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\TopicFR{Analyse harmonique}
\TopicEN{Harmonic analysis}

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\title{On the boundedness of a family of oscillatory singular integrals}

\author{\firstname{Hussain} \lastname{Al-Qassem}\IsCorresp}
\address{Mathematics Program, Department of Mathematics, Statistics and Physics, College of Arts and Sciences, Qatar University, 2713, Doha, Qatar}
\email{husseink@qu.edu.qa}

\author{\firstname{Leslie} \lastname{Cheng}}
\address{Department of Mathematics, Bryn Mawr College, Bryn Mawr, PA 19010, U.S.A.}
\email{lcheng@brynmawr.edu}

\author{\firstname{Yibiao} \lastname{Pan}}
\address{Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.}
\email{yibiao@pitt.edu}

\subjclass{42B20, 42B30, 42B35}
\keywords{\kwd{oscillatory integrals}
\kwd{singular integrals}
\kwd{Calder\'on--Zygmund kernels}
\kwd{Hardy spaces}}

\begin{abstract}
Let $\Omega \in H^1(\mathbb{S}^{n-1})$ with mean value zero, $P$ and $Q$ be polynomials in $n$ variables with real coefficients and $Q(0)=0$. We prove that
\[
\Biggl|\mbox{p.v.}\int_{\mathbb{R}^n}e^{i(P(x)+1/Q(x))}\frac{\Omega(x/|x|)}{|x|^n}\mathrm{d}x\Biggr| \le A \|\Omega\|_{H^1(\mathbb{S}^{n-1})}
\]
where $A$ may depend on $n$, $\deg(P)$ and $\deg(Q)$, but not otherwise on the coefficients of $P$ and $Q$.

The above result answers an open question posed in~\cite{WW}. Additional boundedness results of similar nature are also obtained.
\end{abstract}


\begin{document}
\maketitle
\section{Introduction}\label{sec1}


The study of oscillatory singular integrals has a long-standing history (\cite{CA, FE, Gr, PS, RS, St, SW}). For the specific topic considered in this paper, we shall begin with a well-known result of Stein and Wainger in~\cite{SW} and its extension by Stein in~\cite{St1}.


Let $n \ge 2$, $K(x)$ be a Calder\'on--Zygmund kernel given by
\begin{equation}\label{101}
K(x) = \frac{\Omega(x/|x|)}{{|x|^n}}
\end{equation}
where $\Omega: \sn \to \C$ is integrable over the unit sphere $\sn$ with respect to the induced Lebesgue measure $\sigma$ and satisfies
\begin{equation}\label{102}
\int_{\sn}\Omega(x) \dsigma(x) = 0.
\end{equation}

For $d \in \N$, let $\mcp_{n,d}$ denote the space of real-valued polynomials in $n$ variables whose degrees do not exceed $d$. It was proves in~\cite{St1} that, if $\Omega \in L^\infty(\sn)$ and $P \in \mcp_{n,d}$, then
\[
\Bigg|\mbox{p.v.}\int_{\Rn}e^{iP(x)} K(x) \dx\Bigg| \le C_{n,d} \|\Omega\|_{L^\infty(\sn)}
\]
where $C_{n,d}$ is independent of the coefficients of $P$.

In the recent paper~\cite{WW} the authors obtained an extension of the above result in which the phase functions belong to a certain class of rational functions while $\Omega$ is allowed to be in a block space $B^{0,0}_q(\sn)$. Their result can be described as follows.

\begin{theo}[\cite{WW}]\label{Th1}
Let $q>1 $ and $K(x)$ be a Calder\'on--Zygmund kernel given by~\eqref{101}{\rm--}\eqref{102}. Let $P(x), Q(x) \in \mcp_{n,d}$ such that $Q(0)=0$ and $\Omega \in B^{0,0}_q(\sn)$. Then
\begin{equation}\label{103}
\Bigg|\emph{p.v.}\int_{\Rn}e^{i(P(x)+1/Q(x))}K(x)\dx\Bigg| \le A
\end{equation}
where $A$ may depend on $\|\Omega\|_{B^{0,0}_q(\sn) }$, $n$ and $d$ but not otherwise on the coefficients of $P$ and $Q$.
\end{theo}

The definition of $B^{0, \nu}_q(\sn)$ for $\nu > -1$ and $q > 1$ can be found in~\cite{WW}. It had been known that the bound~\eqref{103} also holds for all $\Omega \in L \log L(\sn)$, which was proved by Folch-Gabayet and Wright in~\cite{FW}.

Let $H^1(\sn)$ denote the Hardy space over the unit sphere. An important question, posed by the authors of~\cite{WW}, is whether the bound~\eqref{103} continues to hold under the condition $\Omega
\in H^1(\sn)$ (with the same phase functions $P(x)+1/Q(x)$). This is a very natural question because both $ B^{0,0}_q(\sn)$ and $L \log L(\sn)$ are proper subspaces of $H^1(\sn)$.

Our first result answers the above question in the affirmative.

\begin{theo}\label{Th2}
Let $K(x)$ be a Calder\'on--Zygmund kernel given by~\eqref{101}{\rm--}\eqref{102}. Let $P(x), Q(x) \in \mcp_{n,d}$ such that $Q(0)=0$. Suppose that $\Omega \in H^1(\sn)$. Then
\begin{equation}\label{104}
\Bigg|\emph{p.v.}\int_{\Rn}e^{i(P(x)+1/Q(x))}K(x)\dx\Bigg| \le A \|\Omega\|_{H^1(\sn)}
\end{equation}
where $A$ may depend on $n$ and $d$ but not otherwise on the coefficients of $P$ and $Q$.
\end{theo}

As usual, Theorem~\ref{Th2} implies the uniform boundedness of oscillatory singular integral operators of the following type on $L^2(\R^{m})$:
\[
f \to \mbox{p.v.}\int_{\Rn} f(u_1-P_1(y), \dots, u_m-P_m(y))e^{i/Q(y)}|y|^{-n}\Omega(y/|y|)\dy,
\]
where $P_1, \dots, P_m, Q$ are polynomials and $\Omega$ is a function in $H^1(\sn)$ with a zero mean-value. The proof of Theorem~\ref{Th2} will be given in Section~2.

The general question about whether the condition $Q(0) =0$ can be removed is open. But for $\deg(Q) \le 1$, this is known to be the case.

\begin{theo}[\cite{FW, WW}]\label{Th3}
Let $K(x)$ be a Calder\'on--Zygmund kernel given by~\eqref{101}{\rm--}\eqref{102}. Let $P(x) \in \mcp_{n,d}$ and $Q(x)= a+ v\cdot x$ where $a \in \R$ and $v \in \Rn$. Suppose that $\Omega \in L\log L(\sn)$ or $\Omega \in B^{0,0}_q(\sn)$ for some $q>1$. Then
\begin{equation}\label{108}
\Bigg|\emph{p.v.}\int_{\Rn}e^{i(P(x)+1/Q(x))}K(x)\dx\Bigg| \le A
\end{equation}
where $A$ may depend on $n$, $d$ and the respective norm of $\Omega$, but not otherwise on $a$, $v$ and the coefficients of $P$.
\end{theo}

We have the following extension of Theorem~\ref{Th3}:

\begin{theo}\label{Th4}
Let $P(x) \in \mcp_{n,d}$. Let $l \in \N$, $h(x)$ be a nonzero real-valued homogeneous polynomial of degree $l$ and $Q(x) = a + h(x)$. Then for every Calder\'on--Zygmund kernel $K(x)$ given by~\eqref{101}{\rm--}\eqref{102} with an $\Omega(\,\cdot\,)$ in $ H^1(\sn)$,
\begin{equation}\label{109}
\Bigg|\emph{p.v.}\int_{\Rn}e^{i(P(x)+1/Q(x))}K(x)\dx\Bigg| \le A
\end{equation}
where $A$ may depend on $\|\Omega\|_{H^1(\sn)}$, $n$, $d$ and $l$ but not otherwise on the coefficients of $P(x)$ and~$Q(x)$.
\end{theo}

The proof of Theorem~\ref{Th4} will be given in Section~\ref{sec3}.

The following is an important estimate due to E. M. Stein:

\begin{theo}\label{Th5}
Let $\Omega \in L\log L(\sn)$ and $d \in \N$. For every homogeneous polynomial of degree $d$ on $\Rn$ ${P(x)=\sum_{|\alpha|=d}a_\alpha x^\alpha}$, let $m_P = {\sum_{|\alpha|=d}|a_\alpha|} $. Then there exists a constant $C_{n, d,\Omega} > 0$ which is independent of $\{a_\alpha\}$ such that
\begin{equation}\label{105}
\int_{\sn}|\Omega(x)|\Biggl|\log\Biggl(\frac{|P(x)|}{m_P}\Bigg)\Bigg|\dsigma(x) \le C_{n,d,\Omega}
\end{equation}
holds whenever $m_P \ne 0$.
\end{theo}

What happens if $P(x)$ is a general polynomial instead of a homogeneous polynomial? For $ {P(x)=\sum_{|\alpha|\le d}a_\alpha x^\alpha}$, the direct analogue of~\eqref{105}, where $m_P$ is replaced by ${\sum_{|\alpha|\le d}|a_\alpha|}$, is clearly false. This is due to the fact that, unlike $P \to m_P$ for homogeneous polynomials of a fixed degree,
\[
\sum_{|\alpha|\le d}a_\alpha x^\alpha \to \sum_{|\alpha|\le d}|a_\alpha|
\]
is not a norm on $\mcp_{n,d}\big\vert_{\sn}$. To remedy this situation, we can simply replace the above with any norm on $\mcp_{n,d}\big\vert_{\sn}$ (e.g. $\|{\,\cdot\,}\|_\infty$) to arrive at the following extension of Theorem~\ref{Th5}:

\begin{theo}\label{Th6}
Let $\|{\,\cdot\,}\|$ be a norm on $\mcp_{n,d}\big\vert_{\sn}$. Then there exists a positive constant $C$ which depends on $n, d$ and $\|{\,\cdot\,}\|$ only such that
\begin{equation}\label{106}
\int_{\sn}|\Omega(y)|\Biggl|\log\Biggl(\frac{|P(x)|}{\|P\big\vert_{\sn}\|}\Biggr)\Biggr|\dsigma(x) \le C (1+\|\Omega\|_{L\log L(\sn)})
\end{equation}
holds for all $\Omega \in L\log L(\sn)$ and all $P \in \mcp_{n,d}$ not vanishing identically over $\sn$.
\end{theo}

Since any two norms on a finite dimensional space are equivalent, one recovers~\eqref{105} when applying~\eqref{106} to homogeneous polynomials.

More broadly, results such as Theorem~\ref{Th6} can be framed in terms of functions of finite type and compactness, as is done in the theorem below.

\begin{theo}\label{Th7}
Let $M$ be a compact smooth submanifold of $\Rn$, $\sigma=\sigma_M$ be the induced Lebesgue measure on $M$ and $U$ be an open subset of\/ $\R^m$. Let $f \in C^\infty(M\times U)$ such that, for every $ u \in U$, $f(\,\cdot\,, u)$ does not vanish to infinite order at any point in $M$. Then, for every compact subset $W$ of\/ $U$, there exists a positive constant $C= C(M, n, m, f, W)$ such that
\begin{equation}\label{107}
\sup_{u \in W}\int_{M}|\Omega(y)|\log(|f(y, u)|)|\dsigma(y) \le C (1+\|\Omega\|_{L\log L(M)})
\end{equation}
holds for all $\Omega \in L\log L(M)$.
\end{theo}

The proof of Theorem~\ref{Th7} will be based on Malgrange preparation theorem. It will be given in Section~4 where one will also see how Theorem~\ref{Th6} follows as a simple consequence. As an application of Theorem~\ref{Th7}, we have the following:

\begin{theo}\label{Th8}
Let $b \in \R\backslash\{0\}$ and $P(x) \in \mcp_{n,d}$. Let $\rho \in \R^+$, $U$ be an open subset of $\R^m$ and $W$ be a compact subset of $U$. Let $\psi \in C^\infty(\sn\times U)$ such that, for every $ u \in U$, $\psi(\,\cdot\,, u)$ does not vanish to infinite order at any point in $\sn$. For $(x, u) \in (\Rn\backslash\{0\})\times U$, let
\begin{equation}\label{009}
\Phi(x, u) = b |x|^\rho\psi(x/|x|, u).
\end{equation}
Then for every Calder\'on--Zygmund kernel $K(x)$ given by~\eqref{101}{\rm--}\eqref{102} with an $\Omega(\,\cdot\,)$ in $ L\log L(\sn)$,
\begin{equation}\label{010}
\sup_{u\in W}\Bigg|\emph{p.v.}\int_{\Rn}e^{i(P(x)+1/\Phi(x,u))}K(x)\dx\Bigg| \le A
\end{equation}
where $A$ may depend on $\|\Omega\|_{L\log L(\sn)}$, $\psi$, $W$, $n$, $d$ and $\rho$ but not otherwise on $b$ and the coefficients of $P(x)$.
\end{theo}

In the rest of the paper we shall use $A \lesssim B$ to mean that $A \le c B$ for a certain constant $c$ which depends on some essential parameters only.

\section{Proof of Theorem~\ref{Th2}}\label{sec2}

We shall now prove~\eqref{104} under the assumptions of Theorem~\ref{Th2}. By~\eqref{102} and the atomic decomposition of $H^1(\sn)$ (see~\cite{CW} and~\cite{CO}), it suffices to prove that
\begin{equation}\label{201}
\Bigg|\mbox{p.v.}\int_{\Rn}e^{i(P(x)+1/Q(x))}\frac{\Omega(x/|x|)}{|x|^n}\dx\Bigg| \le A
\end{equation}
under the assumption that $\Omega(\,\cdot\,)$ is a regular atom on $\sn$, i.e. $\Omega(\,\cdot\,)$ enjoys the following additional properties:
\begin{enumerate}\alphenumi
\item \label{a} $\supp(\Omega) \subseteq \sn \cap B(\zeta_0, \delta)$ for some $\zeta_0 \in \sn$ and $\delta > 0$ where $B(\zeta_0, \delta) =\{y \in \Rn: |y-\zeta_0| < \delta\}$; and
\item \label{b} $\|\Omega\|_\infty \le \delta^{-n+1}.$
\end{enumerate}

If $\delta \ge 1/4$, \eqref{201} follows from \eqref{b} and Theorem~\ref{Th1}. Thus, we may assume that $0< \delta < 1/4$. By using a rotation if necessary, we may also assume that $\zeta_0= (0, \dots, 0, 1)$. For any $x=(x_1, \dots, x_{n-1}, x_n) \in \Rn$, we write $x=(\tilde{x}, x_n)$ where $\tilde{x}=(x_1, \dots, x_{n-1})$. We also extend the definition of $\Omega(\,\cdot\,)$ from $\sn$ to $\Rn\backslash\{0\}$ by using $\Omega(x)=\Omega(x/|x|)$. We define $\Omega_\delta: \Rn\backslash\{0\} \to \C$ by
\[
\Omega_\delta(x) = (\delta^{n-1}|x|^n)\frac{\Omega(\delta \tilde{x}, x_n)}{|(\delta \tilde{x}, x_n)|^{n}}.
\]
Then $\Omega_\delta(\,\cdot\,)$ is homogeneous of degree~$0$. It is well-known that, by the theory of Calder\'on--Zygmund operators, \eqref{102} implies that
\begin{equation}\label{202}
\int_{\sn}\Omega_\delta(y) \dsigma(y) = 0.
\end{equation}

Next we will show that $\|\Omega_\delta\|_\infty \lesssim 1$. To see this, we assume that $\Omega_\delta(x) \ne 0$ for some $x \in \Rn\backslash\{0\}$. Then
\[
\eta \coloneqq \Bigg|\frac{(\delta \tilde{x}, x_n)}{|(\delta \tilde{x}, x_n)|}- \zeta_0\Bigg| < \delta.
\]
By \eqref{b} and $x_n = |(\delta \tilde{x}, x_n)| (1 - \eta^2/2)$,
\begin{align*}
|\Omega_\delta(x)| &\le \Biggl(\frac{|x|}{|(\delta \tilde{x}, x_n)|}\Biggr)^n
\\
&= (\delta |(\delta \tilde{x}, x_n)|)^{-n}\big(|(\delta \tilde{x}, x_n)|^2+(\delta^2 -1) x_n^2\big)^{n/2}
\\
&= \delta^{-n} \big(1 + (\delta^2-1)(1-\eta^2/2)^2\big)^{n/2}
\\
&= \delta^{-n}\big(\delta^2(1-\eta^2/2)^2+ \eta^2(1-\eta^2/4)\big)^{n/2} \lesssim 1.
\end{align*}

Let $P_\delta(x)= P(\delta \tilde{x}, x_n)$ and $Q_\delta(x)= Q(\delta \tilde{x}, x_n)$. Then $P_\delta(\,\cdot\,), Q_\delta(\,\cdot\,) \in \mcp_{n,d}$, $\deg(P_\delta)= \deg(P)$, $\deg(Q_\delta)= \deg(Q)$ and $Q_\delta(0)= Q(0) = 0$. It follows from Theorem~\ref{Th1} that
\begin{align*}
\Biggl|\mbox{p.v.}\int_{\Rn}e^{i(P(x)+1/Q(x))}\frac{\Omega(x)}{|x|^n}\dx\Biggr|
&= \Biggl|\mbox{p.v.}\int_{\Rn}e^{i(P_\delta(x)+1/Q_\delta(x))}\frac{\Omega(\delta \tilde{x}, x_n)}{|(\delta \tilde{x}, x_n)|^{n}}\delta^{n-1}\dx\Biggr|
\\
&=\Biggl|\mbox{p.v.}\int_{\Rn}e^{i(P_\delta(x)+1/Q_\delta(x))}\frac{\Omega_\delta(x)}{|x|^n}\dx\Biggr| \le A
\end{align*}
where $A$ depends on $n$ and $d $ only. This proves Theorem~\ref{Th2}.


\section{Proof of Theorem~\ref{Th4}}\label{sec3}

First let us recall the following version of van der Corput's lemma.

\begin{lemm}\label{Lem2}\ 
\begin{enumerate}\romanenumi
\item \label{9i} Let $\phi$ be a real-valued $C^k$ function on $[a, b]$ satisfying $|\phi^{(k)}(x)| \ge 1$ for every $x \in [a, b]$. Suppose that $k \ge 2$, or that $k = 1$ and $\phi^\prime$ is monotone on $[a, b]$. Then there exists a positive constant $c_k$ such that
\[
\Bigg|\int_a^b e^{i \lambda \phi(x)} \,\dx \Bigg| \le c_k |\lambda|^{-1/k}
\]
for all $\lambda \in \R$. The constant $c_k$ is independent of $\lambda, a, b$ and $\phi$.
\item \label{9ii} Let $\phi$ and $c_k$ be the same as in \eqref{9i}. If $\psi \in C^1([a, b])$, then
\[
\Bigg|\int_a^b e^{i \lambda \phi(x)} \psi(x) \,\dx \Bigg| \le c_k |\lambda|^{-1/k} (\|\psi\|_{L^\infty([a, b])} + \|\psi^\prime\|_{L^1([a, b])})
\]
holds for all $\lambda \in \R$.
\end{enumerate}
\end{lemm}

We will now give the proof of Theorem~\ref{Th4}. Since the case $a = 0$ is already covered by Theorem~\ref{Th5}, we shall assume that $a \ne 0$. Initially we will assume that $\Omega \in L^\infty(\sn)$.

For $\omega \in \sn$, let
\[
\theta = \theta(\omega) =\Bigg|\frac{a}{h(\omega)}\Bigg|^{1/l}.
\]
Then
\[
\mbox{p.v.}\int_{\Rn}e^{i(P(x)+1/Q(x))}K(x)\,\dx = \int_{\sn}\Omega(\omega) I(\omega) \,\dsigma(\omega)
\]
where
\[
I(\omega) = \int_0^\infty e^{i(P(r\omega) + 1/(a + h(\omega)r^l))} \frac{\dr}{r} = I_1(\omega) + I_2(\omega) + I_3(\omega)
\]
where
\begin{align*}
I_1(\omega)&= \int_0^{\alpha\theta} e^{i(P(r\omega) + 1/(a + h(\omega)r^l))} \frac{\dr}{r},
\\
I_2(\omega)&= \int_{\alpha\theta}^{\beta\theta} e^{i(P(r\omega) + 1/(a + h(\omega)r^l))} \frac{\dr}{r}
\intertext{and}
I_3(\omega)&= \int^{\infty}_{\beta\theta} e^{i(P(r\omega) + 1/(a + h(\omega)r^l))} \frac{\dr}{r}
\end{align*}
for some suitable constants $\alpha $ and $\beta$. Since $|I_2(\omega)| \le \ln(\beta/\alpha)$, it suffices to show that there exist $ 0< \alpha < \beta$ such that
\begin{equation}\label{402}
\int_{\sn}\Omega(\omega)I_j(\omega)\,\dsigma(\omega) = O(1)
\end{equation}
for $j = 1$ and $j = 3$.

The estimate~\eqref{402} for $j=3$ follows from a slight modification of the proof of Theorem~\ref{Th1} in~\cite{FW}. Details will be omitted. Below we shall show how to obtain~\eqref{402} for $j=1$ with an appropriate selection of $\alpha$.

Let
\[
\phi_{\omega}(r) = P(r\omega) + \frac{1}{a + h(\omega)r^l}.
\]
In order to apply van der Corput's lemma, we shall need to obtain appropriate lower bounds for at least one of the derivatives of $\phi_{\omega}(\,\cdot\,)$ near $0$. When $l = 1$, this can be done with any derivative of $\phi_{\omega}(\,\cdot\,)$ whose order exceeds the degree of $P(\,\cdot\,)$. When $l > 1$, one needs to be more selective as demonstrated below.

Let $g(t)= (1\pm t^l)^{-1}$. Then, for $k=0, 1, 2, \dots,$
\begin{equation*}
\Bigg|\frac{\dd^sg(0)}{\dd t^s}\Bigg| =
\begin{cases}
s! &\text{if $l \, \big\vert\MK s$}\\
0 &\text{if $l\mathbin{\not\big\vert} s$.}
\end{cases}
\end{equation*}
Let $k_0 \in \N$ such that $k_0l > \max\{\deg(P), 1\}$. By $|g^{(k_0l)}(0)| = (k_0l)! \ge 1$, there exists an $\alpha \in (0, 1) $ such that $|g^{(k_0l)}(0)| \ge 1/2$ for $|t| \le \alpha$. By
\[
\phi_{\omega}(r) = P(r\omega) + a^{-1} g(r/\theta),
\]
we have
\[
|\phi_\omega^{(k_0l)}(r)| = (|a|\theta^{k_0l})^{-1}|g^{(k_0l)}(r/\theta)|
\]
\[
\ge (2|a|\theta^{k_0l})^{-1}
\]
for $ r \in (0, \alpha\theta]$.

Let $b= \min\{|a|, 1\}$. If $|a| \ge 1$, then
\[
\int_{(b^{1/(k_0l)}\alpha\theta, \, \alpha\theta]} e^{i\phi_{\omega}(r)} \frac{\dr}{r} = 0.
\]
If $|a| < 1$, then $b=|a|$ and by Lemma~\ref{Lem2},
\[
\Bigg|\int_{(b^{1/(k_0l)}\alpha\theta, \, \alpha\theta]} e^{i\phi_{\omega}(r)} \frac{\dr}{r}\Bigg|
\lesssim \frac{1}{((2|a|\theta^{k_0l})^{-1})^{1/(k_0l)}}\cdot\frac{1}{|a|^{1/(k_0l)}\alpha\theta} \lesssim 1.
\]
Thus, we always have
\begin{equation}\label{403}
\Bigg|\int_{(b^{1/(k_0l)}\alpha\theta, \, \alpha\theta]} e^{i\phi_{\omega}(r)} \frac{\dr}{r}\Bigg| \lesssim 1.
\end{equation}

Therefore, it suffices to show that
\begin{equation}\label{404}
\int_{\sn} \Omega(\omega)\Biggl(\int_0^{\alpha\theta b^{1/(k_0l)}} e^{i\phi_\omega(r)} \frac{\dr}{r}\Biggr)\,\dsigma(\omega)= O(1).
\end{equation}

Let
\[
q(x) = \Biggl(\frac{1}{a}\Biggr)\sum_{j=0}^{k_0-1}\Biggl(-\frac{h(x)}{a}\Biggr)^j.
\]
For any $\omega \in \sn$ and $0< r \le \alpha\theta b^{1/(k_0l)}$, by $0 \le b \le 1$,
\[
\Bigg|\frac{h(\omega) r^l}{a}\Bigg| \le b^{1/k_0}\alpha^l \Biggl(\Biggl|\frac{h(w)}{a}\Biggr|\theta^l\Biggr)
\le \alpha^l < 1,
\]
which implies that
\begin{align*}
|\phi_\omega(r) - (P(r\omega)+q(rw))| &= |a|^{-1}\Bigg|\Bigg(1 + \frac{h(w)r^l}{a}\Bigg)^{-1} - \sum_{j=0}^{k_0-1}\Bigg(-\frac{h(w)r^l}{a}\Bigg)^j\Bigg|
\\
&\lesssim |a|^{-1} \Bigg|\frac{h(w)r^l}{a}\Bigg|^{k_0} = |a|^{-1}\theta^{-k_0l}r^{k_0l}.
\end{align*}
Thus,
\[
\Bigg|\int_0^{\alpha\theta b^{1/(k_0l)}} \bigg(e^{i\phi_\omega(r)} -e^{i(P(r\omega)+q(r\omega))}\bigg) \frac{\dr}{r}\Bigg|
\lesssim |a|^{-1}\theta^{-k_0l} \int_0^{\alpha\theta b^{1/(k_0l)}} r^{k_0l-1}\dr
\lesssim \alpha^{k_0l}|a|^{-1}b \lesssim 1,
\]
which immediately gives
\begin{equation}\label{405}
\int_{\sn} \Omega(\omega)\int_0^{\alpha\theta b^{1/(k_0l)}} \bigg(e^{i\phi_\omega(r)} -e^{i(P(r\omega)+q(r\omega))}\bigg) \frac{\dr}{r}\dsigma(\omega)= O(1).
\end{equation}

By an inequality on page 334 of~\cite{St1},
\[
\Bigg|\mbox{p.v.}\int_{|x|\le \alpha |a|^{1/l}b^{1/(k_0l)}m_h^{-1/l}}e^{i (P(x)+q(x))}K(x)\dx\Bigg| \le A,
\]
i.e.
\begin{equation}\label{406}
\int_{\sn} \Omega(\omega)\int_0^{\alpha |a|^{1/l}b^{1/(k_0l)}m_h^{-1/l}} e^{i(P(r\omega)+ q(r\omega))} \frac{\dr}{r}\dsigma(\omega)= O(1).
\end{equation}
Trivially, we have
\[
\Bigg|\int^{\alpha\theta b^{1/(k_0l)}}_{\alpha |a|^{1/l}b^{1/(k_0l)}m_h^{-1/l}} e^{i(P(r\omega)+ q(r\omega))} \frac{\dr}{r}\Bigg| \lesssim \Bigg|\ln\Bigg(\frac{|h(\omega)|}{m_h}\Bigg)\Bigg|.
\]
It follows from Theorem~\ref{Th5} that
\begin{equation}\label{407}
\int_{\sn} \Omega(\omega)\int^{\alpha\theta b^{1/(k_0l)}}_{\alpha |a|^{1/l}b^{1/(k_0l)}m_h^{-1/l}} e^{i(P(r\omega)+ q(r\omega))} \frac{\dr}{r}\dsigma(\omega)= O(1).
\end{equation}
By \eqref{405}--\eqref{407}, we obtain~\eqref{404}. This proves~\eqref{109} for $\Omega \in L^\infty(\sn)$. By applying the argument used in the proof of Theorem~\ref{Th2}, one then obtains~\eqref{109} for $\Omega \in H^1(\sn)$. Details are omitted.

\section{Nonvanishing of infinite order}\label{sec4}

Let $M$ be a smooth $k$-dimensional submanifold of $\Rn$, $f: M \rightarrow \R$ be a $C^\infty$ function and $p \in M$. We say that $f$ does not vanish to infinite order at $p$ if there is a chart $ (U_p, \varphi) $ around $p$ such that $\varphi(p) = 0$ and $D^\alpha(f\circ\varphi^{-1})(0) \ne 0$ for some $\alpha \in (\N\cup\{0\})^k$. For $r>0$, let $B_k(r)$ denote the open ball in $\R^k$ which is centered at the origin and has radius $r$. We begin with the following:

\begin{lemm}\label{Lem1}
Let $k, m \in \N$, $x \in \R^k$, $y \in \R^m$ and $R > 0$. Let $g(x, y) \in C^\infty(B_k(R)\times B_m(R))$ such that
\begin{equation}\label{301}
\frac{\partial^\alpha g(0,0)}{\partial x^\alpha} \ne 0
\end{equation}
holds for some $\alpha \in (\N\cup\{0\})^k$. Then there exists an $r \in (0, R/3)$ such that, for every $\delta \in \big(0, (\max\{|\alpha|, 1\})^{-1}\big)$ and every $C^\infty$ function $h: \R^k \rightarrow \R$ which is supported in $B_k(r)$,
\begin{equation}\label{302}
\sup_{y \in B_m(r)} \int_{B_k(r)} |g(x, y)|^{-\delta} |h(x)|\dx < \infty.
\end{equation}
\end{lemm}

\begin{proof}
If~\eqref{301} holds with $|\alpha|=0$, i.e. $g(0,0) \ne 0$, then~\eqref{302} follows trivially by continuity.

Now suppose that $g(0, 0) = 0$ and let
\begin{equation}\label{303}
l = \min\Bigg\{|\alpha|:\space\alpha\in(\N\cup\{0\})^k \,\, \text{and}\,\, \frac{\partial^\alpha g(0,0)}{\partial x^\alpha} \ne 0 \Bigg\}.
\end{equation}
By an argument on p.~317 of~\cite{St1}, we may assume that
\[
\frac{\partial^lg(0,0)}{\partial x^l_k} \ne 0.
\]
By~\eqref{303} we also have
\[
\frac{\partial^jg(0,0)}{\partial x^j_k} = 0
\]
for $j = 0, 1, \dots, l-1$. Let $\tilde{x} = (x_1, \dots, x_{k-1})$. By Malgrange preparation theorem (\cite{GG}), there exist $r \in (0, R/3)$, $\eta_0 > 0$, $a_0(\tilde{x}, y), \dots, a_{l-1}(\tilde{x}, y) \in C^\infty(B_{k-1}(r)
\times B_m(r))$ and $c(x,y) \in C^\infty(B_{k}(r)
\times B_m(r))$ such that, for all $(x, y) \in B_{k}(r)
\times B_m(r) $,
\[
g(x, y) = c(x, y)(x^l_k+ a_{l-1}(\tilde{x}, y) x^{l-1}_k + \dots + a_{0}(\tilde{x}, y))
\]
and $|c(x, y)| \ge \eta_0$. Thus, for any $\delta \in (0, 1/l)$ and any $C^\infty$ function $h(x)$ supported on $B_k(r)$,
\begin{align*}
&\sup_{y \in B_m(r)} \int_{B_k(r)} |g(x, y)|^{-\delta} |h(x)|\dx
\\
&\mkern 100mu\lesssim \sup_{y \in B_m(r)}\int_{B_{k-1}(r)} \int_{|x_k| < r} (x^l_k+ a_{l-1}(\tilde{x}, y) x^{l-1}_k + \dots + a_{0}(\tilde{x}, y))^{-\delta} \dx_k \dd\tilde{x} < \infty.\qedhere
\end{align*}
\end{proof}

\begin{proof}[Proof of Theorem~\ref{Th7}]
Let $M$ be a compact smooth submanifold of $\Rn$ and $U$ be an open subset of $\R^m$. Let $f \in C^\infty(M\times U)$ such that, for every $ u \in U$, $f(\,\cdot\,, u)$ does not vanish to infinite order at any point in $M$. Suppose that $W$ is a compact subset of $U$. By Lemma~\ref{Lem1} and the compactness of $M$ and $W$, there exist $\delta = \delta_{f, W} > 0$ and $C= C(M, n, m, f, W)$ such that
\begin{equation}\label{304}
\sup_{u \in W}\int_{M}|f(y, u)|^{-\delta} \dsigma(y) \le C.
\end{equation}
For any $\Omega \in L\log L(M)$ and $u \in W$, it follows from~\eqref{304} that
\[
\int_{\{y\in M:\, |\Omega(y)| < |f(y,u)|^{-\delta/2}\}}|\Omega(y)||\log(|f(y, u)|)|\dsigma(y)
\lesssim \int_{M} |f(y, u)|^{-\delta} \dsigma(y) \lesssim 1.
\]
On the other hand, we have trivially that
\[
\int_{\{y\in M:\, |\Omega(y)| \ge |f(y,u)|^{-\delta/2}\}}|\Omega(y)||\log(|f(y, u)|)|\dsigma(y)
\lesssim \|\Omega\|_{L\log L(M)}.
\]
Thus~\eqref{107} holds and the proof of Theorem~\ref{Th7} is now complete.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{Th6}]
Let $m=\dim(\mcp_{n,d}\vert_{\sn})$ and $p_1(x), \dots, p_m(x) \in \mcp_{n,d}$ such that $\{p_j\vert_{\sn}: 1 \le j \le m\}$ forms a basis for $\mcp_{n,d}\vert_{\sn}$. Define $f: \sn\times (\R^m\backslash\{0\}) \rightarrow \R$ by
\[
f(x, u) = \sum_{j=1}^m u_j p_j(x)
\]
for $x \in \sn$ and $u = (u_1, \dots, u_m) \in \R^m\backslash\{0\}$. For each $u \in \R^m\backslash\{0\}$, $f(\,\cdot\,, u)$ is not identically zero on $\sn$ which, by real-analyticity, implies that it does not vanish to infinite order at any point in $\sn$. By Theorem~\ref{Th7},
\begin{equation}\label{305}
\sup_{u \in \sm}\int_{\sn}|\Omega(x)||\log(|f(x, u)|)|\dsigma(x) \le C (1+\|\Omega\|_{L\log L(\sn)}),
\end{equation}
which implies that~\eqref{106} holds when the norm is given by
\[
\sum_{j=1}^m u_j p_j\big\vert_{\sn} \rightarrow \Bigg(\sum_{j=1}^mu_j^2\Bigg)^{1/2}.
\]
Since any two norms on $\mcp_{n,d}\big\vert_{\sn}$ are equivalent, Theorem~\ref{Th6} is proved.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{Th8}]
By assumption and Theorem~\ref{Th7},
\begin{equation}\label{306}
\sup_{u \in W}\int_{\sn}|\Omega(\omega)||\log(|\psi(\omega, u)|)|\dsigma(\omega) \le C (1+\|\Omega\|_{L\log L(\sn)}).
\end{equation}
One can then adopt the arguments in the proof of Theorem~\ref{Th1} in~\cite{FW}, at times using~\eqref{306} instead of~\eqref{105}, to finish the proof. Details are omitted.
\end{proof}

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