$L^p(\mathbb {R}^d)$ Boundedness for the Calderón Commutator with Rough Kernel

Abstract

Let $k\in \mathbb {N}$, $\Omega$ be homogeneous of degree zero, integrable on $S^{d-1}$ and have vanishing moment of order k, a be a function on $\mathbb {R}^d$ such that $\nabla a\in L^{\infty }(\mathbb {R}^d)$, and $T_{\Omega ,\,a;k}$ be the d-dimensional Calderón commutator defined by

$$\begin{aligned} T_{\Omega ,\,a;k}f(x)=\mathrm{p.\,v.}\int _{\mathbb {R}^d}\frac{\Omega (x-y)}{|x-y|^{d+k}}\big (a(x)-a(y)\big )^kf(y){d}y. \end{aligned}$$

In this paper, the authors prove that if

$$\begin{aligned} \sup _{\zeta \in S^{d-1}}\int _{S^{d-1}}|\Omega (\theta )|\log ^{\beta } \Bigg (\frac{1}{|\theta \cdot \zeta |}\Bigg )d\theta <\infty , \end{aligned}$$

with $\beta \in (1,\,\infty )$, then for $\frac{2\beta }{2\beta -1}<p<2\beta$, $T_{\Omega ,\,a;\,k}$ is bounded on $L^p(\mathbb {R}^d)$.

1 Introduction

We will work on $\mathbb {R}^d$, $d\ge 2$. Let $k\in \mathbb {N}$, $\Omega$ be homogeneous of degree zero, integrable on $S^{d-1}$, the unit sphere in $\mathbb {R}^d$, and have vanishing moment of order k, that is, for all multi-indices $\gamma \in \mathbb {Z}_+^d$,

$$\begin{aligned} \int _{S^{d-1}}\Omega (\theta )\theta ^{\gamma }d\theta =0,\,\,\,|\gamma |=k. \end{aligned}$$

(1.1)

Let a be a function on $\mathbb {R}^d$ such that $\nabla a\in L^{\infty }(\mathbb {R}^d)$. Define the d-dimensional Calderón commutator $T_{\Omega , a;\,k}$ by

$$\begin{aligned} T_{\Omega , a;k}f(x)=\mathrm{p. v.}\int _{\mathbb {R}^d}\frac{\Omega (x-y)}{|x-y|^{d+k}}\big (a(x)-a(y)\big )^kf(y)dy. \end{aligned}$$

(1.2)

For simplicity, we denote $T_{\Omega ,a;\,1}$ by $T_{\Omega ,a}$. Commutators of this type were introduced by Calderón [1], who proved that if $\Omega \in L\log L(S^{d-1})$, then $T_{\Omega ,a}$ is bounded on $L^p(\mathbb {R}^d)$ for all $p\in (1,\,\infty )$. It should be pointed out that Calderón’s result in [1] also holds for $T_{\Omega ,a;\,k}$. Pan et al. [13] improved Calderón’s result, and obtained the following conclusion.

Theorem 1.1

Let $\Omega$ be homogeneous of degree zero, satisfy the vanishing moment (1.1) with $k=1$, a be a function on $\mathbb {R}^d$ such that $\nabla a\in L^{\infty }(\mathbb {R}^d)$. Suppose that $\Omega \in H^1(S^{d-1})$ (the Hardy space on $S^{d-1}$), then $T_{\Omega ,a}$ is bounded on $L^p(\mathbb {R}^d)$ for all $p\in (1,\,\infty )$.

Chen et al. [4] showed that the converse of Theorem 1.1 is also true. Precisely, Chen at al. [4, p. 1501] established the following result.

Theorem 1.2

Let $\Omega$ be homogeneous of degree zero, $\Omega \in \mathrm{Lip}_{\alpha }(S^{d-1})$ for some $\alpha \in (0,\,1]$, and satisfy the vanishing moment (1.1) with $k=1$, $a\in L_{\mathrm{loc}}^1(\mathbb {R}^d)$. If $T_{\Omega ,\,a}$ is bounded on $L^p(\mathbb {R}^d)$ for some $p\in (1,\,\infty )$, then $\nabla a\in L^{\infty }(\mathbb {R}^d)$.

Hofmann [10] considered the weighted $L^p$ boundedness with $A_p$ weights for $T_{\Omega ,a;\,k}$, and proved that if $\Omega \in L^{\infty }(S^{d-1})$ and satisfies (1.1), then for $p\in (1,\,\infty )$ and $w\in A_p(\mathbb {R}^d)$, $T_{\Omega ,a;k}$ is bounded on $L^p(\mathbb {R}^d,\,w)$, where and in the following, $A_p(\mathbb {R}^d)$ denotes the weight function class of Muckenhoupt, see [7, Chap. 9] for the definition and properties of $A_p(\mathbb {R}^d)$. Ding and Lai [5] considered the weak type endpoint estimate for $T_{\Omega , a}$, and proved that $\Omega \in L\log L(S^{d-1})$ is a sufficient condition such that $T_{\Omega ,a}$ is bounded from $L^1(\mathbb {R}^d)$ to $L^{1,\,\infty }(\mathbb {R}^d)$.

For $\beta \in [1,\,\infty )$, we say that $\Omega \in GS_{\beta }(S^{d-1})$ if $\Omega \in L^1(S^{d-1})$ and

$$\begin{aligned} \sup _{\zeta \in S^{d-1}}\int _{S^{d-1}}|\Omega (\theta )|\log ^{\beta }\Bigg (\frac{1}{|\zeta \cdot \theta |}\Bigg )d\theta <\infty . \end{aligned}$$

(1.3)

The condition (1.3) was introduced by Grafakos and Stefanov [8] in order to study the $L^p(\mathbb {R}^d)$ boundedness for the homogeneous singular integral operator defined by

$$\begin{aligned} T_{\Omega }f(x)=\mathrm{p.\, v.}\int _{\mathbb {R}^d}\frac{\Omega (x-y)}{|x-y|^d}f(y)dy, \end{aligned}$$

(1.4)

where $\Omega$ is homogeneous of degree zero and has mean value zero on $S^{d-1}$. Obviously, $L(\log L)^{\beta }(S^{d-1})\subset GS_{\beta }(S^{d-1})$. On the other hand, as it was pointed out in [8], there exist integrable functions on $S^{d-1}$ which are not in $H^1(S^{d-1})$ but satisfy (1.3) for all $\beta \in (1,\,\infty )$. Thus, it is of interest to consider the $L^p(\mathbb {R}^d)$ boundedness for operators such as $T_{\Omega }$ and $T_{\Omega ,a;\,k}$ when $\Omega \in GS_{\beta }(S^{d-1})$. Grafakos and Stefanov [8] proved that if $\Omega \in GS_{\beta }(S^{d-1})$ for some $\beta \in (1,\,\infty ]$, then $T_{\Omega }$ is bounded on $L^p(\mathbb {R}^d)$ for $1+1/\beta<p<1+\beta$. Fan et al. [6] improved the result of [8], and proved the following result.

Theorem 1.3

Let $\Omega$ be homogeneous of degree zero, integrable and have mean value zero on $S^{d-1}$. Suppose that $\Omega \in GS_{\beta }(S^{d-1})$ with $\beta \in (1,\,\infty )$, then for $\frac{2\beta }{2\beta -1}<p<2\beta$, $T_{\Omega }$ is bounded on $L^p(\mathbb {R}^d)$.

The purpose of this paper is to establish the $L^p(\mathbb {R}^d)$ boundedness of $T_{\Omega ,\,a;k}$ when $\Omega \in GS_{\beta }(S^{d-1})$ for some $\beta >1$. Our main result can be stated as follows.

Theorem 1.4

Let $k\in \mathbb {N}$, $\Omega$ be homogeneous of degree zero, satisfy the vanishing moment (1.1), a be a function on $\mathbb {R}^d$ such that $\nabla a\in L^{\infty }(\mathbb {R}^d)$. Suppose that $\Omega \in GS_{\beta }(S^{d-1})$ with $\beta \in (1,\,\infty )$, Then for $\frac{2\beta }{2\beta -1}<p<2\beta$, $T_{\Omega ,\,a;\,k}$ is bounded on $L^p(\mathbb {R}^d)$.

Different from the operator $T_{\Omega }$ defined by (1.4), $T_{\Omega ,a;\,k}$ is not a convolution operator, and the argument in [6, 8] does not apply to $T_{\Omega ,\,a;\,k}$ directly. To prove Theorem 1.4, we will first prove the $L^2(\mathbb {R}^d)$ boundedness of $T_{\Omega ,a;k}$ by employing the ideas used in [10], together with some new localizations and decompositions. The argument in the proof of $L^2(\mathbb {R}^d)$ boundedness is based on a refined decomposition appeared in (2.10). To prove the $L^p(\mathbb {R}^d)$ boundedness of $T_{\Omega ,a;\,k}$, we will introduce a suitable approximation to $T_{\Omega ,a;\,k}$ by a sequence of integral operators, whose kernels enjoy Hörmander’s condition. We remark that the idea approximating rough convolution operators by smooth operators was originated by Watson [16].

In what follows, C always denotes a positive constant that is independent of the main parameters involved but whose value may differ from line to line. We use the symbol $A\lesssim B$ to denote that there exists a positive constant C such that $A\le CB$. Constant with subscript such as $C_1$, does not change in different occurrences. For any set $E\subset \mathbb {R}^d$, $\chi _E$ denotes its characteristic function. For a cube $Q\subset \mathbb {R}^d$ and $\lambda \in (0,\,\infty )$, $\lambda Q$ denotes the cube with the same center as Q whose side length is $\lambda$ times that of Q. For a suitable function f, we denote ${\widehat{f}}$ the Fourier transform of f. For $p\in [1,\,\infty ]$, $p'$ denotes the dual exponent of p, namely, $p'=p/(p-1)$.

2 Proof of Theorem 1.4: $L^2(\mathbb {R}^d)$ Boundedness

This section is devoted to the proof of the $L^2(\mathbb {R}^d)$ boundedness of $T_{\Omega ,a;\,k}$. For simplicity, we only consider the case $k=1$. As it was pointed out in [10, Sect. 2], the argument in this section still works for all $k\in \mathbb {N}$, if we proceed by induction on the order k.

Let $\phi \in C^{\infty }_0(\mathbb {R}^d)$ be a radial function, $\mathrm{supp}\,\phi \subset B(0,\,2)$, $\phi (x)=1$ when $|x|\le 1$. Set $\varphi (x)=\phi (x)-\phi (2x)$. We then have that

$$\begin{aligned} \sum _{j\in \mathbb {Z}}\varphi (2^{-j}x)\equiv 1,\,\,|x|>0. \end{aligned}$$

(2.1)

Let $\varphi _j(x)=\varphi (2^{-j}x)$ for $j\in \mathbb {Z}$.

For a function $\Omega \in L^1(S^{d-1})$, define the operator $W_{\Omega , j}$ by

$$\begin{aligned} W_{\Omega ,j}h(x)=\int _{\mathbb {R}^d}\frac{\Omega (x-y)}{|x-y|^{d+1}}\varphi _j(x-y)h(y)dy. \end{aligned}$$

(2.2)

Lemma 2.1

Let $\Omega$ be homogeneous of degree zero, integrable on $S^{d-1}$, satisfy the vanishing moment (1.1) with $k=1$ and $\Omega \in GS_{\beta }(S^{d-1})$ for some $\beta \in (1,\,\infty )$, a be a function on $\mathbb {R}^d$ such that $\nabla a\in L^{\infty }(\mathbb {R}^d)$. Then, for any $r\in (0,\,\infty )$, functions ${\eta }_1,\,{\eta }_2\in C^{\infty }_0(\mathbb {R}^d)$ which are supported on balls of radius no larger than r,

$$\begin{aligned} \Big |\int _{\mathbb {R}^d}\eta _2(x)T_{\Omega ,\,a}\eta _1(x)dx\Big |\lesssim \Vert \Omega \Vert _{L^1(S^{d-1})} r^{-d}\prod _{j=1}^2\big (\Vert \eta _j\Vert _{L^{\infty }(\mathbb {R}^d)}+r\Vert \nabla \eta _j\Vert _{L^{\infty }(\mathbb {R}^d)}\big ). \end{aligned}$$

Recall that under the hypothesis of Lemma 2.1, the operator $T_{\Omega ,m}$ defined by

$$\begin{aligned} T_{\Omega ,m}f(x)=\mathrm{p.\,v.}\int _{\mathbb {R}^d}\frac{\Omega (x-y)(x_m-y_m)}{|x-y|^{d+1}}f(y)dy,\,\,1\le m\le d \end{aligned}$$

(2.3)

is bounded on $L^2(\mathbb {R}^d)$ (see [8]). Lemma 2.1 can be proved by repeating the proof of Lemma 2.5 in [10].

Let $\psi \in C^{\infty }_0(\mathbb {R}^d)$ be a radial function, have integral zero and $\mathrm{supp}\,\psi \subset B(0,\,1)$. Let $Q_s$ be the operator defined by $Q_sf(x)=\psi _s*f(x)$, where $\psi _s(x)=s^{-d}\psi (s^{-1}x)$. We assume that

$$\begin{aligned} \int ^{\infty }_0[{\widehat{\psi }}(s)]^4\frac{ds}{s}=1. \end{aligned}$$

Then, the Calderón reproducing formula

$$\begin{aligned} \int ^{\infty }_0Q_s^4\frac{ds}{s}=I \end{aligned}$$

(2.4)

holds true. In addition, the Littlewood–Paley theory tells us that

$$\begin{aligned} \Big \Vert \Big (\int _{0}^{\infty }|Q_sf|^2\frac{ds}{s}\Big )^{1/2}\Big \Vert _{L^2(\mathbb {R}^d)}\lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

(2.5)

For each fixed $j\in \mathbb {Z}$, set

$$\begin{aligned} T_{\Omega ,\,a}^{j}f(x)=\int _{\mathbb {R}^d}K_{j}(x,\,y)f(y)dy, \end{aligned}$$

where

$$\begin{aligned} K_{j}(x,\,y)=\frac{\Omega (x-y)}{|x-y|^{d+1}}(a(x)-a(y)\big )\varphi _j(|x-y|). \end{aligned}$$

Lemma 2.2

Let $\Omega$ be homogeneous of degree zero, integrable on $S^{d-1}$ and $\Omega \in GS_{\beta }(S^{d-1})$ for some $\beta \in (1,\,\infty )$, then for $j\in \mathbb {Z}$ and $0<s\le 2^j$,

$$\begin{aligned} \Vert Q_sW_{\Omega ,\,j}f\Vert _{L^2(\mathbb {R}^d)}\lesssim 2^{-j}\log ^{-\beta }(2^j/s+1)\Vert f\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

Proof

Let $K_{\Omega ,j}(x)=\frac{\Omega (x)}{|x|^{d+1}}\varphi _j(|x|)$. By Plancherel’s theorem, it suffices to prove that

$$\begin{aligned} |\widehat{\psi _s}(\xi )\widehat{K_{\Omega ,j}}(\xi )|\lesssim 2^{-j}\log ^{-\beta }(2^j/s+1). \end{aligned}$$

(2.6)

As it was proved by Grafakos and Stefanov [8, p. 458], we know that

$$\begin{aligned} |\widehat{K_{\Omega ,j}}(\xi )|\lesssim 2^{-j}\log ^{-\beta }(|2^j\xi |+1). \end{aligned}$$

On the other hand, it is easy to verify that

$$\begin{aligned} |\widehat{\psi _s}(\xi )|\lesssim \min \{1,\,|s\xi |\}. \end{aligned}$$

Observe that (2.6) holds true when $|2^j\xi |\le 1$, since

$$\begin{aligned} |s\xi |\log ^{-\beta }(2^j|\xi |+1)=\frac{s}{2^j}|2^j\xi |\log ^{-\beta }(|2^j\xi |+1)\lesssim \frac{s}{2^j}\lesssim \log ^{-\beta }(2^j/s+1). \end{aligned}$$

If $|s\xi |\ge 1$, we certainly have that

$$\begin{aligned} |\widehat{\psi _s}(\xi )\widehat{K_{\Omega ,j}}(\xi )|\lesssim 2^{-j}\log ^{-\beta }(2^j|\xi |+1)\lesssim 2^{-j}\log ^{-\beta }(2^j/s+1). \end{aligned}$$

Now, we assume that $s|\xi |<1$ and $|2^j\xi |>1$, and

$$\begin{aligned} 2^{-k}2^j<s\le 2^{-k+1}2^j,\,\,\,2^{k_1-1}<|\xi |\le 2^{k_1} \end{aligned}$$

for $k\in \mathbb {N}$ and $k_1\in \mathbb {Z}$ respectively. Then $j+k_1\in \mathbb {N}$, $j+k_1\le k$ and

$$\begin{aligned} |s\xi |\log ^{-\beta }(2^j|\xi |+1)\lesssim 2^{j-k+k_1}(j+k_1)^{-\beta }\lesssim k^{-\beta }\lesssim \log ^{-\beta }(2^j/s+1). \end{aligned}$$

This verifies (2.6). $\square$

Lemma 2.3

Let $\Omega$ be homogeneous of degree zero, satisfy the vanishing moment (1.1) with $k=1$ and $\Omega \in GS_{\beta }(S^{d-1})$ for some $\beta \in (1,\,\infty )$, a be a function on $\mathbb {R}^d$ with $\nabla a\in L^{\infty }(\mathbb {R}^d)$. Then

• (i) $T_{\Omega , a}1\in \mathrm{BMO}(\mathbb {R}^d)$;

• (ii) for any $j\in \mathbb {Z}$ and $s\in (0,\,2^j]$;

  $$\begin{aligned} \Vert Q_sT_{\Omega ,a}^{j}1\Vert _{L^{\infty }(\mathbb {R}^d)}\lesssim \Vert \Omega \Vert _{L^1(S^{d-1})}2^{-j}s. \end{aligned}$$

Conclusion (ii) is just Lemma 2.4 in [10], while (i) of Lemma 2.3 can be proved by mimicking the proof of Lemma 2.3 in [10], since for all $1\le m\le d$, $T_{\Omega , m}$ defined by (2.3) is bounded on $L^2(\mathbb {R}^d)$ when $\Omega \in GS_{\beta }(S^{d-1})$ for $\beta >1$. We omit the details for brevity.

Proof of Theorem 1.4

$L^2(\mathbb {R}^d)$ boundedness. By (2.4), it suffices to prove that for $f,\,g\in C^{\infty }_0(\mathbb {R}^d)$,

$$\begin{aligned} \Big |\int ^{\infty }_0\int _{0}^t\int _{\mathbb {R}^d}Q_s^4{T}_{\Omega ,\,a}Q_t^4f(x)g(x)dx \frac{ds}{s}\frac{dt}{t}\Big |\lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}\Vert g\Vert _{L^{2}(\mathbb {R}^d)}, \end{aligned}$$

(2.7)

and

$$\begin{aligned} \Big |\int ^{\infty }_0\int _{t}^{\infty }\int _{\mathbb {R}^d}Q_s^4{T}_{\Omega ,\,a}Q_t^4f(x)g(x)dx \frac{ds}{s}\frac{dt}{t}\Big |\lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}\Vert g\Vert _{L^{2}(\mathbb {R}^d)}. \end{aligned}$$

(2.8)

Observe that (2.8) can be deduced from (2.7) and a standard duality argument. Thus, we only need to prove (2.7).

We now prove (2.7). Without loss of generality, we assume that $\Vert \nabla a\Vert _{L^{\infty }(\mathbb {R}^d)}=1$. Write

$$\begin{aligned}&\int ^{\infty }_0\int _{0}^t\int _{\mathbb {R}^d}Q_s^4{T}_{\Omega ,\,a} Q_t^4f(x)g(x)dx\frac{ds}{s}\frac{dt}{t}\\&\quad =\sum _{j\in \mathbb {Z}}\int ^{2^j}_0\int _{0}^t\int _{\mathbb {R}^d}Q_s{T}_{\Omega ,\,a}^{j}Q_t^4f(x)Q_s^3g(x)dx \frac{ds}{s}\frac{dt}{t}\\&\qquad +\sum _{j\in \mathbb {Z}}\int ^{\infty }_{2^j}\int _{0}^{(2^jt^{\alpha -1})^{\frac{1}{\alpha }}} \int _{\mathbb {R}^d}Q_s{T}_{\Omega ,\,a}^{j}Q_t^4f(x)Q_s^3g(x)dx\frac{ds}{s}\frac{dt}{t}\\&\qquad +\sum _{j\in \mathbb {Z}}\int ^{\infty }_{2^j}\int _{(2^jt^{\alpha -1})^{\frac{1}{\alpha }}}^t \int _{\mathbb {R}^d}Q^4_s{T}_{\Omega ,\,a}^{j}Q_t^4f(x)g(x)dx\frac{ds}{s}\frac{dt}{t}:=\mathrm{D}_1+\mathrm{D}_2+\mathrm{D}_3, \end{aligned}$$

where $\alpha \in \Big (\frac{d+1}{d+2},\,1\Big )$ is a constant.

We first consider term $\mathrm{D}_2$. For each fixed $j\in \mathbb {Z}$, let $\{I_{j,l}\}_{l}$ be a sequence of cubes having disjoint interiors and side length $2^j$, such that $\mathbb {R}^d=\cup _{l}I_{j,l}.$ For each fixed j, l, let $\omega _{j,l}\in C^{\infty }_0(\mathbb {R}^d)$ such that $\mathrm{supp}\,\omega _{j,l}\subset 48dI_{j,l}$, $0\le \omega _{j,l}\le 1$ and $\omega _{j,l}(x)\equiv 1$ when $x\in 32dI_{j,l}$. Let $I_{j,l}^*=64dI_{j,l}$ and $x_{j,l}$ be the center of $I_{j,l}$. For each l, set $a_{j,l}(y)=(a(y)-a(x_{j,l}))\omega _{j,l}(y)$, and $h_{s,j,l}(y)=Q_s^2g(y)\chi _{I_{j,l}}(y)$. It is obvious that for all l,

$$\begin{aligned} \Vert a_{j,l}\Vert _{L^{\infty }(\mathbb {R}^d)}\lesssim 2^j,\,\,\Vert \nabla a_{j,l}\Vert _{L^{\infty }(\mathbb {R}^d)}\lesssim 1, \end{aligned}$$

and for $s\in (0,\,2^j]$ and $x\in \mathrm{supp}\,Q_sh_{s,j,l}$,

$$\begin{aligned} T_{\Omega ,\,a}^{j}h(x)=a_{j,l}(x)W_{\Omega ,j}h(x)-W_{\Omega ,j}(a_{j,l}h)(x). \end{aligned}$$

For each fixed j and l, let

$$\begin{aligned}&\mathrm{D}_{j,l,1}(s,t)=-\int _{\mathbb {R}^d}[a_{j,l},Q_s]W_{\Omega ,j}Q_t^4f(x)Q_sh_{s,j,l}(x)dx,\\&\mathrm{D}_{j,l,2}(s,t)=\int _{\mathbb {R}^d}a_{j,l}(x)Q_sW_{\Omega ,j}Q_t^4f(x)Q_sh_{s,j,l}(x)dx,\\&\mathrm{D}_{j,l,3}(s,t)=\int _{\mathbb {R}^d}Q_sW_{\Omega ,\,j}[a_{j,l},Q_s]Q_t^4f(x)h_{s,j,l}(x)dx, \end{aligned}$$

and

$$\begin{aligned} \mathrm{D}_{j,l,4}(s,\,t)=-\int _{\mathbb {R}^d}Q_sW_{\Omega ,\,j}(a_{j,l}Q_sQ_t^4f)(x)h_{s,j,l}(x)dx, \end{aligned}$$

where and in the following, for a locally integrable function b and an operator U, $[b,\,U]$ denotes the commutator of U with symbol b, namely,

$$\begin{aligned}{}[b,\,U]h(x)=b(x)Uh(x)-U(bh)(x). \end{aligned}$$

(2.9)

Observe that both of $Q_s$ and $W_{\Omega ,j}$ are convolution operators and $Q_sW_{\Omega ,j}=W_{\Omega ,j}Q_s$. For $j\in \mathbb {Z}$ and $s\in (0,\,2^j]$, we have that

$$\begin{aligned}&\int _{\mathbb {R}^d}Q_s^4T_{\Omega ,\,a}^{j}Q_t^4f(x)g(x)dx\nonumber \\&\quad =\sum _l\int _{\mathbb {R}^d}Q_sT_{\Omega ,\,a}^{j} Q_t^4f(x)Q_sh_{s,j,l}(x)dx\nonumber \\&\quad =\sum _{n=1}^4\sum _{l}\mathrm{D}_{j,l,n}(s,t). \end{aligned}$$

(2.10)

It now follows from Hölder’s inequality that

$$\begin{aligned}&\Big |\sum _{j}\sum _l\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}\mathrm{D}_{j,l,1}(s,t)\frac{ds}{s}\frac{dt}{t} \Big |\\&\quad \le \Big \Vert \Big (\sum _j\sum _l\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}|\chi _{I_{j,l}^*}Q_t^4f|^2 2^{-j}s \frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}}\Big \Vert _{L^2(\mathbb {R}^d)}\\&\qquad \times \Big \Vert \Big (\sum _j\sum _l\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }} |W_{\Omega ,j}[a_{j,l},Q_s]Q_sh_{s,j,l} |^2\frac{1}{2^{-j}s} \frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}}\Big \Vert _{L^{2}(\mathbb {R}^d)}. \end{aligned}$$

Invoking the fact that $\sum _{l}\chi _{I_{j,l}^*}\lesssim 1$, we deduce that

$$\begin{aligned}&\Big \Vert \Big (\sum _j\sum _l\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}|\chi _{I_{j,l}^*}Q_t^4f|^2 2^{-j}s \frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}}\Big \Vert _{L^2(\mathbb {R}^d)}\\&\quad \lesssim \Big \Vert \Big ( \int ^{\infty }_{0}|Q_t^4f|^2\int _{0}^{t}\sum _{j:\, 2^j\ge s^{\alpha }t^{1-\alpha }}2^{-j}s \frac{ds}{s} \frac{dt}{t}\Big )^{1/2}\Big \Vert _{L^2(\mathbb {R}^d)}\lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

Let $M_{\Omega }$ be the operator defined by

$$\begin{aligned} M_{\Omega }f(x)=\sup _{r>0}r^{-d}\int _{|x-y|<r}|\Omega (x-y)||f(y)|dy. \end{aligned}$$

The method of rotation of Calderón and Zygmund states that

$$\begin{aligned} \Vert M_{\Omega }f\Vert _{L^p(\mathbb {R}^d)}\lesssim \Vert \Omega \Vert _{L^1(S^{d-1})}\Vert f\Vert _{L^p(\mathbb {R}^d)},\,\,p\in (1,\,\infty ). \end{aligned}$$

(2.11)

Let M be the Hardy–Littlewood maximal operator. Observe that when $s\in (0,\,2^j]$,

$$\begin{aligned} \big |[a_{j,l},\,Q_s]h(x)\big |\le \int _{\mathbb {R}^d}|\psi _s(x-y)||a_{j,l}(x)-a_{j,l}(y)||h(y)|dy\lesssim sMh(x). \end{aligned}$$

This, together with (2.11), yields

$$\begin{aligned}&\Big \Vert \Big (\sum _j\sum _l\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}|W_{\Omega ,j} [a_{j,l},Q_s]Q_sh_{s,j,l} |^2(2^{-j}s)^{-1} \frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}}\Big \Vert ^2_{L^{2}(\mathbb {R}^d)}\\&\quad \lesssim \sum _j\sum _l\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }} \Vert M_{\Omega }MQ_sh_{s,j,l} \Vert ^2_{L^2(\mathbb {R}^d)}2^{-j}s \frac{ds}{s}\frac{dt}{t}\\&\quad \lesssim \sum _j\sum _l\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }} \Vert h_{s,j,l} \Vert ^2_{L^2(\mathbb {R}^d)}2^{-j}s \frac{ds}{s}\frac{dt}{t}\lesssim \Vert g\Vert ^2_{L^{2}(\mathbb {R}^d)}, \end{aligned}$$

where the last inequality follows from the fact that

$$\begin{aligned} \int _{s}^{\infty }\sum _{j:2^j\ge s^{\alpha }t^{1-\alpha }}2^{-j}s\frac{dt}{t}\lesssim 1. \end{aligned}$$

Therefore,

$$\begin{aligned}&\Big |\sum _{j}\sum _l\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{\frac{1}{\alpha }}}\mathrm{D}_{j,l,1}(s,t)\frac{ds}{s}\frac{dt}{t} \Big |\nonumber \\&\quad \lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}\Vert g\Vert _{L^{2}(\mathbb {R}^d)}. \end{aligned}$$

(2.12)

Similar to the estimate (2.12), we have that

$$\begin{aligned}&\Big |\sum _{j}\sum _l\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{\frac{1}{\alpha }}}\mathrm{D}_{j,l,3}(s,t)\frac{ds}{s}\frac{dt}{t} \Big |\lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}\Vert g\Vert _{L^{2}(\mathbb {R}^d)}. \end{aligned}$$

(2.13)

To estimate the term $\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}\mathrm{D}_{j,l,2}(s,t)\frac{ds}{s}\frac{dt}{t}$, we write

$$\begin{aligned} \mathrm{D}_{j,l,2}(s,t)= & {} \int _{\mathbb {R}^d}Q_sW_{\Omega ,j}Q_t^4f(x)[a_{j,l},Q_s]h_{s,j,l}(x)dx\\&+\int _{\mathbb {R}^d}Q_sW_{\Omega ,j}Q_t^4f(x)Q_s(a_{j,l}h_{s,j,l})(x)dx\\= & {} \mathrm{D}_{j,l,2}^1(s,t)+\mathrm{D}_{j,l,2}^2(s,t). \end{aligned}$$

Repeating the estimate for $\mathrm{D}_{j,l,1}$, we have that

$$\begin{aligned}&\Big |\sum _{j}\sum _l\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{\frac{1}{\alpha }}}\mathrm{D}_{j,l,2}^1(s,t)\frac{ds}{s}\frac{dt}{t} \Big |\lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}\Vert g\Vert _{L^{2}(\mathbb {R}^d)}. \end{aligned}$$

(2.14)

Write

$$\begin{aligned}&\Big |\sum _{j}\sum _l\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{\frac{1}{\alpha }}}\mathrm{D}_{j,l,2}^2(s,t)\frac{ds}{s}\frac{dt}{t} \Big |\\&\quad \le \Big \Vert \Big (\sum _{j}\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{\frac{1}{\alpha }}} |Q_s^2(2^jW_{\Omega ,j})Q_t^3f|^2\log ^{\sigma }(2^j/s+1) \frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}} \Big \Vert _{L^2(\mathbb {R}^d)}\\&\qquad \times \Big \Vert \Big (\sum _{j}\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{\frac{1}{\alpha }}} \big |2^{-j}Q_t\big (\sum _la_{j,l}h_{s,j,l}\big )\big |^2 \log ^{-\sigma }\Big (\frac{2^j}{s}+1\Big )\frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}} \Big \Vert _{L^{2}(\mathbb {R}^d)}\\&\quad :=\mathrm{I}_1\mathrm{I}_2, \end{aligned}$$

where $\sigma >1$ is a constant such that $2\beta -\sigma >1$. Invoking the estimate (2.5), we obtain that

$$\begin{aligned} \quad \quad \mathrm{I}_2\lesssim & {} \Big (\sum _{j}\int _{0}^{2^j} \big \Vert 2^{-j}\sum _la_{j,l}h_{s,j,l}\big \Vert ^2_{L^2(\mathbb {R}^d)} \log ^{-\sigma }(2^j/s+1)\frac{ds}{s}\Big )^{\frac{1}{2}}\\\lesssim & {} \Big (\sum _{j}\int _{0}^{2^j} \big \Vert \sum _l|h_{s,j,l}|\big \Vert _{L^2(\mathbb {R}^d)}^2 \log ^{-\sigma }(2^j/s+1)\frac{ds}{s}\Big )^{\frac{1}{2}}\\= & {} \Big (\int _{0}^{\infty } \Vert Q_s^2g\Vert ^2_{L^2(\mathbb {R}^d)} \sum _{j:2^j\ge s}\log ^{-\sigma }(2^j/s+1)\frac{ds}{s}\Big )^{\frac{1}{2}}\lesssim \Vert g\Vert _{L^{2}(\mathbb {R}^d)}. \end{aligned}$$

Note that $Q_s^2(2^jW_{\Omega ,j})=Q_s(2^jW_{\Omega ,j})Q_s$. It follows from Lemma 2.2 and (2.5) that

$$\begin{aligned} \mathrm{I}_1= & {} \Big (\sum _{j}\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{\frac{1}{\alpha }}} \Vert Q_s^2(2^jW_{\Omega ,j})Q_t^3f\Vert ^2_{L^2(\mathbb {R}^d)}\log ^{\sigma }\Big (\frac{2^j}{s}+1\Big ) \frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}}\\\lesssim & {} \Big (\sum _{j}\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{\frac{1}{\alpha }}} \Vert Q_sQ_t^3f\Vert ^2_{L^2(\mathbb {R}^d)}\log ^{-2\beta +\sigma }(2^j/s+1) \frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}}\\\lesssim & {} \Big \Vert \Big (\int _{0}^{\infty }\int _{0}^{\infty } |Q_sQ_t^3f|^2 \frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}} \Big \Vert ^2_{L^2(\mathbb {R}^d)}\lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

The estimates for $\mathrm{I}_1$ and $\mathrm{I}_2$ show that

$$\begin{aligned}&\Big |\sum _{j}\sum _l\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}\mathrm{D}_{j,l,2}^2(s,t)\frac{ds}{s}\frac{dt}{t} \Big |\lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}\Vert g\Vert _{L^{2}(\mathbb {R}^d)}. \end{aligned}$$

This, together with (2.14), gives us that

$$\begin{aligned}&\Big |\sum _{j}\sum _l\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}\mathrm{D}_{j,l,2}(s,t)\frac{ds}{s}\frac{dt}{t} \Big |\lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}\Vert g\Vert _{L^{2}(\mathbb {R}^d)}. \end{aligned}$$

(2.15)

We now estimate term corresponding to $\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}\mathrm{D}_{j,l,4}(s,t)\frac{ds}{s}\frac{dt}{t}$. Write

$$\begin{aligned}&\Big |\sum _{j}\sum _l\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}\mathrm{D}_{j,l,4}(s,t)\frac{ds}{s}\frac{dt}{t}\Big |\\&\quad \le \Big \Vert \Big (\sum _j\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}|Q_sQ_t^3f|^2\log ^{-\sigma }(2^j/s+1)\frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}} \Big \Vert _{L^2(\mathbb {R}^d)}\\&\qquad \times \Big \Vert \Big (\sum _j\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}\Big |Q_t\Big (\sum _la_{j,l} W_{\Omega ,j}Q_sh_{s,j,l}\Big )\Big |^2\log ^{\sigma } \Big (\frac{2^j}{s}+1\Big )\frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}} \Big \Vert _{L^{2}(\mathbb {R}^d)}\\&\quad :=\mathrm{I}_3\mathrm{I_4}. \end{aligned}$$

Obviously,

$$\begin{aligned} \mathrm{I}_3\lesssim \Big \Vert \Big (\int _{0}^{\infty }\int _{0}^{\infty }|Q_sQ_t^3f|^2\frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}} \Big \Vert _{L^2(\mathbb {R}^d)}\lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

On the other hand, it follows from Littlewood–Paley theory and Lemma 2.2 that

$$\begin{aligned} \mathrm{I}_4\lesssim & {} \Big (\sum _j\int _{0}^{2^j}\Big \Vert \sum _la_{j,l} W_{\Omega ,\,j}Q_sh_{s,j,l}\Big \Vert ^2_{L^2(\mathbb {R}^d)}\log ^{\sigma } \Big (\frac{2^j}{s}+1\Big )\frac{ds}{s}\Big )^{\frac{1}{2}}\\\lesssim & {} \Big (\sum _j\int _0^{2^j}2^{2j}\sum _l\Vert W_{\Omega ,j}Q_sh_{s,j,l}\Vert ^2_{L^2(\mathbb {R}^d)}\log ^{\sigma } \Big (\frac{2^j}{s}+1\Big )\frac{ds}{s}\Big )^{\frac{1}{2}}\\\lesssim & {} \Big (\sum _j\int _0^{2^j}\sum _l\Vert h_{s,j,l}\Vert ^2_{L^2(\mathbb {R}^d)}\log ^{-2\beta +\sigma } \Big (\frac{2^j}{s}+1\Big )\frac{ds}{s}\Big )^{\frac{1}{2}}\lesssim \Vert g\Vert _{L^2(\mathbb {R}^d)}, \end{aligned}$$

since $\Vert a_{j,l}\Vert _{L^{\infty }(\mathbb {R}^d)}\lesssim 2^j$, and the supports of functions $\{a_{j,l} W_{\Omega ,\,j}Q_sh_{s,j,l}\}$ have bounded overlaps. The estimate for $\mathrm{I}_4$, together with the estimate for $\mathrm{I}_3$, gives us that

$$\begin{aligned} \Big |\sum _{j}\sum _l\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}\mathrm{D}_{j,l,4}(s,t)\frac{ds}{s}\frac{dt}{t}\Big |\lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}\Vert g\Vert _{L^{2}(\mathbb {R}^d)}. \end{aligned}$$

(2.16)

Combining inequalities (2.12), (2.13), (2.15) and (2.16) leads to that

$$\begin{aligned} |\mathrm{D}_2|\lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}\Vert g\Vert _{L^{2}(\mathbb {R}^d)}. \end{aligned}$$

The estimate for $\mathrm{D}_1$ is fairly similar to the estimate $\mathrm{D}_2$. For example, since

$$\begin{aligned} \int _{0}^{t}\sum _{j:\, 2^j\ge t}2^{-j}s\frac{ds}{s}\lesssim 1, \,\, \int _{s}^{\infty }\sum _{j:2^j\ge t}2^{-j}s\frac{dt}{t}\lesssim 1, \end{aligned}$$

we have that

$$\begin{aligned}&\Big |\sum _{j}\sum _l\int _0^{2^j}\int _{0}^{t}\mathrm{D}_{j,l,1}(s,t)\frac{ds}{s}\frac{dt}{t} \Big |\\&\quad \le \Big \Vert \Big (\sum _j\sum _l\int _0^{2^j}\int _{0}^{t}|\chi _{I_{j,l}^*}Q_t^4f|^2 2^{-j}s \frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}}\Big \Vert _{L^2(\mathbb {R}^d)}\\&\qquad \times \Big \Vert \Big (\sum _j\sum _l\int _0^{2^j}\int _{0}^{\infty } |W_{\Omega ,j}[a_{j,l},Q_s]Q_sh_{s,j,l} |^2(2^{-j}s)^{-1} \frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}}\Big \Vert _{L^{2}(\mathbb {R}^d)}\\&\quad \lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}\Vert g\Vert _{L^{2}(\mathbb {R}^d)}. \end{aligned}$$

The estimates for terms $\sum _{j}\sum _l\int _0^{2^j}\int _{0}^{t}\mathrm{D}_{j,l,i}(s,t)\frac{ds}{s}\frac{dt}{t}$ $(i=2,3,4)$ are parallel to the estimates for $\sum _{j}\sum _l\int _{2^j}^{\infty }\int _{0}^{(2^jt^{\alpha -1})^{1/\alpha }}\mathrm{D}_{j,l,i}(s,t)\frac{ds}{s}\frac{dt}{t}$. Altogether, we have that

$$\begin{aligned} |\mathrm{D}_1|\lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}\Vert g\Vert _{L^{2}(\mathbb {R}^d)}. \end{aligned}$$

It remains to consider $\mathrm{D}_3$. This was essentially proved in [10, pp. 1281–1283]. For the sake of self-contained, we present the details here. Set

$$\begin{aligned} h(x,y)=\int \int \psi _s(x-z)\sum _{j:2^j\le s^\alpha t^{1-\alpha }}K_j(z,u)[\psi _t(u-y)-\psi _t(x-y)]dudz. \end{aligned}$$

Let H be the operator with integral kernel h. It then follows that

$$\begin{aligned} |\mathrm{D}_3|\lesssim & {} \Big |\int _0^\infty \int _0^t\int _{\mathbb {R}^d} HQ_t^3f(x)Q_s^3g(x)dx\frac{ds}{s}\frac{dt}{t}\Big |\\&+\Big |\sum _{j\in \mathbb {Z}}\int ^{\infty }_{2^j}\int _{(2^jt^{\alpha -1})^{\frac{1}{\alpha }}}^t \int _{\mathbb {R}^d}(Q_s{T}_{\Omega ,\,a}^{j}1)(x)Q_t^4f(x)Q_s^3g(x)dx\frac{ds}{s}\frac{dt}{t}\Big |\\= & {} |\mathrm{D}_{31}|+|\mathrm{D}_{32}|. \end{aligned}$$

As in [10, p. 1282], we obtain by Lemma 2.1 and the mean value theorem that

$$\begin{aligned} |h(x,y)|\lesssim \Big (\frac{s}{t}\Big )^{\varrho } t^{-d}\chi _{\{(x,y):|x-y|\le Ct\}}(x,y), \end{aligned}$$

where $\varrho =(d+2)\alpha -d-1\in (0,\,1)$. Then we have

$$\begin{aligned} |HQ_t^3f(x)|\lesssim \Big (\frac{s}{t}\Big )^{\varrho } M(Q_t^3f)(x), \end{aligned}$$

and

$$\begin{aligned} |\mathrm{D}_{31}|\lesssim & {} \int _0^\infty \int _0^t\int _{\mathbb {R}^d}|M(Q_t^3f)(x)||Q_s^3g(x)|dx\Big (\frac{s}{t}\Big )^{\varrho }\frac{ds}{s}\frac{dt}{t}\\\lesssim & {} \Big \Vert \Big (\int _0^\infty \int _0^t|M(Q_t^3f)|^{2}\Big (\frac{s}{t}\Big )^{\varrho } \frac{ds}{s}\frac{dt}{t}\Big )^{\frac{1}{2}}\Big \Vert _{L^2(\mathbb {R}^d)}\\&\times \Big \Vert \Big (\int _0^\infty \int _s^\infty |Q_s^3g|^{2}\Big (\frac{s}{t}\Big )^{\varrho }\frac{dt}{t}\frac{ds}{s} \Big )^{\frac{1}{2}}\Big \Vert _{L^{2}(\mathbb {R}^d)}\\\lesssim & {} \Big \Vert \Big (\int _0^\infty |M(Q_t^3f)|^2\frac{dt}{t}\Big )^{\frac{1}{2}}\Big \Vert _{L^2(\mathbb {R}^d)} \Big \Vert \Big (\int _0^\infty |Q_s^3g|^{2} \frac{ds}{s} \Big )^{\frac{1}{2}}\Big \Vert _{L^{2}(\mathbb {R}^d)}\\\lesssim & {} \Vert f\Vert _{L^2(\mathbb {R}^d)}\Vert g\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

As for $\mathrm{D}_{32}$, we split it into three parts as follows:

$$\begin{aligned} \mathrm{D}_{32}=\sum _{j\in \mathbb {Z}}\int ^{\infty }_{0}\int _{0}^t- \sum _{j\in \mathbb {Z}}\int ^{2^j}_{0}\int _{0}^t-\sum _{j\in \mathbb {Z}} \int _{2^j}^\infty \int _{0}^{(2^jt^{\alpha -1})^{\frac{1}{\alpha }}}=\mathrm{D}_{321}-\mathrm{D}_{322}-\mathrm{D}_{323}. \end{aligned}$$

Let

$$\begin{aligned} \zeta (x)=\int ^{\infty }_1\psi _t*\psi _t*\psi _t*\psi _t(x)\frac{dt}{t},\,\,P_s=\int _s^\infty Q_t^4\frac{dt}{t}. \end{aligned}$$

Han and Sawyer [9] proved that $\zeta$ is a radial function which is supported on a ball having radius C and has mean value zero. Observe that $P_sf(x)=\zeta _s*f(x)$ with $\zeta _s(x)=s^{-d}\zeta (s^{-1}x)$. The Littlewood–Paley theory tells us that

$$\begin{aligned} \Big \Vert \Big (\int ^{\infty }_0|P_sf|^2\frac{ds}{s}\Big )^{\frac{1}{2}}\Big \Vert _{L^2(\mathbb {R}^d)}\lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

(i) of Lemma 2.3 states that $T_{\Omega ,a}1\in \mathrm{BMO}(\mathbb {R}^d)$. Recall that $\mathrm{supp}\,\psi \subset B(0,\,1)$ and $\psi$ has integral zero. Thus for $x\in \mathbb {R}^d$,

$$\begin{aligned} |Q_s(T_{\Omega ,a}1)(x)|\le s^{-d}\int _{|x-y|\le s}|\psi (s^{-1}(x-y))||T_{\Omega ,a}1(y)-\langle T_{\Omega ,a}1\rangle _{B(x,s)}|dy\lesssim 1, \end{aligned}$$

where $\langle T_{\Omega ,a}1\rangle _{B(x,s)}$ denotes the mean value of $T_{\Omega ,a}1$ on the ball centered at x and having radius s. Therefore,

$$\begin{aligned} |\mathrm{D}_{321}|= & {} \Big |\int _{\mathbb {R}^d}\int _0^\infty Q_sT_{\Omega ,a}1(x)P_sf(x)Q_s^3g(x)\frac{ds}{s}dx\Big |\\\lesssim & {} \Big \Vert \Big (\int ^{\infty }_0|P_sf|^2\frac{ds}{s}\Big )^{\frac{1}{2}}\Big \Vert _{L^2(\mathbb {R}^d)} \Big \Vert \Big (\int ^{\infty }_0|Q_s^3g|^{2}\frac{ds}{s}\Big )^{\frac{1}{2}}\Big \Vert _{L^{2}(\mathbb {R}^d)}\\\lesssim & {} \Vert f\Vert _{L^2(\mathbb {R}^d)}\Vert g\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

From (ii) of Lemma 2.3 and Hölder’s inequality, we obtain that

$$\begin{aligned} |\mathrm{D}_{322}|\lesssim & {} \Big \Vert \Big (\sum _j\int _{0}^{2^j}\int _0^t 2^{-j}s|Q_t^4f|^2\frac{ds}{s}\frac{dt}{t}\Big )^{1/2}\Big \Vert _{L^2(\mathbb {R}^d)}\\&\times \Big \Vert \Big (\sum _j\int _0^{2^j}\int _0^t 2^{-j}s|Q_s^3g|^{2}\frac{ds}{s}\frac{dt}{t}\Big )^{1/2}\Big \Vert _{L^2(\mathbb {R}^d)}\\\lesssim & {} \Vert f\Vert _{L^2(\mathbb {R}^d)}\Vert g\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

The same result holds true for $\mathrm{D}_{323}$. Combining the estimates for terms $\mathrm{D}_{321}$, $\mathrm{D}_{322}$ and $\mathrm{D}_{323}$ give us that

$$\begin{aligned} |\mathrm{D}_3|\lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}\Vert g\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

This leads to (2.7) and then establishes the $L^2(\mathbb {R}^d)$ boundedness of $T_{\Omega ,a}$. $\square$

3 Proof of Theorem 1.4: $L^p$ Boundedness

We begin with some lemmas.

Lemma 3.1

Let $\varpi \in C^{\infty }_0(\mathbb {R}^d)$ be a radial function such that $\mathrm{supp}\, \varpi \subset \{1/4\le |\xi |\le 4\}$ and

$$\begin{aligned} \sum _{l\in \mathbb {Z}}\varpi ^3(2^{-l}\xi )=1,\,\,\,|\xi |>0, \end{aligned}$$

and $S_l$ be the multiplier operator defined by

$$\begin{aligned} \widehat{S_lf}(\xi )=\varpi (2^{-l}\xi ){\widehat{f}}(\xi ). \end{aligned}$$

Let $k\in \mathbb {Z}_+$, a be a function on $\mathbb {R}^d$ such that $\nabla a\in L^{\infty }(\mathbb {R}^d)$. Then

$$\begin{aligned} \Big \Vert \Big (\sum _{l\in \mathbb {Z}}\big |2^{kl}[a,\,S_l]^kf\big |^2\Big )^{\frac{1}{2}}\Big \Vert _{L^2(\mathbb {R}^d)}\lesssim \Vert f\Vert _{L^2(\mathbb {R}^d)}, \end{aligned}$$

(3.1)

and

$$\begin{aligned} \Big \Vert \sum _{l\in \mathbb {Z}}2^{kl}[a,\,S_l]^kf_l\Big \Vert _{L^2(\mathbb {R}^d)}\lesssim \Big \Vert \Big (\sum _{l}|f_l|^2\Big )^{1/2}\Big \Vert _{L^2(\mathbb {R}^d)}, \end{aligned}$$

(3.2)

where and in the following, for a locally integrable function a and an operator U, $[a,\,U]^0f=Uf$, while for $k\in \mathbb {N}$ $[a,\,U]^k$ denotes the commutator of $[a,\,U]^{k-1}$ and a, defined as (2.9).

Note that (3.2) follows from (3.1) and a duality argument. For the case of $k=0$, (3.1) follows from Littlewood–Paley theory. Inequality (3.1) with $k=1$ was proved in [3, Lemma 2.3], while for the case of $k\ge 2$, the proof of (3.1) is similar to the proof of [3, Lemma 2.3].

Lemma 3.2

Let $k\in \mathbb {N}$, $n\in \mathbb {Z}_+$ with $n\le k$, D, E be positive constants and $E\le 1$, m be a multiplier such that $m\in L^1(\mathbb {R}^d)$, and

$$\begin{aligned} \Vert m\Vert _{L^{\infty }(\mathbb {R}^d)}\le D^{-k}E \end{aligned}$$

and for all multi-indices $\gamma \in \mathbb {Z}_+^d$,

$$\begin{aligned} \Vert \partial ^{\gamma }m\Vert _{L^{\infty }(\mathbb {R}^d)}\le D^{|\gamma |-k}. \end{aligned}$$

Let a be a function on $\mathbb {R}^d$ with $\nabla a\in L^{\infty }(\mathbb {R}^d)$, and $T_m$ be the multiplier operator defined by

$$\begin{aligned} \widehat{T_{m}f}(\xi ) = m(\xi ){\widehat{f}}(\xi ). \end{aligned}$$

Then for any $\varepsilon \in (0,\,1)$,

$$\begin{aligned} \Vert [a,\,T_{m}]^nf\Vert _{L^2(\mathbb {R}^d)}\lesssim D^{n-k}E^{\varepsilon }\Vert f\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

Proof

Our argument here is a generalization of the proof of Lemma 2 in [11], together with some more refined estimates, see also [12, Lemma 2.3] for the original version. We only consider the case $1\le n\le k$, since

$$\begin{aligned} \Vert [a,\,T_{m}]^0f\Vert _{L^2(\mathbb {R}^d)}\lesssim D^{-k}E^{\varepsilon }\Vert f\Vert _{L^2(\mathbb {R}^d)} \end{aligned}$$

holds obviously.

Let $\varphi \in C^{\infty }_0(\mathbb {R}^d)$ be the same as in (2.1). Recall that $\mathrm{supp}\,\varphi \subset \{1/4\le |x|\le 4\}$, and

$$\begin{aligned} \sum _{j\in \mathbb {Z}}\varphi (2^{-j}x)\equiv 1,\,\,|x|>0. \end{aligned}$$

Let $\varphi _{l,D}(x)=\varphi (2^{-l}D^{-1}x)$ for $l\in \mathbb {Z}$. Set

$$\begin{aligned} W_{l}(x)=K(x)\varphi _{l,D}(x),\,\,l\in \mathbb {Z}, \end{aligned}$$

where K is the inverse Fourier transform of m. Observing that for all multi-indices $\gamma \in \mathbb {Z}_+^d$, $\partial ^{\gamma }\varphi (0)=0$, we thus have that

$$\begin{aligned} \int _{\mathbb {R}^d}{\widehat{\varphi }}(\xi )\xi ^{\gamma }d\xi =0. \end{aligned}$$

This, in turn, implies that for all $N\in \mathbb {N}$ and $\xi \in \mathbb {R}^d$,

$$\begin{aligned} |\widehat{W_l}(\xi )|= & {} \Big |\int _{\mathbb {R}^d}\Big (m(\xi -\frac{\eta }{2^lD})-\sum _{|\gamma |\le N}\frac{1}{\gamma !}\partial ^{\gamma }m(\xi )\Big (\frac{\eta }{2^lD}\Big )^{\gamma }\Big ){\widehat{\varphi }}(\eta )d\eta \Big |\nonumber \\\lesssim & {} 2^{-l(N+1)}D^{-(N+1)}\sum _{|\gamma |=N+1}\Vert \partial ^{\gamma }m\Vert _{L^{\infty }(\mathbb {R}^d)} \int _{\mathbb {R}^d}|\eta |^{N+1}|{\widehat{\varphi }}(\eta )|d\eta \nonumber \\\lesssim & {} 2^{-l(N+1)}D^{-k}. \end{aligned}$$

(3.3)

On the other hand, a trivial computation gives that for $l\in \mathbb {Z}$,

$$\begin{aligned} \Vert \widehat{W_l}\Vert _{L^{\infty }(\mathbb {R}^d)}\le \Vert m\Vert _{L^{\infty }(\mathbb {R}^d)}\Vert \widehat{\varphi _{l,D}}\Vert _{L^1(\mathbb {R}^d)}\lesssim D^{-k}E. \end{aligned}$$

(3.4)

Combining the inequalities (3.3) and (3.4) shows that for any $l\in \mathbb {Z}$, $N\in \mathbb {N}$ and $\varepsilon \in (0,\,1)$,

$$\begin{aligned}&\Vert \widehat{W_l}\Vert _{L^{\infty }(\mathbb {R}^d)}\lesssim 2^{-l(N+1)(1-\varepsilon )}D^{-k} E^{\varepsilon }. \end{aligned}$$

(3.5)

Let $T_{m,l}$ be the convolution operator with kernel $W_l$. Inequality (3.5), via Plancherel’s theorem, tells us that for $l\in \mathbb {Z}$ and $N\in \mathbb {N}$,

$$\begin{aligned} \Vert T_{m,l}f\Vert _{L^2(\mathbb {R}^d)}\lesssim 2^{-l(N+1)(1-\varepsilon )} D^{-k} E^{\varepsilon }\Vert f\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

(3.6)

We claim that for all $l\in \mathbb {Z}$, $N\in \mathbb {N}$ and $\varepsilon \in (0,\,1)$,

$$\begin{aligned} \Vert [a,\,T_{m,l}]^nf\Vert _{L^2(\mathbb {R}^d)}\lesssim 2^{-l(N+1)(1-\varepsilon )+ln}D^{n-k}E^{\varepsilon }\Vert f\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

(3.7)

Observe that $\mathrm{supp}\,W_l\subset \{x:\,|x|\le D2^{l+2}\}$. If I is a cube having side length $2^lD$, and $f\in L^2(\mathbb {R}^d)$ with $\mathrm{supp}\,f\subset I$, then $T_{m,l}f\subset 100dI$. Therefore, to prove (3.7), we may assume that $\mathrm{supp}\, f\subset I$ with I a cube having side length $2^lD$. Let $x_0\in I$ and $a_I(y)=(a(y)-a(x_0))\chi _{100dI}(y)$. Then

$$\begin{aligned} \Vert a_I\Vert _{L^{\infty }(\mathbb {R}^d)}\lesssim 2^lD. \end{aligned}$$

Write

$$\begin{aligned} {[}a,\,T_{m,l}]^nf(x)=\sum _{i=0}^n(a_I(x))^iC_n^i T_{m,l}\big ((-a_I)^{k-i}f\big )(x). \end{aligned}$$

It then follows from (3.6) that

$$\begin{aligned} \Vert [a,\,T_{m,\,l}]^nf\Vert _{L^2(\mathbb {R}^d)}\lesssim & {} \sum _{i=0}^n2^{il}D^i\Vert T_{m,l}\big ((-a_I)^{n-i}f\big )\Vert _{L^2(\mathbb {R}^d)} \\\lesssim & {} 2^{nl-l(N+1)(1-\varepsilon )}D^{n-k} E^{\varepsilon }\Vert f\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

This yields (3.7).

We now conclude the proof of Lemma 3.2. Recall that $E\in (0,\,1]$. It suffices to prove Lemma 3.2 for the case of $\varepsilon \in (2/3,\,1)$. For fixed $\varepsilon \in (2/3,\,1)$, we choose $N_1\in \mathbb {N}$ such that $(N_1+1)(1-\varepsilon )>n$, $N_2\in \mathbb {N}$ such that $(N_2+1)(1-\varepsilon )<n$. It follows from (3.7) that

$$\begin{aligned} \Vert [a,\,T_{m}]^nf\Vert _{L^2(\mathbb {R}^d)}\le & {} \sum _{l\le 0}\Vert [a,\,T_{m,\,l}]^nf\Vert _{L^2(\mathbb {R}^d)} + \sum _{l\in \mathbb {N}}\Vert [a,\,T_{m,l}]^nf\Vert _{L^2(\mathbb {R}^d)}\\\lesssim & {} D^{n-k}E^{\varepsilon }\sum _{l\in \mathbb {N}} 2^{-l(N_1+1)(1-\varepsilon )+ln}\Vert f\Vert _{L^2(\mathbb {R}^d)}\\&+D^{n-k}E^{\varepsilon }\sum _{l\le 0} 2^{-l(N_2+1)(1-\varepsilon )+ln}\Vert f\Vert _{L^2(\mathbb {R}^d)}\\\lesssim & {} D^{n-k}E^{\varepsilon }\Vert f\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

This completes the proof of Lemma 3.2. $\square$

Lemma 3.3

Let $k\in \mathbb {N}$, $n\in \mathbb {Z}_+$ with $n\le k$, D, A and B be positive constants with $A,\,B<1$, m be a multiplier such that $m\in L^1(\mathbb {R}^d)$, and

$$\begin{aligned} \Vert m\Vert _{L^{\infty }(\mathbb {R}^d)}\le D^{-k}(AB)^{k+1}, \end{aligned}$$

and for all multi-indices $\gamma \in \mathbb {Z}^d_+$,

$$\begin{aligned} \Vert \partial ^{\gamma }m\Vert _{L^{\infty }(\mathbb {R}^d)}\le D^{|\gamma |-k}B^{-|\gamma |}. \end{aligned}$$

Let $T_m$ be the multiplier operator defined by

$$\begin{aligned} \widehat{T_{m}f}(\xi ) = m(\xi ){\widehat{f}}(\xi ). \end{aligned}$$

Let a be a function on $\mathbb {R}^d$ such that $\nabla a\in L^{\infty }(\mathbb {R}^d)$. Then for any $\sigma \in (0,\,1)$,

$$\begin{aligned} \big \Vert [a,\,T_{m}]^nf\big \Vert _{L^2(\mathbb {R}^d)}\lesssim D^{n-k}A^{\sigma }B^{k-n+\sigma }\Vert f\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

(3.8)

Proof

Let $T_{m,l}$ be the same as in the proof of Lemma 3.2. As in the proof of Lemma 3.2, we know that for all $l\in \mathbb {Z}$, $N\in \mathbb {N}$ and $\varepsilon \in (0,\,1)$,

$$\begin{aligned} \big \Vert [a,\,T_{m,l}]^nf\big \Vert _{L^2(\mathbb {R}^d)}\lesssim & {} 2^{-l(N+1)(1-\varepsilon )+nl}D^{n-k}\nonumber \\&\times B^{-(N+1)(1-\varepsilon )+(k+1)\varepsilon }A^{(k+1)\varepsilon }\Vert f\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

(3.9)

For each fixed $\sigma \in (0,\,1)$, we choose $\varepsilon \in (0,\,1)$ such that

$$\begin{aligned} (k+1)\varepsilon -k-\sigma >1-\varepsilon , \end{aligned}$$

and choose $N_1\in \mathbb {N}$ such that

$$\begin{aligned} (N_1+1)(1-\varepsilon )>n,\,\, -(N_1+1)(1-\varepsilon )+(k+1)\varepsilon >k-n+\sigma . \end{aligned}$$

Also, we choose $N_2\in \mathbb {N}$ such that $(N_2+1)(1-\varepsilon )<n$. Note that such a $N_2$ satisfies

$$\begin{aligned} -(N_2+1)(1-\varepsilon )+(k+1)\varepsilon >k-n+\sigma . \end{aligned}$$

Recalling that $B<1$, we have that

$$\begin{aligned} B^{-(N_1+1)(1-\varepsilon )+(k+1)\varepsilon }\le B^{k-n+\sigma },\,\,B^{-(N_2+1)(1-\varepsilon )+(k+1)\varepsilon } \le B^{k-n+\sigma }. \end{aligned}$$

Our desired estimate (3.8) now follows (3.9) by

$$\begin{aligned} \Vert [a,\,T_{m}]^nf\Vert _{L^2(\mathbb {R}^d)}\lesssim & {} D^{n-k}A^{\sigma }B^{k-n+\sigma }\sum _{l\in \mathbb {N}} 2^{-l(N_1+1)(1-\varepsilon )+ln}\Vert f\Vert _{L^2(\mathbb {R}^d)}\\&+D^{n-k}A^{\sigma }B^{k-n+\sigma }\sum _{l\le 0} 2^{-l(N_2+1)(1-\varepsilon )+ln}\Vert f\Vert _{L^2(\mathbb {R}^d)}\\\lesssim & {} D^{n-k}B^{k-n+\sigma }A^{\sigma }\Vert f\Vert _{L^2(\mathbb {R}^d)}, \end{aligned}$$

since $(k+1)\varepsilon >\sigma$ and $A<1$. This completes the proof of Lemma 3.3. $\square$

The following conclusion is a variant of Theorem 1 in [11], and will be useful in the proof of Theorem 1.4.

Theorem 3.4

Let $k\in \mathbb {N}$, $A\in (0,\,1/2)$ be a constant, $\{\mu _j\}_{j\in \mathbb {Z}}$ be a sequence of functions on $\mathbb {R}^d\backslash \{0\}$. Suppose that for some $\beta \in (1,\,\infty )$,

$$\begin{aligned} \Vert \mu _j\Vert _{L^1(\mathbb {R}^d)}\lesssim 2^{-jk},\, |\widehat{\mu _j}(\xi )|\lesssim 2^{-jk}\min \{|A2^j\xi |^{k+1},\,\log ^{-\beta }(2+|2^{j}\xi |)\}, \end{aligned}$$

and for all multi-indices $\gamma \in \mathbb {Z}_+^d$,

$$\begin{aligned} \Vert \partial ^{\gamma }\widehat{\mu _j}\Vert _{L^{\infty }(\mathbb {R}^d)}\lesssim 2^{j(|\gamma |-k)}. \end{aligned}$$

Let $K(x)=\sum _{j\in \mathbb {Z}}\mu _j(x)$ and T be the convolution operator with kernel K. Then for any $\varepsilon \in (0,\,1)$, function a on $\mathbb {R}^d$ with $\nabla a\in L^{\infty }(\mathbb {R}^d)$,

$$\begin{aligned} \Vert [a,\,T]^kf\Vert _{L^2(\mathbb {R}^d)}\lesssim \log ^{-\varepsilon \beta +1}\big (\frac{1}{A}\big )\Vert f\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

Proof

At first, we claim that for $k_1\in \mathbb {Z}$ with $0\le k_1\le k$,

$$\begin{aligned} \Vert Tf\Vert _{L^2_{k_1-k}(\mathbb {R}^d)}\lesssim \Vert f\Vert _{L^2_{k_1}(\mathbb {R}^d)}, \end{aligned}$$

(3.10)

where $\Vert f\Vert _{L^2_{k_w}(\mathbb {R}^d)}$ for $k_2\in \mathbb {Z}$ is the Sobolev norm defined as

$$\begin{aligned} \Vert f\Vert _{L^2_{k_2}(\mathbb {R}^d)}^2=\int _{\mathbb {R}^d}|\xi |^{2k_2}|{\widehat{f}}(\xi )|^2d\xi . \end{aligned}$$

In fact, by the Fourier transfrom estimate of $\mu _j$, we have that for each fixed $\xi \in \mathbb {R}^d\backslash \{0\}$,

$$\begin{aligned} \sum _{j\in \mathbb {Z}}|\widehat{\mu _j}(\xi )|\lesssim \sum _{j:\, 2^j\ge |\xi |^{-1}}2^{-jk}+|\xi |^{k+1}\sum _{j:\,2^j\le |\xi |^{-1}}2^{j}\lesssim |\xi |^k. \end{aligned}$$

This, together with Plancherel’s theorem, gives (3.10).

Let $U_j$ be the convolution operator with kernel $\mu _j$, and $\varpi \in C^{\infty }_0(\mathbb {R}^d)$ such that $0\le \varpi \le 1$, $\mathrm{supp}\,\varpi \subset \{1/4\le |\xi |\le 4\}$ and

$$\begin{aligned} \sum _{l\in \mathbb {Z}}\varpi ^3(2^{-l}\xi )=1,\,\,|\xi |>0. \end{aligned}$$

Set $m_j(x)=\widehat{\mu _j}(\xi )$, and $m_j^l(\xi )=m_j(\xi )\varpi (2^{j-l}\xi )$. Define the operator $U_j^l$ by

$$\begin{aligned} \widehat{U_j^lf}(\xi )=m_j^l(\xi )\varpi (2^{j-l}\xi ){\widehat{f}}(\xi ). \end{aligned}$$

Now let $S_l$ be the multiplier operator defined as in Lemma 3.1. Let $f\in C^{\infty }_0(\mathbb {R}^d)$, $B=B(0,\,R)$ be a ball such that $\mathrm{supp}\,f\subset B$, and let $x_0\in B$. We can write

$$\begin{aligned} \quad [a,\,T]^kf= & {} \sum _{n=0}^kC_k^n(a-a(x_0))^{k-n}T\big ((a(x_0)-a)^nf)(x)\nonumber \\= & {} \sum _{n=0}^kC_k^n(a-a(x_0))^{k-n}\sum _{l}\sum _j(S_{l-j}U_{j}^lS_{l-j})\big ((a(x_0)-a)^nf)\nonumber \\= & {} \sum _{l}\sum _j[a,\,S_{l-j}U_{j}^lS_{l-j}]^kf. \end{aligned}$$

(3.11)

We now estimate $\big \Vert [a,\,S_{l-j}U_j^lS_{l-j}]^kf\big \Vert _{L^2(\mathbb {R}^d)}.$ At first, we have that $m_j^l\in L^1(\mathbb {R}^d)$ and

$$\begin{aligned} |m_j^l(\xi )|\lesssim 2^{-jk}\min \{A^{k+1}2^{l(k+1)},\,\log ^{-\beta } (2+2^l)\}. \end{aligned}$$

Furthermore, by the fact that

$$\begin{aligned} |\partial ^{\gamma }\phi (2^{j-l}\xi )|\lesssim 2^{(j-l)|\gamma |},\,\,|\partial ^{\gamma }m_j(\xi )|\lesssim 2^{j(|\gamma |-k)}, \end{aligned}$$

it then follows that for all $\gamma \in \mathbb {Z}_+^d$,

$$\begin{aligned} |\partial ^{\gamma }m_j^l(\xi )|\lesssim \Big \{\begin{array}{ll}2^{j(|\gamma |-k)}\,\,&{}\hbox {if}\,\,l\in \mathbb {N}\\ 2^{j(|\gamma |-k)}2^{-|\gamma |l},\,\,&{}\hbox {if}\,\,l\le 0.\end{array} \end{aligned}$$

An application of Lemma 3.2 (with $D=2^j$, $E=\min \{(A2^l)^{k+1},\,l^{-\beta }\}$) yields

$$\begin{aligned} \Vert [a,\,U_j^l]^nf\Vert _{L^2(\mathbb {R}^d)}\lesssim 2^{j(n-k)}\min \{(A2^l)^{k+1},\, l^{-\beta }\}^{\varepsilon }\Vert f\Vert _{L^2(\mathbb {R}^d)},\,\,l\in \mathbb {N}. \end{aligned}$$

(3.12)

On the other hand, we deduce from Lemma 3.3 (with $D=2^j$ and $B=2^l$) that for some $\sigma \in (0,\,1)$,

$$\begin{aligned} \Vert [a,\,U_j^l]^nf\Vert _{L^2(\mathbb {R}^d)}\lesssim 2^{j(n-k)}2^{l(k-n)}A^{\sigma }2^{\sigma l}\Vert f\Vert _{L^2(\mathbb {R}^d)},\,\,l\le 0. \end{aligned}$$

(3.13)

Write

$$\begin{aligned} {[}a,\,S_{l-j}U_j^lS_{l-j}]^k=\sum _{n_1=0}^kC_{k}^{n_1}[a,\,S_{l-j}]^{n_1}\sum _{n_2=0}^{k-n_1}C_{k-n_1}^{n_2} [a,\,U_j^l]^{n_2}[a,\,S_{l-j}]^{k-n_1-n_2}. \end{aligned}$$

For fixed $n_1,\,n_2,\,n_3\in \mathbb {Z}_+$ with $n_1+n_2+n_3=k$, a standard computation involving Lemma 3.1, estimates (3.12) and (3.13) leads to that for $l\in \mathbb {N}$,

$$\begin{aligned}&\big \Vert \sum _{j\in \mathbb {Z}}[a,\,S_{l-j}]^{n_1}[a,\,U_j^l]^{n_2}[a,\,S_{l-j}]^{n_3} f\big \Vert ^2_{L^2(\mathbb {R}^d)}\\&\quad \lesssim \sum _{j\in \mathbb {Z}}2^{2(j-l)n_1}\Vert [a,\,U_{j}^l]^{n_2}[a,\,S_{l-j}]^{n_3}f\Vert ^2_{L^2 (\mathbb {R}^d)}\\&\quad \lesssim \min \{(A2^l)^{k+1},\, l^{-\beta }\}^{2\varepsilon } \Vert f\Vert ^2_{L^2(\mathbb {R}^d)}; \end{aligned}$$

and for $l\in \mathbb {Z}_-$,

$$\begin{aligned}&\big \Vert \sum _{j\in \mathbb {Z}}[a,\,S_{l-j}]^{n_1}[a,\,U_j^l]^{n_2}[a,\,S_{l-j}]^{n_3} f\big \Vert ^2_{L^2(\mathbb {R}^d)}\\&\quad \lesssim \sum _{j\in \mathbb {Z}}2^{2(j-l)n_1}\Vert [a,\,U_{j}^l]^{n_2}[a,\,S_{l-j}]^{n_3}f\Vert ^2_{L^2 (\mathbb {R}^d)}\\&\quad \lesssim A^{2\sigma }2^{2\sigma l} \Vert f\Vert ^2_{L^2(\mathbb {R}^d)}. \end{aligned}$$

Therefore,

$$\begin{aligned} \sum _{l}\Vert [a,\,S_{l-j}U_j^lS_{l-j}]^kf\Vert _{L^2(\mathbb {R}^d)}= & {} \sum _{l:\,l>\log (\frac{1}{\sqrt{A} })}\Vert [a,\,S_{l-j}U_j^lS_{l-j}]^kf\Vert _{L^2(\mathbb {R}^d)}\\&+\sum _{l:\,0\le l\le \log (\frac{1}{\sqrt{ A}})}\Vert [a,\,S_{l-j}U_j^lS_{l-j}]^kf\Vert _{L^2(\mathbb {R}^d)}\\&+\sum _{l:\,l<0}\Vert [a,\,S_{l-j}U_j^lS_{l-j}]^kf\Vert _{L^2(\mathbb {R}^d)}\\\lesssim & {} \Big (\sum _{l:\,l>\log (\frac{1}{\sqrt{ A}})}l^{-\varepsilon \beta }+A^{\sigma }\sum _{l:\,l<0}2^{\sigma l}\Big )\Vert f\Vert _{L^2(\mathbb {R}^d)}\\&+A^{(k+1)\varepsilon }\sum _{l:\,0\le l\le \log (\frac{1}{\sqrt{ A}})}2^{(k+1)l\varepsilon }\Vert f\Vert _{L^2(\mathbb {R}^d)}\\\lesssim & {} \log ^{-\varepsilon \beta +1}(\frac{1}{A})\Vert f\Vert ^2_{L^2(\mathbb {R}^d)}. \end{aligned}$$

This, via (3.11), leads to our desired conclusion. $\square$

Proof of Theorem 1.4

$L^p(\mathbb {R}^d)$ boundedness. By duality, it suffices to prove that $T_{\Omega ,a;\,k}$ is bounded on $L^p(\mathbb {R}^d)$ for $2<p<2\beta$.

For $j\in \mathbb {Z}$, let $K_j(x)=\frac{\Omega (x)}{|x|^{d+k}}\chi _{\{2^{j-1}\le |x|<2^j\}}(x)$. Let $\omega \in C^{\infty }_0(\mathbb {R}^d)$ be a nonnegative radial function such that

$$\begin{aligned} \mathrm{supp}\, \omega \subset \{x:\,|x|\le 1/4\},\,\,\,\int _{\mathbb {R}^d}\omega (x)dx=1, \end{aligned}$$

and

$$\begin{aligned} \int _{\mathbb {R}^d}x^{\gamma }\omega (x)dx=0,\,\,\,1\le |\gamma |\le k. \end{aligned}$$

For $j\in \mathbb {Z}$, set $\omega _j(x) = 2^{-dj}\omega (2^{-j}x)$. For a positive integer l, define

$$\begin{aligned} H_l(x)=\sum _{j\in \mathbb {Z}}K_j*\omega _{j-l}(x). \end{aligned}$$

Let $R_l$ be the convolution operator with kernel $H_l$. For a function a on $\mathbb {R}^d$ such that $\nabla a\in L^{\infty }(\mathbb {R}^d)$, recall that $[a,\,R_l]^k$ denotes the k-th commutator of $R_l$ with symbol a.

We claim that for each fixed $\varepsilon \in (0,\,1)$, $l\in \mathbb {N}$,

$$\begin{aligned} \Vert T_{\Omega , a;\,k}f-[a,\,R_{l}]^kf\Vert _{L^2(\mathbb {R}^d)}\lesssim l^{-\varepsilon \beta +1}\Vert f\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

(3.14)

To prove this, write

$$\begin{aligned} H_l(x)-\sum _{j\in \mathbb {Z}}K_j(x)=\sum _{j\in \mathbb {Z}}\big (K_j(x)-K_j*\omega _{j-l}(x)\big )=:\sum _{j\in \mathbb {Z}}\mu _{j,l}(x). \end{aligned}$$

By the vanishing moment of $\omega$, we know that for all multi-indices $\gamma \in \mathbb {Z}_+^d$ with $1\le |\gamma |\le k$, $\partial ^{\gamma }{\widehat{\omega }}(0)=0.$ By Taylor series expansion and the fact that ${\widehat{\omega }}(0)=1$, we deduce that

$$\begin{aligned} |{\widehat{\omega }}(2^{j-l}\xi )-1|\lesssim \min \{1,\,|2^{j-l}\xi |^{k+1}\}. \end{aligned}$$

When $\Omega \in GS_{\beta }(S^{d-1})$ for some $\beta \in (1,\,\infty )$, it was proved in [8, p. 458] that

$$\begin{aligned} |\widehat{K_j}(\xi )|\lesssim 2^{-jk}\min \{1,\,\log ^{-\beta }(2+|2^j\xi |)\}. \end{aligned}$$

Thus, the Fourier transform estimate

$$\begin{aligned}&|\widehat{\mu _{j,l}}(\xi )|=|\widehat{K_j}(\xi )||{\widehat{\omega }}(2^{j-l}\xi )-1|\lesssim 2^{-jk}\min \{\log ^{-\beta }(2+|2^j\xi |),\,|2^{j-l}\xi |^{k+1}\}\nonumber \\ \end{aligned}$$

(3.15)

holds true. On the other hand, a trivial computation shows that for all multi-indices $\gamma \in \mathbb {Z}_+^d$,

$$\begin{aligned} \Vert \partial ^{\gamma }\widehat{K_j}\Vert _{L^{\infty }(\mathbb {R}^{d})}\lesssim \Vert \Omega \Vert _{L^1(S^{d-1})}2^{(|\gamma |-k)j}, \end{aligned}$$

and so for all $\xi \in \mathbb {R}^d$,

$$\begin{aligned}&|\partial ^{\gamma }\widehat{\mu _{j,l}}(\xi )|\lesssim \sum _{\gamma _1+\gamma _2=\gamma }|\partial ^{\gamma _1}\widehat{K_j}(\xi )||\partial ^{\gamma _2}{\widehat{\omega }}(2^{j-l}\xi )|\lesssim \Vert \Omega \Vert _{L^1(S^{d-1})}2^{j(|\gamma |-k)}.\nonumber \\ \end{aligned}$$

(3.16)

The Fourier transforms (3.15) and (3.16), via Theorem 3.4 with $A=2^{-l}$, lead to (3.14) immediately.

Let $\varepsilon \in (0,\,1)$ be a constant which will be chosen later. An application of (3.14) gives us that

$$\begin{aligned} \big \Vert [a,\,R_{2^l}]^kf-[a,\, R_{2^{l+1}}]^kf\big \Vert _{L^2(\mathbb {R}^d)}\lesssim 2^{(-\varepsilon \beta +1)l}\Vert f\Vert _{L^2(\mathbb {R}^d)}. \end{aligned}$$

(3.17)

Therefore, the series

$$\begin{aligned} T_{\Omega ,a;\,k}=[a,\,R_2]^k+\sum _{l=1}^{\infty }([a,\,R_{2^{l+1}}]^k-[a,\,R_{2^{l}}]^k) \end{aligned}$$

(3.18)

converges in $L^2(\mathbb {R}^d)$ operator norm.

For $l\in \mathbb {N}$, let $L_l(x,\,y)=H_l(x-y)(a(x)-a(y))^k$. We claim that for any $y,\,y'\in \mathbb {R}^d$,

$$\begin{aligned}&\int _{|x-y|\ge 2|y-y'|}|L_l(x,y)-L_l(x,y')|dx\nonumber \\&\quad +\int _{|x-y|\ge 2|y-y'|}|L_l(y,x)-L_l(y',x)|dx\lesssim l. \end{aligned}$$

(3.19)

To prove this, let $|y-y'|=r$. A trivial computation yields

$$\begin{aligned} \int _{|x-y|\ge 2r}\big |H_l(x-y)(a(y)-a(y'))^k\big |dx\lesssim & {} r\sum _{j}\int _{|x|\ge 2r}|K_j*\omega _{j-l}(x)|dx\\\lesssim & {} r^k\sum _{j:\,2^{j-2}\ge r}\Vert K_j\Vert _{L^1(\mathbb {R}^d)}\Vert \omega _{j-l}\Vert _{L^1(\mathbb {R}^d)}\lesssim 1, \end{aligned}$$

since $\Vert K_j\Vert _{L^1(\mathbb {R}^d)}\lesssim 2^{-j}$. For each fixed $j\in \mathbb {Z}$, observe that

$$\begin{aligned} \Vert \omega _{j-l}(\cdot -y)-\omega _{j-l}(\cdot -y')\Vert _{L^1(\mathbb {R}^d)}\lesssim \min \{1,\,2^{l-j}|y-y'|\}. \end{aligned}$$

It then follows from Young’s inequality that

$$\begin{aligned}&\int _{|x-y\ge 2r}|H_l(x-y)-H_l(x-y')||a(x)-a(y)|^kdx\\&\quad =\sum _{n=1}^{\infty }\int _{2^nr\le |x-y\le 2^{n+1}r}|H_l(x-y)-H_l(x-y')||a(x)-a(y)|^kdx\\&\quad \lesssim \sum _{n=1}^{\infty }(2^nr)^k\sum _{j:\,2^j\approx 2^nr}\Vert K_j\Vert _{L^1(\mathbb {R}^d)} \Vert \omega _{j-l}(\cdot -y)-\omega _{j-l}(\cdot -y')\Vert _{L^1(\mathbb {R}^d)}\\&\quad \lesssim \sum _{k=1}^{\infty }\min \{1,\,2^{-k}2^{l}\}\lesssim l. \end{aligned}$$

Combining the estimates above gives us that

$$\begin{aligned}&\int _{|x-y|\ge 2|y-y'|}|L_l(x,y)-L_l(x,y')|dx\\&\quad \le \int _{|x-y|\ge 2r}\big |H_l(x-y)(a(y)-a(y'))^k\big |dx\\&\qquad +\int _{|x-y\ge 2r}|H_l(x-y)-H_l(x-y')||a(x)-a(y)|^kdx\lesssim l. \end{aligned}$$

Similarly, we can verify that

$$\begin{aligned} \int _{|x-y|\ge 2|y-y'|}|L_l(y,x)-L_l(y',x)|dx\lesssim l. \end{aligned}$$

This establishes (3.19).

Recall that $T_{\Omega ,\,a;k}$ is bounded on $L^2(\mathbb {R}^d)$. It follows from (3.14) that $[a,\,R_l]^k$ is also bounded on $L^2(\mathbb {R}^d)$ with bound independent of l. This, along with (3.19) and Calderón-Zygmud theory, tells us that

$$\begin{aligned} \big \Vert [a,\,R_l]^kf-[a,\, R_{l+1}]^kf\big \Vert _{L^p(\mathbb {R}^d)}\lesssim l\Vert f\Vert _{L^p(\mathbb {R}^d)},\,\,p\in (1,\,\infty ), \end{aligned}$$

and so

$$\begin{aligned} \big \Vert [a,\,R_{2^l}]^kf-[a,\, R_{2^{l+1}}]^kf\big \Vert _{L^p(\mathbb {R}^d)}\lesssim 2^l\Vert f\Vert _{L^p(\mathbb {R}^d)},\,\,p\in (1,\,\infty ). \end{aligned}$$

(3.20)

Interpolating inequalities (3.17) and (3.20) shows that for any $\varrho \in (0,\,1)$ and $p\in (2,\,\infty )$,

$$\begin{aligned} \big \Vert [a,\,R_{2^l}]^kf-[a,\, R_{2^{l+1}}]^kf\big \Vert _{L^p(\mathbb {R}^d)}\lesssim 2^{(-2\varepsilon \beta /p+1+\varrho )l}\Vert f\Vert _{L^p(\mathbb {R}^d)}. \end{aligned}$$

For each p with $2<p<2\beta$, we can choose $\varepsilon >0$ close to 1 sufficiently, and $\varrho >0$ close to 0 sufficiently, such that $2\varepsilon \beta /p-1-\varrho >0$. This, in turn, shows that

$$\begin{aligned} \sum _{l=1}^{\infty }\big \Vert [a,\,R_{2^l}]^kf-[a,\, R_{2^{l+1}}]^kf\big \Vert _{L^p(\mathbb {R}^d)}\lesssim \Vert f\Vert _{L^p(\mathbb {R}^d)}, \end{aligned}$$

and the series (3.18) converges in the $L^p(\mathbb {R}^d)$ operator norm. Therefore, $T_{\Omega ,\,a;k}$ is bounded on $L^p(\mathbb {R}^d)$ for $2<p<2\beta$. This finishes the proof of Theorem 1.4. $\square$

Remark 3.5

Let $\Omega$ be homogeneous of degree zero, integrable and have mean value zero on ${S}^{d-1}$, $T_{\Omega }$ be the homogeneous singular integral operator defined by (1.4). For $b\in \mathrm{BMO}(\mathbb {R}^d)$, define the commutator of $T_{\Omega }$ and b by

$$\begin{aligned} {[}b,\,T_{\Omega }]f(x)=b(x)T_{\Omega }f(x)-T_{\Omega }(bf)(x). \end{aligned}$$

When $\Omega \in \mathrm{Lip}_{\alpha }(S^{d-1})$ with $\alpha \in (0,\,1]$, Uchiyama [15] proved that $[b,\,T_{\Omega }]$ is a compact operator on $L^p(\mathbb {R}^d)$ ($p\in (1,\,\infty )$) if and only if $b\in \mathrm{CMO}(\mathbb {R}^d)$, where $\mathrm{CMO}(\mathbb {R}^d)$ is the closure of $C^{\infty }_0(\mathbb {R}^d)$ in the $\mathrm{BMO}(\mathbb {R}^d)$ topology, which coincide with the space of functions of vanishing mean oscillation. When $\Omega \in GS_{\beta }(S^{d-1})$ for $\beta \in (2,\,\infty )$, Chen and Hu [2] considered the compactness of $[b,\,T_{\Omega }]$ on $L^p(\mathbb {R}^d)$ with $\beta /(\beta -1)<p<\beta$. For other work about the compactness of $[b,\,T_{\Omega }]$, see [14] and the references therein. It is of interest to characterize the compactness of Calderón commutator $T_{\Omega ,\,a;\,k}$ on $L^p(\mathbb {R}^d)$ ($p\in (1,\,\infty )$). We will consider this in a forthcoming paper.