Linear and bilinear $T(b)$ theorems à la Stein

Abstract

In this work, we state and prove versions of the linear and bilinear $T(b)$ theorems involving quantitative estimates, analogous to the quantitative linear $T(1)$ theorem due to Stein.

1. Introduction

The impact of the classical Calderón-Zygmund theory permeates through analysis and PDEs. Nowadays, both the linear and multilinear aspects of this theory are well understood and continue to be intertwined with aspects of analysis that are beyond their reach, such as those considering the bilinear Hilbert transform.

Two fundamental results in the linear theory from the 1980s are the celebrated $T(1)$ theorem of David and Journé 4 and $T(b)$ theorem of David, Journé, and Semmes 5. Both results were strongly motivated by the study of the Cauchy integral on a Lipschitz curve and the related Calderón commutators. Their gist lies in understanding the boundedness of a singular operator via appropriate simpler testing conditions.

In the $T(1)$ theorem, one needs to test a singular operator and its transpose on the constant function 1. If both the operator and its transpose were $L^\infty \to \mathit{BMO}$ bounded, then by duality and interpolation 6, the operator would be bounded on $L^2$. The remarkable aspect of the $T(1)$ theorem is that one does not need to test the operator on the whole $L^\infty$, but just on one special element in it. Going back to the Cauchy integral operator associated to a Lipschitz function $A$, it turns out that it is not necessarily easy to test the operator on 1. It is, however, much easier to test the operator on the $L^\infty$ function $1+iA'$. Thus, as the name suggests, the $T(b)$ theorem extends the $T(1)$ theorem by replacing the constant function 1 with a suitable $L^\infty$ function $b$ or, to be more precise, by replacing 1 with two suitable functions $b_0$ and $b_1$ in $L^\infty$ on which we test an operator and its transpose. The bilinear Calderón-Zygmund theory has its own versions of the $T(1)$ and $T(b)$ theorems, such as those proved by Grafakos and Torres 8 and by Hart 12, respectively. See Theorems D and E below.

In this work, we revisit the $T(b)$ theorem, both in the linear and bilinear settings, through the lens of a gem due to Stein 15. We are alluding to his formulation of the $T(1)$ theorem involving quantitative estimates for a singular operator and its transpose when tested now on normalized bump functions. Our goal is to prove that an analogous natural formulation à la Stein can be given for the $T(b)$ theorems in the linear and bilinear settings. We note that, while for the sake of clarity in our presentation we have chosen to delineate the linear and bilinear settings, a unified discussion is certainly possible under the encompassing more general multilinear setting.

2. Linear Calderón-Zygmund theory

In this section, we consider a linear singular operator $T$ a priori defined from $\mathcal{S}$ into $\mathcal{S}'$ of the form

$$\begin{align} T(f)(x) = \int _{\mathbb{R}^d} K(x, y) f(y) dy . {\tag{2.1}} \end{align}$$

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 $\mathbb{R}^{2d} \setminus \Delta$. 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 (linear) Calderón-Zygmund kernel if it satisfies the following conditions:

(i)

For all $x, y \in \mathbb{R}^d$, we have $|K(x, y)| \lesssim |x-y|^{-d}$.

(ii)

There exists $\delta \in (0, 1]$ such that$$\begin{align} |K(x, y) - K(x', y)| & + |K(y, x) - K(y, x')| \lesssim \frac{|x - x'|^\delta }{|x-y|^{d+\delta }} {\tag{2.2}} \end{align}$$

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

We say that a linear singular operator $T$ of the form 2.1 with a Calderón-Zygmund kernel is a linear Calderón-Zygmund operator if $T$ extends to a bounded operator on $L^{p_0}$ for some $1< p_0 < \infty$. It is well known 14 that if $T$ is a linear Calderón-Zygmund operator, then it is bounded on $L^p$ for all $1< p < \infty$. Hence, in the following, we restrict our attention to the $L^2$-boundedness of such linear operators. We point out that the Calderón-Zygmund operator $T$ is also $L^\infty \to \mathit{BMO}$ bounded. Here, $\mathit{BMO}$ 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 $\mathit{BMO}$-seminorm by

$$\begin{equation*} \|f\|_{\mathit{BMO}} := \sup _{Q} \frac{1}{|Q|} \int _Q|f(x) - \operatorname *{ave}_Q f|dx, \end{equation*}$$

where the supremum is taken over all cubes $Q \subset \mathbb{R}^d$ and

$$\begin{equation*} \operatorname *{ave}_Q f : = \frac{1}{|Q|} \int _Q f(x) dx. \end{equation*}$$

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

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

2.1. Classical linear $T(1)$ and $T(b)$ theorems

In this subsection, we provide a brief discussion of the classical $T(1)$ and $T(b)$ theorems proved in 4 and 5, respectively. In order to do so, we need to define a few more notions.

Definition 2.3.

We say that a function $\phi \in \mathcal{D}$ 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, $B_x(r)$ denotes the ball of radius $r$ centered at $x$. Given $x_0 \in \mathbb{R}^d$ and $R>0$, we set

$$\begin{align} \phi ^{x_0, R}(x) = \phi \Big (\frac{x-x_0}{R}\Big ). {\tag{2.3}} \end{align}$$

Definition 2.4.

We say that a linear singular integral operator $T: \mathcal{S}\to \mathcal{S}'$ has the weak boundedness property if there exists $M \in \mathbb{N}\cup \{0\}$ such that we have

$$\begin{align} \big | \big \langle T(\phi _1^{x_1, R}), \phi _2^{x_2, R}\big \rangle \big |\lesssim R^d {\tag{2.4}} \end{align}$$

for all normalized bump functions $\phi _1$ and $\phi _2$ of order $M$, $x_1, x_2 \in \mathbb{R}^d$, and $R>0$.

We note that it suffices to verify 2.4 for $x_1 = x_2$; see 11. The statement of the $T(1)$ theorem of David and Journé 4 is the following.

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

Let $T: \mathcal{S}\to \mathcal{S}'$ 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

(i)

$T$ satisfies the weak boundedness property,

(ii)

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

Since $T$ is a priori defined only in $\mathcal{S}$, the expressions $T(1)$ and $T^*(1)$ are, of course, not well defined and need to be interpreted carefully. The same comment applies to the corresponding theorems in the bilinear setting.

The main concept needed in extending the $T(1)$ theorem to the $T(b)$ theorem is that of para-accretive functions.

Definition 2.5.

We say that a function $b \in L^\infty$ is para-accretive⁠¹ if there exists $c_0> 0$ such that, for every cube $Q$, there exists a subcube $\widetilde{Q} \subset Q$ such that

¹

An extra condition that $b^{-1}\in L^\infty$ is sometimes included in the definition of para-accretivity. This, however, is not necessary. Indeed, it follows from 2.6 and the Lebesgue differentiation theorem that $|b(x)| \geq c_0$ almost everywhere. In particular, we have $b^{-1} \in L^\infty$.

✖

$$\begin{align} \frac{1}{|Q|} \bigg |\int _{\widetilde{Q}} b(x) dx \bigg | \geq c_0. {\tag{2.5}} \end{align}$$

It follows from 2.5 that

$$\begin{align} |\widetilde{Q}| \geq \frac{c_0}{\|b\|_{L^\infty }}|Q|. {\tag{2.6}} \end{align}$$

In particular, the function 1 is automatically para-accretive. It is also worth pointing out that the definition of para-accretivity in Definition 2.5 is not the same as the one used in the classical $T(b)$ theorem of David, Journé, and Semmes 5. The notion of para-accretivity stated here is borrowed from 1012; for a similar definition in which cubes are replaced by balls, see Christ’s monograph 2. The two definitions of para-accretivity are nevertheless equivalent. Since this natural observation seems to be missing from the literature, for the convenience of the reader we have included its proof in the appendix.

Before giving a meaning to operators to which the $T(b)$ theorem applies, we need one more definition.

Definition 2.6.

Given $0< \eta \leq 1$, let $C^\eta$ be the collection of all functions from $\mathbb{R}^d\to \mathbb{C}$ such that $\|f\|_{C^\eta } < \infty$, where the $C^\eta$-norm is given by

$$\begin{equation*} \|f\|_{C^\eta } = \sup _{x\ne y}\frac{|f(x) - f(y)|}{|x-y|^\eta }. \end{equation*}$$

We also denote by $C^\eta _0$ the subspace of all compactly supported functions in $C^\eta$.

Definition 2.7.

Let $b_0$ and $b_1$ be para-accretive functions. A linear singular operator $T:b_1C_0^\eta \to (b_0C_0^\eta )'$ is called a linear singular integral operator of Calderón-Zygmund type associated to $b_0$ and $b_1$ if $T$ is continuous from $b_1C_0^\eta$ into $(b_0C_0^\eta )'$ for some $\eta >0$ and there exists a Calderón-Zygmund kernel $K$ such that

$$\begin{equation*} \langle T (M_{b_1} f), M_{b_0}g \rangle = \int _{\mathbb{R}^{2d}} K(x, y) b_1(y) f(y) b_0(x) g(x) dx dy, \end{equation*}$$

for all $f, g \in C_0^\eta$ such that $\operatorname *{supp}f \cap \operatorname *{supp}g = \emptyset$. Here, $M_b$ denotes the operation of multiplication by $b$.

With these preparations, we are now ready to state the classical $T(b)$ theorem 5.

Theorem B ($T(b)$ theorem).

Let $b_0$ and $b_1$ be para-accretive functions. Suppose that $T$ is a linear singular integral operator of Calderón-Zygmund type associated to $b_0$ and $b_1$. Then, $T$ can be extended to a bounded operator on $L^2$ if and only if the following conditions hold:

(i)

$M_{b_0} T M_{b_1}$ satisfies the weak boundedness property,

(ii)

$M_{b_0}T(b_1)$ and $M_{b_1}T^*(b_0)$ are in $\mathit{BMO}$.

In the special case when $b_0$ and $b_1$ are accretive⁠² and $Tb_1 = T^*b_0 = 0$, the $T(b)$ theorem was independently proved by McIntosh and Meyer 13.

²

A function $b \in L^\infty$ is called accretive if there exists $\delta > 0$ such that $\operatorname *{Re}b \geq \delta$ for all $x \in \mathbb{R}^d$. Note that an accretive function is para-accretive.

✖

Remark 2.8.

In 5, condition (ii) of Theorem B is stated slightly differently; it was assumed that $T(b_1), T^*(b_0)\in \mathit{BMO}$. We note that this is just a matter of notation. For example, the condition $T(b_1)\in \mathit{BMO}$ in 5 means that that there exists $\beta \in \mathit{BMO}$ such that

$$\begin{equation*} \langle T(b_1), f \rangle = \langle \beta , f \rangle \quad \text{for all mean-zero }f \in b_0 C_0^\eta . \end{equation*}$$

This is clearly equivalent to

$$\begin{align} \langle T(b_1), b_0 f \rangle = \langle \beta ,b_0 f \rangle \quad \text{for all }f \in C_0^\eta \text{ such that } \int b_0 f dx = 0. {\tag{2.7}} \end{align}$$

Here, we used the fact that $b_0 f \leftrightarrow f$ is a one-to-one correspondence since $b_0$ is para-accretive and thus, in particular, is bounded away from zero almost everywhere. In Theorem B, we followed the notation from 12 to signify the fact that the condition indeed depends on both $b_0$ and $b_1$, and what we mean by condition (ii) in Theorem B is precisely the statement 2.7. See also Theorem E below in the bilinear setting.

Lastly, note that, as in the $T(1)$ theorem, expressions $M_{b_0}T(b_1)$ and $M_{b_1} T^*(b_0)$ are not a priori well defined, and thus some care must be taken.

2.2. Formulations of the $T(1)$ and $T(b)$ theorems à la Stein

There is another formulation of the $T(1)$ theorem due to Stein 15 in which conditions (i) and (ii) in Theorem A are replaced by the quantitative estimate 2.8 involving normalized bump functions.

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

$$\begin{align} \|T(\phi ^{x_0, R})\|_{L^2}+\|T^*(\phi ^{x_0, R})\|_{L^2}\lesssim R^\frac{d}{2} {\tag{2.8}} \end{align}$$

for any normalized bump function $\phi$ of order $M$, $x_0 \in \mathbb{R}^d$, and $R>0$.

By viewing the expressions $T(\phi ^{x_0, R})$ and $T^*(\phi ^{x_0, R})$ as $T(1\cdot \phi ^{x_0, R})$ and $T^*(1\cdot \phi ^{x_0, R})$, it is natural to extend this result by replacing the constant function 1 by para-accretive functions $b_0$ and $b_1$. This is the first result of our paper.

Theorem 1 ($T(b)$ theorem à la Stein).

Let $T$, $b_0$, and $b_1$ be as in Theorem B. 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 the following two inequalities hold for any normalized bump function $\phi$ of order $M$, $x_0 \in \mathbb{R}^d$, and $R>0$:

$$\begin{align} \|T(b_1\phi ^{x_0, R})\|_{L^2}\lesssim R^\frac{d}{2}, {\tag{2.9}}\\ \|T^*(b_0\phi ^{x_0, R})\|_{L^2}\lesssim R^\frac{d}{2} . {\tag{2.10}} \end{align}$$

We present the proof of Theorem 1 in Section 4.

As an application of this result, one could recover the well-known fact that the commutator of a pseudodifferential operator with symbol in the Hörmander class $S_{1, 0}^1$ and the multiplication operator of a Lipschitz function $a$ is bounded on $L^2$. Indeed, suppose that for all $x, \xi \in \mathbb{R}^d$ and all multi-indices $\alpha , \beta$ we have

$$\begin{equation*} |\partial _x^\alpha \partial _\xi ^\beta \sigma (x, \xi )|\lesssim (1+|\xi |)^{1-|\beta |}, \end{equation*}$$

and let

$$\begin{equation*} T_\sigma (f)(x)=\int _{\mathbb{R}^d} \sigma (x, \xi )f(\xi )e^{ix\cdot \xi }\,d\xi \end{equation*}$$

be the corresponding pseudodifferential operator. Also, given $a$ such that $\partial a/\partial x_j\in L^\infty (\mathbb{R}^d)$ for $1\leq j\leq d$, let

$$\begin{equation*} [T_\sigma , M_a]=T_\sigma (af)-aT(f) \end{equation*}$$

be the commutator of $T_\sigma$ and the multiplication operator $M_a$. It is straightforward to check that the kernel of $[T_\sigma , M_a]$ is Calderón-Zygmund and, by a similar computation to the one in 15, pp. 309-310, 2.9 and 2.10 hold as well, thus proving $[T_\sigma , M_a]: L^2\to L^2$.

3. Bilinear Calderón-Zygmund theory

Next, we turn our attention to the bilinear setting and consider the corresponding extensions of the results in Section 2. Namely, we consider a bilinear singular operator $T$ a priori defined from $\mathcal{S}\times \mathcal{S}$ into $\mathcal{S}'$ of the form

$$\begin{align} T(f, g)(x) = \int _{\mathbb{R}^{2d}} K(x, y, z) f(y) g(z) dy dz, {\tag{3.1}} \end{align}$$

where we assume that, away from the diagonal $\Delta = \{ (x, y, z) \in \mathbb{R}^{3d}:\, x = y = z\}$, the distributional kernel $K$ coincides with a function that is locally integrable on $\mathbb{R}^{3d} \setminus \Delta$. The formal transposes $T^{*1}$ and $T^{*2}$ are defined in an analogous manner with the kernels $K^{*1}$ and $K^{*2}$ given by $K^{*1}(x, y, z) : = K(y, x, z)\,\, \text{and}\,\, K^{*2}(x, y, z) : = K(z, y, x).$

Definition 3.1.

A locally integrable function $K$ on $\mathbb{R}^{3d} \setminus \Delta$ is called a (bilinear) Calderón-Zygmund kernel if it satisfies the following conditions:

(i)

For all $x, y, z \in \mathbb{R}^d$, we have$$\begin{equation*} |K(x, y, z)| \lesssim \big (|x-y|+ |x-z|\big )^{-2d}. \end{equation*}$$

(ii)

There exists $\delta \in (0, 1]$ such that$$\begin{align} |K(x, y, z) - K(x', y, z)| \lesssim \frac{|x - x'|^\delta }{\big (|x-y|+ |x-z|\big )^{2d+\delta }} {\tag{3.2}} \end{align}$$

for all $x, x', y, z\in \mathbb{R}^d$ satisfying $|x - x'| < \frac{1}{2} \max \big (|x-y|, |x-z|\big )$. Moreover, we assume that the formal transpose kernels $K^{*1}$ and $K^{*2}$ also satisfy the regularity condition 3.2.

We say that a bilinear singular operator $T$ of the form 3.1 with a bilinear Calderón-Zygmund kernel is a bilinear Calderón-Zygmund operator if $T$ extends a bounded operator on $L^{p_0}\times L^{q_0}$ into $L^{r_0}$ for some $1< p_0 , q_0 < \infty$ with $\frac{1}{p_0} + \frac{1}{q_0} = \frac{1}{r_0} \leq 1$.

Similarly to the linear case, the crux of the bilinear Calderón-Zygmund theory is contained in the fact that if $T$ is a bilinear Calderón-Zygmund operator, then it is bounded on $L^{p}\times L^{q}$ into $L^{r}$ for all $1< p , q < \infty$ with $\frac{1}{p} + \frac{1}{q} = \frac{1}{r} \leq 1$ (with the appropriate statements at the endpoints); see Grafakos and Torres 8. Therefore, the main question is to prove that there exists at least one triple $(p_0, q_0, r_0$) with $1< p_0 , q_0 < \infty$ and $\frac{1}{p_0} + \frac{1}{q_0} = \frac{1}{r_0} \leq 1$ such that $T$ is bounded from $L^{p_0}\times L^{q_0}$ into $L^{r_0}$.

The weak boundedness property for bilinear singular operators has a similar flavor as the one in the linear case.

Definition 3.2.

We say that a bilinear singular integral operator $T: \mathcal{S}\times \mathcal{S}\to \mathcal{S}'$ has the (bilinear) weak boundedness property if there exists $M \in \mathbb{N}\cup \{0\}$ such that we have

$$\begin{align} \big | \langle T(\phi _1^{x_1, R}, \phi _2^{x_2, R}), \phi _3^{x_3, R} \rangle \big |\lesssim R^d {\tag{3.3}} \end{align}$$

for all normalized bump functions $\phi _1, \phi _2, \phi _3$ of order $M$, $x_1, x_2, x_3 \in \mathbb{R}^d$, and $R>0$.

Remark 3.3.

It follows from 1, Lemma 9 that it suffices to verify 3.3 for $x_1 = x_2 = x_3$.

3.1. Bilinear $T(1)$ and $T(b)$ theorems

We now state the bilinear $T(1)$ theorem in the form given by Hart 11.

Theorem D (Bilinear $T(1)$ theorem).

Let $T: \mathcal{S}\times \mathcal{S}\to \mathcal{S}'$ be a bilinear singular integral operator with a standard Calderón-Zygmund kernel. Then, $T$ can be extended to a bounded operator on $L^p \times L^q \to L^r$ for all $1 < p, q < \infty$ with $\frac{1}{p} + \frac{1}{q} = \frac{1}{r}$ if and only if

(i)

$T$ satisfies the weak boundedness property,

(ii)

$T(1, 1), \, T^{*1}(1, 1),$ and $T^{*2}(1, 1)$ are in $\mathit{BMO}$.

We chose this formulation since it closely follows the statement of the classical linear $T(1)$ theorem given in the previous section. Further, note that Theorem D is equivalent to the formulation of Grafakos and Torres 8; see also Christ and Journé 3.

Next, we turn our attention to the bilinear version of the $T(b)$ theorem.

Definition 3.4.

Let $b_0$, $b_1$, and $b_2$ be para-accretive functions. A bilinear singular operator $T:b_1C_0^\eta \times b_2C^\eta _0\to (b_0C_0^\eta )'$ is called a bilinear singular integral operator of Calderón-Zygmund type associated to $b_0$, $b_1$, and $b_2$ if $T$ is continuous from $b_1 C_0^\eta \times b_2 C_0^\eta$ into $(b_0 C_0^\eta )'$ for some $\eta > 0$ and there exists a bilinear Calderón-Zygmund kernel $K$ such that

$$\begin{equation} \langle T (M_{b_1} f_1), M_{b_2} f_2),M_{b_0}f_0 \rangle = \int _{\mathbb{R}^{3d}} K(x, y, z) b_0(x) f_0(x) b_1(y) f_1(y) b_2(z) f_2(z) dx dy dz, {\tag{3.4}} \end{equation}$$

for all $f_0, f_1, f_2 \in C_0^\eta$ with $\operatorname *{supp}f_0\cap \operatorname *{supp}f_1\cap \operatorname *{supp}f_2= \emptyset$.

Hart 12 proved the following result.

Theorem E (Bilinear $T(b)$ theorem).

Let $b_0, b_1$, and $b_2$ be para-accretive functions. Suppose that $T$ is a bilinear singular integral operator of Calderón-Zygmund type associated to $b_0, b_1$, and $b_2$. Then, $T$ can be extended to a bounded operator on $L^p \times L^q \to L^r$ for all $p, q < \infty$ satisfying $\frac{1}{p} + \frac{1}{q} = \frac{1}{r}$ if and only if

(i)

$M_{b_0} T \big (M_{b_1}(\cdot ), M_{b_2}(\cdot )\big )$ satisfies the weak boundedness property,

(ii)

$M_{b_0} T(b_1, b_2), \, M_{b_1} T^{*1}(b_0, b_2)$, and $M_{b_2} T^{*2}(b_1, b_0)$ are in $\mathit{BMO}$.

As in Theorem B, we used the notation such as $M_{b_0} T(b_1, b_2) \in \mathit{BMO}$ rather than $T(b_1, b_2)\in \mathit{BMO}$ to signify the fact that each of the three statements in condition (ii) of Theorem E involves $b_0, b_1$ and $b_2$. See Remark 2.8.

3.2. Formulation of the bilinear $T(b)$ theorem à la Stein

As in the linear setting, we consider the formulation after Stein (Theorem C) involving quantitative estimates. In the following, we only state and prove the formulation after Stein in the context of the bilinear $T(b)$ theorem. The corresponding version for the bilinear $T(1)$ theorem follows by setting $b_0 = b_1 = b_2 = 1$; this result already appears in 8.

Theorem 2 (Bilinear $T(b)$ theorem à la Stein).

Let $b_0, b_1$, and $b_2$ be para-accretive functions. Suppose that $T$ is a bilinear singular integral operator of Calderón-Zygmund type associated to $b_0, b_1$, and $b_2$. Then, $T$ can be extended to a bounded operator on $L^p \times L^q \to L^r$ for all $p, q < \infty$ satisfying $\frac{1}{p} + \frac{1}{q} = \frac{1}{r}$ if and only if there exists $M \in \mathbb{N} \cup \{0\}$ such that we have

$$\begin{align} \|T(b_1\phi ^{x_1, R}, b_2\phi ^{x_2, R})\|_{L^2}\lesssim R^\frac{d}{2}, {\tag{3.5}}\\ \|T^{*1}(b_0\phi ^{x_0, R}, b_2\phi ^{x_2, R})\|_{L^2}\lesssim R^\frac{d}{2}, {\tag{3.6}}\\ \|T^{*2}(b_1\phi ^{x_1, R}, b_0\phi ^{x_0, R})\|_{L^2}\lesssim R^\frac{d}{2} {\tag{3.7}} \end{align}$$

for any normalized bump function $\phi$ of order $M$, $x_0, x_1, x_2 \in \mathbb{R}^d$, and $R>0$.

We prove this result in Section 5.

4. Proof of Theorem 1

Suppose that $T$ is bounded on $L^2$. Let $\phi$ be a normalized bump function. Then, given any $x_0 \in \mathbb{R}^d$ and $R>0$, we have

$$\begin{align*} \|T(b_1\phi ^{x_0, R})\|_{L^2} \lesssim \|b_1\|_{L^\infty }\|\phi ^{x_0, R}\|_{L^2} \lesssim R^\frac{d}{2}. \end{align*}$$

This proves 2.9. Condition 2.10 follows from a similar computation.

Next, we assume that conditions 2.9 and 2.10 hold. It suffices to show that conditions 2.9 and 2.10 imply conditions (i) and (ii) in Theorem B.

We first prove condition (i) in Theorem B. Let $\phi _1$ and $\phi _2$ be normalized bump functions of order 0. Then, it follows from 2.9 and 2.3 that we have

$$\begin{align*} \big | \big \langle M_{b_0}T M_{b_1}( \phi _1^{x_1, R}), \phi _2^{x_2, R}\big \rangle \big | \lesssim \|b_0\|_{L^\infty } \|T (b_1 \phi _1^{x_1, R})\|_{L^2}\|\phi _2^{x_2, R}\|_{L^2} \lesssim R^d \end{align*}$$

for all $x_1, x_2 \in \mathbb{R}^d$ and $R>0$. This proves the weak boundedness property of $M_{b_0}TM_{b_1}$.

Next, we prove condition (ii) in Theorem B. In the following, we only show $M_{b_0}T(b_1) \in \mathit{BMO}$, assuming 2.9. The proof of $M_{b_1} T^*(b_0) \in \mathit{BMO}$ follows from 2.10 in an analogous manner.

We first recall from 5 how to extend the definition of $T$ to $b_1 C^\eta _b$, where $C^\eta _b := C^\eta \cap L^\infty$. Denote by $\{b_0C^\eta _0\}_0$ the subspace of mean-zero functions in $b_0C^\eta _0$. Given $f\in b_1 C^\eta _b$ and $g \in \{b_0C^\eta _0\}_0$, let $\psi \in C^\eta _0$ with $0 \leq \psi \leq 1$ and $\psi \equiv 1$ in a neighborhood of $\operatorname *{supp}g$. Then, we define the action of $T(f)$ on $g$ by

$$\begin{align} \langle T(f), g \rangle :& = \langle T(f \psi ), g \rangle + \langle T(f(1-\psi )), g \rangle \\ & = \langle T(f \psi ), g \rangle + \int _{\mathbb{R}^{2d}} \big [K(x, y) - K(x_0, y)\big ] f(y)(1-\psi (y)) g(x)dx dy. {\tag{4.1}} \end{align}$$

Note that this definition is independent of the choice of $\psi$. Here, the last equality in 4.1 holds for any $x_0 \in \operatorname *{supp}g$.

Let $\phi \in C^\infty _0$ with $0 \leq \phi \leq 1$ such that $\phi (x) = 1$ for $|x| \leq \frac{1}{2}$ and $\operatorname *{supp}\phi \subset B_0(1)$. Let $\phi _R(x) = \phi (R^{-1} x)$. Then, $T(b_1\phi _R)$ converges to $T(b_1)$ in the weak-$\ast$ topology of $(\{b_0C^\eta _0\}_0)'$. Namely, for all $g \in \{b_0C^\eta _0\}_0$, we have

$$\begin{align} \langle T(b_1), g \rangle = \lim _{R \to \infty } \langle T(b_1\phi _R), g \rangle . {\tag{4.2}} \end{align}$$

Indeed, letting $\psi \in C^\infty _0$ such that $\psi \equiv 1$ on $\operatorname *{supp}g$ as before, we have

$$\begin{align} \langle T(b_1\phi _R), g \rangle = \langle T(b_1\psi \phi _R), g \rangle + \langle T(b_1(1-\psi ) \phi _R), g \rangle . {\tag{4.3}} \end{align}$$

First, note that

$$\begin{align} \langle T(b_1\psi \phi _R), g \rangle = \langle T(b_1\psi ), g \rangle {\tag{4.4}} \end{align}$$

for all sufficiently large $R$. In view of 2.2, it follows from Lebesgue dominated convergence theorem that

$$\begin{align} \lim _{R \to \infty } \langle T& (b_1 (1-\psi ) \phi _R), g \rangle = \lim _{R \to \infty }\int _{\mathbb{R}^{2d}} K(x, y) b_1(y) (1-\psi (y)) \phi _R(y) g(x) dy dx \\ & = \lim _{R \to \infty }\int _{\mathbb{R}^{2d}} \big [K(x, y)- K(x_0, y)\big ] b_1(y) (1-\psi (y)) \phi _R(y) g(x) dy dx \\ & = \int _{\mathbb{R}^{2d}} \big [K(x, y)- K(x_0, y)\big ] b_1(y) (1-\psi (y)) g(x) dy dx \\ & = \langle T(b_1 (1-\psi )), g \rangle , {\tag{4.5}} \end{align}$$

where $x_0 \in \operatorname *{supp}g$. Then, 4.2 follows from 4.1, 4.3, 4.4, and 4.5.

Suppose now that we have

$$\begin{align} \|T(b_1 \phi _R)\|_{\mathit{BMO}} \lesssim 1, {\tag{4.6}} \end{align}$$

uniformly in $R>0$. Then, by Banach-Alaoglu theorem with $\mathit{BMO}= (H^1)'$, there exists a sequence $\{R_j\}_{j = 1}^\infty$ such that $T(b_1 \phi _{R_j})$ converges in the weak-$\ast$ topology to some function $\beta$ in $\mathit{BMO}$. Namely,

$$\begin{align} \lim _{j \to \infty } \langle T(b_1\phi _{R_j}), g \rangle = \langle \beta , g \rangle {\tag{4.7}} \end{align}$$

for all $g \in H^1$. In particular, 4.7 holds for all $g \in \{ b_0 C_0^\eta \}_0$. Then, from 4.2 and 4.7 with the uniqueness of a limit, we can identify $T(b_1)$ (or rather $M_{b_0} T(b_1)$) with $\beta \in \mathit{BMO}$. See Remark 2.8.

Therefore, it remains to prove 4.6. Let $M \in \mathbb{N}\cup \{0\}$ be as in Theorem 1. Then, by imposing that $\|\partial ^\alpha \phi \|_{L^\infty } \leq 1$ 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 $Q$ of side length $\ell > 0$ with center $x_0 \in \mathbb{R}^d$. Set $\phi _Q: = \phi ^{x_0, r}$, where $r := 6 \operatorname {diam}(Q) = 6 \sqrt d \, \ell$. By writing $T(b_1 \phi _R)$ as

$$\begin{align} T(b_1\phi _R) = T(b_1 \phi _Q\phi _R) + T\big (b_1 (1-\phi _Q)\phi _R\big ), {\tag{4.8}} \end{align}$$

we consider the first and second terms separately.

On the one hand, when $R \leq r$, write $\phi _Q\phi _R$ as

$$\begin{align} \phi _Q(x)\phi _R(x) = \phi (\tfrac{R}{r}\tfrac{x}{R}-\tfrac{x_0}{r}\big )\phi \big (\tfrac{x}{R}\big ) = [ \psi _1 \phi ]^{0, R}(x) {\tag{4.9}} \end{align}$$

with $\psi _1 (x) := \phi (\tfrac{R}{r}x-\tfrac{x_0}{r}\big )$. Note that $\psi _1 \phi$ is a normalized bump function. Then, by the Cauchy-Schwarz inequality and 2.9, we have

$$\begin{align} \int _{Q} \big |T(b_1 \phi _Q \phi _R) \big | dx \leq |Q|^\frac{1}{2} \big \|T\big (b_1 [\psi _1 \phi ]^{0, R}\big )\big \|_{L^2} \lesssim R^\frac{d}{2} |Q|^\frac{1}{2} \lesssim |Q|. {\tag{4.10}} \end{align}$$

On the other hand, when $R > r$, write $\phi _Q\phi _R$ as

$$\begin{align} \phi _Q(x) \phi _R(x) = \phi (\tfrac{x-x_0}{r}\big ) \phi \big (\tfrac{r}{R} \tfrac{x-x_0}{r} + \tfrac{x_0}{R}\big ) = [\phi \psi _2]^{x_0, r}(x) {\tag{4.11}} \end{align}$$

with $\psi _2 (x) := \phi \big (\tfrac{r}{R} x + \tfrac{x_0}{R}\big )$. Then, noting that $\phi \psi _2$ is a normalized bump function, it follows from the Cauchy-Schwarz inequality and 2.9 that

$$\begin{align} \int _{Q} \big |T(b_1 \phi _Q \phi _R) \big | dx \leq |Q|^\frac{1}{2} \big \|T\big (b_1 [\phi \psi _2]^{x_0, r}\big )\big \|_{L^2} \lesssim R^\frac{d}{2} |Q|^\frac{1}{2} \lesssim |Q|. {\tag{4.12}} \end{align}$$

Next, we estimate the second term in 4.8. From the support condition

$$\begin{align} \operatorname *{supp}(1- \phi _Q) \subset \mathbb{R}^d \setminus B_{x_0}( 3 \operatorname {diam}(Q)) \subset \mathbb{R}^d \setminus Q, {\tag{4.13}} \end{align}$$

we have

$$\begin{equation*} T\big (b_1 (1-\phi _Q)\phi _R\big )(x) = \int _{\mathbb{R}^d} K(x, y) b_1(y) \big (1- \phi _Q(y) \big ) \phi _R(y) dy, \end{equation*}$$

for all $x\in Q$. Define $c_{Q, R}$ by

$$\begin{equation*} c_{Q, R} : = \int _{\mathbb{R}^d} K(x_0, y) b_1(y) \big (1- \phi _Q(y)\big )\phi _R(y) dy, \end{equation*}$$

where $x_0$ is the center of the cube $Q$. Then, it follows from 2.2 with 4.13 that, for $x \in Q$, we have

$$\begin{align} \big | T\big (b_1 (1-\phi _Q) \phi _R\big )(x) - c_{Q, R}\big | & \leq \int _{|x - x_0| \leq \operatorname {diam}(Q) \leq \frac{1}{2} |x - y|} |K(x, y) - K(x_0, y)| dy \\ & \lesssim 1 {\tag{4.14}} \end{align}$$

uniformly in $R> 0$.

Hence, putting 4.8, 4.10, 4.12, and 4.14 together, we conclude that there exists $A> 0$ such that for each cube $Q$ and $R> 0$, there exists a constant $c_{Q, R}$ such that

$$\begin{align} \frac{1}{|Q|}\int _Q | T(b_1\phi _R)(x) - c_{Q, R}\big | dx \leq A. \tag{4.15} \end{align}$$

Therefore, it follows from Proposition 7.1.2 in 7 that

$$\begin{equation*} \sup _{R>0} \|T(b_1\phi _R) \|_{\mathit{BMO}} \leq 2A. \end{equation*}$$

This proves 4.7 and thus completes the proof of Theorem 1.

5. Proof of Theorem 2

Suppose that $T$ is bounded on $L^4\times L^4 \to L^2$. Then, given a normalized bump function $\phi$, we have

$$\begin{align*} \|T(b_1\phi ^{x_1, R}, b_2\phi ^{x_2, R} )\|_{L^2} \lesssim \|b_1\|_{L^\infty }\|\phi ^{x_1, R}\|_{L^4} \|b_2\|_{L^\infty }\|\phi ^{x_2, R}\|_{L^4} \lesssim R^\frac{d}{2} \end{align*}$$

for any $x_1, x_2 \in \mathbb{R}^d$ and $R>0$. This proves 3.5. A similar computation yields 3.6 and 3.7.

Next, we assume that conditions 3.5, 3.6, and 3.7 hold. It suffices to show that conditions 3.5, 3.6, and 3.7 imply conditions (i) and (ii) in Theorem E.

We first prove condition (i) in Theorem E. Let $\phi _j$, $j = 0, 1, 2$, be normalized bump functions of order $M$. Then, it follows from the Cauchy-Schwarz inequality, 3.5, and 2.3 that

$$\begin{equation*} \big | \langle M_{b_0}T (b_1 \phi _1^{x_1, R}, b_2\phi _2^{x_2, R}), \phi _0^{x_0, R} \rangle \big | \lesssim \|b_0\|_{L^\infty } \| T (b_1 \phi _1^{x_1, R}, b_2\phi _2^{x_2, R})\|_{L^2} \| \phi _0^{x_0, R} \|_{L^2} \lesssim R^d \end{equation*}$$

for all $x_0, x_1, x_2 \in \mathbb{R}^d$ and $R>0$. This proves condition (i) in Theorem E.

Next, we prove condition (ii) in Theorem E. As in the proof of Theorem 1, we only show $M_{b_0}T(b_1, b_2) \in \mathit{BMO}$, assuming 3.5. The proof of the other two conditions follows in a similar manner in view of the symmetric condition in Definition 3.1.

Since $T$ is a priori defined only on $b_1 C^\eta _0 \times b_2 C_0^\eta$, we first extend $T$ to $b_1 C^\eta _b \times b_2 C_b^\eta$. Fix $f_j\in b_j C^\eta _b$ , $j = 1, 2$. Given $g \in \{b_0C^\eta _0\}_0$, let $\psi \in C^\eta _0$ with $0 \leq \psi \leq 1$ and $\psi \equiv 1$ in a neighborhood of $\operatorname *{supp}g$. Then, we define the action of $T(f_1, f_2)$ on $g$ by

$$\begin{align} \langle T(f_1, f_2), g \rangle & := \langle T(f_1 \psi , f_2 \psi ), g \rangle + \langle T(f_1 (1-\psi ), f_2 \psi ), g \rangle \\ & \hphantom {XX} + \langle T(f_1 \psi , f_2 (1-\psi ) ), g \rangle + \langle T(f_1 (1-\psi ), f_2 (1-\psi ) ), g \rangle . {\tag{5.1}} \end{align}$$

Note that the last three terms can be written as triple integrals of the form 3.4. From this, we see that this definition is independent of the choice of $\psi$.

Let $\phi \in C^\infty _0$ with $0 \leq \phi \leq 1$ such that $\phi (x) = 1$ for $|x| \leq \frac{1}{2}$ and $\operatorname *{supp}\phi \subset B_0(1)$. Let $\phi _R(x) = \phi (R^{-1} x)$. Then, $T(b_1\phi _R, b_2\phi _R)$ converges to $T(b_1, b_2)$ in the weak-$\ast$ topology of $(\{b_0C^\eta _0\}_0)'$. Namely, we have

$$\begin{align} \langle T(b_1, b_2), g \rangle = \lim _{R \to \infty } \langle T(b_1\phi _R, b_2\phi _R), g \rangle {\tag{5.2}} \end{align}$$

for all $g \in \{b_0C^\eta _0\}_0$. See 12 for the proof of 5.2.

Suppose that we have

$$\begin{align} \|T(b_1 \phi _R, b_2 \phi _R)\|_{\mathit{BMO}} \lesssim 1, {\tag{5.3}} \end{align}$$

uniformly in $R>0$. Then, as in the proof of Theorem 1, it follows from the Banach-Alaoglu theorem that there exists a sequence $\{R_j\}_{j = 1}^\infty$ and $\beta \in \mathit{BMO}$ such that

$$\begin{align} \lim _{j \to \infty } \langle T(b_1\phi _{R_j}, b_2 \phi _{R_j}), g \rangle = \langle \beta , g \rangle {\tag{5.4}} \end{align}$$

for all $g \in H^1$, in particular for all $g \in \{ b_0 C_0^\eta \}_0$. Hence, from 5.2 and 5.4, we conclude that $M_{b_0} T(b_1, b_2) \in \mathit{BMO}$.

Therefore, it remains to prove 5.3. By imposing that $\|\partial ^\alpha \phi \|_{L^\infty } \leq 1$ for all multi-indices $\alpha$ with $|\alpha | \leq M$, the function $\phi$ defined above is a normalized bump function of order $M$. As in the proof of Theorem 1, let $Q$ be the cube of side length $\ell > 0$ with center $x_0 \in \mathbb{R}^d$. Set $\phi _Q = \phi ^{x_0, r}$, where $r = 6 \operatorname {diam}(Q)$. Then, write $T(b_1 \phi _R, b_2 \phi _R)$ as

$$\begin{align} T(b_1\phi _R, b_2 \phi _R) & = T(b_1 \phi _Q\phi _R, b_2 \phi _Q\phi _R) + T\big (b_1 (1-\phi _Q)\phi _R, b_2 \phi _Q\phi _R\big )\\ & \hphantom {X} + T\big (b_1 \phi _Q\phi _R, b_2 (1-\phi _Q)\phi _R\big ) + T\big (b_1 (1-\phi _Q)\phi _R, b_2 (1-\phi _Q)\phi _R\big ) \\ & := \operatorname {I}+\operatorname {II}+ \operatorname {III}+ \operatorname {IV}. {\tag{5.5}} \end{align}$$

It follows from the Cauchy-Schwarz inequality and 3.5 with 4.9 and 4.11 that

$$\begin{align} \int _{Q} |\operatorname {I}| dx \leq \begin{cases} \vphantom {\Big |} |Q|^\frac{1}{2} \big \|T\big (b_1 [\psi _1\phi ]^{0, R}, b_2 [\psi _1\phi ]^{0, R}\big )\big \|_{L^2} \lesssim |Q|, & \text{when }R\leq r, \\ \vphantom {\Big |} |Q|^\frac{1}{2} \big \|T\big (b_1 [\phi \psi _2]^{x_0, r}, b_2 [\phi \psi _2]^{x_0, r}\big )\big \|_{L^2} \lesssim |Q|, & \text{when }R> r.\\ \end{cases} {\tag{5.6}} \end{align}$$

Next, we consider the terms $\operatorname {II}$, $\operatorname {III}$, and $\operatorname {IV}$. Let $\phi _Q^c := 1 - \phi _Q$. Then, from the support condition 4.13, we have

$$\begin{equation*} \operatorname {II}(x) = \int _{\mathbb{R}^{2d}} K(x, y, z) b_1(y) \phi ^c_Q(y)\phi _R(y) b_2(z) \phi _Q(z) \phi _R(z) dydz \end{equation*}$$

for $x\in Q$. Define $c^{(2)}_{Q, R}$ by

$$\begin{equation*} c^{(2)}_{Q, R} : = \int _{\mathbb{R}^{2d}} K(x_0, y, z) b_1(y) \phi ^c_Q(y)\phi _R(y) b_2(z) \phi _Q(z) \phi _R(z) dydz, \end{equation*}$$

where $x_0$ is the center of the cube $Q$. Then, it follows from 3.2 with 4.13 that, for $x \in Q$, we have

$$\begin{align} | \operatorname {II}(x) - c^{(2)}_{Q, R}\big | & \leq \int _{\operatorname *{supp}\phi _Q}\int _{|x - x_0| \leq \operatorname {diam}(Q) \leq \frac{1}{2} |x - y|} |K(x, y, z) - K(x_0, y, z)| dy dz \\ & \lesssim 1 {\tag{5.7}} \end{align}$$

uniformly in $R> 0$. By symmetry, the same estimate holds for $\operatorname {III}$. As for $\operatorname {IV}$, by letting

$$\begin{equation*} c^{(4)}_{Q, R} : = \int _{\mathbb{R}^{2d}} K(x_0, y, z) b_1(y) \phi ^c_Q(y)\phi _R(y) b_2(z) \phi _Q^c(z) \phi _R(z) dydz, \end{equation*}$$

we have, for $x \in Q$,

$$\begin{align} | \operatorname {IV}(x) - c^{(4)}_{Q, R}\big | & \leq \int _{|x - x_0| \leq \operatorname {diam}(Q) \leq \frac{1}{2} \min ( |x - y|, |x-z|)} |K(x, y, z) - K(x_0, y, z)| dy dz \\ & \lesssim 1 {\tag{5.8}} \end{align}$$

uniformly in $R> 0$.

Hence, putting 5.5, 5.6, 5.7, and 5.8 together, we conclude that there exists $A> 0$ such that for each cube $Q$ and $R> 0$, there exists a constant $\widetilde{c}_{Q, R}$ such that

$$\begin{align} \frac{1}{|Q|}\int _Q | T(b_1\phi _R, b_2\phi _R)(x) - \widetilde{c}_{Q, R}\big | dx \leq A, \tag{5.9} \end{align}$$

thus yielding 5.3. This completes the proof of Theorem 2.