; Widman, K.O.  An existence theorem for harmonic mappings of Riemannian manifolds .  Acta Math., 138(1-2):1–16, 1977.
  * [21] Jin, B.J.; Kang, K.  Liouville theorem for the steady-state non-Newtonian Navier-Stokes equations in two dimensions.  J. Math. Fluid Mech., 16(2):275–292, 2014.
  * [22] Jost, J.  Partial differential equations.  Graduate Texts in Mathematics. Springer, New York, 2013.
  * [23] Koch, G.; Nadirashvili, N.; Seregin, G.; Šverák, V.  Liouville theorems for the Navier-Stokes equations and applications.  Acta Math., 203(1):83–105, 2009.
  * [24] Moser, J.  On Harnack’s theorem for elliptic differential equations.  Comm. Pure Appl. Math., 14:577–591, 1961.
  * [25] Nečas, J.  Introduction to the theory of nonlinear elliptic equations., volume 52 of Teubner-Texte zur Mathematik (Teubner Texts in Mathematics).  Teubner Verlagsgesellschaft, Leipzig, 1983.
  * [26] Nitsche Johannes; Nitsche Joachim.  Ein Kriterium für die Existenz nicht-linearer ganzer Lösungen elliptischer Differentialgleichungen.  Arch. Math., 10:294–297, 1959.
  * [27] Schoen, R.  Analytic aspects of the harmonic map problem. Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983).  Sci. Res. Inst. Publ., 2, Springer, New York, pages 321–358, 1984.
  * [28] Schoen, R.; Yau, S.T.  Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature.  Comment. Math. Helv., pages 333–341, 1976.
  * [29] Zhang, G.  A note on Liouville theorem for stationary flows of shear thickening fluids in the plane.  J. Math. Fluid Mech., 15(4):771–782, 2013.
  * [30] Zhang, G.  Liouville theorems for stationary flows of shear thickening fluids in 2D.  Ann. Acad. Sci. Fenn. Math., 40(2):889–905, 2015.

Michael Bildhauer | e-mail: |<EMAIL_ADDRESS>
---|---|---
Martin Fuchs | e-mail: |<EMAIL_ADDRESS>
Department of Mathematics |  |
Saarland University |  |
P.O. Box 15 11 50 |  |
66041 Saarbrücken |  |
Germany |  |


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21  11p 0.50.5 100

# Compact $T(1)$ theorem à la Stein

Árpád Bényi, Guopeng Li, Tadahiro Oh, and Rodolfo H. Torres  Árpád Bényi,
Department of Mathematics
516 High St, Western Washington University
Bellingham, WA 98225, USA<EMAIL_ADDRESS>Guopeng Li, School of Mathematics
The University of Edinburgh
and The Maxwell Institute for the Mathematical Sciences
James Clerk Maxwell Building
The King’s Buildings
Peter Guthrie Tait Road
Edinburgh
EH9 3FD
United Kingdom<EMAIL_ADDRESS>Tadahiro Oh, School of Mathematics
The University of Edinburgh
and The Maxwell Institute for the Mathematical Sciences
James Clerk Maxwell Building
The King’s Buildings
Peter Guthrie Tait Road
Edinburgh
EH9 3FD
United Kingdom<EMAIL_ADDRESS>Rodolfo H. Torres, Department of Mathematics
University of California
900 University Ave., Riverside, CA 92521, USA<EMAIL_ADDRESS>
###### Abstract.

We prove a compact $T(1)$ theorem, involving quantitative estimates, analogous
to the quantitative classical $T(1)$ theorem due to Stein. We also discuss the
$C_{c}^{\infty}$-to-CMO mapping properties of non-compact Calderón-Zygmund
operators.

###### Key words and phrases:

$T(1)$ theorem; Calderón-Zygmund operator; compactness, bounded mean
oscillation, vanishing mean oscillation, continuous mean oscillation, BMO ,
VMO , CMO

###### 2020 Mathematics Subject Classification:

42B20, 47B07

###### Contents

  1. 1 Introduction
  2. 2 Calderón-Zygmund operators, BMO, and CMO
  3. 3 $T(1)$ theorem and main result
    1. 3.1 Classical and compact $T(1)$ theorems
    2. 3.2 $T(1)$ theorems à la Stein
  4. 4 Proof of Theorem 1
  5. 5 Mapping properties of Calderón-Zygmund operators into CMO
  6. A Lack of compactness of convolution operators in $L^{p}(\mathbb{R}^{d})$
  7. B BMO , VMO , and CMO are not weakly sequentially complete

##  1\. Introduction

The classical $T(1)$ theorem of David and Journé [10] is a fundamental result
in harmonic analysis that characterizes the $L^{2}$-boundedness of operators
with Calderón-Zygmund kernels. A remarkable insight of this theorem is that
one only needs to test whether the actions (properly defined) of the operator
and its formal transpose on the constant function 1 belong to BMO and to
verify the so-called weak boundedness property.

There are other equally useful versions of the $T(1)$ theorem that avoid the
introduction of the weak boundedness property. For example, in [10], the
$L^{2}$-boundedness is also shown to be equivalent to the uniform boundedness
(with respect to $\xi$) in BMO of the operator and its transpose acting on the
character functions $x\mapsto e^{ix\cdot\xi}$. In a very elegant formulation,
Stein [28] proved a quantitative version of the $T(1)$ theorem that entirely
avoids à priori mentioning BMO in the statement of the theorem or defining the
operator on smooth $L^{\infty}$-functions. He instead considered appropriate
uniform $L^{2}$-estimates for the operator and its transpose acting on
normalized bump functions; see Theorem C below. Although the actual proof of
Stein’s version does use the definition of $T(1)$ and equivalent formulations
of the weak boundedness property, the advantage of his formulation is that,
once the theorem has been established, the definitions of such concepts do not
need to be considered when applying the theorem to specific applications. See
also [4] for a further discussion on this result by Stein.

Much more recently, several authors have considered versions of the $T(1)$
theorem characterizing the situation when operators with Calderón-Zygmund
kernels are not only bounded but also compact. Such operators do exist and,
beyond some simple ones defined on compact sets whose kernels have integrable
singularities, an interesting example is provided by a certain class of
pseudodifferential operators in $\mathbb{R}^{d}$; see Cordes [9]. See also [7]
for a version of the result for pseudodifferential operators even in a
weighted setting. Other examples are provided by appropriate paraproducts, but
such operators are usually employed in the proof of compact versions of the
$T(1)$ theorem and thus their compactness need to be proved by different
methods.

In [30], Villarroya proved a compactness version of the $T(1)$ theorem by
replacing the weak boundedness property and the condition
$T(1),T^{*}(1)\in\textit{BMO}$ in the classical $T(1)$ theorem (see Theorem A)
with an appropriate weak compactness property and the stronger requirement
that $T(1)$ and $T^{*}(1)$ belong to CMO $\subsetneq$ BMO (CMO denotes the
space of functions of continuous mean oscillation; see Definition 2.2 below),
and by imposing some additional conditions on the kernel of the operator. More
recently, Mitkovski and Stockdale [23] in the linear case and Fragkos, Green,
and Wick [13] in the multilinear setting, established compactness versions of
the $T(1)$ theorem, avoiding the additional conditions on the kernel of a
Calderón-Zygmund operator that appeared in the original version of Villarroya.
We note that the space CMO is a celebrity in harmonic analysis in its own
right, appearing naturally in compactness results ever since the classical
compactness result of Uchiyama [29] on commutators of Calderón-Zygmund
operators with CMO functions.

The main goal of this article is to present a version of a compact $T(1)$
theorem in the spirit of Stein’s $T(1)$ theorem on boundedness (Theorem C),
without any reference to the actions of $T$ and $T^{*}$ on the constant
function $1$ in the statement of the result. A secondary goal of this article
is to discuss the $C_{c}^{\infty}$-to-CMO mapping properties of non-compact
Calderón-Zygmund operators, both of convolution and non-convolution types. For
this purpose, we also revisit some delicate topological properties of BMO
-type spaces; see Appendix B.

This article is organized as follows. In Section 2, we go over some background
and definitions needed for the remainder of the paper. After recalling the
classical $T(1)$ theorem and its compact counterpart in Subsection 3.1, we
state our main result, a compact $T(1)$ theorem à la Stein (Theorem 1) in
Subsection 3.2. Section 4 is dedicated to the proof of Theorem 1. In Section
5, we then consider the $C_{c}^{\infty}$-to-CMO mapping properties of non-
compact Calderón-Zygmund operators. As a byproduct of our analysis, we also
present a direct proof that, under the cancellation assumption
$T(1)=T^{*}(1)=0$, a compact Calderón-Zygmund operator is compact on the Hardy
space $H^{1}$; see Proposition 5.3. In the appendices, we discuss non-
compactness of convolution operators as well as the fact about BMO and some of
its subspaces not being weakly sequentially complete, which are of interest on
their own. The results in the appendices are probably known to a more
specialized functional analysis audience or may follow from more abstract
facts. Hence, we do not want to claim authorship on them but they are included
here for completeness. We provide references when we could locate them in the
literature, but when we could not, we present some essentially self-contained
arguments for some results for the benefit of readers.

##  2\. Calderón-Zygmund operators, BMO, and CMO

A linear singular integral operator $T$ is a map, a priori defined and
continuous from $\mathcal{S}(\mathbb{R}^{d})$ into
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$ (the usual Schwartz space and its
dual), that takes the form

$\displaystyle T(f)(x)=\int_{\mathbb{R}^{d}}K(x,y)f(y)dy$ (2.1)

when the point $x$ is not in the support of $f$. Here, we assume that, away
from the diagonal $\Delta=\\{(x,y)\in\mathbb{R}^{2d}:\,x=y\\}$, the
distributional kernel $K$ of $T$ coincides with a function that is locally
integrable on $\mathbb{R}^{2d}\setminus\Delta$, which we still call $K$. The
formal transpose $T^{*}$ of $T$ is defined similarly with the kernel $K^{*}$
given by $K^{*}(x,y)=K(y,x)$.

######  Definition 2.1.

A locally integrable function $K$ on $\mathbb{R}^{2d}\setminus\Delta$ is
called a Calderón-Zygmund kernel if it satisfies the following two conditions:

  * (i)

For all $x,y\in\mathbb{R}^{d}$, we have

$\displaystyle|K(x,y)|\lesssim|x-y|^{-d}.$ (2.2)

  * (ii)

There exists $\delta\in(0,1]$ such that

$\displaystyle|K(x,y)-K(x^{\prime},y)|$
$\displaystyle+|K(y,x)-K(y,x^{\prime})|\lesssim\frac{|x-x^{\prime}|^{\delta}}{|x-y|^{d+\delta}}$
(2.3)

for all $x,x^{\prime},y\in\mathbb{R}^{d}$ satisfying
$|x-x^{\prime}|<\frac{1}{2}|x-y|$.

We say that a linear operator $T$ of the form (2.1) with a Calderón-Zygmund
kernel $K$ is a Calderón-Zygmund operator if $T$ extends to a bounded operator
on $L^{p_{0}}(\mathbb{R}^{d})$ for some $1<p_{0}<\infty$. It is well-known
[27] that if $T$ is a Calderón-Zygmund operator, then it is bounded on
$L^{p}(\mathbb{R}^{d})$ for all $1<p<\infty$. Hence, in the following, we
restrict our attention to the $L^{2}$-boundedness of such linear operators. It
is also well known that a Calderón-Zygmund operator is bounded from
$L^{\infty}(\mathbb{R}^{d})$ to $\textit{BMO}\,(\mathbb{R}^{d})$ and also from
$H^{1}(\mathbb{R}^{d})$ to $L^{1}(\mathbb{R}^{d})$. Here,
$H^{1}(\mathbb{R}^{d})$ denotes the Hardy space, which is the predual of BMO
[11], and BMO denotes the space of functions of _bounded mean oscillation_ ,
which we now recall.

######  Definition 2.2.

Given a locally integrable function $f$ on $\mathbb{R}^{d}$, define the BMO
-seminorm by

$\displaystyle\|f\|_{\textit{BMO}}=\sup_{Q}M_{Q}(f)=\sup_{Q}\frac{1}{|Q|}\int_{Q}|f(x)-\operatorname*{ave}_{Q}f|\,dx,$
(2.4)

where the supremum is taken over all cubes $Q$ in $\mathbb{R}^{d}$ with sides
parallel to the coordinate axes,111In the following, it is understood that all
the cubes have sides parallel to the coordinate axes. and

$\displaystyle\operatorname*{ave}_{Q}f=\frac{1}{|Q|}\int_{Q}f(x)dx.$ (2.5)

Then, we say that $f$ is of bounded mean oscillation if
$\|f\|_{\textit{BMO}}<\infty$ and we define $\textit{BMO}\,(\mathbb{R}^{d})$
by

$\textit{BMO}\,(\mathbb{R}^{d})=\big{\\{}f\in
L^{1}_{\text{loc}}(\mathbb{R}^{d}):\,\|f\|_{\textit{BMO}}<\infty\big{\\}}.$

As usual, the elements of this space need to be considered as equivalent
classes of functions modulo additive constants.

We denote by $C^{\infty}_{c}(\mathbb{R}^{d})$ (and by $C_{0}(\mathbb{R}^{d})$,
respectively) the space of $C^{\infty}$-functions with compact supports
(continuous functions vanishing at infinity, respectively). Also, we denote by
$(C^{\infty}_{c})_{0}(\mathbb{R}^{d})$ the subspace of
$C^{\infty}_{c}(\mathbb{R}^{d})$ consisting of mean-zero functions. We will
often suppress the underlying space $\mathbb{R}^{d}$ from our notation.

The closure of $C_{c}^{\infty}$ in the BMO topology is called the space of
functions of _continuous mean oscillation_ , and it is denoted by CMO . By
definition CMO is a closed subspace of BMO , where the topology of CMO is
inherited from that of BMO , induced by the BMO semi-norm in (2.4).

We also recall the following characterization of the Hardy space (see, for
example, [27, Corollary 1 on p. 221]). Let $d\geq 2$.222When $d=1$, we replace
the Riesz transforms by the Hilbert transform. Given $j=1,\dots,d$, let
$R_{j}$ be the Riesz transform defined by
$\widehat{R_{j}(f)}(\xi)=i\frac{\xi_{j}}{|\xi|}\widehat{f}(\xi)$, where
$\widehat{~{}}$ denotes the Fourier transform. Then, the Hardy space $H^{1}$
consists of functions $f\in L^{1}$ such that $R_{j}f\in L^{1}$, $j=1,\dots,d$,
endowed with the norm:

$\displaystyle\|f||_{H^{1}}=\|f\|_{L^{1}}+\sum_{j=1}^{d}\|R_{j}f\|_{L^{1}}.$
(2.6)

Next, we recall a characterization of CMO ; see [29, Lemma on p. 166]. We
follow here Bourdaud’s formulation in [5, Théorème 7 (iii) on p. 1198]. Given
$f\in L^{1}_{\text{loc}}$, we define

$\displaystyle\Gamma_{1}(f)=\lim_{r\to
0}\bigg{(}\sup_{\begin{subarray}{c}|Q|\leq
r\end{subarray}}\frac{1}{|Q|}\int_{Q}|f(x)-\operatorname*{ave}_{Q}f|dx\bigg{)},$
(2.7)

where the supremum is taken over all cubes of volume $|Q|\leq r$ and
$\displaystyle\operatorname*{ave}_{Q}f$ is as in (2.5). Similarly, we define

$\displaystyle\Gamma_{2}(f)=\lim_{r\to\infty}\bigg{(}\sup_{|Q|\geq
r}\frac{1}{|Q|}\int_{Q}|f(x)-\operatorname*{ave}_{Q}f|dx\bigg{)},$ (2.8)

where the supremum is taken over all cubes of volume $|Q|\geq r$. Lastly,
given a cube $Q\subset\mathbb{R}^{d}$, we define

$\displaystyle\Gamma_{3}^{Q}(f)=\limsup_{\begin{subarray}{c}x_{0}\in\mathbb{R}^{d}\\\
|x_{0}|\to\infty\end{subarray}}\bigg{(}\frac{1}{|Q|}\int_{Q+x_{0}}|f(x)-\operatorname*{ave}_{Q+x_{0}}f|dx\bigg{)}.$
(2.9)

Then, we have the following characterization of CMO by the quantities
$\Gamma_{1}$, $\Gamma_{2}$, and $\Gamma_{3}^{Q}$.

######  Lemma 2.3.

Let $f\in\textit{BMO}$. Then, $f\in\textit{CMO}$ if and only if

$\displaystyle\Gamma_{1}(f)=\Gamma_{2}(f)=\Gamma_{3}^{Q}(f)=0$ (2.10)

for any cube $Q\subset\mathbb{R}^{d}$.

In [5], Lemma 2.3 was stated in terms of the analogues of the quantities
$\Gamma_{1}$, $\Gamma_{2}$, and $\Gamma_{3}^{Q}$, where cubes are replaced by
balls. A straightforward modification of the argument in [5] yields Lemma 2.3,
where we work with cubes.

##  3\. $T(1)$ theorem and main result

###  3.1. Classical and compact $T(1)$ theorems

In this subsection, we provide a brief discussion on the classical $T(1)$
theorem (Theorem A) and its compact counterpart (Theorem B). In order to do
so, we need a few more definitions.

######  Definition 3.1.

We say that a function $\phi\in C_{c}^{\infty}$ is a _normalized bump
function_ of order $M$ if $\operatorname{supp}\phi\subset B_{0}(1)$ and
$\|\partial^{\alpha}\phi\|_{L^{\infty}}\leq 1$ for all multi-indices $\alpha$
with $|\alpha|\leq M$. Here, given $r>0$ and $x\in\mathbb{R}^{d}$, $B_{x}(r)$
denotes the ball of radius $r$ centered at $x$.

Given $x_{0}\in\mathbb{R}^{d}$ and $R>0$, we set

$\displaystyle\phi^{x_{0},R}(x)=\phi\Big{(}\frac{x-x_{0}}{R}\Big{)}.$ (3.1)

The statement of the $T(1)$ theorem of David and Journé [10] is the following.

######  Theorem A ($T(1)$ theorem).

Let $T:\mathcal{S}\to\mathcal{S}^{\prime}$ be a linear singular integral
operator with a Calderón-Zygmund kernel. Then, $T$ can be extended to a
bounded operator on $L^{2}$ if and only if it satisfies the following two
conditions:

  * (A.i)

$T$ satisfies the weak boundedness property; there exists
$M\in\mathbb{N}\cup\\{0\\}$ such that we have

$\displaystyle\big{|}\big{\langle}T(\phi_{1}^{x_{1},R}),\phi_{2}^{x_{2},R}\big{\rangle}\big{|}\lesssim
R^{d}$ (3.2)

for any normalized bump functions $\phi_{1}$ and $\phi_{2}$ of order $M$,
$x_{1},x_{2}\in\mathbb{R}^{d}$, and $R>0$.

  * (A.ii)

$T(1)$ and $T^{*}(1)$ are in BMO .

Since $T$ is a priori defined only in $\mathcal{S}$, the expressions $T(1)$
and $T^{*}(1)$ need to be interpreted carefully; see (4.2) below.

######  Definition 3.2.

We say that a linear singular integral operator
$T:\mathcal{S}\to\mathcal{S}^{\prime}$ with a Calderón-Zygmund kernel has the
weak compactness property if there exists $M\in\mathbb{N}\cup\\{0\\}$ such
that we have

$\displaystyle\lim_{|x_{0}|+R+R^{-1}\to\infty}R^{-d}\big{|}\big{\langle}T(\phi_{1}^{x_{0}+x_{1},R}),\phi_{2}^{x_{0}+x_{2},R}\big{\rangle}\big{|}=0$
(3.3)

for any normalized bump functions $\phi_{1}$ and $\phi_{2}$ of order $M$ and
$x_{1},x_{2}\in\mathbb{R}^{d}$.

The following version of the compact counterpart of the $T(1)$ theorem
(Theorem A) was recently proved in [23]; see also [30]. For a version of a
compact $T(b)$ theorem, see [31].

######  Theorem B (compact $T(1)$ theorem).

Let $T$ be as in Theorem A. Then, $T$ can be extended to a compact operator on
$L^{2}$ if and only if it satisfies the following two conditions:

  * (B.i)

$T$ satisfies the weak compactness property.

  * (B.ii)

$T(1)$ and $T^{*}(1)$ are in CMO .

We say that a linear singular integral operator $T$ with a Calderón-Zygmund
kernel is a compact Calderón-Zygmund operator if it is compact on $L^{2}$.

######  Remark 3.3.

If $T$ is a compact Calderón-Zygmund operator, then it is also compact

  * (a)

on $L^{p}$ for any $1<p<\infty$; this follows from interpolation of
compactness [19],

  * (b)

from $L^{\infty}$ to CMO ; see [25],

  * (c)

from the Hardy space $H^{1}$ to $L^{1}$; this follows from (b) applied to
$T^{*}$ (which is compact on $L^{2}$ in view of Lemma 3.5 (ii)) and
duality;333Although $L^{1}$ is not the dual of $L^{\infty}$, an argument using
duality can still be applied because the $L^{1}$-norm can be computed by
pairing with $L^{\infty}$-functions. see [24, p. 1285],

  * (d)

from $L^{1}$ to $L^{1,\infty}$; see [24],

  * (e)

from BMO to CMO ([25]) under the cancellation assumption that
$T(1)=T^{*}(1)=0$,

  * (f)

from $H^{1}$ to $H^{1}$ under the cancellation assumption that
$T(1)=T^{*}(1)=0$; this claim follows from (e) and duality; see also
Proposition 5.3.

######  Remark 3.4.

Our definitions of the weak boundedness property (3.2) and the weak
compactness property (Definition 3.3) are slightly different from those in
[10, 23], but one can easily check that they are equivalent.

We also note that, as for the weak boundedness property, it suffices to verify
(3.2) for $x_{1}=x_{2}$; see [18]. However, it does not seem to be the case
for the weak compactness property (3.3).

###  3.2. $T(1)$ theorems à la Stein

We first recall a quantitative formulation of the classical $T(1)$ theorem due
to Stein; see [28, Theorem 3 on p. 294].

######  Theorem C ($T(1)$ theorem à la Stein).

Let $T$ be as in Theorem A. Then, $T$ can be extended to a bounded operator on
$L^{2}$ if and only if there exists $M\in\mathbb{N}\cup\\{0\\}$ such that we
have

$\displaystyle\|T(\phi^{x_{0},R})\|_{L^{2}}+\|T^{*}(\phi^{x_{0},R})\|_{L^{2}}\lesssim
R^{\frac{d}{2}}$ (3.4)

for any normalized bump function $\phi$ of order $M$,
$x_{0}\in\mathbb{R}^{d}$, and $R>0$.

Here, the conditions (A.i) and (A.ii) in Theorem A are replaced by the
quantitative estimate (3.4) involving normalized bump functions. In the
following, we seek for a suitable replacement of the condition (3.4) for a
compact $T(1)$ theorem à la Stein. Before proceeding further, we first recall
the following basic properties of compact operators.

######  Lemma 3.5.

Let $X$ and $Y$ be Banach spaces and $T:X\to Y$ be a continuous linear
operator.

  * (i)

If $T$ is compact, then $T$ maps weakly convergent sequences to strongly
convergent sequences.

  * (ii)

The operator $T$ is compact if and only if its transpose $T^{*}$ is compact.

Lemma 3.5 is well known and its proof can be found in many textbook covering
the subject; see, for example, Reed and Simons [26, Theorems VI.11 and VI.12
(c)].

We now derive the $L^{2}$\- and $L^{\infty}$-conditions for our formulation of
a compact $T(1)$ theorem à la Stein.

$\boldsymbol{L^{2}}$-condition: We first rewrite the condition (3.4) as

$\displaystyle\|T(R^{-\frac{d}{2}}\phi^{x_{0},R})\|_{L^{2}}+\|T^{*}(R^{-\frac{d}{2}}\phi^{x_{0},R})\|_{L^{2}}\lesssim
1.$

Note that $R^{-\frac{d}{2}}\phi^{x_{0},R}$ converges weakly to $0$ in $L^{2}$
as $|x_{0}|+R+R^{-1}\to\infty$. Suppose now that $T$ is compact on $L^{2}$.
Then, from Lemma 3.5 (i), we see that $R^{-\frac{d}{2}}T(\phi^{x_{0},R})$ and
$R^{-\frac{d}{2}}T^{*}(\phi^{x_{0},R})$ converge strongly to $0$ in $L^{2}$ as
$|x_{0}|+R+R^{-1}\to\infty$, namely,

$\displaystyle\lim_{|x_{0}|+R+R^{-1}\to\infty}\bigg{(}R^{-\frac{d}{2}}\|T(\phi^{x_{0},R})\|_{L^{2}}+R^{-\frac{d}{2}}\|T^{*}(\phi^{x_{0},R})\|_{L^{2}}\bigg{)}=0.$
(3.5)

$\boldsymbol{L^{\infty}}$-condition: Let $T$ be a Calderón-Zygmund operator
that is compact on $L^{2}$. From Remark 3.3 (b), we see that $T$ is compact
from $L^{\infty}$ to CMO . Let $\\{f_{n}\\}_{n\in\mathbb{N}}$ be a weak-$\ast$
convergent sequence in $L^{\infty}$.444Recall that
$L^{\infty}(\mathbb{R}^{d})=(L^{1}(\mathbb{R}^{d}))^{*}$. Then, it follows
from the Banach-Steinhaus theorem that the sequence
$\\{f_{n}\\}_{n\in\mathbb{N}}$ is bounded in $L^{\infty}$, and, hence, from
the compactness of $T$ from $L^{\infty}$ to CMO , we see that
$\\{T(f_{n})\\}_{n\in\mathbb{N}}$ is precompact in CMO . In view of Lemma 3.5
(ii), the same conclusion holds for $T^{*}$. This leads to the
$L^{\infty}$-condition (ii) in Theorem 1 below.

We are now ready to state our main result, a compact counterpart of the $T(1)$
theorem à la Stein.

######  Theorem 1 (compact $T(1)$ theorem à la Stein).

Let $T$ be as in Theorem A. Then, $T$ can be extended to a compact operator on
$L^{2}$ if and only if the following two conditions hold:

  * (i)

($L^{2}$-condition). There exists $M\in\mathbb{N}\cup\\{0\\}$ such that (3.5)
holds for any normalized bump function $\phi$ of order $M$.

  * (ii)

($L^{\infty}$-condition). Given any weak-$\ast$ convergent sequence
$\\{f_{n}\\}_{n\in\mathbb{N}}$ in $L^{\infty}$ such that
$\\{T(f_{n})\\}_{n\in\mathbb{N}}$ and $\\{T^{*}(f_{n})\\}_{n\in\mathbb{N}}$
are bounded in CMO , both the sequences $\\{T(f_{n})\\}_{n\in\mathbb{N}}$ and
$\\{T^{*}(f_{n})\\}_{n\in\mathbb{N}}$ are precompact in CMO .

Note that in the derivation of the $L^{\infty}$-condition and the
$L^{2}$-condition before the statement of Theorem 1, we have already
established their necessity, assuming that $T$ is compact. Hence, it remains
to prove their sufficiency. Our proof of Theorem 1 follows the lines in the
proof of [4, Theorem 1] on the classical $T(1)$ theorem à la Stein for
boundedness of linear singular integral operators. The main task is to verify
the conditions (B.i) and (B.ii) in Theorem B by assuming (i) and (ii) in
Theorem 1. In particular, for verifying the condition (B.ii), we make use of
the characterization of CMO functions given in Lemma 2.3 and the condition
(ii) in Theorem 1.

######  Remark 3.6.

It is not clear at this point if the $L^{\infty}$-condition (ii) in Theorem 1
can be removed. See Remark 4.1 below.

######  Remark 3.7.

In view of a compact bilinear $T(1)$ theorem in [13] and other properties of
compact bilinear operators recently studied in [3], it would be of interest to
investigate an analogue of Theorem 1 in the bilinear setting. We leave this to
interested readers.

##  4\. Proof of Theorem 1

In this section, we present a proof of Theorem 1. As mentioned above, it
suffices to prove sufficiency of the conditions (i) and (ii). In the
following, by assuming the conditions (i) and (ii) in Theorem 1, we verify the
conditions (B.i) and (B.ii) in Theorem B, which will in turn imply that $T$ is
compact on $L^{2}$.

We first prove (B.i): the weak compactness property. Let $\phi_{1}$ and
$\phi_{2}$ be normalized bump functions of order 0. From (3.1), we have

$\displaystyle\|R^{-\frac{d}{2}}\phi_{1}^{x_{0}+x_{1},R}\|_{L^{2}}+\|R^{-\frac{d}{2}}\phi_{2}^{x_{0}+x_{2},R}\|_{L^{2}}\lesssim
1,$ (4.1)

uniformly in $x_{0},x_{1},x_{2}\in\mathbb{R}^{d}$ and $R>0$. Then, from (3.5)
and (4.1), we have

$\displaystyle\lim_{|x_{0}|+R+R^{-1}\to\infty}R^{-d}\big{|}\big{\langle}T(\phi_{1}^{x_{0}+x_{1},R}),\phi_{2}^{x_{0}+x_{2},R}\big{\rangle}\big{|}$
$\displaystyle\hphantom{XXXX}\leq\lim_{|x_{0}|+R+R^{-1}\to\infty}\min\Big{(}\|T(R^{-\frac{d}{2}}\phi_{1}^{x_{0}+x_{1},R})\|_{L^{2}}\|R^{-\frac{d}{2}}\phi_{2}^{x_{0}+x_{2},R}\|_{L^{2}},$
$\displaystyle\hphantom{lXXXXXXXX}\|R^{-\frac{d}{2}}\phi_{1}^{x_{0}+x_{1},R}\|_{L^{2}}\|T^{*}(R^{-\frac{d}{2}}\phi_{2}^{x_{0}+x_{2},R})\|_{L^{2}}\Big{)}$
$\displaystyle\hphantom{XXXX}=0$

for any $x_{1},x_{2}\in\mathbb{R}^{d}$. This proves the condition (B.i) in
Theorem B.

Next, we prove the condition (B.ii) in Theorem B. In the following, we only
show $T(1)\in\textit{CMO}$ , since $T^{*}(1)\in\textit{CMO}$ follows
analogously by symmetry. Since (3.5) implies the bound (3.4), it follows from
Theorem C that $T(1)\in\textit{BMO}$ , where $T(1)$ is interpreted as

$\displaystyle\langle T(1),g\rangle=\lim_{R\to\infty}\langle
T(\phi_{R}),g\rangle$ (4.2)

for any $g\in(C^{\infty}_{c})_{0}$.555 Recall that $(C^{\infty}_{c})_{0}$ is
dense in the Hardy space $H^{1}$, which is a predual of BMO ; see [10, p.
372]. Namely, $T(1)$ is defined as the weak-$\ast$ limit (in BMO ) of
$T(\phi_{R})$.  Here, $\phi$ is a test function in $C^{\infty}_{c}$ with
$0\leq\phi\leq 1$ such that $\phi(x)=1$ for $|x|\leq\frac{1}{2}$ and
$\operatorname{supp}\phi\subset B_{0}(1)$, and

$\displaystyle\phi_{R}(x)=\phi(R^{-1}x).$ (4.3)

Moreover, proceeding as in [4, Section 4] with (3.4), we have

$\displaystyle\|T(\phi_{R})\|_{\textit{BMO}}\lesssim 1,$ (4.4)

uniformly in $R>0$; see [4, (4.6)] with $b_{1}=1$.

In the following, we show that $T(\phi_{R})$ indeed belongs to CMO for each
$R\gg 1$ by verifying (2.10) in Lemma 2.3. Fix $M\in\mathbb{N}$. Then, by
imposing that

$\displaystyle\|\partial^{\alpha}\phi\|_{L^{\infty}}\leq 1$ (4.5)

for all multi-indices $\alpha$ with $|\alpha|\leq M$, the function $\phi$
defined above is a normalized bump function of order $M$.

Fix a cube $Q=Q(\ell,x_{0})$ of side length $\ell>0$ with center
$x_{0}\in\mathbb{R}^{d}$. Set

$\displaystyle r=6\operatorname{diam}(Q)=6\sqrt{d}\ell.$ (4.6)

$\bullet$ Case 1: $r\to\infty$.
By Cauchy-Schwarz’s inequality with (3.5) and (4.6), we have

$\displaystyle\operatorname*{ave}_{Q}|T(\phi_{R})|=\frac{1}{|Q|}\int_{Q}|T(\phi_{R})(x)|dx\leq\frac{1}{|Q|^{\frac{1}{2}}}\|T(\phi_{R})\|_{L^{2}}\lesssim\frac{R^{\frac{d}{2}}}{r^{\frac{d}{2}}}\longrightarrow
0,$

as $r\to\infty$, from which we obtain

$\displaystyle\frac{1}{|Q|}\int_{Q}|T(\phi_{R})-\operatorname*{ave}_{Q}T(\phi_{R})|dx\leq
2\operatorname*{ave}_{Q}|T(\phi_{R})|\longrightarrow 0,$

as $r\to\infty$. In view of (2.8), this shows

$\displaystyle\Gamma_{2}(T(\phi_{R}))=\lim_{r\to\infty}\bigg{(}\sup_{|Q|\geq
r}\frac{1}{|Q|}\int_{Q}|T(\phi_{R})(x)-\operatorname*{ave}_{Q}T(\phi_{R})|dx\bigg{)}=0.$
(4.7)

$\bullet$ Case 2: $|x_{0}|\to\infty$.
Fix $R,r>0$. With $\phi_{Q}=\phi^{x_{0},r}$ as in (3.1), write $T(\phi_{R})$
as

$\displaystyle
T(\phi_{R})=T(\phi_{Q}\phi_{R})+T\big{(}(1-\phi_{Q})\phi_{R}\big{)}=:\hskip
0.56905pt\text{I}\hskip 0.56905pt+\text{I I}.$ (4.8)

By choosing $|x_{0}|\gg R$, we have $\phi_{Q}\phi_{R}=0$ and thus $\hskip
0.56905pt\text{I}\hskip 0.56905pt=0$. Hence, we only need to estimate the
second term I I on the right-hand side of (4.8). From the support condition

$\displaystyle\operatorname{supp}(1-\phi_{Q})\subset\mathbb{R}^{d}\setminus
B_{x_{0}}(3\operatorname{diam}(Q))\subset\mathbb{R}^{d}\setminus Q$

and (2.1), we have

$\displaystyle\text{I
I}(x)=\int_{\mathbb{R}^{d}}K(x,y)\big{(}1-\phi_{Q}(y)\big{)}\phi_{R}(y)dy$
(4.9)

for any $x\in Q$. By taking an average over $Q$, we then have

$\displaystyle\operatorname*{ave}_{Q}\text{I
I}=\frac{1}{|Q|}\int_{Q}\int_{\mathbb{R}^{d}}K(z,y)\big{(}1-\phi_{Q}(y)\big{)}\phi_{R}(y)dydz.$
(4.10)

By taking the difference of (4.9) and (4.10), we have

$\displaystyle\text{I I}(x)-\operatorname*{ave}_{Q}\text{I
I}=\frac{1}{|Q|}\int_{Q}\int_{\mathbb{R}^{d}}\big{(}K(x,y)-K(z,y)\big{)}\big{(}1-\phi_{Q}(y)\big{)}\phi_{R}(y)dydz$

for any $x\in Q$. In particular, from (2.3), we have

$\displaystyle\begin{split}&\big{|}\text{I
I}(x)-\operatorname*{ave}_{Q}\text{I I}\big{|}\\\
&\quad\lesssim\frac{1}{|Q|}\int_{Q}\operatorname*{\int}_{|x-z|\leq\operatorname{diam}(Q)<\frac{1}{2}|x-y|}\frac{|x-z|^{\delta}}{|x-y|^{d+\delta}}\big{|}1-\phi_{Q}(y)\big{|}|\phi_{R}(y)|dydz,\end{split}$
(4.11)

uniformly in $x\in Q$.

By considering the support of $\phi_{R}$, we have $|y|\lesssim R$ in (4.11).
Thus, for $|x_{0}|\gg R+r$, it follows from $|x-x_{0}|\leq r$ that
$|x-y|\sim|x|\sim|x_{0}|$. Then, by performing the $y$-integration in (4.11)
on $\operatorname{supp}(\phi_{R})$, we obtain

$\displaystyle\big{|}\text{I I}(x)-\operatorname*{ave}_{Q}\text{I
I}\big{|}\lesssim\frac{r^{\delta}R^{d}}{|x_{0}|^{d+\delta}}\longrightarrow 0,$

as $|x_{0}|\to\infty$, uniformly in $x\in Q$. Hence, together with $\hskip
0.56905pt\text{I}\hskip 0.56905pt=0$ for $|x_{0}|\gg R$, we obtain

$\displaystyle\frac{1}{|Q|}\int_{Q}|T(\phi_{R})-\operatorname*{ave}_{Q}T(\phi_{R})|dx\longrightarrow
0,$ (4.12)

as $|x_{0}|\to\infty$.

Let $\widetilde{Q}=\widetilde{Q}(\ell,x_{1})$ be a cube of side length
$\ell>0$ with center $x_{1}\in\mathbb{R}^{d}$. Then, given
$\widetilde{x}_{0}\in\mathbb{R}^{d}$, we have

$\widetilde{Q}+\widetilde{x}_{0}=Q(\ell,x_{0}),$

where $Q(\ell,x_{0})$ denotes the cube of side length $\ell>0$ with center
$x_{0}:=x_{1}+\widetilde{x}_{0}$. For fixed $x_{1}\in\mathbb{R}^{d}$, we have
$|x_{0}|\to\infty$ as $|\widetilde{x}_{0}|\to\infty$. Therefore, from (2.9)
and (4.12), we obtain

$\displaystyle\Gamma_{3}^{\widetilde{Q}}(T(\phi_{R}))=\limsup_{\begin{subarray}{c}\widetilde{x}_{0}\in\mathbb{R}^{d}\\\
|\widetilde{x}_{0}|\to\infty\end{subarray}}\bigg{(}\frac{1}{|\widetilde{Q}|}\int_{\widetilde{Q}+\widetilde{x}_{0}}|T(\phi_{R})(x)-\operatorname*{ave}_{\widetilde{Q}+\widetilde{x}_{0}}T(\phi_{R})|dx\bigg{)}=0$
(4.13)

for any cube $\widetilde{Q}\subset\mathbb{R}^{d}$.

$\bullet$ Case 3: $r\to 0$.
Fix $R>0$ and $x_{0}\in\mathbb{R}^{d}$. In view of (3.5), we have

$\displaystyle\lim_{r\to 0}r^{-\frac{d}{2}}\|T(\phi^{x_{0},r})\|_{L^{2}}=0.$

Then, by setting
$A_{r}=\min\big{(}r^{\frac{1}{2}}\|T(\phi^{x_{0},r})\|_{L^{2}}^{-\frac{1}{d}},r^{-\frac{1}{d+2}}\big{)}$,
we have

$\displaystyle A_{r}$ $\displaystyle\longrightarrow\infty,$ (4.14)
$\displaystyle A_{r}^{\frac{d+2}{2}}r$ $\displaystyle\longrightarrow 0,$
(4.15)

and

$\displaystyle
A_{r}^{\frac{d}{2}}r^{-\frac{d}{2}}\|T(\phi^{x_{0},r})\|_{L^{2}}\longrightarrow
0\quad\text{or equivalently}\quad
r^{-\frac{d}{2}}\|T(\phi^{x_{0},A_{r}r})\|_{L^{2}}\longrightarrow 0,$ (4.16)

as $r\to 0$, where, in (4.16), we used the fact that $A_{r}r\to 0$ as $r\to
0$. In the following, we assume $r>0$ is sufficiently small such that
$A_{r}>1$.

We only consider the case $x_{0}\in\operatorname{supp}\phi_{R}$, since the
case $x_{0}\notin\operatorname{supp}\phi_{R}$ can be treated in an analogous
but easier manner. Define $\widetilde{\phi}_{Q}$ by

$\displaystyle\widetilde{\phi}_{Q}=\phi^{x_{0},A_{r}r}.$ (4.17)

Then, we can write $T(\phi_{R})$ as

$\displaystyle\begin{split}T(\phi_{R})&=\phi_{R}(x_{0})T(\widetilde{\phi}_{Q})+T\big{(}\widetilde{\phi}_{Q}(-\phi_{R}(x_{0})+\phi_{R})\big{)}+T\big{(}(1-\widetilde{\phi}_{Q})\phi_{R}\big{)}\\\
&=:\widetilde{\hskip 0.56905pt}\text{I}\hskip 0.56905pt+\widetilde{\text{I
I}}+\widetilde{\text{I I I}}.\end{split}$ (4.18)

From (4.17) and (4.16), we have

$\displaystyle\operatorname*{ave}_{Q}|\widetilde{\hskip
0.56905pt}\text{I}\hskip
0.56905pt|\leq\frac{|\phi_{R}(x_{0})|}{|Q|}\int_{Q}|T(\widetilde{\phi}_{Q})|dx\lesssim\frac{1}{r^{\frac{d}{2}}}\big{\|}T\big{(}\phi^{x_{0},A_{r}r}\big{)}\big{\|}_{L^{2}}\longrightarrow
0,$ (4.19)

as $r\to 0$.

By the mean value theorem with (4.3), (4.5), and (4.17) (which implies
$|x-x_{0}|\lesssim A_{r}r$ on $\operatorname{supp}\widetilde{\phi}_{Q}$), we
have

$\displaystyle|\widetilde{\phi}_{Q}(x)(-\phi_{R}(x_{0})+\phi_{R}(x))|\lesssim
R^{-1}|\widetilde{\phi}_{Q}(x)||x-x_{0}|\lesssim
R^{-1}A_{r}r|\widetilde{\phi}_{Q}(x)|.$ (4.20)

Then, from the boundedness of $T$ on $L^{2}(\mathbb{R}^{d})$, (4.20), (4.17),
and (4.15), we have

$\displaystyle\operatorname*{ave}_{Q}|\widetilde{\text{I
I}}|\leq\frac{1}{|Q|^{\frac{1}{2}}}\|\widetilde{\text{I
I}}\|_{L^{2}(Q)}\lesssim\frac{A_{r}r}{Rr^{\frac{d}{2}}}\|\widetilde{\phi}_{Q}\|_{L^{2}(\mathbb{R}^{d})}\sim\frac{A_{r}^{\frac{d+2}{2}}r}{R}\longrightarrow
0,$ (4.21)

as $r\to 0$.

As for $\widetilde{\text{I I I}}$, proceeding as in Case 2, we have

$\displaystyle\widetilde{\text{I I
I}}(x)-\operatorname*{ave}_{Q}\widetilde{\text{I I
I}}=\frac{1}{|Q|}\int_{Q}\int_{\mathbb{R}^{d}}\big{(}K(x,y)-K(z,y)\big{)}\big{(}1-\widetilde{\phi}_{Q}(y)\big{)}\phi_{R}(y)dydz$

for any $x\in Q$. Then, from (2.3) with $A_{r}>1$, we have

$\displaystyle\big{|}\widetilde{\text{I I
I}}(x)-\operatorname*{ave}_{Q}\widetilde{\text{I I I}}\big{|}$
$\displaystyle\quad\lesssim\frac{1}{|Q|}\int_{Q}\operatorname*{\int}_{|x-z|\leq\operatorname{diam}(Q)<\frac{1}{2A_{r}}|x-y|}\frac{|x-z|^{\delta}}{|x-y|^{d+\delta}}\big{|}1-\phi_{Q}(y)\big{|}|\phi_{R}(y)|dydz,$

uniformly in $x\in Q$. By integrating in $y$ and $z$ in the polar coordinates
(centered at $x$ with $r_{1}=|x-y|\gtrsim A_{r}r$ and $r_{2}=|x-z|\lesssim r$)
with $|Q|\sim r^{d}$ and (4.14), we have

$\displaystyle\begin{split}\big{|}\widetilde{\text{I I
I}}(x)-\operatorname*{ave}_{Q}\widetilde{\text{I I
I}}\big{|}\lesssim\frac{1}{r^{d}}\int_{0}^{c_{2}r}\int_{c_{1}A_{r}r}^{\infty}\frac{1}{r_{1}^{1+\delta}}dr_{1}r_{2}^{d-1+\delta}dr_{2}\sim\frac{1}{A_{r}^{\delta}}\longrightarrow
0,\end{split}$ (4.22)

as $r\to 0$, uniformly in $x\in Q$.

Hence, putting (4.18), (4.19), (4.21), and (4.22) together,

$\displaystyle\frac{1}{|Q|}\int_{Q}|T(\phi_{R})-\operatorname*{ave}_{Q}T(\phi_{R})|dx\longrightarrow
0,$ (4.23)

as $r\to 0$, uniformly in $R\gg 1$. From the definition (2.7) of $\Gamma_{1}$,
this shows

$\displaystyle\Gamma_{1}(T(\phi_{R}))=\lim_{r\to
0}\bigg{(}\sup_{\begin{subarray}{c}|Q|\leq
r\end{subarray}}\frac{1}{|Q|}\int_{Q}|T(\phi_{R})(x)-\operatorname*{ave}_{Q}T(\phi_{R})|dx\bigg{)}=0.$
(4.24)

Therefore, from Lemma 2.3 with (4.7), (4.13), and (4.24), we conclude that

$\displaystyle T(\phi_{R})\in\textit{CMO}$ (4.25)

for each $R>0$.

Let $\\{R_{j}\\}_{j\in\mathbb{N}}$ be an increasing sequence of positive
numbers such that $R_{j}\to\infty$ as $j\to\infty$. Then, the sequence
$\\{\phi_{R_{j}}\\}_{j\in\mathbb{N}}$ converges to the constant function
$\phi(0)=1$ in the weak-$\ast$ topology of $L^{\infty}$. Moreover, from (4.4)
and (4.25), the sequence $\\{T(\phi_{R_{j}})\\}_{j\in\mathbb{N}}$ is bounded
in CMO . Therefore, from the $L^{\infty}$-condition (ii), we conclude that
there exists a subsequence $\\{T(\phi_{R_{j_{k}}})\\}_{k\in\mathbb{N}}$
converging strongly to $T(1)$ in CMO , which in particular implies
$T(1)\in\textit{CMO}$ .

We have now verified (B.i) and (B.ii) in Theorem B, which in turn implies that
$T$ is compact on $L^{2}$. This concludes the proof of Theorem 1.

######  Remark 4.1.

In view of (4.23), the convergence in (4.24) is uniform in $R\gg 1$. However,
the convergence in (4.7) and (4.13) in Cases 1 and 2, respectively, is not
uniform in $R\gg 1$. This is the reason that we needed to introduce the
$L^{\infty}$-condition (ii) in Theorem 1.

######  Remark 4.2.

In view of $(\textit{CMO}\,)^{*}=H^{1}$ (see [8, Theorem (4.1) on p. 638]666We
note that the space VMO in [8] corresponds to CMO in the current work.) and
that $T(\phi_{R})\in\textit{CMO}$ , $R>0$, it follows from (4.2) (which holds
for any $g\in H^{1}$) that $T(1)$ is a weak limit of CMO functions with
respect to the weak topology on CMO . Hence, if CMO were weakly sequentially
complete, we would be able to conclude $T(1)\in\textit{CMO}$ without the
$L^{\infty}$-condition (ii) in Theorem 1. However, this is not the case. See
Appendix B.

##  5\. Mapping properties of Calderón-Zygmund operators into CMO

While every Calderón-Zygmund operator maps $L^{\infty}$ to BMO, compact ones
in particular map $L^{\infty}$ to CMO as pointed out in Remark 3.3 (b). We
will show that, in general, without the compactness assumption, Calderón-
Zygmund operators fail to have this property. Moreover, this failure persists
even if we restrict input functions to smaller subspaces of $L^{\infty}$,
consisting of smooth functions; see Proposition 5.5.

In the case of Calderón-Zygmund operators of convolution type, we do have that
they map $C_{0}$ into CMO . This claim appears in [28, p. 180]777The space VMO
in [28] also corresponds to CMO in our discussion. without a proof. In
Proposition 5.4, we prove this claim in a little more generality. We note
that, in [16, p. 207] and [23, p. 10], analogous claims are made for general
Calderón-Zygmund operators (in particular for those of non-convolution type),
which are unfortunately incorrect. In Proposition 5.5, we construct an example
of a Calderón-Zygmund operator $T$ such that
$T(C^{\infty}_{c})\not\subset\textit{CMO}$ .

We first recall some further properties of Calderón-Zygmund operators. Given a
Calderón-Zygmund operator $T$, we have $T(f)\in L^{1}$ for
$f\in(C^{\infty}_{c})_{0}$. Here, $(C^{\infty}_{c})_{0}$ denotes the subspace
of $C^{\infty}_{c}$ consisting of mean-zero functions, which is dense in
$H^{1}$. If, in addition, $T$ maps $H^{1}$ into itself, then we have

$0=\int_{\mathbb{R}^{d}}T(f)(x)\,dx=\langle 1,T(f)\rangle=\langle
T^{*}(1),f\rangle$

for any $f\in(C^{\infty}_{c})_{0}$. Namely, we have $T^{*}(1)=0$. In fact,
such a condition is sufficient for boundedness of $T$ on $H^{1}$.

######  Lemma 5.1.

Let $T$ be a Calderón-Zygmund operator of convolution type888Namely, the
kernel $K$ of the operator $T$ is of the form $K(x,y)=k(x-y)$ such that
$Tf=k*f$, at least in the distributional sense. Note that, for a Calderón-
Zygmund operator $T$ of convolution type, we have $T(1)=T^{*}(1)=0$ (as
elements in BMO , namely, modulo additive constants). or, more generally, let
$T$ be a Calderón-Zygmund operator with $T^{*}(1)=0$. Then, $T$ maps $H^{1}$
into itself.

In the convolution case, this result goes back to Fefferman and Stein [11,
Corollary 1]. For non-convolution operators with $T^{*}(1)=0$, this result was
proved by Alvarez and Milman [1, Theorem 1.1]. Under the additional assumption
that $T(1)=0$, this result was shown by Frazier et al. [14, 15]. This case
subsumes the Fefferman-Stein result on Calderón-Zygmund operators of
convolution type; see, for example, the comments in [15, p. 67]. Note that if
$T(1)=T^{*}(1)=0$, then we have both $T,\,T^{*}:H^{1}\to H^{1}$ and
$\textit{BMO}\to\textit{BMO}$ . The next two results show that the
cancellation condition $T(1)=T^{*}(1)=0$ also provides additional mapping
properties.

######  Lemma 5.2.

Let $T$ be a Calderón-Zygmund operator of convolution type or, more generally,
let $T$ be a Calderón-Zygmund operator with $T(1)=T^{*}(1)=0$. Then, for any
$f\in C^{\infty}_{c}$, we have $T(f)\in C_{0}$.

###### Proof.

We note that this is very easy to prove in the convolution case. Since $T$ is
bounded on $L^{2}$, there exists $m\in L^{\infty}$ such that
$\widehat{T(f)}=m\widehat{f}$. Since $f\in C^{\infty}_{c}$, we have
$\widehat{f}\in\mathcal{S}$ and thus $m\widehat{f}\in L^{1}$. Then, by the
Riemann-Lebesgue lemma, we obtain $T(f)\in C_{0}$.

In the more general case, we can not take such direct advantage of the Fourier
transform. Using certain molecular decompositions, Meyer [21, Proposition 2]
showed that, under the hypothesis of the lemma, $T$ maps Hölder continuous
functions into Hölder continuous functions. In particular, for $f\in
C^{\infty}_{c}$, $T(f)$ is continuous.

Suppose that $\operatorname{supp}f\subset B_{0}(R)\subset\mathbb{R}^{d}$.
Then, if $|x|\geq 2R$, we have $|x-y|\sim|x|$ for any
$y\in\operatorname{supp}f$. Hence, from (2.2), we have

$|T(f)(x)|=\left|\int_{\operatorname{supp}f}K(x,y)f(y)\,dy\right|\lesssim\int_{\operatorname{supp}f}\frac{1}{|x|^{d}}|f(y)|\,dy\lesssim\frac{\|f\|_{L^{1}}}{|x|^{d}}\longrightarrow
0$ (5.1)

as $|x|\to\infty$. Therefore, we conclude that $T(f)\in C_{0}$. ∎

As stated in Remark 3.3 (e) and (f), under the cancellation condition
$T(1)=T^{*}(1)=0$, a compact Calderón-Zygmund operator $T$ is compact from BMO
to CMO and also from $H^{1}$ into itself, where the latter claim follows from
the former via duality. In fact, a compact Calderón-Zygmund operator $T$ is
compact from CMO to itself and thus from Lemma 3.5 (ii), we see that $T$ is
compact from $H^{1}$ into itself. Nonetheless, using the characterization
(2.6) of the Hardy space $H^{1}$, we present a direct proof of the compactness
on $H^{1}$ of a compact Calderón-Zygmund operator $T$ under the cancellation
condition $T(1)=T^{*}(1)=0$ without using such a duality argument.

######  Proposition 5.3.

Let $T$ be a Calderón-Zygmund operator of convolution type or, more generally,
let $T$ be a Calderón-Zygmund operator with $T(1)=T^{*}(1)=0$. Suppose that
$T$ is compact on $L^{2}$. Then, $T$ is compact from $H^{1}$ to itself.

###### Proof.

Let $T$ be a compact Calderón-Zygmund operator with $T(1)=T^{*}(1)=0$. By
Remark 3.3 (c), we know that $T:H^{1}\to L^{1}$ is compact. On the other hand,
recalling from [22, Section 8.9] that the collection of Calderón-Zygmund
operators $S$ satisfying the cancellation condition $S(1)=S^{*}(1)=0$ forms an
algebra, we see that $R_{j}T$ is also a Calderón-Zygmund operator,
$j=1,\dots,d$ (since $R_{j}(1)=R_{j}^{*}(1)=0$ as a convolution operator).
Moreover, since $T$ is compact on $L^{2}$ and $R_{j}$ is bounded on $L^{2}$,
the composition $R_{j}T$ is compact on $L^{2}$. Therefore, the desired
compactness on $H^{1}$ now follows from the definition (2.6) of the
$H^{1}$-norm:

$\|Tf||_{H^{1}}=\|Tf\|_{L^{1}}+\sum_{j=1}^{d}\|(R_{j}T)f\|_{L^{1}},$

since each operator appearing on the right-hand side is compact from $H^{1}$
into $L^{1}$. ∎

We now state the first main result of this section which extends the
aforementioned claim in [28, p. 180] to a slightly general setting.

######  Proposition 5.4.

Let $T$ be a Calderón-Zygmund operator of convolution type or, more generally,
let $T$ be a Calderón-Zygmund operator with $T(1)=T^{*}(1)=0$. Then, $T$ maps
$C_{0}$ to CMO .

###### Proof.

Let $f\in C_{0}$. Then, there exists $\\{f_{n}\\}_{n\in\mathbb{N}}\subset
C^{\infty}_{c}$ such that $f_{n}$ converges to $f$ in $L^{\infty}$ (and thus
in BMO ) as $n\to\infty$. By the boundedness of $T:L^{\infty}\to BMO$, we have
$T(f),\,T(f_{n})\in BMO$ for any $n\in\mathbb{N}$. Then, from the definition
of the transpose and Lemma 5.1, we have

$\displaystyle\begin{split}|\langle T(f)-T(f_{n}),g\rangle|&=|\langle
f-f_{n},T^{*}(g)\rangle|\leq\|f-f_{n}\|_{\textit{BMO}}\|T^{*}(g)\|_{H^{1}}\\\
&\lesssim\|f-f_{n}\|_{\textit{BMO}}\|g\|_{H^{1}}\end{split}$ (5.2)

for any $g\in H^{1}$. Recalling

$\|b\|_{BMO}=\sup_{\|g\|_{H^{1}}=1}|\langle b,g\rangle|,$

we see from (5.2) that $T(f_{n})$ converges to $T(f)$ in BMO as $n\to\infty$.
On the other hand, it follows from Lemma 5.2 that $T(f_{n})\in C_{0}$ for each
$n\in\mathbb{N}$. Therefore, recalling from [5, Théorème 7] that the closure
of $C_{0}$ in BMO is CMO, we conclude that $T(f)\in\textit{CMO}$. ∎

Our next goal is to show that Proposition 5.4 does not hold in general without
the cancellation condition $T(1)=T^{*}(1)=0$; see Proposition 5.5. For this
purpose, we first need to recall the notion of a paraproduct.

Let $\varphi,\psi\in C_{c}^{\infty}$ be radial functions such that
$\operatorname{supp}(\varphi)\subset B_{0}(1)$, $\widehat{\psi}(0)=0$, and

$\displaystyle\int_{0}^{\infty}|\widehat{\psi}(te_{1})|^{2}\frac{dt}{t}=1,$

where $e_{1}=(1,0,...,0)\in\mathbb{R}^{d}$. For $t\in\mathbb{R}_{+}$, we also
define the linear convolution operators $P_{t}$ and $Q_{t}$ by

$\displaystyle P_{t}f=\varphi_{t}*f\qquad\text{and}\qquad Q_{t}f=\psi_{t}*f,$
(5.3)

where $h_{t}=t^{-d}h(t^{-1}\cdot)$ for a function $h$ on $\mathbb{R}^{d}$.
Then, the Calderón reproducing formula [6] states that

$\displaystyle\int_{0}^{\infty}Q_{t}^{2}f\frac{dt}{t}=f$

holds in $L^{2}$, where $Q_{t}^{2}f=Q_{t}(Q_{t}f)=\psi_{t}*\psi_{t}*f$. Given
$b\in\textit{BMO}$, we define a paraproduct $\Pi_{b}$ by

$\displaystyle\Pi_{b}(f)=\int_{0}^{\infty}Q_{t}\big{(}(Q_{t}b)(P_{t}f)\big{)}\frac{dt}{t},$
(5.4)

a priori defined for $f\in\mathcal{S}$. It is well known by now that if
$b\in\textit{BMO}$ then $\Pi_{b}$ is a Calderón-Zygmund operator, satisfying

$\displaystyle\Pi_{b}(1)=b\qquad\text{and}\qquad\Pi^{*}_{b}(1)=0,$ (5.5)

where once again the equalities are understood by testing against functions in
$(C^{\infty}_{c})_{0}$; see, for example, [10, Lemmas 1 and 2]. See also
(4.2).

In what follows, we will prove that, for an appropriate symbol
$b\in\textit{BMO}\setminus\textit{CMO}$ , the paraproduct $\Pi_{b}$ does not
map $C_{c}^{\infty}$ into CMO . A naive approach would be to proceed as
follows. Fix $b\in\textit{BMO}\setminus\textit{CMO}$ . Assuming that $\Pi_{b}$
maps $C_{c}^{\infty}$ into CMO, we would have
$\Pi_{b}(\phi_{R})\in\textit{CMO}$, where $\phi_{R}$ is as in (4.3). If CMO
were weakly sequentially complete, we would then have
$b=\Pi_{b}(1)\in\textit{CMO}$, since, as seen in (4.2), $\Pi_{b}(1)$ is
defined as the limit of $\Pi_{b}(\phi_{R})$ by testing against
$H^{1}$-functions, namely as a limit with respect to the weak topology in CMO
, in view of the duality $(\textit{CMO})^{*}=H^{1}$. This would be a
contradiction with the fact that $b\in\textit{BMO}\setminus\textit{CMO}$ .
However, this naive approach breaks down precisely because CMO is _not_
sequentially complete; see again Appendix B and Remark 4.2. Therefore, a more
delicate argument and a more careful choice of the symbol $b$ is needed here.

######  Proposition 5.5.

There exists a Calderón-Zygmund operator, in particular a paraproduct
$\Pi_{b}$ with $b\in\textit{BMO}\setminus\textit{CMO}$ such that
$\Pi_{b}(C^{\infty}_{c})\notin\textit{CMO}$ .

We begin by showing that, for our argument, the CMO space can be “replaced” by
VMO, the space of functions of _vanishing mean oscillation_ which is defined
as the closure of $C^{\infty}\cap\textit{BMO}\,$ in the BMO topology. In
particular, we have

$\displaystyle\textit{CMO}\subset\textit{VMO}\subset\textit{BMO}.$ (5.6)

We recall the following characterization of VMO ; see, for example, [5,
Théorème 5].

######  Lemma 5.6.

Let $f\in\textit{BMO}$ . Then, $f\in\textit{VMO}$ if and only if
$\Gamma_{1}(f)=0$, where $\Gamma_{1}$ is as in (2.7).

The next lemma shows that CMO is the subspace of function in VMO which
“vanish” at infinity; see [5, Théorème 7].

######  Lemma 5.7.

Let $\textit{BMO}_{c}$ denote the subspace of BMO consisting of functions with
compact support. Then, we have

$\textit{CMO}=\textit{VMO}\cap\overline{\textit{BMO}_{c}},$

where the closure is with respect to the BMO -norm.

Let $f\in C^{\infty}_{c}(\mathbb{R}^{d})$. As we saw in (5.1), we have that
$T(f)$ (which is in BMO ) vanishes at infinity. Hence, from [5, Théorème 6],
we conclude that $T(f)\in\overline{\textit{BMO}_{c}}$. Therefore, it follows
from Lemma 5.7 that $T(f)\in\textit{CMO}$ if and only if $T(f)\in\textit{VMO}$
.

We are now ready to prove Proposition 5.5.

###### Proof of Proposition 5.5.

In the following, we work on $\mathbb{R}$. For a suitable choice of
$b\in\textit{BMO}\setminus\textit{VMO}$, we show that
$\Pi_{b}(\phi_{R})\notin\textit{VMO}$ for any finite $R\gg 1$ (and hence
$\Pi_{b}(\phi_{R})\notin\textit{CMO}$ in view of (5.6)), where $\phi_{R}$ is
as in (4.3).

From (5.3), we have

$\displaystyle Q_{t}\big{(}(Q_{t}b)(P_{t}\phi_{R})\big{)}(x)$
$\displaystyle=\frac{1}{t^{2d}}\int_{\mathbb{R}^{2}}\psi\Big{(}\frac{x-y}{t}\Big{)}Q_{t}b(y)\varphi\Big{(}\frac{y-z}{t}\Big{)}\phi_{R}(z)dzdy.$
(5.7)

Suppose that $\operatorname{supp}\psi$,
$\operatorname{supp}\varphi\subset[-C_{0},C_{0}]$ for some $C_{0}>0$. Then, we
have $|x-y|,|y-z|\leq tC_{0}$ on the right-hand side of (5.7). Thus, by the
triangle inequality, we have

$\displaystyle|z|\leq 2tC_{0}+|x|\leq 2C_{0}+1$ (5.8)

for any $0<t\leq 1$ and $|x|\leq 1$. Hence, from (5.7) with (4.3) and (5.8),
we have

$\displaystyle\begin{split}Q_{t}\big{(}(Q_{t}b)(P_{t}\phi_{R})\big{)}(x)&=Q_{t}\big{(}(Q_{t}b)(P_{t}1)\big{)}(x)\end{split}$
(5.9)

for any $0<t\leq 1$, $|x|\leq 1$, and $R\geq 4C_{0}+2$.

We now choose a suitable function $b\in\textit{BMO}\setminus\\!\textit{VMO}$ .
Define an odd function $b\in C^{\infty}((-\infty,0)\cup(0,\infty);\mathbb{R})$
by setting

$\displaystyle b(x)=\begin{cases}1,&\text{for $0<x\leq 1$},\\\ -1,&\text{for
$-1\leq x<0$},\\\ 0,&\text{for $x=0$ and $|x|\geq 2$}\end{cases}$

and smoothly interpolating on the intervals $[-2,-1]$ and $[1,2]$ in such a
way that $b$ is odd. While we have $b\in L^{\infty}\subset\textit{BMO}$ , we
see that $b\notin\textit{VMO}$ due to the jump at $x=0$. Indeed, with
$Q=[-a,a]$, we have

$\displaystyle
M_{Q}(b)=\frac{1}{|Q|}\int_{Q}|b(x)-\operatorname*{ave}_{Q}b|\,dx=\frac{1}{|Q|}\int_{Q}|b(x)|\,dx=1$

for any $0<a<1$, which implies $b\notin\textit{VMO}$ in view of Lemma 5.6.

From (5.4) and (5.5), we have

$\displaystyle
b=\Pi_{b}(1)=\int_{0}^{\infty}Q_{t}\big{(}(Q_{t}b)(P_{t}1)\big{)}\frac{dt}{t},$

where the second inequality holds as elements in the dual space of $H^{1}$
(namely, as elements in BMO ). Write $b=b_{1}+b_{2}$, where $b_{1}$ is defined
by

$\displaystyle
b_{1}=\int_{0}^{1}Q_{t}\big{(}(Q_{t}b)(P_{t}1)\big{)}\frac{dt}{t}.$ (5.10)

It is easy to see that $b_{2}$ is a smooth function on $\mathbb{R}$ since it
essentially corresponds to the low frequency part of the paraproduct
$\Pi_{b}(1)$. Moreover, since $\varphi$ and $\psi$ are even and $b$ is odd, it
follows from (5.3) that $b_{2}$ is a smooth odd function which in particular
implies $b_{2}(0)=0$. Hence, we conclude that $b_{1}$ has a jump discontinuity
at $x=0$:

$\displaystyle b_{1}(0+)-b_{1}(0-)=2.$ (5.11)

Fix $R\geq 4C_{0}+2$. Write $\Pi_{b}(\phi_{R})=A_{1}+A_{2}$, where $A_{1}$ is
defined by

$\displaystyle
A_{1}=\int_{0}^{1}Q_{t}\big{(}(Q_{t}b)(P_{t}\phi_{R})\big{)}\frac{dt}{t}.$
(5.12)

Then, from (5.9), (5.10), and (5.12), we have $A_{1}(x)=b_{1}(x)$ for any
$|x|\leq 1$. Hence, from (5.11), we obtain

$\displaystyle A_{1}(0+)-A_{1}(0-)=2,$

where we used the fact that $A_{2}(0)=0$ just as for $b_{2}$ discussed above.
By noting that $A_{2}$ is a smooth function on $\mathbb{R}$, we then have

$\displaystyle\Pi_{b}(\phi_{R})(0+)-\Pi_{b}(\phi_{R})(0-)=2.$

Hence, by computing $M_{Q}(\Pi_{b}(\phi_{R}))$ for $Q=(-a,a)$ with $a$ tending
to $0$, we see that $\Gamma_{1}(\Pi_{b}(\phi_{R}))\neq 0$, where $M_{Q}$ and
$\Gamma_{1}$ are as in (2.4) and (2.7), respectively. Therefore, we conclude
from Lemma 5.6 that $\Pi_{b}(\phi_{R})\notin\textit{VMO}$ . This shows that,
in general, a Calderón-Zygmund operator does not map $C_{c}^{\infty}$ into VMO
or CMO. ∎

######  Remark 5.8.

As subspaces of BMO, the difference between CMO and VMO appears only in the
global aspect of their elements, namely the decay at spatial infinity of their
elements (just like the difference between $C_{0}$ and $C$ as subspaces of
$L^{\infty}$). In the counterexample presented above, the failure of
$C^{\infty}_{c}$-to-CMO mapping property comes from the local property of
elements shared by CMO and VMO , namely, they can not have an isolated jump
discontinuity, while an element in BMO can have an isolated jump
discontinuity,

##  Appendix A Lack of compactness of convolution operators in
$L^{p}(\mathbb{R}^{d})$

From the result in the previous section, it may be tempting to think that some
Calderón-Zygmund operators of convolution type may be compact. However it is
known that that cannot be true. See for example, Feitchinger [12, p. 310]
where it is stated “that in the case of a non-compact group there do not exist
compact multipliers between two isometrically translation invariant spaces”.
We will provide a simple proof of this fact in the context of the Lebesgue
spaces $L^{p}(\mathbb{R}^{d})$.

Consider first the following simple but illustrative example. Given
$\varepsilon>0$, let $J^{-\varepsilon}$ be the Bessel potential of order
$\varepsilon>0$. Namely, $J^{-\varepsilon}$ is a Fourier multiplier operator
with a bounded multiplier $(1+|\xi|^{2})^{-\frac{\varepsilon}{2}}$. It is not
difficult to show that $J^{-\varepsilon}$ is a Calderón-Zygmund operator. If
$J^{-\varepsilon}$ were compact on $L^{2}(\mathbb{R}^{d})$, then this would
imply that $L^{2}(\mathbb{R}^{d})$ compactly embeds into
$H^{-\varepsilon}(\mathbb{R}^{d})$, which is false. However, the situation is
different in the compact setting. We point out that if the underlying space is
the $d$-dimensional torus $\mathbb{T}^{d}=(\mathbb{R}/\mathbb{Z})^{d}$, then
Rellich’s lemma states that the embedding
$L^{2}(\mathbb{T}^{d})\hookrightarrow H^{-\varepsilon}(\mathbb{T}^{d})$ is
indeed compact. Namely, the lack of compactness of the Bessel potential
$J^{-\varepsilon}$ on $L^{2}(\mathbb{R}^{d})$ comes from the fact that
functions on $\mathbb{R}^{d}$ can run off to the spatial infinity.

The example above is representative of the following general fact.

######  Proposition A.1.

Let $1\leq p<\infty.$ A non-trivial999Namely, $T\neq 0$. translation invariant
operator $T$ is not a compact operator on $L^{p}(\mathbb{R}^{d})$.

###### Proof.

Without loss of generality, we assume that $T$ is bounded on
$L^{p}(\mathbb{R}^{d})$ since, otherwise, $T$ is obviously not compact. Since
$T$ is translation invariant, we have

$\displaystyle T(\tau_{y}f)=\tau_{y}T(f),$ (A.1)

where $\tau_{y}f(x)=f(x-y)$.

Fix $f\in L^{p}$ such that $T(f)\neq 0$. Given a sequence
$\\{y_{n}\\}_{n\in\mathbb{N}}\subset\mathbb{R}^{d}$ with $|y_{n}|\to\infty$,
let $f_{n}=\tau_{y_{n}}f$. Clearly, the sequence
$\\{f_{n}\\}_{n\in\mathbb{N}}$ is bounded in $L^{p}$. If $T$ were compact on
$L^{p}$, then it would follow from the Kolmogorov-Riesz theorem (see [32,
Theorem on p. 275]; see also [17, Theorem 5]) that there exists a subsequence
$\\{f_{n_{k}}\\}_{k\in\mathbb{N}}$ such that, given any $\varepsilon>0$, there
exists $R=R(\varepsilon)>0$ such that

$\displaystyle\int_{|x|>R}|T(f_{n_{k}})(x)|^{p}dx<\varepsilon^{p}$ (A.2)

for any $k\in\mathbb{N}$.

Fix $0<\varepsilon<\frac{1}{2}\|Tf\|_{L^{p}}$. Suppose that there exists
$R=R(\varepsilon)>0$ such that (A.2) holds. Then, from the definition of
$f_{n_{k}}$, (A.1), a change of variables, and the dominated convergence
theorem together with the $L^{p}$-boundedness of $T$, we have

$\displaystyle\lim_{k\to\infty}\int_{|x|>R}|T(f_{n_{k}})(x)|^{p}dx$
$\displaystyle=\lim_{k\to\infty}\int_{|x|>R}|T(f)(x-y_{n_{k}})|^{p}dx$
$\displaystyle=\lim_{k\to\infty}\int_{|x+y_{n_{k}}|>R}|T(f)(x)|^{p}dx=\|Tf\|_{L^{p}}^{p}>2^{p}\varepsilon^{p},$

which is a contradiction to (A.2). Therefore, we conclude that a non-trivial
translation invariant $T$ is not compact on $L^{p}$. ∎

##  Appendix B BMO , VMO , and CMO are not weakly sequentially complete

In this appendix, we study weakly sequential completeness properties for the
BMO -type spaces which is closely related to the need for the
$L^{\infty}$-condition in the proof of Theorem 1; see Remark 4.1 (ii).

Weakly sequential completeness of the Hardy space $H^{1}$ follows as an
immediate consequence of [8, Lemma (4.2)]. A natural question is then to ask
whether the dual and pre-dual of $H^{1}$ (BMO and CMO , respectively) are also
weakly sequential complete. We will show that the answer is negative.

######  Proposition B.1.

BMO, VMO, and CMO are not weakly sequentially complete.

Before proceeding to a proof of Proposition B.1, we state several lemmas. We
first recall the following lemma from [2, Proposition 2.3].

######  Lemma B.2.

Let $X$ be a weakly sequentially complete Banach space. Then, any closed
subspace $Y$ of $X$ is a weakly sequentially complete Banach space.

Let $c_{0}=c_{0}(\mathbb{N})\subset\ell^{\infty}(\mathbb{N})$ denote the space
of bounded sequences ${\bf a}=\\{a_{n}\\}_{n\in\mathbb{N}}$ such that
$\displaystyle\lim_{n\to\infty}a_{n}=0$. If we endow $c_{0}$ with the supremum
norm, then we have $(c_{0})^{*}=\ell^{1}(\mathbb{N})=\ell^{1}$, where the
duality pairing is given by

$\langle{\bf a},{\bf b}\rangle=\sum_{n=1}^{\infty}a_{n}b_{n}$

for ${\bf a}\in c_{0}$ and ${\bf b}\in\ell^{1}$. The space $c_{0}$ provides a
useful example for many functional analytic situations related to weak
sequential (non-)completeness. For the readers convenience, we include a proof
of the following fact.

######  Lemma B.3.

$c_{0}$ is not weakly sequentially complete.

###### Proof.

Given $k\in\mathbb{N}$, define a sequence ${\bf
a}_{k}=\\{a_{kn}\\}_{n\in\mathbb{N}}$ by setting $a_{kn}=1$ for $n\leq k$ and
$a_{kn}=0$ for $n>k$. Then, it is easy to verify that ${\bf a}_{k}\in c_{0}$
for each $k\in\mathbb{N}$ and that ${\bf a}_{k}$ is weakly Cauchy. However,
its pointwise limit is a constant sequence with value 1 which is not in
$c_{0}$. This proves the claim. ∎

Given a Banach space $X$, we say that $X$ contains a copy of $c_{0}$, if there
exist a sequence $\\{x_{n}\\}_{n\in\mathbb{N}}$ of unit vectors and constants
$C_{1},C_{2}>0$ such that

$\displaystyle
C_{1}\sup_{n\in\mathbb{N}}|a_{n}|\leq\bigg{\|}\sum_{n=1}^{\infty}a_{n}x_{n}\bigg{\|}_{X}\leq
C_{2}\sup_{n\in\mathbb{N}}|a_{n}|$ (B.1)

for any ${\bf a}=\\{a_{n}\\}_{n\in\mathbb{N}}\in c_{0}$.

Let

$\displaystyle
Y={\bigg{\\{}x=\sum_{n=1}^{\infty}a_{n}x_{n}:\\{a_{n}\\}_{n\in\mathbb{N}}\in
c_{0}\bigg{\\}}}.$ (B.2)

In view of (B.1), we see that every element in $Y$ is uniquely represented as

${\bf a}\cdot{\bf x}=\sum_{n=1}^{\infty}a_{n}x_{n}$

for some ${\bf a}=\\{a_{n}\\}_{n\in\mathbb{N}}\in c_{0}$. Moreover, $Y$ is
isomorphic and “almost isometric” to $c_{0}$, meaning that

$\|{\bf a}\|_{c_{0}}\approx\|{\bf a}\cdot{\bf x}\|_{X}$

in the sense of (B.1). Once again, in view of (B.1), we see that $Y$ is closed
in $X$ because $c_{0}$ is a complete Banach space.

######  Lemma B.4.

Let $X$ be a Banach space. If $X$ contains a copy of $c_{0}$, then $X$ is not
weakly sequentially complete.

Under an additional assumption that the space $X$ has an unconditional basis,
the converse of Lemma B.4 also holds true; see [20, Theorem 1.c.10].

###### Proof of Lemma B.4.

Since $X$ contains a copy of $c_{0}$, there exists
$\\{x_{n}\\}_{n\in\mathbb{N}}\subset X$, satisfying (B.1). Define a subspace
$Y$ of $X$ as in (B.2). Recalling that $c_{0}^{*}=\ell^{1}$, we see that
$Y^{*}\cong\ell^{1}$. Indeed, given ${\bf
b}=\\{b_{n}\\}_{n\in\mathbb{N}}\in\ell^{1}$, we set ${\bf
b}(x)=\sum_{n=1}^{\infty}a_{n}b_{n}$ for $x={\bf a}\cdot{\bf x}\in Y$. By
noting that the $Y$-$Y^{*}$ duality pairing agrees with the $c_{0}$-$\ell^{1}$
duality pairing, it follows from Lemma B.3 that $Y$ is not weakly sequentially
complete. Finally, by recalling that $Y$ is a closed subspace of $X$, we
conclude from Lemma B.2 that $X$ is not weakly sequentially complete. ∎

We are now ready to present a proof of Proposition B.1.

###### Proof of Proposition B.1.

We only consider the $d=1$ case. Define a function $\varphi$ on $\mathbb{R}$
by setting

$\displaystyle\varphi(x)=\begin{cases}1,&0\leq x\leq 1,\\\ -1,&-1\leq x<0,\\\
0,&\text{otherwise}.\end{cases}$

Then, a direct computation shows that $\|f\|_{\textit{BMO}}=1$. While the
function $f$ suffices for showing weakly sequential non-completeness of BMO ,
we need to smooth it out to treat the VMO and CMO cases since
$\varphi\notin\textit{VMO}$ .

Let $\rho:\mathbb{R}\to[0,1]$ be a smooth even function with compact support,
and set $\rho_{\varepsilon}(x)=\varepsilon^{-1}\rho(\varepsilon^{-1}x)$. Fix
small $\varepsilon>0$ and set
$\varphi_{\varepsilon}=\rho_{\varepsilon}*\varphi$. Then, we have

  * •

$\operatorname{supp}\varphi_{\varepsilon}\subset[-2,2]$,

  * •

$\|\varphi_{\varepsilon}\|_{\textit{BMO}}\sim 1$,

  * •

$\varphi_{\varepsilon}\in\textit{VMO}$ and hence
$\varphi_{\varepsilon}\in\textit{CMO}$ (in view of its compact support).

Now, given $n\in\mathbb{N}$, we set
$f_{n}(x)=\|\varphi_{\varepsilon}\|_{\textit{BMO}}^{-1}\cdot\varphi_{\varepsilon}(x-10n)$.
Then, we define a subspace $Y$ of BMO by

$\displaystyle Y={\bigg{\\{}f=\sum_{n=1}^{\infty}a_{n}f_{n}:{\bf
a}=\\{a_{n}\\}_{n\in\mathbb{N}}\in c_{0}\bigg{\\}}}.$

We claim that that there exist constants $C_{1},C_{2}>0$ such that

$\displaystyle
C_{1}\sup_{n\in\mathbb{N}}|a_{n}|\leq\bigg{\|}\sum_{n=1}^{\infty}a_{n}f_{n}\bigg{\|}_{\textit{BMO}}\leq
C_{2}\sup_{n\in\mathbb{N}}|a_{n}|$ (B.3)

for any ${\bf a}=\\{a_{n}\\}_{n\in\mathbb{N}}\in c_{0}$. In fact, it follows
from [16, Proposition 3.1.2 (2)], the disjointness of the supports of $f_{n}$,
and the fact that the $L^{\infty}$-norm of $f_{n}$ is constant in
$n\in\mathbb{N}$ that

$\bigg{\|}\sum_{n=1}^{\infty}a_{n}f_{n}\bigg{\|}_{\textit{BMO}}\leq
2\bigg{\|}\sum_{n=1}^{\infty}a_{n}f_{n}\bigg{\|}_{L^{\infty}}\leq
C_{2}\sup_{n\in\mathbb{N}}|a_{n}|.$

On the other hand, there exists $C_{1}$, independent of $k\in\mathbb{N}$, such
that

$\bigg{\|}\sum_{n=1}^{\infty}a_{n}f_{n}\bigg{\|}_{\textit{BMO}}\geq\frac{1}{|\operatorname{supp}f_{k}|}\int_{\operatorname{supp}f_{k}}|a_{k}f_{k}(x)|\,dx=C_{1}|a_{k}|$

for any $k\in\mathbb{N}$, where we used the fact that $f_{k}$ has mean zero in
the first step and the second step follows from noting that the size of the
support and the $L^{1}$-norm of $f_{k}$ do not depend on $k\in\mathbb{N}$.
This proves (B.3).

From the right-hand side of (B.3), we see that every element in $Y$ is the
limit in BMO of $C_{c}^{\infty}$-functions (namely their partial sums). Hence,
from [5, Théorème 7 (ii)], we see that $Y\subset\textit{CMO}$. This shows that
CMO contains a copy of $c_{0}$. Hence, from Lemma B.4, we conclude that CMO is
not weakly sequentially complete. By applying Lemma B.2, we also see that BMO
and VMO are not weakly sequentially complete. ∎

We conclude this section by recalling that BMO is sequentially complete with
respect to the weak-$\ast$ topology as the dual of $H^{1}$. In fact, this is a
particular case of the following general result. While the proof is standard,
we include it for readers’ convenience.

######  Lemma B.5.

Let $X$ be a Banach space. Then, $X^{*}$ is weak-$\ast$ sequentially complete.

###### Proof.

Let $\\{f_{n}\\}_{n\in\mathbb{N}}$ be a weak-$\ast$ Cauchy sequence in
$X^{*}$. From the Banach-Steinhaus theorem, we see that
$\sup_{n\in\mathbb{N}}\|f_{n}\|_{X^{*}}<\infty$. Then, it follows from the
Banach-Alaoglu theorem that there exists a subsequence
$\\{f_{n_{j}}\\}_{j\in\mathbb{N}}$ and $f\in X^{*}$ such that, as
$j\to\infty$, $f_{n_{j}}\to f$ with respect to the weak-$\ast$ topology. That
is,

$\langle f,h\rangle=\lim_{k\to\infty}\langle
f_{n_{j_{k}}},h\rangle=\lim_{n\to\infty}\langle f_{n},h\rangle$

for any $h\in X$, where the second equality follows from the fact that
$\\{f_{n}\\}_{n\in\mathbb{N}}$ is weak-$\ast$ Cauchy in $X^{*}$ (which in
particular guarantees uniqueness of the weak-$*$ limit). The proof is
complete. ∎

######  Acknowledgements.

Á.B. acknowledges the support from an AMS-Simons Research Enhancement Grant
for PUI Faculty. G.L. and T.O. were supported by the European Research Council
(grant no. 864138 “SingStochDispDyn”). The first three authors would like to
thank the West University of Timişoara for its hospitality, where part of this
paper was prepared.

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