# Comptes Rendus
Mathématique

Chenyan Wang and Huoxiong Wu

A note on singular oscillatory integrals with certain rational phases

Volume 361 (2023), p. 363-370

https://doi.org/10.5802/crmath.418

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This article is licensed under the

CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.

http://creativecommons.org/licenses/by/4.0/

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In 2003, Folch-Gabayet and Wright [3] showed that for general rational phases, the estimate (1) is not true. Meanwhile, they considered the rational phases of the form $P(x)+1/Q(x)$, where P and Q are real polynomials with $Q(0) = 0$, and for $\Omega \in L \log L(S^{n-1})$, obtained the following estimate:

$$ \left| \text{p. v.} \int_{\mathbb{R}^n} e^{i(P(x)+1/Q(x))} K(x) dx \right| \le A, \quad (2) $$

where A depends on $\|\Omega\|_{L\log L(S^{n-1})}$, n and the degrees of P and Q, but not otherwise on the coefficients of P and Q. It is well known that

$$ L^{\infty}(S^{n-1}) \subsetneq \bigcup_{r>1} L^r(S^{n-1}) \subsetneq L \log L(S^{n-1}). $$

Therefore, the estimate (2) essentially improved and generalized the corresponding results in [6, 8, 9].

On the other hand, to study the mapping properties of singular integrals with rough kernels on $L^p(\mathbb{R}^n)$, Jiang and Lu introduced the following block spaces $B_q^{0,v}(S^{n-1})$ for $v > -1$ and $q > 1$ (see [5] for the details of block spaces).

**Definition 1 ([5]).** A q-block on $S^{n-1}$ is an $L^q$-function $b$ ($1 < q \le \infty$) that satisfies

$$ \text{supp}(b) \subseteq Q, \quad (i) $$

$$ \|b\|_{L^q(S^{n-1})} \le |Q|^{1/q-1}, \quad (ii) $$

where $Q = S^{n-1} \cap \{y \in \mathbb{R}^n : |y - \zeta| < \rho \text{ for some } \zeta \in S^{n-1} \text{ and } \rho \in (0, 1]\}$.

**Definition 2 ([5]).** For $v > -1$ and $q > 1$, the block spaces $B_q^{0,v}$ on $S^{n-1}$ are defined by

$$ B_q^{0,v}(S^{n-1}) = \{ \Omega \in L^1(S^{n-1}) : \Omega(y') = \sum_s \lambda_s b_s(y'), M_q^{0,v}(\{\lambda_s\}) < \infty \}. $$

where each $\lambda_s$ is a complex number, each $b_s$ is a q-block supported in $Q_s$,

$$ M_q^{0,v}(\{\lambda_s\}) = \sum_s |\lambda_s| \left\{ 1 + \left( \log^+ \frac{1}{|Q_s|} \right)^{1+v} \right\}, $$

and

$$ \|\Omega\|_{B_q^{0,v}(S^{n-1})} = \inf \left\{ M_q^{0,v}(\{\lambda_s\}) : \Omega(x') = \sum_s \lambda_s b_s(x') \right\}. $$

It is easy to check that

$$ B_q^{0,v_1}(S^{n-1}) \subsetneq B_q^{0,v_2}(S^{n-1}), \quad \forall v_1 > v_2 > -1. $$

Moreover, it follows from [4, 10] that for any $q > 1$,

$$ \bigcup_{r>1} L^r(S^{n-1}) \subsetneq B_q^{0,v}(S^{n-1}) \subset H^1(S^{n-1}) + L(\log^+ L)^{1+v}(S^{n-1}), \quad \forall v > -1, $$

and $B_q^{0,v}(S^{n-1}) \not\subsetneq L \log^+ L(S^{n-1})$ for any $v \in (-1, 0)$, in particular,

$$ \bigcup_{r>1} L^r(S^{n-1}) \subsetneq B_q^{0,0}(S^{n-1}), L \log^+ L(S^{n-1}) \subsetneq H^1(S^{n-1}), $$

but the relationship between $B_q^{0,0}(S^{n-1})$ and $L \log^+ L(S^{n-1})$ remains open. Therefore, it is interesting to establish the estimate (2) under the assumption of that $\Omega \in B_q^{0,0}(S^{n-1})$, and more generally $\Omega \in H^1(S^{n-1})$.

In this paper, we will establish the estimate (2) provided that $\Omega \in B_q^{0,0}(S^{n-1})$. As for the case of that $\Omega \in H^1(S^{n-1})$, which is more interesting, it is still open. Our results can be formulated as follows.
**Theorem 3.** Suppose that $K(x) = \Omega(x)/|x|^n$, where $\Omega \in B_q^{0,0}(S^{n-1})$ is homogeneous of degree zero with mean value zero and $q > 1$, $P$ and $Q$ are real polynomials with $Q(0) = 0$. Then

$$ \left| p \cdot v \cdot \int_{\mathbb{R}^n} e^{i(P(x)+1/Q(x))} K(x) dx \right| \le B, \qquad (3) $$

where $B$ depends on $\|\Omega\|_{B_q^{0,0}(S^{n-1})}$, $n$ and the degrees of $P$ and $Q$, but not otherwise on the coefficients of $P$ and $Q$.

**Remark 4.** Employing the arguments in [3, Proposition 1.4], the following result shows that the requirement $Q(0) = 0$ in Theorem 3 is most likely not necessary.

**Theorem 5.** With $K$ and $P$ as in Theorem 3, but now $Q(x) = a + v \cdot x$, where $a \in \mathbb{R}$ and $v \in \mathbb{R}^n$

$$ \left| p \cdot v \cdot \int_{\mathbb{R}^n} e^{i(P(x)+1/Q(x))} K(x) dx \right| \le B, \qquad (4) $$

where $B$ depends on $\|\Omega\|_{B_q^{0,0}(S^{n-1})}$, $n$ and the degree of $P$ but not otherwise on the coefficients of $P$.

As an immediate consequence of Theorems 3 and 5, we have the following result.

**Corollary 6.** With $K$ and $Q$ as in Theorem 3 or 5, but now $P: \mathbb{R}^n \to \mathbb{R}^m$ being a polynomial mapping, convolution with the distribution

$$ L(\phi) = p \cdot v \cdot \int_{\mathbb{R}^n} \phi(P(x)) e^{i/Q(x)} K(x) dx, \quad \phi \in \mathcal{S}(\mathbb{R}^m), $$

is bounded on $L^2(\mathbb{R}^m)$.

Employing the arguments in proving [3, Theorem 1.1], the main ingredient of the proof of Theorem 3 or 5 in the current paper is to establish the following integral estimate, which has an independent interest.

**Proposition 7.** Let $\Omega \in B_q^{0,0}(S^{n-1})$ for some $q > 1$, $P(x) = \sum_{|\alpha|=d} c_\alpha x^\alpha$ be a homogeneous polynomial of degree $d$ on $\mathbb{R}^n$. Write $m_P = \sum_{|\alpha|=d} |c_\alpha|$. Then

$$ \int_{S^{n-1}} |\Omega(\omega)| \cdot \left| \log \left( \frac{|P(\omega)|}{m_P} \right) \right| d\sigma(\omega) \lesssim C_d \|\Omega\|_{B_q^{0,0}(S^{n-1})}. \qquad (5) $$

The rest of this paper is organized as follows. In Section 2 we will give some auxiliary lemmas. The proofs of our main results will be given in Section 3. We remark that some ideas in our arguments are taken from [3].

Finally, we make some conventions on notation. Throughout this paper, we denote by $C$ a positive constant which is independent of the main parameters, but it may vary from line to line. Let $A, B$ be complex-valued quantities. We use $A \lesssim B$ or $A = O(B)$ to denote the estimate $|A| \le C|B|$. We use $A \sim B$ to denote the estimate $A \lesssim B \lesssim A$. For $1 \le \gamma \le \infty$, $\gamma'$ is the conjugate index of $\gamma$, and $1/\gamma + 1/\gamma' = 1$.

## 2. Preliminaries

In this section, we recall and establish some auxiliary lemmas, which will be used in our arguments later.

**Lemma 8 ([3]).** For any polynomial $Q(r) = \sum_{j=1}^d b_k r^k$ on $\mathbb{R}^+$, there is a finite collection $\{G_j\}_{j=1}^M$ of disjoint intervals, called “gaps”, of $\mathbb{R}^+$ with $M = O(1)$ such that

(1) The complement $\mathbb{R}^+ \cup \bigcup_{j=1}^M \{G_j\}$ is the union of $M-1$ disjoint “dyadic” intervals, that is, the ratio of the endpoints of such intervals is $\sim 1$.

(2) For each $G_j$, there is a $k = k_j$, $1 \le k_j \le d$, such that for $r \in G_j$,

$$ |Q(r)| \sim |b_{k_j}| r^{k_j} \quad \text{and} \quad |Q'(r)| \sim |b_{k_j}| r^{k_j-1} $$
Also if $L_j$ and $R_j$ denote the left and right endpoints of $G_j$ respectively, then

(i) if $R_j < \infty$, then $R_j = C_d ||b_l||/||b_m||^{1/m-l}$ for some $1 \le l < m \le d$ and

(ii) if $L_j > 0$, then $L_j = C_d ||b_r||/||b_s||^{1/s-r}$ for some $1 \le r < s \le d$.

**Lemma 9 ([8]).** Suppose $\phi$ is real-valued and smooth on $(a, b)$, and that $|\phi^{(k)}| \ge \lambda > 0$ for all $t \in (a, b)$. Then

$$ \left| \int_a^b e^{i\phi(t)} dt \right| \le C_k \lambda^{-1/k} $$

when either $k \ge 2$, or $k = 1$ and $\phi'(t)$ is monotonic.

**Lemma 10.** Let $\gamma > 1, \Omega \in L^\gamma(S^{n-1})$ and $P(x) = \sum_{|\alpha|=d} c_\alpha x^\alpha$ be a homogeneous polynomial of degree $d$ on $\mathbb{R}^n$. Write $m_P = \sum_{|\alpha|=d} |c_\alpha|$. Then

$$ \int_{S^{n-1}} |\Omega(\omega)| \cdot \left| \log \left( \frac{|P(\omega)|}{m_P} \right) \right| d\sigma(\omega) \lesssim C_d \gamma' \|\Omega\|_{L^\gamma(S^{n-1})}. \quad (6) $$

**Proof.** We may assume $m_P = 1$. By the Hölder inequality,

$$ \int_{S^{n-1}} |\Omega(\omega)| \cdot \left| \log(|P(\omega)|) \right| d\sigma(\omega) \le \|\Omega\|_{L^\gamma(S^{n-1})} \left( \int_{S^{n-1}} \left| \log(|P(\omega)|) \right|^{\gamma'} d\sigma(\omega) \right)^{1/\gamma'} $$

And

$$ \begin{aligned} \left( \int_{S^{n-1}} |\log(|P(\omega)|)|^{\gamma'} d\sigma(\omega) \right)^{1/\gamma'} & \lesssim \left( \int_{1/2}^1 r^{n-1} \int_{S^{n-1}} |\log(|P(\omega)|)|^{\gamma'} d\sigma(\omega) dr \right)^{1/\gamma'} \\ & \lesssim \left( \sum_{k \ge 0} (k\gamma')^{\gamma'} \int_{1/2}^1 r^{n-1} dr \int_{\{\omega \in S^{n-1}: r^d |P(\omega)| \le 2^{-k\gamma'}\}} d\sigma(\omega) \right)^{1/\gamma'} \\ & \le \gamma' \left( \sum_{k \ge 0} k\gamma' \int_{\left\{\frac{1}{2} \le |x| \le 1: 2^{-k\gamma'} - 1 \le |P(x)| \le 2^{-k\gamma'}\right\}} dx \right)^{1/\gamma'} \\ & \le \gamma' \left( \sum_{k \ge 0} k\gamma' \left| \left\{ \frac{1}{2} \le |x| \le 1: |P(x)| \le 2^{-k\gamma'} \right\} \right| \right)^{1/\gamma'} \end{aligned} $$

which reduces matters to obtaining uniform sublevel set estimates for $P$ under the normalisation $m_P = 1$. Using the fact that all norms are equivalent on the space of polynomials of degree at most $d$, we can find a derivative $\partial^\alpha$, $0 \le |\alpha| \le d$, such that $1 \lesssim |\partial^\alpha P(x)|$ on $|x| \le 1$. If $\alpha = 0$, then the above sublevel sets are empty for large $k$ and so we may assume $\alpha > 0$. In this case, using the mean value theorem, one can show that $|\{|x| \le 1: |P(x)| \le 2^{-k\gamma'}\}| \lesssim 2^{-k\gamma'/|\alpha|}$ (see e.g. [2]). Thus

$$ \left( \sum_{k \ge 1} k^{\gamma'} \left| \left\{ \frac{1}{2} \le |x| \le 1: |P(x)| \le 2^{-k\gamma'} \right\} \right| \right)^{1/\gamma'} \le \left( \sum_{k \ge 1} k^{\gamma'} 2^{-k\gamma' / |\alpha|} \right)^{1/\gamma'} \\ \le \sum_{k \ge 1} k 2^{-k/d} \lesssim C_d. $$

This implies the desired conclusion and completes the proof of Lemma 10. $\square$

### 3. Proofs of Main Results

In this section, we present the proofs of Proposition 7 and Theorem 3.
**Proof of Proposition 7.** Since $\Omega \in B_q^{0,0}(S^{n-1})$, we know by Definition 2 that there is a decomposition: $\Omega(x') = \sum_s \lambda_s b_s(x')$, where each $b_s$ is a $q$-block, supported in $Q_s$ and

$$ \sum_s |\lambda_s| \left( 1 + \log^+ \frac{1}{|Q_s|} \right) < \infty. $$

Therefore,

$$ \begin{aligned} \int_{S^{n-1}} |\Omega(\omega)| \cdot \left| \log\left(\frac{|P(\omega)|}{m_P}\right) \right| d\sigma(\omega) &\le \sum_s |\lambda_s| \int_{S^{n-1}} |b_s(\omega)| \cdot \left| \log\left(\frac{|P(\omega)|}{m_P}\right) \right| d\sigma(\omega) \\ &\le \left( \sum_{|Q_s| \ge e^{-q'}} + \sum_{|Q_s| < e^{-q'}} \right) |\lambda_s| \int_{S^{n-1}} |b_s(\omega)| \cdot \left| \log\left(\frac{|P(\omega)|}{m_P}\right) \right| d\sigma(\omega) \\ &=: I + II. \end{aligned} $$

Recall that for each $b_s$, $\text{supp}(b_s) \subset Q_s$ and $\|b_s\|_{L^q(S^{n-1})} \le |Q_s|^{1/q-1} = |Q_s|^{-1/q'}$. If $|Q_s| \ge e^{-q'}$, we take $\gamma = q$ and obtain

$$ \gamma' \|b_s\|_{L^Y(S^{n-1})} \le q' \|b_s\|_{L^q(S^{n-1})} \le q' |Q_s|^{-1/q'} \lesssim 1; $$

if $|Q_s| < e^{-q'}$, we take $\gamma = \log|Q_s|/(1+\log|Q_s|)$, then $1 < \gamma < q$, $\gamma' = \log(1/|Q_s|)$ and

$$ \gamma' \|b_s\|_{L^Y(S^{n-1})} \le \log \frac{1}{|Q_s|} \|b_s\|_{L^q(S^{n-1})} |Q_s|^{1/\gamma-1/q} \le \log \frac{1}{|Q_s|} |Q_s|^{-1/\gamma'} \lesssim \log \frac{1}{|Q_s|}. $$

These, combining with Lemma 10, lead to

$$ I \le C_d \sum_{|Q_s| \ge e^{-q'}} |\lambda_s| q' \|b_s\|_{L^q(S^{n-1})} \lesssim C_d \sum_{|Q_s| \ge e^{-q'}} |\lambda_s|, $$

and for $\gamma = \log|Q_s|/(1+\log|Q_s|)$,

$$ II \le C_d \sum_{|Q_s| < e^{-q'}} |\lambda_s| \gamma' \|b_s\|_{L^Y(S^{n-1})} \lesssim C_d \sum_{|Q_s| > e^{-q'}} |\lambda_s| \log \frac{1}{|Q_s|}. $$

Consequently,

$$ \int_{S^{n-1}} |\Omega(\omega)| \cdot \left| \log \left( \frac{|P(\omega)|}{m_P} \right) \right| d\sigma(\omega) \lesssim C_d \sum_s |\lambda_s| \left( 1 + \log \frac{1}{|Q_s|} \right), $$

which completes the proof of Proposition 7. $\square$

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**Proof of Theorem 3.** The arguments are completely similar to those in proving [3, Theorem 1.1]. The only difference is replacing [3, Lemma 2.2] by Proposition 7 in the current setting. For completeness, we present the details as follows.

We may assume $P(0) = 0$. Using polar coordinates write the integral in (3) as

$$ I = \int_{S^{n-1}} \Omega(\omega) \int_0^\infty e^{i[P_\omega(r)+1/Q_\omega(r)]} \frac{1}{r} dr d\sigma(\omega), $$

where $Q(x) = Q_\omega(r) = \sum_{j=1}^{d'} q_j(\omega) r^j$, $P(x) = P_\omega(r) = \sum_{k=1}^d p_k(\omega) r^k$ and $p_j, q_k$ are homogeneous polynomials of degree $j$ and $k$. Using Lemma 8, we may write $I = \sum I_{j,k} + O(1)$, where

$$ I_{j,k} = \int_{S^{n-1}} \Omega(\omega) \int_{G_j \cap F_k} e^{i[P_\omega(r)+1/Q_\omega(r)]} \frac{1}{r} dr d\sigma(\omega). $$

Here $\{G_j\}$ and $\{F_k\}$ are the “gaps” of $Q_\omega(r)$ and $P_\omega(r)$, respectively. Note that although the inner integral of $I_{j,k}$ depends on $\omega$ in a complicated way, we know the form of the endpoints of $G_j$ and $F_k$ as given by Lemma 8 and so it is at least measurable as a function of $\omega$. It suffices to bound each $I_{j,k}$ separately.

We have $|Q_\omega(r)| \sim |q_{j,l}(\omega)|r^{j,l}$ and $|Q'_\omega(r)| \sim |q_{j,l}(\omega)|r^{j,l-1}$ on $G_j$, for some $1 \le j,l \le d'$, and $|P_\omega(r)| \sim |p_{k,m}(\omega)|r^{k,m}$ and $|P'_\omega(r)| \sim |p_{k,m}(\omega)|r^{k,m-1}$ on $F_k$, for some $1 \le k,m \le d$.
Therefore away from where $r^{j_l+k_m} \sim (|p_{k_m}(\omega)| \cdot |q_{j_l}(\omega)|)^{-1}$ the size of the phase $\phi_\omega(r) = P_\omega(r) + 1/Q_\omega(r)$ and its derivative is understood. In fact, on $R_{j,k} = G_j \cap F_k \cap [C(|p_{k_m}(\omega)| \cdot |q_{j_l}(\omega)|)^{-1/(k_m+j_l)}, \infty)$ (for C large enough), we have

$$|\phi_{\omega}(r)| \sim |p_{k_m}(\omega)|^{k_m} \quad \text{and} \quad |\phi'_{\omega}(r)| \sim |p_{k_m}(\omega)|^{k_m-1}.$$

An application of van der Corput's Lemma 9 shows

$$\left| \int_{\{r \in R_{j,k} : r \ge \Theta\}} e^{i\phi_{\omega}(r)} \frac{1}{r} dr \right| = O(1),$$

where $\Theta = |p_{k_m}(\omega)|^{1/k_m}$. Since we are applying Lemma 9 with $k=1$, we need to first split the integration of the above integral into $O(1)$ intervals, where $\phi'_\omega(r)$ is monotone. In the complementary interval, $r < \Theta$, due to the size of $\phi_\omega(r)$ on $R_{j,k}$, we see that

$$\left| \int_{\{r \in R_{j,k} : r < \Theta\}} \left[ e^{i\phi_{\omega}(r)} \frac{1}{r} - 1 \right] dr \right| = O(1).$$

Therefore for the part of $I_{j,k}$ over $R_{j,k}$,

$$\int_{S^{n-1}} \Omega(\omega) \int_{R_{j,k}} e^{i\phi_{\omega}(r)} \frac{1}{r} dr d\sigma(\omega) = \int_{S^{n-1}} \Omega(\omega) \int_{\{r \in R_{j,k} : r < \Theta\}} \frac{dr}{r} d\sigma(\omega) + O(1).$$

Similarly for $L_{j,k} = G_j \cap F_k \cap (-\infty, \delta(|p_{k_m}(\omega)| \cdot |q_{j_l}(\omega)|)^{-1/(k_m+j_l)}]$ (for $\delta$ small enough), we have

$$\int_{S^{n-1}} \Omega(\omega) \int_{L_{j,k}} e^{i\phi_{\omega}(r)} \frac{1}{r} dr d\sigma(\omega) = \int_{S^{n-1}} \Omega(\omega) \int_{\{r \in L_{j,k} : r \ge \Lambda\}} \frac{dr}{r} d\sigma(\omega) + O(1),$$

where $\Lambda = |q_{j_l}(\omega)|^{1/j_l}$. Therefore

$$\begin{aligned} I_{j,k} &= \int_{S^{n-1}} \Omega(\omega) \int_{G_j \cap F_k} e^{i[P_\omega(r)+1/Q_\omega(r)]} \frac{1}{r} dr d\sigma(\omega) \\ &= \int_{S^{n-1}} \Omega(\omega) \int_{\{r \in R_{j,k} : r \le \Theta\}} \frac{dr}{r} d\sigma(\omega) \\ &\quad + \int_{S^{n-1}} \Omega(\omega) \int_{\{r \in L_{j,k} : r \ge \Lambda\}} \frac{dr}{r} d\sigma(\omega) + O(1) \end{aligned} \quad (7)$$

and these two last integrals can be shown to be $O(1)$ by repeatedly applying Proposition 7.

In fact, by the structures of $R_{j,k}$ and $L_{jk}$, the integrals in (7):

$$\int_{S^{n-1}} \Omega(\omega) \int_{\{r \in R_{j,k} : r \le \Theta\}} \frac{dr}{r} d\sigma(\omega), \quad \text{and} \quad \int_{S^{n-1}} \Omega(\omega) \int_{\{r \in L_{j,k} : r \ge \Lambda\}} \frac{dr}{r} d\sigma(\omega)$$

can be written in the form

$$\int_{S^{n-1}} \Omega(\omega) \int_{E(\omega)} \frac{dr}{r} d\sigma(\omega),$$

where $E(\omega)$ is the intersection of $O(1)$ intervals of the form $[a(\omega), \infty)$ or $(-\infty, a(\omega)]$ with

$$a(\omega) \in \left\{ \left( |p_{k_1}(\omega)| / |p_{k_2}(\omega)| \right)^{1/(k_2-k_1)}, \left( |q_{j_1}(\omega)| / |q_{j_2}(\omega)| \right)^{1/(j_2-j_1)}, |p_k(\omega)|^{-1/k}, |q_j(\omega)|^{-1/j}, \left( |p_k(\omega)| \cdot |q_j(\omega)| \right)^{-1/(j+k)} \right\}.$$

Without loss of the generality, we may say that $E(\omega)$ is the intersection of $M$ half infinite intervals such as $[a(\omega), \infty)$ or $(-\infty, a(\omega)]$. Let us write $E(\omega) = [a(\omega), \infty) \cap E'(\omega)$, where $a(\omega)$, say, is $(|p_{k_1}(\omega)|/|p_{k_2}(\omega)|)^{1/(k_2-k_1)}$ and $E'(\omega)$ is the intersection of $M-1$ half infinite intervals as in $E(\omega)$. We can then use Proposition 7 to write

$$\left| \int_{S^{n-1}} \Omega(\omega) \int_{E(\omega)} \frac{dr}{r} d\sigma(\omega) \right| \le \left| \int_{S^{n-1}} \Omega(\omega) \int_{[A_1, \infty) \cap E'(\omega)} \frac{dr}{r} d\sigma(\omega) \right| + O(1),$$
where $A_1 = (m_{p_{k_1}} / m_{p_{k_2}})^{1/(k_2-k_1)}$, which is independent of $\omega$. Indeed,

$$ \left| \int_{S^{n-1}} \Omega(\omega) \int_{[a(\omega), \infty) \cap E'(\omega)} \frac{dr}{r} d\sigma(\omega) \right| \le \left| \int_{S^{n-1}} \Omega(\omega) \int_{[A_1, \infty) \cap E'(\omega)} \frac{dr}{r} d\sigma(\omega) \right| + \left| \int_{S^{n-1}} \Omega(\omega) \int_{[A_1, a(\omega)] \cap E'(\omega)} \frac{dr}{r} d\sigma(\omega) \right|, $$

and for the second integral, by Lemma 9 and Proposition 7, we have

$$ \begin{aligned} & \left| \int_{S^{n-1}} \Omega(\omega) \int_{[A_1, a(\omega)] \cap E'(\omega)} \frac{dr}{r} d\sigma(\omega) \right| \\ & \le \left| \int_{S^{n-1}} \Omega(\omega) \int_{[A_1, a(\omega)]} \frac{dr}{r} d\sigma(\omega) \right| \\ & \le \frac{1}{|k_2 - k_1|} \left| \int_{S^{n-1}} |\Omega(\omega)| \log\left(\frac{|p_{k_1}(\omega)|}{m_{p_{k_1}}}\right) d\sigma(\omega) + \int_{S^{n-1}} |\Omega(\omega)| \log\left(\frac{|p_{k_2}(\omega)|}{m_{p_{k_2}}}\right) d\sigma(\omega) \right| \\ & \lesssim O(1). \end{aligned} $$

For the other forms of $a(\omega)$, the argument is similar.

Similarly, we have

$$ \left| \int_{S^{n-1}} \Omega(\omega) \int_{[A_1, \infty) \cap E'(\omega)} \frac{dr}{r} d\sigma(\omega) \right| \le \left| \int_{S^{n-1}} \Omega(\omega) \int_{[A_1, \infty) \cap [A_2, \infty) \cap E''(\omega)} \frac{dr}{r} d\sigma(\omega) \right| + O(1), $$

or

$$ \left| \int_{S^{n-1}} \Omega(\omega) \int_{[A_1, \infty) \cap E'(\omega)} \frac{dr}{r} d\sigma(\omega) \right| \le \left| \int_{S^{n-1}} \Omega(\omega) \int_{[A_1, \infty) \cap (-\infty, A_2] \cap E''(\omega)} \frac{dr}{r} d\sigma(\omega) \right| + O(1), $$

where $A_2$ is independent of $\omega$, and $E''(\omega)$ is the intersection of $M-2$ half infinite intervals as in $E(\omega)$. Continuing this process, after $M$ iterations, we obtain that

$$ \left| \int_{S^{n-1}} \Omega(\omega) \int_{E(\omega)} \frac{dr}{r} d\sigma(\omega) \right| \le \left| \int_{S^{n-1}} \Omega(\omega) \int_{E} \frac{dr}{r} d\sigma(\omega) \right| + O(1), $$

where $E$ is the intersection of $M$ intervals of the form $[A, \infty)$ or $(-\infty, A]$, where $A$ is independent of $\omega$. Therefore, by the mean value zero of $\Omega$ on $S^{n-1}$, we have

$$ \int_{S^{n-1}} \Omega(\omega) \int_{E} \frac{dr}{r} d\sigma(\omega) = 0. $$

This implies that

$$ \left| \int_{S^{n-1}} \Omega(\omega) \int_{E(\omega)} \frac{dr}{r} d\sigma(\omega) \right| \le \left| \int_{S^{n-1}} \Omega(\omega) \int_{E} \frac{dr}{r} d\sigma(\omega) \right| + O(1) = O(1), $$

and completes the proof of Theorem 3. $\square$

<figure><img src="image_4.png" /></figure>

**Proof of Theorem 5.** By replacing [3, Lemma 2.2] by Proposition 7, the proof is similar to the proof of [3, Proposition 1.4]. We omit the details here. $\square$

<figure><img src="image_5.png" /></figure>

## References

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