On a generalization of the Hörmander condition

Abstract

We consider a natural generalization of the classical Hörmander condition in the Calderón–Zygmund theory. Recently the author [J. Fourier Anal. Appl. 27 (2021)] proved the $L^p$ boundedness of singular integral operators under the $L^1$ mean Hörmander condition, which was originally introduced by Grafakos and Stockdale [Bull. Hellenic Math. Soc. 63 (2019), pp. 54–63]. In this paper, we show that the $L^1$ mean condition actually coincides with the classical one. On the other hand, we introduce a new variant of the Hörmander condition, which is strictly weaker than the classical one but still enough for the $L^p$ boundedness. Moreover, it still works in the non-doubling setting with a little modification.

1. Introduction

Our aim is to generalize Theorem A, which is one of the most famous results in the classical Calderón–Zygmund theory of singular integral operators.

Theorem A (1, Lemma 2, Theorem 1, 2, (1.24), 7, Theorem 2.2).

Let $T$ be a singular integral operator associated with a kernel $K$. Suppose that $T$ is bounded on $L^{p_0}(\mathbb{R}^d)$ for some $1 < p_0 < \infty$ and its kernel $K$ satisfies the Hörmander condition:

$$\begin{equation} [K]_{H_\infty } \coloneq \sup _{B \subset \mathbb{R}^d} \sup _{y \in B} \int _{x \in \mathbb{R}^d \setminus 2B} \lvert K(x, y) - K(x, c(B)) \rvert \, dx < \infty , {\tag{1.1}} \end{equation}$$

where the supremum $\sup _{B \subset \mathbb{R}^d}$ is taken over all balls $B \subset \mathbb{R}^d$, $c(B)$ is the center of $B$, $2B$ denotes the ball with the same center as B and whose radius is twice as long. Then $T$ is bounded from $L^1(\mathbb{R}^d)$ to $L^{1, \infty }(\mathbb{R}^d)$ and from $H^1(\mathbb{R}^d)$ to $L^1(\mathbb{R}^d)$, thus on $L^p(\mathbb{R}^d)$ for any $1 < p < p_0$.

In 2019, Grafakos and Stockdale 6 introduced an $L^q$ mean Hörmander condition

$$\begin{equation} [K]_{H_q} \coloneq \sup _{B \subset \mathbb{R}^d} \left( \frac{1}{\lvert B \rvert } \int _{ y \in B } \left( \int _{x \in \mathbb{R}^d \setminus 2B} \lvert K(x, y) - K(x, c(B)) \rvert \, dx \right)^q dy \right)^{1/q} < \infty {\tag{1.2}} \end{equation}$$

in order to establish a “limited-range” version of Theorem A. The author 11 improved Theorem A by assuming the $L^1$ mean Hörmander condition using an idea inspired by Fefferman 4, THEOREM 2’.

Theorem B (11, Theorem 1, Theorem 3).

Let $T$ be a singular integral operator associated with a kernel $K$. Suppose that $T$ is bounded on $L^{p_0}(\mathbb{R}^d)$ for some $1 < p_0 < \infty$ and its kernel $K$ satisfies the $L^1$ mean Hörmander condition:

$$\begin{equation} [K]_{H_1} = \sup _{B \subset \mathbb{R}^d} \frac{1}{\lvert B \rvert } \int _{ y \in B } \int _{x \in \mathbb{R}^d \setminus 2B} \lvert K(x, y) - K(x, c(B)) \rvert \, dx \, dy < \infty . {\tag{1.3}} \end{equation}$$

Then $T$ is bounded from $L^1(\mathbb{R}^d)$ to $L^{1, \infty }(\mathbb{R}^d)$ and from $H^1(\mathbb{R}^d)$ to $L^1(\mathbb{R}^d)$, thus on $L^p(\mathbb{R}^d)$ for any $1 < p < p_0$.

In this paper, we show that the $L^1$ mean Hörmander condition 1.3 is the same as the classical one 1.1 and therefore Theorems A and B are equivalent.

Theorem 1.

The inequality

$$\begin{equation} [K]_{H_1} \leq [K]_{H_\infty } \lesssim [K]_{H_1} {\tag{1.4}} \end{equation}$$

holds for any $K \in L^1_{\mathrm{loc}}( (\mathbb{R}^d \times \mathbb{R}^d) \setminus \Delta )$, where $\Delta$ denotes the diagonal set $\{ \, (x, x) \, : \, x \in \mathbb{R}^d \, \}$.

Now we introduce a new variant of the Hörmander condition:

$$\begin{equation} [K]_{H_*} \coloneq \sup _{B \subset \mathbb{R}^d} \frac{1}{ \lvert B \rvert } \int _{y \in B} \int _{x \in \mathbb{R}^d \setminus 2B} \left\lvert \, K(x, y) - \frac{1}{ \lvert B \rvert } \int _{z \in B} K(x, z) \, dz \, \right\rvert \, dx \, dy < \infty , {\tag{1.5}} \end{equation}$$

which is a natural generalization of the $L^1$ mean Hörmander condition in terms of $\mathrm{BMO}$. Recall that a function $f \in L^1_{\mathrm{loc}}(\mathbb{R}^d)$ is in $\mathrm{BMO}(\mathbb{R}^d)$ if

$$\begin{equation} \lVert f \rVert _{\mathrm{BMO}} \coloneq \sup _{B \subset \mathbb{R}^d} \frac{1}{ \lvert B \rvert } \int _{y \in B} \left\lvert \, f(y) - \frac{1}{ \lvert B \rvert } \int _{z \in B} f(z) \, dz \, \right\rvert \, dy < \infty . {\tag{1.6}} \end{equation}$$

We call 1.5 a $\mathrm{BMO}$ Hörmander condition. Note that we can easily see $[K]_{H_*} \leq 2[K]_{H_\infty }$, which is analogous to $\lVert f \rVert _\mathrm{BMO}\leq 2 \lVert f \rVert _\infty$. We will show the following:

Theorem 2.

Let $T$ be a singular integral operator associated with a kernel $K$. Suppose that $T$ is bounded on $L^{p_0}(\mathbb{R}^d)$ for some $1 < p_0 < \infty$ and its kernel $K$ satisfies $[K]_{H_*} < \infty$. Then $T$ is bounded from $H^1(\mathbb{R}^d)$ to $L^1(\mathbb{R}^d)$, thus on $L^p(\mathbb{R}^d)$ for any $1 < p < p_0$. On the other hand, $T$ is not bounded from $L^1(\mathbb{R}^d)$ to $L^{1, \infty }(\mathbb{R}^d)$ in general. In particular, the $\mathrm{BMO}$ Hörmander condition is strictly weaker than the classical one.

Moreover, Theorem 2 still holds in the non-doubling setting with an appropriate modification. Let $\mu$ be a Radon measure on $\mathbb{R}^d$ which satisfies the polynomial growth condition: there exists a constant $C_\mu > 0$ and $0 < n \leq d$ such that

$$\begin{equation} \mu ( B(c, r) ) \leq C_\mu r^n {\tag{1.7}} \end{equation}$$

for any balls $B(c, r)$. In this case, the following generalization of Theorem A is known.

Theorem C (9, Theorem 6.1, 12, Theorem 4.2).

Let $T$ be a singular integral operator associated with a kernel $K$. Suppose that $T$ is bounded on $L^{p_0}(\mu )$ for some $1 < p_0 < \infty$ and its kernel $K$ satisfies the Hörmander condition with respect to $\mu$:

$$\begin{equation} [K]_{H_\infty } \coloneq \sup _{B \subset \mathbb{R}^d} \sup _{y \in B} \int _{x \in \mathbb{R}^d \setminus 2B} \lvert K(x, y) - K(x, c(B)) \rvert \, d\mu (x) < \infty , {\tag{1.8}} \end{equation}$$

and $\lvert K(x, y) \rvert \lesssim \lvert x-y \rvert ^{-n}$. Then $T$ is bounded from $L^1(\mu )$ to $L^{1, \infty }(\mu )$ and from $H^1_{\mathrm{atb}}(\mu )$ to $L^1(\mu )$, thus on $L^p(\mu )$ for any $1 < p < p_0$, where $H^1_{\mathrm{atb}}(\mu )$ is the atomic block Hardy space introduced by Tolsa 12.

The $L^1(\mu ) \to L^{1, \infty }(\mu )$ boundedness was proved by Nazarov, Treil and Volberg 9, and the $H^1_{\mathrm{atb}}(\mu ) \to L^1(\mu )$ boundedness by Tolsa 12. Also Tolsa 13 gave another proof of the $L^1(\mu ) \to L^{1, \infty }(\mu )$ boundedness. We will give a natural generalization of Theorem C in the sense of Theorem 2 (see Theorem 3 in Section 4 for details). To establish the theorem, we modify the $\mathrm{BMO}$ Hörmander condition into an $\mathrm{RBMO}$ version, where $\mathrm{RBMO}$ is the Regularized Bounded Mean Oscillation space, which is also introduced by Tolsa 12.

This paper is organized as follows. In Section 2, we discuss the equivalence of the $L^1$ mean Hörmander condition and the classical one (Theorem 1). In Section 3, we prove the $H^1$-$L^1$ estimate and the failure of the $L^1$-$L^{1, \infty }$ estimate under the $\mathrm{BMO}$ Hörmander condition (Theorem 2). In Section 4, we show that Theorem 2 still works in the non-doubling setting with a little modification (Theorem 3).

2. The equivalence of $[K]_{H_1}$ and $[K]_{H_\infty }$

In this section, we prove the equivalence of $[K]_{H_1}$ and $[K]_{H_\infty }$. The argument here is based on the idea suggested by Professor Akihiko Miyachi. At first, we prepare an elementary fact.

Proposition 1.

The inequality

$$\begin{equation} \frac{1}{2} [K]_{H_\infty } \leq \sup _{B \subset \mathbb{R}^d} \sup _{ y \in B } \int _{x \in \mathbb{R}^d \setminus 3B} \lvert K(x, y) - K(x, c(B)) \rvert \, dx \leq [K]_{H_\infty } \tag{2.1} \end{equation}$$

holds for any $K \in L^1_{\mathrm{loc}}( (\mathbb{R}^d \times \mathbb{R}^d) \setminus \Delta )$.

Proof of Proposition 1.

We consider the first inequality. Fix a ball $B = B(c, 2r) \subset \mathbb{R}^d$, $y \in B$ and write $z \coloneq (y + c)/2$. Since $B(z, 3r) \subset 2B$ and $c, y \in B(z, r)$, we have

$$\begin{align*} \int _{x \in \mathbb{R}^d \setminus 2B} \lvert K(x, y) - K(x, c) \rvert \, dx &\leq \int _{x \in \mathbb{R}^d \setminus 2B} \lvert K(x, y) - K(x, z) \rvert \, dx + \int _{x \in \mathbb{R}^d \setminus 2B} \lvert K(x, z) - K(x, c) \rvert \, dx \\[3.0pt] &\leq \int _{x \in \mathbb{R}^d \setminus B(z, 3 r)} \lvert K(x, y) - K(x, z) \rvert \, dx + \int _{x \in \mathbb{R}^d \setminus B(z, 3 r)} \lvert K(x, c) - K(x, z) \rvert \, dx \\[3.0pt] &\leq 2 \sup _{B \subset \mathbb{R}^d} \sup _{ y \in B } \int _{x \in \mathbb{R}^d \setminus 3B} \lvert K(x, y) - K(x, c(B)) \rvert \, dx . \end{align*}$$

The second inequality is trivial.

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Proof of Theorem 1.

Fix $r > 0$ and $c, y \in \mathbb{R}^d$ with $\lvert y-c \rvert \leq r$. We write

$$\begin{gather*} I(c, y) \coloneq \int _{x \in \mathbb{R}^d \setminus B(c, 2r)} \lvert K(x, y) - K(x, c) \rvert \, dx , \\ J(c, y) \coloneq \int _{x \in \mathbb{R}^d \setminus B(c, 3r)} \lvert K(x, y) - K(x, c) \rvert \, dx , \\ A \coloneq B(c, r) \cap B(y, r) . \end{gather*}$$

Since $B(y, 2r) \cup B(c, 2r) \subset B(c, 3r)$, we obtain

$$\begin{align*} J(c, y) &= \int _{x \in \mathbb{R}^d \setminus B(c, 3r)} \lvert K(x, y) - K(x, c) \rvert \, dx \\ &\leq \int _{x \in \mathbb{R}^d \setminus B(c, 3r)} \lvert K(x, y) - K(x, z) \rvert \, dx + \int _{x \in \mathbb{R}^d \setminus B(c, 3r)} \lvert K(x, z) - K(x, c) \rvert \, dx \\ &\leq \int _{x \in \mathbb{R}^d \setminus B( y, 2 r )} \lvert K(x, z) - K(x, y) \rvert \, dx + \int _{x \in \mathbb{R}^d \setminus B( c, 2 r )} \lvert K(x, z) - K(x, c) \rvert \, dx \\ &= I(y, z) + I(c, z) \end{align*}$$

for any $z \in A$. Therefore, letting $J = J(c, y)$, we have

$$\begin{equation*} A \subset \{ \, z \in A \, : \, I(c, z) \geq J/2 \, \} \cup \{ \, z \in A \, : \, I(y, z) \geq J/2 \, \} , \end{equation*}$$

which implies at least one of the following:

$$\begin{gather} \lvert A \rvert / 2 \leq \lvert \{ \, z \in A \, : \, I(c, z) \geq J/2 \, \} \rvert , {\tag{2.2}}\\ \lvert A \rvert / 2 \leq \lvert \{ \, z \in A \, : \, I(y, z) \geq J/2 \, \} \rvert . {\tag{2.3}} \end{gather}$$

By the symmetry between $c$ and $y$, we assume that 2.2 holds without loss of generality. Now we have

$$\begin{equation*} \begin{split} \lvert A \rvert J / 4 &\leq \int _{ z \in A : I(c, z) \geq J/2 } J/2\, dz \leq \int _{z \in A} I(c, z)\, dz \leq \int _{z \in B(c, r)} I(c, z)\, dz\\ &\leq \lvert B(c, r) \rvert [K]_{H_1} . \end{split} \end{equation*}$$

Since

$$\begin{equation} \lvert A \rvert \geq \lvert B( (c + y) / 2 , r / 2 ) \rvert = 2^{-d} \lvert B(c, r) \rvert , {\tag{2.4}} \end{equation}$$

we get $J \leq 2^{d+2} [K]_{H_1}$. By Proposition 1, we conclude that the inequality

$$\begin{equation} [K]_{H_\infty } \leq 2^{d+3} [K]_{H_1} {\tag{2.5}} \end{equation}$$

holds.

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Remark.

Let $\mu$ be a Radon measure on $\mathbb{R}^d$ which satisfies the doubling property: there exists a constant $C_\mu > 0$ such that

$$\begin{equation} \mu ( B(c, 2r) ) \leq C_\mu \mu ( B(c, r) ) {\tag{2.6}} \end{equation}$$

for any balls $B(c, r)$, and consider $[K]_{H_\infty }$ and $[K]_{H_1}$ with respect to $\mu$:

$$\begin{gather*} [K]_{H_\infty } \coloneq \sup _{B \subset \mathbb{R}^d} \sup _{y \in B} \int _{x \in \mathbb{R}^d \setminus 2B} \lvert K(x, y) - K(x, c(B)) \rvert \, d\mu (x) , \\ [K]_{H_1} \coloneq \sup _{B \subset \mathbb{R}^d} \frac{1}{\mu (B)} \int _{ y \in B } \int _{x \in \mathbb{R}^d \setminus 2B} \lvert K(x, y) - K(x, c(B)) \rvert \, d\mu (x) \, d\mu (y) . \end{gather*}$$

We can show the inequality $[K]_{H_\infty } \lesssim [K]_{H_1}$ still holds by the same argument. To see this, note that $\mu$ satisfies

$$\begin{equation*} \mu (A) \geq \mu ( B( (c+y)/2, r/2 ) ) \gtrsim \mu ( B( (c+y)/2, 3r/2 ) ) \geq \mu (B(c, r)) , \end{equation*}$$

which can be a replacement of 2.4.

3. A $\mathrm{BMO}$ Hörmander condition

In this section, we study the $\mathrm{BMO}$ Hörmander condition,

$$\begin{equation} [K]_{H_*} \coloneq \sup _{B \subset \mathbb{R}^d} \frac{1}{ \lvert B \rvert } \int _{y \in B} \int _{x \in \mathbb{R}^d \setminus 2B} \left\lvert \, K(x, y) - \frac{1}{ \lvert B \rvert } \int _{z \in B} K(x, z) \, dz \, \right\rvert \, dx \, dy < \infty , \tag{\xhref[disp-formula]{#texmlid3}{1.5}} \end{equation}$$

which is a natural generalization of the $L^1$ mean Hörmander condition 1.3 in the sense of $\mathrm{BMO}$. At first we observe that $[K]_{H_*}$ satisfies an elemental property, which is analogous to that of the $\mathrm{BMO}$ norm (see 3, Proposition 6.5, 5, Proposition 3.1.2 for example).

Proposition 2.

The following are equivalent:

(i)

$[K]_{H_*} < \infty$.

(ii)

There exists a collection of functions $\{ m_B \}_{B}$ such that$$\begin{equation} \sup _{ B \subset \mathbb{R}^d } \frac{1}{ \lvert B \rvert } \int _{y \in B} \int _{ x \in \mathbb{R}^d \setminus 2B } \lvert K(x, y) - m_B (x) \rvert \, dx < \infty . \tag{3.1} \end{equation}$$

Proof of Proposition 2.

(i)$\Rightarrow$(ii) is obvious: consider $m_B(x) = \frac{1}{\lvert B \rvert } \int _{z \in B} K(x, z) \, dz$. (ii)$\Rightarrow$(i) is also easy to check:

$$\begin{align*} &\frac{1}{ \lvert B \rvert } \int _{y \in B} \int _{x \in \mathbb{R}^d \setminus 2B} \left\lvert \, K(x, y) - \frac{1}{ \lvert B \rvert } \int _{z \in B} K(x, z) \, dz \, \right\rvert \, dx \, dy \\ &\quad \leq \frac{1}{ \lvert B \rvert } \int _{y \in B} \int _{x \in \mathbb{R}^d \setminus 2B} \lvert K(x, y) - m_B(x) \rvert \, dx \, dy \\ & \qquad + \frac{1}{ \lvert B \rvert } \int _{y \in B} \int _{x \in \mathbb{R}^d \setminus 2B} \left\lvert \, m_B(x) - \frac{1}{ \lvert B \rvert } \int _{z \in B} K(x, z) \, dz \, \right\rvert \, dx \, dy \\ &\leq \frac{1}{ \lvert B \rvert } \int _{y \in B} \int _{x \in \mathbb{R}^d \setminus 2B} \lvert K(x, y) - m_B(x) \rvert \, dx \, dy \\ & \qquad + \frac{1}{ \lvert B \rvert } \int _{y \in B} \left( \frac{1}{ \lvert B \rvert } \int _{z \in B} \int _{x \in \mathbb{R}^d \setminus 2B} \lvert K(x, z) - m_B(x) \rvert \, dx \, dz \right) dy \\ &= \frac{2}{ \lvert B \rvert } \int _{y \in B} \int _{x \in \mathbb{R}^d \setminus 2B} \lvert K(x, y) - m_B(x) \rvert \, dx \, dy . \end{align*}$$ ■

We write

$$\begin{equation} [K]_{H_{**}} \coloneq \inf _{\{m_B\}_{B}} \sup _{ B \subset \mathbb{R}^d } \frac{1}{ \lvert B \rvert } \int _{y \in B} \int _{ x \in \mathbb{R}^d \setminus 2B } \lvert K(x, y) - m_B (x) \rvert \, dx \, dy , {\tag{3.2}} \end{equation}$$

where the infimum is taken over all collections $\{m_B\}_{B}$. By the proof of Proposition 2, we have

$$\begin{equation} [K]_{H_{**}} \leq [K]_{H_{*}} \leq 2 [K]_{H_{**}} . \tag{3.3} \end{equation}$$

Also note that $[K]_{H_{**}} \leq [K]_{H_1}$: consider $m_B(x) = K( x, c(B) )$. Therefore, we have

$$\begin{equation} [K]_{H_{*}} \leq 2 [K]_{H_1} , {\tag{3.4}} \end{equation}$$

it means that the classical Hörmander condition implies the $\mathrm{BMO}$ version. We will show later that the converse is not true.

Now we are going to discuss the boundedness of singular integral operators under the $\mathrm{BMO}$ Hörmander condition.

Proposition 3.

Let $T$ be a singular integral operator associated with a kernel $K$. Suppose that $T$ is bounded on $L^{p_0}(\mathbb{R}^d)$ for some $1 < p_0 < \infty$ and its kernel $K$ satisfies $[K]_{H_{**}} < \infty$. Then $T$ is bounded from $H^1(\mathbb{R}^d)$ to $L^1(\mathbb{R}^d)$ with a constant proportional to $\lVert T \rVert _{L^{p_0} \to L^{p_0}} + [K]_{H_{**}}$.

Proof of Proposition 3.

Let $a \in H^1(\mathbb{R}^d)$ be an atom, that is,

$$\begin{equation*} \operatorname {supp}a \subset B , \quad \lVert a \rVert _\infty \leq \lvert B \rvert ^{-1} , \quad \int _B a = 0 \end{equation*}$$

for some ball $B \subset \mathbb{R}^d$. Since $T$ is bounded on $L^{p_0}(\mathbb{R}^d)$, it is enough to show that

$$\begin{equation*} \lVert Ta \rVert _1 \lesssim \lVert T \rVert _{L^{p_0} \to L^{p_0}} + [K]_{H_{**}}. \end{equation*}$$

We decompose $\lVert Ta \rVert _1$ as

$$\begin{equation*} \lVert Ta \rVert _1 = \lVert Ta \rVert _{L^1(2B)} + \lVert Ta \rVert _{L^1(\mathbb{R}^d \setminus 2B)} \end{equation*}$$

and prove

$$\begin{gather} \lVert Ta \rVert _{L^1(2B)} \lesssim \lVert T \rVert _{L^{p_0} \to L^{p_0}} , {\tag{3.5}}\\ \lVert Ta \rVert _{L^1(\mathbb{R}^d \setminus 2B)} \lesssim [K]_{H_{**}} . {\tag{3.6}} \end{gather}$$

3.5. By the Hölder inequality and the $L^{p_0}$ boundedness of $T$, we have

$$\begin{align*} \lVert Ta \rVert _{L^1(2B)} &\leq \lvert 2B \rvert ^{1/{p_0'}} \lVert Ta \rVert _{p_0} \\ &\leq 2^{d/p_0'} \lVert T \rVert _{L^{p_0} \to L^{p_0}} \lvert B \rvert ^{1/p_0'} \lVert a \rVert _{p_0} \\ &\leq 2^{d/p_0'} \lVert T \rVert _{L^{p_0} \to L^{p_0}} \lvert B \rvert \lVert a \rVert _\infty \\ &\leq 2^{d/p_0'} \lVert T \rVert _{L^{p_0} \to L^{p_0}} . \end{align*}$$

3.6. Since

$$\begin{align*} \lVert Ta \rVert _{L^1(\mathbb{R}^d \setminus 2B)} &= \int _{x \in \mathbb{R}^d \setminus 2B} \left\lvert \, \int _{y \in B} K(x, y) a(y) \, dy \, \right\rvert \, dx \\ &= \int _{x \in \mathbb{R}^d \setminus 2B} \left\lvert \, \int _{y \in B} K(x, y) a(y) \, dy - m_B(x) \int _{y \in B} a(y) \, dy \, \right\rvert \, dx \\ &\leq \lVert a \rVert _\infty \int _{y \in B} \int _{x \in \mathbb{R}^d \setminus 2B} \lvert K(x, y) - m_B(x) \rvert \, dx \, dy \\ &\leq \frac{1}{\lvert B \rvert } \int _{y \in B} \int _{x \in \mathbb{R}^d \setminus 2B} \lvert K(x, y) - m_B(x) \rvert \, dx \, dy \end{align*}$$

for any collections $\{m_B\}_B$, we have $\lVert Ta \rVert _{L^1(\mathbb{R}^d \setminus 2B)} \leq [K]_{H_{**}}$.

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We can see that the assumption $[K]_{H_{**}} < \infty$ is reasonable (thus $[K]_{H_*} < \infty$ is) for the $H^1$-$L^1$ estimate. On the other hand, it is not enough for the $L^1$-$L^{1, \infty }$ estimate. It follows from a simple example.

Proposition 4.

Let $\varphi \in ( L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d) ) \setminus \{ 0 \}$, $\psi \in ( L^2(\mathbb{R}^d) \cap \mathrm{BMO}(\mathbb{R}^d) ) \setminus L^\infty (\mathbb{R}^d)$ and $K(x, y) \coloneq \varphi (x) \psi (y)$. Then

$$\begin{gather} \text{$T \colon f \mapsto \int _{\mathbb{R}^d} K(\cdot , y) f(y) \, dy$ is bounded on $L^2(\mathbb{R}^d)$,} {\tag{3.7}}\\ [K]_{H_*} < \infty , {\tag{3.8}}\\ \text{$T$ is not bounded from $L^1(\mathbb{R}^d)$ to $L^{1, \infty }(\mathbb{R}^d)$.} {\tag{3.9}} \end{gather}$$

Proof of Proposition 4.

3.7. It is obvious that $\lVert T \rVert _{L^2 \to L^2} \leq \lVert \varphi \rVert _2 \lVert \psi \rVert _2$.

3.8. We have

$$\begin{align*} &\frac{1}{ \lvert B \rvert } \int _{y \in B} \int _{x \in \mathbb{R}^d \setminus 2B} \left\lvert \, K(x, y) - \frac{1}{ \lvert B \rvert } \int _{z \in B} K(x, z) \, dz \, \right\rvert \, dx \, dy \\ &\quad = \bigg ( \int _{x \in \mathbb{R}^d \setminus 2B} \lvert \varphi (x) \rvert \, dx \bigg ) \cdot \bigg ( \frac{1}{\lvert B \rvert } \int _{y \in B} \left\lvert \, \psi (y) - \frac{1}{\lvert B \rvert } \int _{z \in B} \psi (z) \, dz \, \right\rvert \, dy \bigg ) \\ & \quad \leq \lVert \varphi \rVert _1 \lVert \psi \rVert _{\mathrm{BMO}} \end{align*}$$

for any ball $B \subset \mathbb{R}^d$, thus $[K]_{H_*} \leq \lVert \varphi \rVert _1 \lVert \psi \rVert _{\mathrm{BMO}}$.

3.9. For each $f \in L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$, $Tf$ is given by

$$\begin{equation*} Tf(x) = \int _{\mathbb{R}^d} \varphi (x) \psi (y) f(y) \, dy = \varphi (x) \int _{\mathbb{R}^d} \psi (y) f(y) \, dy , \end{equation*}$$

hence

$$\begin{equation*} \lVert Tf \rVert _{1, \infty } = \lVert \varphi \rVert _{1, \infty } \left\lvert \, \int _{\mathbb{R}^d} \psi (y) f(y) \, dy \, \right\rvert . \end{equation*}$$

Since $\psi \not \in L^{\infty }(\mathbb{R}^d)$, there exists a sequence of measurable sets $\{A_j\}_{j \in \mathbb{N}}$ such that

$$\begin{equation*} 0 < \lvert A_j \rvert < \infty , \quad A_j \subset \{ \, y \in \mathbb{R}^d \, : \, \lvert \psi (y) \rvert > j \, \} . \end{equation*}$$

Define $f_j \in L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$ by

$$\begin{equation*} f_j \coloneq \frac{ \chi _{ A_j } }{ \lvert A_j \rvert } \cdot \frac{ \overline{\psi } }{ \lvert \psi \rvert } , \end{equation*}$$

then $f_j$ satisfies $\lVert f_j \rVert _1 = 1$ and

$$\begin{align*} \left\lvert \, \int _{\mathbb{R}^d} \psi (y) f_j(y) \, dy \, \right\rvert = \frac{1}{ \lvert A_j \rvert } \int _{A_j} \lvert \psi (y) \rvert \, dy \geq j \end{align*}$$

for each $j$, thus $T$ is not bounded from $L^1(\mathbb{R}^d)$ to $L^{1, \infty }(\mathbb{R}^d)$.

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4. The $H^1_{\mathrm{atb}}$-$L^1$ estimate with non-doubling measures

In this section, we consider a Radon measure $\mu$ on $\mathbb{R}^d$ which satisfies the polynomial growth condition: there exists a constant $C_\mu > 0$ and $0 < n \leq d$ such that

$$\begin{equation} \mu ( B(c, r) ) \leq C_\mu r^n \tag{\xhref[disp-formula]{#texmlid11}{1.7}} \end{equation}$$

for any balls $B(c, r)$. In this case, unlike in the case of the Lebesgue measure, the Hardy space $H^1(\mu )$ (see Mateu et al. 8) is not suitable for the Calderón–Zygmund theory since Verdera 15 pointed out that the Cauchy integral does not satisfy the $H^1(\mu )$-$L^1(\mu )$ estimate in general. After that, Tolsa 12 developed the atomic block Hardy space $H^1_{\mathrm{atb}}(\mu )$ and established the $H^1_{\mathrm{atb}}(\mu )$-$L^1(\mu )$ estimate of singular integral operators (Theorem C). We will show that our $\mathrm{BMO}$ Hörmander condition still works in this setting with a little modification.

Recall that $H^1_{\mathrm{atb}}(\mu )$ and $\mathrm{RBMO}(\mu )$ are defined as follows.

The coefficient $\delta (B_0, B)$ Let $(B_0, B)$ be a pair of balls such that $B_0 \subset B$. The coefficient $\delta (B_0, B)$ is defined by

$$\begin{equation} \delta (B_0, B) \coloneq \int _{y \in 2B \setminus B_0} \frac{ 1 }{ \lvert y-c(B_0) \rvert ^n } \, d\mu (y) . \tag{4.1} \end{equation}$$

The atomic block A function $b \in L^1_{\mathrm{loc}}(\mu )$ is called an atomic block if there exist a ball $B$, a pair of balls $\{ B_j \}_{j=1}^{2}$, functions $\{ a_j \}_{j=1}^{2}$ and numbers $\{ \lambda _j \}_{j=1}^{2}$ such that

$$\begin{equation} \begin{gathered} \operatorname {supp}b \subset B , \quad \operatorname {supp}a_j \subset B_j , \quad B_j \subset B , \\ \int _B b \, d\mu = 0 , \\ \lVert a_j \rVert _{L^\infty (\mu )} \leq ( ( 1 + \delta (B_j, B) ) \mu ( 2 B_j ) )^{-1} , \\ b = \lambda _1 a_1 + \lambda _2 a_2 \end{gathered} {\tag{4.2}} \end{equation}$$

and write

$$\begin{equation} \lvert b \rvert _{H^1_{\mathrm{atb}}(\mu )} \coloneq \lvert \lambda _1 \rvert + \lvert \lambda _2 \rvert . {\tag{4.3}} \end{equation}$$

The atomic block Hardy space $H^1_{\mathrm{atb}}(\mu )$ The atomic block Hardy space $H^1_{\mathrm{atb}}(\mu )$ is defined by

$$\begin{equation} H^1_{\mathrm{atb}}(\mu ) \coloneq \left\{ \, \sum _{j=1}^{\infty } b_j : \text{$b_j$ are atomic blocks such that } \sum _{j=1}^{\infty } \lvert b_j \rvert _{H^1_{\mathrm{atb}}(\mu )} < \infty \, \right\} {\tag{4.4}} \end{equation}$$

and its norm is

$$\begin{equation} \lVert f \rVert _{H^1_{\mathrm{atb}}(\mu )} \coloneq \inf { \left\{ \, \sum _{j=1}^{\infty } \lvert b_j \rvert _{H^1_{\mathrm{atb}}(\mu )} : \text{$b_j$ are atomic blocks such that } f = \sum _{j=1}^{\infty } b_j \, \right\} }. {\tag{4.5}} \end{equation}$$

The regularized bounded mean oscillation space $\mathrm{RBMO}(\mu )$ A function $f \in L^1_{\mathrm{loc}}(\mu )$ is in $\mathrm{RBMO}(\mu )$ if there exists a collection of numbers $\{ m_B \}_{B}$ such that

$$\begin{equation} \sup _{B \subset \mathbb{R}^d} \frac{1}{\mu (2B)} \int _{y \in B} \lvert f(y) - m_B \rvert \, d\mu (y) + \sup _{B_0 \subset B \subset \mathbb{R}^d} \frac{1}{1 + \delta (B_0, B)} \lvert m_{B_0} - m_B \rvert < \infty , {\tag{4.6}} \end{equation}$$

and its norm is defined by $\lVert f \rVert _{\mathrm{RBMO}(\mu )} \coloneq \inf _{\{m_B\}_{B}} {(\text{LHS of \xhref[disp-formula]{#texmlid12}{4.6}})}$, where supremum $\sup _{B \subset \mathbb{R}^d}$, $\sup _{B_0 \subset B \subset \mathbb{R}^d}$ and infimum $\inf _{\{m_B\}_{B}}$ are taken over all balls $B$ with $\mu (B) > 0$, all pairs of balls $(B_0, B)$ such that $B_0 \subset B$, and all collections $\{m_B\}_{B}$, respectively.

It is known that these spaces $H^1_{\mathrm{atb}}(\mu )$ and $\mathrm{RBMO}(\mu )$ satisfy properties analogous to those of usual $H^1(\mathbb{R}^d)$ and $\mathrm{BMO}(\mathbb{R}^d)$ with the Lebesgue measure, such as the John–Nirenberg inequality, the $H^1_{\mathrm{atb}}(\mu )$-$\mathrm{RBMO}(\mu )$ duality, interpolation inequalities, $T1$ and $Tb$ theorems (see 10, 12, 14, 16).

Now we introduce an $\mathrm{RBMO}$ Hörmander condition: there exists a collection of functions $\{ m_B \}_{B}$ such that

$$\begin{equation} \begin{aligned} &\sup _{B \subset \mathbb{R}^d} \frac{1}{\mu (2B)} \int _{y \in B} \int _{x \in \mathbb{R}^d \setminus 2B} \lvert K(x, y) - m_B(x) \rvert \, d\mu (x) \, d\mu (y) \\ &\quad + \sup _{B_0 \subset B \subset \mathbb{R}^d} \frac{1}{1 + \delta (B_0, B)} \int _{x \in \mathbb{R}^d \setminus 2B} \lvert m_{B_0} (x) - m_B (x) \rvert \, d\mu (x) < \infty , \end{aligned} {\tag{4.7}} \end{equation}$$

and write $[K]_{H_{**}} \coloneq \inf _{\{m_B\}_{B}} {(\text{LHS of \xhref[disp-formula]{#texmlid13}{4.7}})}$. Note that we can easily see $[K]_{H_{**}} \leq 2 [K]_{H_\infty }$. We are going to prove the non-doubling version of Proposition 3.

Theorem 3.

Let $T$ be a singular integral operator associated with a kernel $K$. Suppose that $T$ is bounded on $L^{p_0}(\mu )$ for some $1 < p_0 < \infty$ and its kernel $K$ satisfies the $\mathrm{RBMO}$ Hörmander condition $[K]_{H_{**}} < \infty$ and $\lvert K(x, y) \rvert \leq A \lvert x-y \rvert ^{-n}$ for some constant $A > 0$. Then $T$ is bounded from $H^1_{\mathrm{atb}}(\mu )$ to $L^1(\mu )$ with a constant proportional to $A + \lVert T \rVert _{L^{p_0}(\mu ) \to L^{p_0}(\mu )} + [K]_{H_{**}}$.

Proof of Theorem 3.

Let $b = \sum _{j=1}^{2} \lambda _j a_j \in H^1_{\mathrm{atb}}(\mu )$ be an atomic block. Since $T$ is bounded on $L^{p_0}(\mu )$, it is enough to show that

$$\begin{equation*} \lVert Tb \rVert _{ L^1(\mu ) } \lesssim (A + \lVert T \rVert _{L^{p_0}(\mu ) \to L^{p_0}(\mu )} + [K]_{H_{**}}) \lvert b \rvert _{H^1_{\mathrm{atb}}(\mu )} . \end{equation*}$$

We decompose $\lVert Tb \rVert _{L^1(\mu )}$ as

$$\begin{equation*} \lVert Tb \rVert _{L^1(\mu )} \leq \sum _{j=1}^{2} \lambda _j ( \lVert Ta_j \rVert _{L^1(2B_j, \mu )} + \lVert Ta_j \rVert _{L^1(2B \setminus 2B_j, \mu )} ) + \lVert Tb \rVert _{L^1(\mathbb{R}^d \setminus 2B, \mu )} \end{equation*}$$

and prove

$$\begin{gather} \lVert Ta_j \rVert _{L^1(2B_j, \mu )} \lesssim \lVert T \rVert _{L^{p_0}(\mu ) \to L^{p_0}(\mu )} , {\tag{4.8}}\\ \lVert Ta_j \rVert _{L^1(2B \setminus 2B_j, \mu )} \lesssim A , {\tag{4.9}}\\ \lVert Tb \rVert _{L^1(\mathbb{R}^d \setminus 2B, \mu )} \lesssim [K]_{H_{**}} \lvert b \rvert _{H^1_{\mathrm{atb}}(\mu )} . {\tag{4.10}} \end{gather}$$

4.8. By the Hölder inequality and the $L^{p_0}(\mu )$ boundedness of $T$, we have

$$\begin{align*} \lVert Ta_j \rVert _{L^1(2B_j, \mu )} &\leq \mu (2B)^{1/p_0'} \lVert Ta_j \rVert _{L^{p_0}(\mu )} \\ &\leq \lVert T \rVert _{L^{p_0}(\mu ) \to L^{p_0}(\mu )} \mu (2B_j)^{1/p_0'} \lVert a_j \rVert _{L^{p_0}(\mu )} \\ &\leq \lVert T \rVert _{L^{p_0}(\mu ) \to L^{p_0}(\mu )} \mu (2B_j)^{1/p_0'} \mu (B_j)^{1/p_0} \lVert a_j \rVert _{L^\infty (\mu )} \\ &\leq \lVert T \rVert _{L^{p_0}(\mu ) \to L^{p_0}(\mu )} . \end{align*}$$

4.9. For each $x \in 2B \setminus 2B_j$, we have a pointwise estimate

$$\begin{align*} \lvert Ta_j(x) \rvert &= \left\lvert \, \int _{y \in B_j} K(x, y) a_j(y) \, d\mu (y) \, \right\rvert \\ &\leq \int _{y \in B_j} \lvert K(x, y) a_j(y) \rvert \, d\mu (y) \\ &\leq \lVert a_j \rVert _{L^\infty (\mu )} \int _{y \in B_j} \frac{A}{ \lvert x-y \rvert ^n } \, d\mu (y) \\ &\leq \frac{1}{ ( 1 + \delta (B_j, B) ) \mu ( 2 B_j ) } \int _{y \in B_j} \frac{2^n A}{ \lvert x-c_j \rvert ^n } \, d\mu (y) \\ &\leq \frac{2^n A}{ 1 + \delta (B_j, B) } \frac{1}{ \lvert x-c_j \rvert ^n } . \end{align*}$$

Therefore, we obtain

$$\begin{equation*} \lVert Ta_j \rVert _{L^1(2B \setminus 2B_j, \mu )} \leq \frac{2^n A}{ 1 + \delta (B_j, B) } \delta (B_j, B) \leq 2^n A. \end{equation*}$$

4.10. Since

$$\begin{align*} & \lVert Tb \rVert _{L^1(\mathbb{R}^d \setminus 2B, \mu )} \\ & = \int _{x \in \mathbb{R}^d \setminus 2B} \left\lvert \, \int _{y \in B} K(x, y) b(y) \, d\mu (y) \, \right\rvert \, d\mu (x) \\ & = \int _{x \in \mathbb{R}^d \setminus 2B} \left\lvert \, \int _{y \in B} K(x, y) b(y) \, d\mu (y) - m_B(x) \int _{y \in B} b(y) \, d\mu (y) \, \right\rvert \, d\mu (x) \\ & \leq \int _{y \in B} \int _{x \in \mathbb{R}^d \setminus 2B} \left\lvert \, K(x, y) - m_B(x) \, \right\rvert \, d\mu (x) \, \lvert b(y) \rvert \, d\mu (y) \\ & \leq \sum _{j=1}^{2} \lvert \lambda _j \rvert \lVert a_j \rVert _{L^\infty (\mu )} \int _{y \in B_j} \int _{x \in \mathbb{R}^d \setminus 2B} \left\lvert \, K(x, y) - m_B(x) \, \right\rvert \, d\mu (x) \, d\mu (y) \\ & \leq \sum _{j=1}^{2} \lvert \lambda _j \rvert \bigg ( \frac{1}{ ( 1 + \delta (B_j, B) ) \mu (2B_j) } \int _{y \in B_j} \int _{x \in \mathbb{R}^d \setminus 2B_j} \left\lvert \, K(x, y) - m_{B_j}(x) \, \right\rvert \, d\mu (x) \, d\mu (y) \\ & \qquad \qquad \, + \frac{1}{ ( 1 + \delta (B_j, B) ) \mu (2B_j) } \int _{y \in B_j} \int _{x \in \mathbb{R}^d \setminus 2B} \left\lvert \, m_{B_j}(x) - m_B(x) \, \right\rvert \, d\mu (x) \, d\mu (y) \bigg ) \\ & \leq \sum _{j=1}^{2} \lvert \lambda _j \rvert \bigg ( \frac{1}{ \mu (2B_j) } \int _{y \in B_j} \int _{x \in \mathbb{R}^d \setminus 2B_j} \left\lvert \, K(x, y) - m_{B_j}(x) \, \right\rvert \, d\mu (x) \, d\mu (y) \\ &\qquad \qquad \, + \frac{1}{ 1 + \delta (B_j, B) } \int _{x \in \mathbb{R}^d \setminus 2B} \left\lvert \, m_{B_j}(x) - m_B(x) \, \right\rvert \, d\mu (x) \bigg ) \end{align*}$$

for any collections $\{m_B\}_B$, we have $\lVert Tb \rVert _{L^1(\mathbb{R}^d \setminus 2B)} \leq [K]_{H_{**}} \lvert b \rvert _{H^1_{\mathrm{atb}}(\mu )}$.

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