Uniform $L^p$ Boundedness for Oscillatory Singular Integrals with $C^\infty$ Phases

Abstract

We establish the uniform boundedness of oscillatory singular integral operators on $L^p$ spaces for $C^\infty$ phases and Hölder class singular kernels. Our main result improves and unifies several existing $L^p$ results for oscillatory singular integrals.

1 Introduction

Both oscillatory and singular integrals have played very important roles in the history of harmonic analysis. Oscillatory singular integrals, as a hybrid between the two, have attracted a considerable amount of interest in the past few decades. In this paper we shall focus our attention on the $L^p$ theory for oscillatory singular integral operators. The kernel of such an operator is given by the product of an oscillatory factor $e^{i \Phi (x,y)}$ and a Calderón-Zygmund type kernel function K(x, y). More precisesly, we define $T_{\Phi , K}$ by

$$\begin{aligned} T_{\Phi , K} f (x)= \text{ p.v. } \int _{\mathbb {R}^n} e^{i \Phi (x,y)} K(x,y) f(y) dy . \end{aligned}$$

(1)

The phase function $\Phi$ is assumed to be real-valued. In [11], for any Calderón-Zygmund kernel K(x, y) which is smooth away from $\Delta = \{(x, x):\, x \in \mathbb {R}^n\}$, Phong and Stein established the uniform $L^p$ boundedness for all $T_{\Phi , K}$ with $\Phi$ being in the family of bilinear forms. Subsequently in [12], for any Calderón–Zygmund kernel K(x, y) which is $C^1$ on $\mathbb {R}^n\times \mathbb {R}^n\backslash \Delta$, Ricci and Stein proved the $L^p$ boundedness of $T_{\Phi , K}$ for all polynomial phase functions $\Phi (x,y) = P(x,y)$, with the bound on $\Vert T_{P, K}\Vert _{p,p}$ being uniform as long as a cap is placed on $\deg (P)$. Their result can be stated as follows.

Theorem 1.1

([12]) Let $n \in \mathbb {N}$ and P(x, y) be a real-valued polynomial in $x, y \in \mathbb {R}^n$. Suppose that there is an $A > 0$ such that K(x, y) satisfies

$$\begin{aligned} |K(x,y)| \le \frac{A}{|x - y|^n}; \end{aligned}$$

(2)

$K(\cdot , \cdot ) \in C^1(\mathbb {R}^n\times \mathbb {R}^n\backslash \Delta )$ and

$$\begin{aligned} |\nabla _x K(x,y)| + |\nabla _y K(x,y)| \le \frac{A}{|x - y|^{n+1}} \end{aligned}$$

(3)

for all $(x, y) \in (\mathbb {R}^n\times \mathbb {R}^n)\backslash \Delta$;

$$\begin{aligned} \Vert T_o\Vert _{L^2(\mathbb {R}^n) \rightarrow L^2(\mathbb {R}^n)} \le A \end{aligned}$$

(4)

where

$$\begin{aligned} T_o f(x) = p.v. \int _{\mathbb {R}^n} K(x, y) f(y) dy . \end{aligned}$$

(5)

Then, for $1< p < \infty$, there exists a $C_p > 0$ such that

$$\begin{aligned} \Vert T_{P, K}f\Vert _{L^p(\mathbb {R}^n)} \le C_p \Vert f\Vert _{L^p(\mathbb {R}^n)} \end{aligned}$$

(6)

for all $f \in L^p(\mathbb {R}^n)$. The constant $C_p$ may depend on p, n, A and $\deg (P)$ but is independent of the coefficients of P.

Oscillatory singular integral operators with general $C^\infty$ phase functions were studied in [9] where, among other things, the $L^p$ boundedness was obtained under a “finite-type” phase function condition, both of which are described below.

Definition 1.1

Let $(x_0, y_0) \in \mathbb {R}^n\times \mathbb {R}^n$ and $\Phi (x, y)$ be $C^\infty$ in an open set containing $(x_0, y_0)$. $\Phi$ is said to be of finite type at $(x_0, y_0)$ if there exist two multi-indices $\alpha , \beta \in (\mathbb {N}\cup \{0\})^n$ such that $|\alpha |, |\beta | \ge 1$ and

$$\begin{aligned} \frac{\partial ^{\alpha + \beta } \Phi }{\partial x^{\alpha } \partial y^{\beta }}(x_0, y_0) \ne 0.\end{aligned}$$

Theorem 1.2

([9]) Let $\varphi \in C^\infty _0(\mathbb {R}^n\times \mathbb {R}^n)$ and $\Phi _1(x,y), \ldots , \Phi _m(x,y)$ be $C^\infty$ such that, for every $(u_1, \ldots , u_m) \in {\mathbb {S}}^{m-1}$, $\displaystyle {\sum _{j=1}^mu_j\Phi _j(x,y)}$ is of finite type at every point in $(supp (\varphi ))\cap \Delta$. Let K(x, y) satisfy (2), (3) and (4). Then, for $1< p < \infty$, the operators $T _{\lambda \Phi ,\, \varphi K}$ are uniformly bounded on $L^p(\mathbb {R}^n)$ for all $\Phi (x,y) = \displaystyle {\sum _{j=1}^mu_j\Phi _j(x,y)}$ where $\lambda \in \mathbb {R}$ and $(u_1, \ldots , u_m) \in {\mathbb {S}}^{m-1}$.

For any polynomial phase function P(x, y), if it has at least one nonzero term $a_{\alpha \beta } x^\alpha y^\beta$ with $\min \{|\alpha |, \, |\beta |\} \ge 1$, then the $L^p$ boundedness of the corresponding oscillatory singular integral operators is covered by Theorem 1.2. Otherwise one has $P(x,y) = g(x) + h(y)$, in which case the $L^p$ boundedness follows from $\Vert T_{P,\, K}\Vert _{p,p}=\Vert T_{0, \, K}\Vert _{p,p}$.

On the other hand, it has been well-known that Calderón-Zygmund singular integrals are bounded on $L^p$ spaces even when the $C^1$ assumption and the bounds for $\nabla K$ in (3) are replaced by the following weaker Hölder type condition:

There exists a $\delta > 0$ such that

$$\begin{aligned} \begin{aligned} | K(x,y) - K(x^\prime , y)| \le \frac{A|x - x^\prime |^\delta }{(|x - y| + |x^\prime - y|)^{n+\delta }} \\ \text{ whenever }\,\, |x - x^\prime |< (1/2)\max \{|x-y|, |x^\prime - y|\},\, \, \text{ and } \\ | K(x,y) - K(x, y^\prime )| \le \frac{A|y - y^\prime |^\delta }{(|x - y| + |x - y^\prime |)^{n+\delta }} \\ \text{ whenever }\,\, |y - y^\prime | < (1/2)\max \{|x-y|, |x - y^\prime |\}. \end{aligned} \end{aligned}$$

(7)

In a recent paper [2], the results of Ricci and Stein in Theorem 1.1 were extended to allow K(x, y) to be such a Hölder class kernel.

Theorem 1.3

([2]) Let P(x, y) be a real-valued polynomial. Let K(x, y) be a Hölder class Calderón-Zygmund kernel, i.e. there exist $\delta , A > 0$ such that K(x, y) satisfies (2), (7) and (4). Then, for $1< p < \infty$, there exists a $C_p > 0$ such that

$$\begin{aligned} \Vert T_{P, K}f\Vert _{L^p(\mathbb {R}^n)} \le C_p \Vert f\Vert _{L^p(\mathbb {R}^n)} \end{aligned}$$

(8)

for all $f \in L^p(\mathbb {R}^n)$. The constant $C_p$ may depend on $p, n, \delta , A$ and $\deg (P)$ but is independent of the coefficients of P.

See also [1, 6].

We now state the main result of this paper in which not only the kernels K(x, y) are allowed to be in the Hölder class, but the phase functions can be fairly general.

Theorem 1.4

Let U be an open set in $\mathbb {R}^m$ and G be a compact subset of U. Let $\Phi (x, y, u) \in C^\infty (\mathbb {R}^n\times \mathbb {R}^n\times U)$ and $\varphi (x,y) \in C^\infty _0(\mathbb {R}^n\times \mathbb {R}^n)$ such that, for every $u \in U$, $\Phi (\,\cdot \,, \, \cdot \,, u)$ is of finite type at every point in $(supp (\varphi ))\cap \Delta$. Let K(x, y) be a Hölder class Calderón-Zygmund kernel, i.e. there exist $\delta , A > 0$ such that K(x, y) satisfies (2), (7) and (4). Then, for $1< p < \infty$, there exists a $C_p > 0$ such that

$$\begin{aligned} \Vert T _{\lambda \Phi ,\, \varphi K}f\Vert _{L^p(\mathbb {R}^n)} \le C_p \Vert f\Vert _{L^p(\mathbb {R}^n)} \end{aligned}$$

(9)

for all $f \in L^p(\mathbb {R}^n)$, $\lambda \in \mathbb {R}$ and $u \in G$. The constant $C_p$ may depend on $p, n, m, \delta , A, \varphi$ and G but is independent of $\lambda$ and u.

Remarks.

(i) It is a well-known fact that the conclusion of Theorems 1.2 and 1.4 can fail if the “finite type" assumption for the phase functions is dropped [8, 9, 16].

(ii) The phase functions in Theorem 1.2 are subsumed in the family of phase functions in Theorem 1.4 as one can simply let $U= \mathbb {R}^m\backslash \{0\}$, $G = {\mathbb {S}}^{m-1}$ and

$$\begin{aligned} \Phi (x,y,u)= u \cdot (\Phi _1(x,y), \ldots , \Phi _m(x,y)).\end{aligned}$$

(iii) By (2), it is easy to see that Theorem 1.4 continues to hold if the smooth cut-off function $\varphi (x,y)$ is replaced by, say, $\chi _B(x-y)$, where B is the unit ball in $\mathbb {R}^n$.

(iv) The conclusion of Theorem 1.4 remains valid in the more general context of weighted spaces $L^p(\mathbb {R}^n, w(x)dx)$ with Muckenhoupt $A_p$ weights. See Theorem 4.2.

(v) It follows from Theorem 1.4 that the operators $T _{\lambda \Phi ,\, \varphi K}$ are uniformly bounded on $L^p$ spaces for $\lambda \in \mathbb {R}$ and $u \in G$ if the phase function $\Phi (x, y, u)$ is real-analytic in $\mathbb {R}^n\times \mathbb {R}^n\times U$, where U is an open subset of $\mathbb {R}$ (i.e. m is taken to be 1) and G is a compact subset of U (see Theorem 5.1). It would be interesting to know whether the same holds for $m > 1$.

In the rest of the paper we shall use $A \lesssim B$ ($A \gtrsim B$) to mean that $A \le c B$ ($A \ge cB$) for a certain constant c whose actual value is not essential for the relevant arguments to work. We shall also use $A \approx B$ to mean “$A \lesssim B$ and $B \lesssim A$”.

2 A van der Corput type lemma

A version of the classical van der Corput’s lemma can be stated as follows.

Lemma 2.1

([14]) 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, for every $\psi \in C^1([a, b])$,

$$\begin{aligned} \bigg |\int _a^b e^{i \lambda \phi (x)} \psi (x) dx \bigg | \le c_k |\lambda |^{-1/k} \bigg (|\psi (b)| + \int _a^b|\psi ^\prime (x)|dx\bigg ) \end{aligned}$$

(10)

holds for all $\lambda \in \mathbb {R}$. The constant $c_k$ is independent of $\lambda , a, b$, $\phi$ and $\psi$.

The following lemma, which is in the spirit of Lemma 2.1, is needed in our proof of Theorem 1.4.

Lemma 2.2

Let $\phi \in C^\infty (\mathbb {R}^n)$ be real-valued and $\psi \in C_0^\infty (\mathbb {R}^n)$. Let $M > 0$, $k \in \mathbb {N}$ and $\alpha \in (\mathbb {N}\cup \{0\})^n$ such that $|\alpha |=k$. Suppose that $|\partial ^\beta \phi /\partial x^\beta (x)| \le M$ holds for all $|\beta |=k+1$ and $x \in V_1$, where $V_a$ is defined by

$$\begin{aligned} V_a = \{ x \in \mathbb {R}^n:\, \text{ dist }(x, \text{ supp }(\psi )) \le a \Vert \partial ^\alpha \phi /\partial x^\alpha \Vert _{L^\infty (\text{ supp }(\psi ))}\}\end{aligned}$$

for $a > 0$. Let

$$\begin{aligned} \Vert \psi \Vert _{0,1} = \Vert \psi \Vert _{L^\infty (\mathbb {R}^n)} + \sup _{x\in \mathbb {R}^n, v \in {\mathbb {S}}^{n-1}} \int _{\mathbb {R}}|\nabla \psi (x+tv)|dt. \end{aligned}$$

Then there exists a $c > 0$ such that

$$\begin{aligned} \bigg |\int _{\mathbb {R}^n}e^{i\lambda \phi (x)}\psi (x) dx\bigg | \le c (a^{-n}\Vert \psi \Vert _{0,1}) |\lambda |^{-\varepsilon /k} \int _{V_a} \bigg |\frac{\partial ^\alpha \phi (x)}{\partial x^\alpha }\bigg |^{-\varepsilon (1+1/k)} dx \end{aligned}$$

(11)

for all $a, \varepsilon \in (0, 1]$ and $\lambda \in \mathbb {R}$. The constant c may depend on M, $\alpha$ (and thus k) but is otherwise independent of $a, \varepsilon , \lambda , \psi$ and $\phi$.

The above lemma is a refined version of Lemma 3.2 of [10]. We shall sketch its proof below where our focus will primarily be on providing the necessary details for the current incarnation.

Proof

Without loss of generality we may assume that

$$\begin{aligned} |\{\partial ^\alpha \phi /\partial x^\alpha = 0\}\cap \text{ supp }(\psi )| = 0.\end{aligned}$$

Let $A > 1$ be a suitably chosen constant which depends on M, n and $\alpha$ only, and let $r(x) = A^{-1}|\partial ^\alpha \phi /\partial x^\alpha (x)|$ whenever it is nonzero. By applying the Vitali covering procedure, there exist $x_1, x_2, \ldots \in \{\partial ^\alpha \phi /\partial x^\alpha \ne 0\}\cap \text{ supp }(\psi )$ such that

$$\begin{aligned}{} & {} \{\partial ^\alpha \phi /\partial x^\alpha \ne 0\}\cap \text{ supp }(\psi ) \subseteq \bigcup _j B(x_j, r_j/2) \, \, \, \text{ where } \, r_j=r(x_j), \end{aligned}$$

(12)

$$\begin{aligned}{} & {} \big \{B(x_j, r_j/10)\big \}_{j=1, 2, \ldots } \, \, \, \text{ are } \text{ pairwise } \text{ disjoint. } \end{aligned}$$

(13)

It follows from our selection of A and a packing argument of Sogge and Stein in [13] (see also [14]) that, for each j, there exists a $v_j \in {\mathbb {S}}^{n-1}$ such that

$$\begin{aligned}{} & {} |\partial ^\alpha \phi /\partial x^\alpha (y)| \approx r_j; \end{aligned}$$

(14)

$$\begin{aligned}{} & {} |(v_j \cdot \nabla )^k \phi (y)| \gtrsim r_j \end{aligned}$$

(15)

for all $y \in B(x_j,\, r_j)$ and

$$\begin{aligned} \sum _j \chi _{B(x_j,\, r_j)} \lesssim 1. \end{aligned}$$

(16)

Thus, there exists a partition of unity $\{\eta _j(x)\}_{j=1, 2, \ldots }$ such that each $\eta _j$ is supported in $B(x_j,\, r_j)$, $\sum _j \eta _j(x) = 1$ for $x \in \bigcup _j B(x_j, r_j/2)$, and

$$\begin{aligned} |\partial ^\beta \eta _j/\partial x^\beta | \lesssim r_j^{-|\beta |} \end{aligned}$$

(17)

for all $\beta \in (\mathbb {N}\cup \{0\})^n$.

For $y= (y_1, y_2, \ldots y_n) \in \mathbb {R}^n$, let ${\tilde{y}} = (y_2, \ldots , y_n)$. For each j, let $\Gamma _j$ denote an orthogonal linear transformation on $\mathbb {R}^n$ such that $\Gamma _j((1, 0, \ldots , 0)) = v_j$. Then by (15), for $|y| \le r_j$,

$$\begin{aligned} \bigg |\frac{\partial ^k}{\partial y_1^k}\big (\phi (x_j+ \Gamma _j(y))\big )\bigg | \gtrsim r_j. \end{aligned}$$

(18)

When $k \ge 2$, by using (18), Lemma 2.1 and (17), we have

$$\begin{aligned}{} & {} \bigg |\int _{B(x_j,\, r_j)}e^{i\lambda \phi (x)}\psi (x) \eta _j(x) dx\bigg | \nonumber \\{} & {} \quad \le \int _{|{\tilde{y}}|\le r_j}\bigg |\int _{-(r_j^2-|{\tilde{y}}|^2)^{1/2}}^{(r_j^2-|{\tilde{y}}|^2)^{1/2}} e^{i\lambda \phi (x_j+ \Gamma _j(y))} \psi (x_j + \Gamma _j(y)) \eta _j(x_j + \Gamma _j(y))dy_1\bigg |d{\tilde{y}} \nonumber \\{} & {} \quad \lesssim (\lambda r_j)^{-1/k}\int _{|{\tilde{y}}|\le r_j}\bigg (|\psi (x_j + \Gamma _j(((r_j^2-|{\tilde{y}}|^2)^{1/2}, {\tilde{y}}))) \eta _j(x_j + \Gamma _j(((r_j^2-|{\tilde{y}}|^2)^{1/2}, {\tilde{y}})))| \nonumber \\{} & {} \qquad + \int _{-(r_j^2-|{\tilde{y}}|^2)^{1/2}}^{(r_j^2-|{\tilde{y}}|^2)^{1/2}}\bigg |\frac{\partial }{\partial y_1} \big (\psi (x_j + \Gamma _j(y)) \eta _j(x_j + \Gamma _j(y))\big )\bigg |dy_1 \bigg ) d{\tilde{y}} \nonumber \\{} & {} \quad \lesssim \Vert \psi \Vert _{0,1} (|\lambda |r_j)^{-1/k} r_j^{n-1}. \end{aligned}$$

(19)

For $k=1$, one cannot use Lemma 2.1 because the monotonicity of the first derivative of $\phi (x_j+ \Gamma _j(y))$ in $y_1$ is not known. Fortunately we have the following upper bound for the corresponding second derivative:

$$\begin{aligned}\bigg |\frac{\partial ^2}{\partial y_1^2}( \phi (x_j+ \Gamma _j(y)))\bigg |= |(v_j \cdot \nabla )^2 \phi (x_j+ \Gamma _j(y))| \le M \end{aligned}$$

for $|y| \le r_j$, which allows us to use integration by parts and (15) to get

$$\begin{aligned}{} & {} \bigg |\int _{B(x_j,\, r_j)}e^{i\lambda \phi (x)}\psi (x) \eta _j(x) dx\bigg | \\{} & {} \quad = \bigg |\int _{|{\tilde{y}}|\le r_j} \int _{-(r_j^2-|{\tilde{y}}|^2)^{1/2}}^{(r_j^2-|{\tilde{y}}|^2)^{1/2}} \frac{\partial }{\partial y_1}\bigg (e^{i\lambda \phi (x_j+ \Gamma _j(y))}\bigg ) \frac{\psi (x_j + \Gamma _j(y)) \eta _j(x_j + \Gamma _j(y))}{ (i\lambda )\partial /\partial y_1(\phi (x_j+ \Gamma _j(y)))} dy_1d{\tilde{y}}\bigg | \\{} & {} \quad \lesssim \Vert \psi \Vert _{0,1} (|\lambda |r_j)^{-1} r_j^{n-1}, \end{aligned}$$

which is just (19) for the case $k=1$.

Trivially we have

$$\begin{aligned} \bigg |\int _{B(x_j,\, r_j)}e^{i\lambda \phi (x)}\psi (x) \eta _j(x) dx\bigg | \lesssim \Vert \psi \Vert _{0,1}r_j^n. \end{aligned}$$

(20)

By (19)–(20), for every j and every $\varepsilon \in (0,1]$,

$$\begin{aligned} \bigg |\int _{B(x_j,\, r_j)}e^{i\lambda \phi (x)}\psi (x) \eta _j(x) dx\bigg | \lesssim \Vert \psi \Vert _{0,1}|\lambda |^{-\varepsilon /k}r_j^{-\varepsilon (1+1/k)}r_j^n. \end{aligned}$$

(21)

By (21), (14) and (16), for every $a \in (0, 1]$,

$$\begin{aligned}{} & {} \bigg |\int _{\mathbb {R}^n}e^{i\lambda \phi (x)}\psi (x) dx\bigg | \le \sum _j \bigg |\int _{B(x_j,\, r_j)}e^{i\lambda \phi (x)}\psi (x) \eta _j(x) dx\bigg |\\{} & {} \quad \lesssim (\Vert \psi \Vert _{0,1}|\lambda |^{-\varepsilon /k}a^{-n}) \sum _j r_j^{-\varepsilon (1+1/k)}(ar_j)^n \\{} & {} \quad \lesssim (\Vert \psi \Vert _{0,1}|\lambda |^{-\varepsilon /k}a^{-n}) \int _{\mathbb {R}^n} \bigg |\frac{\partial ^\alpha \phi (x)}{\partial x^\alpha }\bigg |^{-\varepsilon (1+1/k)} \bigg (\sum _j \chi _{B(x_j, \, a r_j)}(x)\bigg ) dx \\{} & {} \quad \lesssim (a^{-n}\Vert \psi \Vert _{0,1}) |\lambda |^{-\varepsilon /k} \int _{V_a} \bigg |\frac{\partial ^\alpha \phi (x)}{\partial x^\alpha }\bigg |^{-\varepsilon (1+1/k)} dx. \end{aligned}$$

$\square$

3 Proof of Theorem 1.4

For $k \in \mathbb {N}$, $r > 0$ and $a \in \mathbb {R}^k$, let $B_k(a, r)= \{x \in \mathbb {R}^k: \, |x-a| < r\}$. For any function F(x, y) defined on a product space $\mathbb {R}^{n_1}\times \mathbb {R}^{n_2}$, where $x \in \mathbb {R}^{n_1}$ and $y \in \mathbb {R}^{n_2}$, and multi-indices $\alpha \in (\mathbb {N}\cup \{0\})^{n_1}$, $\beta \in (\mathbb {N}\cup \{0\})^{n_2}$, we let

$$\begin{aligned} D^\alpha _1 F= \frac{\partial ^{\alpha }F}{\partial x^\alpha }, \, \, D^\beta _2 F= \frac{\partial ^{\beta }F}{\partial y^\beta }. \end{aligned}$$

The same goes for functions defined on more general product spaces $\mathbb {R}^{n_1}\times \cdots \times \mathbb {R}^{n_k}$.

Let K(x, y) be a Hölder class Calderón-Zygmund kernel. Clearly, the three properties (2), (7) and (4) of K(x, y) remain intact under the translation $(x,y) \rightarrow (x-\zeta , \, y-\zeta )$ for any $\zeta \in \mathbb {R}^n$. This observation, together with the compactness of $\text{ supp }(\varphi )$ and G, allows the proof of Theorem 1.4 to be reduced to the task of establishing the following:

Proposition 3.1

Suppose that $\Phi (x, y, u)$ is $C^\infty$ in an open neighborhood of the origin in $\mathbb {R}^n\times \mathbb {R}^n\times \mathbb {R}^m$ and there are two nonzero multi-indices $\alpha _0, \beta _0 \in (\mathbb {N}\cup \{0\})^n$ such that

$$\begin{aligned} D^{\alpha _0}_1 D^{\beta _0}_2 \Phi (0,0,0) \ne 0. \end{aligned}$$

(22)

Then there exists an $r_0 > 0$ such that for every $p \in (1, \infty )$ and every $\varphi \in C_0^\infty (B_{2n}(0, r_0))$, the operator

$$\begin{aligned}T _{\lambda \Phi ,\, \varphi K}: f \rightarrow \text{ p.v. } \int _{\mathbb {R}^n} e^{i \lambda \Phi (x,y, u)} K(x,y) \varphi (x,y) f(y) dy \end{aligned}$$

is uniformly bounded on $L^p(\mathbb {R}^n)$ for $\lambda > 2$ and $u \in B_m(0, r_0)$.

Proof

Let $\lambda > 2$, $k_0= |\alpha _0|$ and $l_0 = |\beta _0|$. Without loss of generality we may assume that

$$\begin{aligned} D^{\alpha _0}_1D^{\beta }_2\Phi (0,0,0) = 0 \end{aligned}$$

(23)

for all $|\beta | < l_0$. By using a transformation $(x, y) \rightarrow (\Gamma (x), \Gamma (y))$ where $\Gamma$ is an orthogonal transformation, if necessary, we may also assume that $\beta _0 = (l_0, 0, \ldots , 0)$. Let

$$\begin{aligned} F(x, y, z, u)= D^{\alpha _0}_1 \Phi (z, x, u) -D^{\alpha _0}_1\Phi (z, y, u).\end{aligned}$$

Then

$$\begin{aligned} \frac{\partial ^j F}{\partial y_1^j}(0, 0, 0, 0) = 0 \end{aligned}$$

for $0 \le j \le l_0 -1$ and

$$\begin{aligned} \frac{\partial ^{l_0} F}{\partial y_1^{l_0}}(0, 0, 0, 0) \ne 0. \end{aligned}$$

By the Malgrange preparation theorem [4], there exist an $r_0 > 0$ and $C^\infty$ functions $a_0(x, {\tilde{y}}, z, u), \ldots , a_{l_0-1} (x, {\tilde{y}}, z, u)$ on $I^{n}\times I^{n-1}\times I^n\times I^m$ and c(x, y, z, u) on $I^{n}\times I^{n}\times I^n\times I^m$, where $I = (-4r_0, 4r_0)$, such that

$$\begin{aligned}{} & {} F(x, y, z, u) = c(x, y, z, u) \nonumber \\{} & {} \quad \times (y_1^{l_0} + a_{l_0-1}(x, {\tilde{y}}, z, u) y_1^{l_0 -1} + \cdots + a_0(x, {\tilde{y}}, z, u)) \end{aligned}$$

(24)

and $|c(x, y, z, u)| \gtrsim 1$ for $(x, y, z, u) \in I^{n}\times I^{n}\times I^n\times I^m$.

Let $\eta \in C^\infty _0(\mathbb {R}^n\times \mathbb {R}^n)$ such that $0 \le \eta (x,y) \le 1$ for $(x,y) \in \mathbb {R}^n\times \mathbb {R}^n$; $\eta (x,y) = 1$ for $|(x,y)| \le 1/2$; and $\eta (x,y) = 0$ for $|(x,y)| \ge 1$. For $t > 0$, let $\eta _t(x,y)= t^{-2n}\eta (x/t, y/t)$.

Also, let $\theta \in C^\infty (\mathbb {R}^n)$ be nonnegative such that $\theta (x)=0$ for $|x| \le 4$ and $\theta (x) = 1$ for $|x| \ge 8$. Let $N_0 =6(2n+1) k_0 l_0$, $\rho = N_0^{-1}$ and

$$\begin{aligned} H_\lambda (x,y) = \frac{\varphi (x,y)}{J(\eta )}\int _{\mathbb {R}^n\times \mathbb {R}^n} \eta _{\lambda ^{-\rho }}(x-v, y-w) K(v, w) \theta (\lambda ^{\rho }(v-w)) dv dw\end{aligned}$$

where $\displaystyle {J(\eta ) = \int _{\mathbb {R}^n\times \mathbb {R}^n}\eta (x,y)dxdy} \gtrsim 1$.

When $H_\lambda (x,y) \ne 0$, there exists a $(v, w) \in \mathbb {R}^n\times \mathbb {R}^n$ such that $|v-w| \ge 4 \lambda ^{-\rho }$ and $|(x, y) - (v, w)| < \lambda ^{-\rho }$. Thus,

$$\begin{aligned} 2\lambda ^{-\rho } \le |v-w|/2 \le |x-y| \le 3|v-w|/2.\end{aligned}$$

By (2),

$$\begin{aligned} |H_\lambda (x,y)| \lesssim \frac{|\varphi (x,y)|}{|x-y|^n}\chi _{[2\lambda ^{-\rho }, \, \,\infty )}(|x-y|). \end{aligned}$$

(25)

Similarly, one can show that, for all $x, y \in \mathbb {R}^n$,

$$\begin{aligned} \Vert H_\lambda (x, \, \cdot \,)\Vert _{0,1} + \Vert H_\lambda (\, \cdot \,, y)\Vert _{0,1} \lesssim \lambda ^{(n+1)\rho }. \end{aligned}$$

(26)

We now decompose $T _{\lambda \Phi ,\, \varphi K}$ as the sum of three operators:

$$\begin{aligned} T _{\lambda \Phi ,\, \varphi K}f = T_1f + T_2f + T_3f \end{aligned}$$

(27)

where

$$\begin{aligned}{} & {} T_1f(x) = \int _{\mathbb {R}^n} e^{i\lambda \Phi (x, y, u)} H_\lambda (x,y) f(y) dy, \end{aligned}$$

(28)

$$\begin{aligned}{} & {} T_2f(x) = \int _{\mathbb {R}^n} e^{i\lambda \Phi (x, y, u)} \big [K(x,y)\theta (\lambda ^\rho (x-y))\varphi (x,y) - H_\lambda (x,y)\big ] f(y) dy,\nonumber \\ \end{aligned}$$

(29)

$$\begin{aligned}{} & {} T_3f(x) = \text{ p.v. }\int _{\mathbb {R}^n} e^{i\lambda \Phi (x, y, u)} K(x,y)(1- \theta (\lambda ^\rho (x-y)))\varphi (x,y) f(y) dy. \end{aligned}$$

(30)

It follows from (25) that

$$\begin{aligned} \Vert T_1\Vert _{L^1(\mathbb {R}^n)\rightarrow L^1(\mathbb {R}^n)} + \Vert T_1\Vert _{L^\infty (\mathbb {R}^n)\rightarrow L^\infty (\mathbb {R}^n)} \lesssim \ln (\lambda ). \end{aligned}$$

(31)

On the other hand, we have

$$\begin{aligned} T_1^*T_1 f(x) = \int _{\mathbb {R}^n} L(x,y) f(y) dy \end{aligned}$$

where

$$\begin{aligned} L(x,y) = \int _{\mathbb {R}^n} e^{i\lambda (\Phi (z, x, u) - \Phi (z, y, u))} H_\lambda (z, x) \overline{H_\lambda (z, y)} dz.\end{aligned}$$

By shrinking the support of $\varphi$ if necessary, we may apply Lemma 2.2 with $\varepsilon = (3l_0)^{-1}$ to get

$$\begin{aligned}{} & {} |L(x,y)| \lesssim \lambda ^{-1/(3k_0l_0)}(\Vert H_\lambda (\,\cdot \,, x) \overline{H_\lambda (\,\cdot \,, y)}\Vert _{0,1})\chi _{[0, 2r_0]}(|x|) \chi _{[0, 2r_0]}(|y|) \nonumber \\{} & {} \quad \times \int _{|z|\le 2r_0}|D^{\alpha _0}_1\Phi (z, x, u) - D^{\alpha _0}_1\Phi (z, y, u)|^{-(k_0+1)/(3k_0l_0)} dz. \end{aligned}$$

(32)

By using (24), (26), (32), $(k_0+1)/(3k_0l_0) < 1$ and the lemma on page 182 of [12], for every $x \in \mathbb {R}^n$,

$$\begin{aligned}{} & {} \int _{\mathbb {R}^n} |L(x,y)|dy \lesssim \lambda ^{-1/(3k_0l_0)} \lambda ^{(2n+1)\rho } \int _{|z| \le 2r_0}\int _{|{\tilde{y}}| \le 2r_0} \bigg (\int _{|y_1| \le 2 r_0}\bigg |y_1^{l_0} \nonumber \\{} & {} \quad + a_{l_0-1}(x, {\tilde{y}}, z, u) y_1^{l_0 -1} + \cdots + a_0(x, {\tilde{y}}, z, u) \bigg |^{-(k_0+1)/(3k_0l_0)} dy_1\bigg )d{\tilde{y}} dz \nonumber \\{} & {} \quad \lesssim \lambda ^{-1/(6k_0l_0)}. \end{aligned}$$

(33)

Similary, we have

$$\begin{aligned} \int _{\mathbb {R}^n} |L(x,y)|dx \lesssim \lambda ^{-1/(6k_0l_0)} \end{aligned}$$

(34)

for all $y \in \mathbb {R}^n$. It follows from (33)–(34) that

$$\begin{aligned} \Vert T_1\Vert _{L^2(\mathbb {R}^n)\rightarrow L^2(\mathbb {R}^n)} \lesssim \lambda ^{-1/(12k_0l_0)}. \end{aligned}$$

(35)

By interpolating between (31) and (35) we obtain

$$\begin{aligned} \Vert T_1\Vert _{L^p(\mathbb {R}^n)\rightarrow L^p(\mathbb {R}^n)} \lesssim (\ln (\lambda ))^{|1-2/p|}\lambda ^{-(1-|1-2/p|)/(12k_0l_0)} \lesssim 1 \end{aligned}$$

(36)

for $1< p < \infty$.

To treat the term $T_2f$, first we observe that

$$\begin{aligned}{} & {} |K(x,y)\theta (\lambda ^\rho (x-y))\varphi (x,y) - H_\lambda (x,y)| \lesssim |\varphi (x, y)|\times \\{} & {} \quad \int _{\mathbb {R}^n\times \mathbb {R}^n}\eta _{\lambda ^{-\rho }}(x-v, y-w) \big |K(x,y)\theta (\lambda ^\rho (x-y)) - K(v, w) \theta (\lambda ^{\rho }(v-w))\big |dvdw \\{} & {} \quad \lesssim |\varphi (x, y)|\bigg [\int _{\mathbb {R}^n\times \mathbb {R}^n}\eta _{\lambda ^{-\rho }}(x-v, y-w) \big |K(x, y) - K(v, w)\big | \theta (\lambda ^\rho (x-y)) dvdw \\{} & {} \qquad + \int _{\mathbb {R}^n\times \mathbb {R}^n}\eta _{\lambda ^{-\rho }}(x-v, y-w) |K(v, w)| \big |\theta (\lambda ^\rho (x-y)) - \theta (\lambda ^{\rho }(v-w))\big |dvdw\bigg ]. \end{aligned}$$

Let the above two integrals be denoted by $I_1(x,y)$ and $I_2(x,y)$, respectively. For $I_1(x, y)$ to be nonzero, there must exist $v, w \in \mathbb {R}^n$ such that $|x - v| < \lambda ^{-\rho }$,   $|y - w| < \lambda ^{-\rho }$, while $|x-y| \ge 4\lambda ^{-\rho }$. Thus, $|v-w| \ge 2\lambda ^{-\rho }$ and $|v - w| \approx |x - y|$. It follows from (7) that

$$\begin{aligned}{} & {} \big |K(x, y) - K(v, w)\big | \le \big |K(x, y) - K(v, y)\big | + \big |K(v, y) - K(v, w)\big | \\{} & {} \quad \lesssim \frac{|x-v|^\delta }{(|x - y| + |v - y|)^{n+\delta }} + \frac{|y-w|^\delta }{(|v - y| + |v - w|)^{n+\delta }} \\{} & {} \quad \lesssim \frac{\lambda ^{-\rho \delta }\chi _{[4\lambda ^{-\rho }, \, \infty )}(|x - y|)}{|x - y|^{n+\delta }},\end{aligned}$$

which implies that

$$\begin{aligned} |I_1(x,y)| \lesssim \frac{\lambda ^{-\rho \delta }\chi _{[4\lambda ^{-\rho }, \, \infty )}(|x - y|)}{|x - y|^{n+\delta }}. \end{aligned}$$

(37)

For $I_2(x, y)$ to be nonzero, there must exist $v, w \in \mathbb {R}^n$ such that $|x - v| < \lambda ^{-\rho }$,   $|y - w| < \lambda ^{-\rho }$, while

$$\begin{aligned} \max \{|x - y|, \, |v - w|\} \ge 4 \lambda ^{-\rho }\end{aligned}$$

and

$$\begin{aligned} \min \{|x - y|, \, |v - w|\} \le 8 \lambda ^{-\rho }.\end{aligned}$$

Thus, $|x - y| \approx |v - w|$ and

$$\begin{aligned} 2\lambda ^{-\rho } \le |x - y| \le 10 \lambda ^{-\rho },\end{aligned}$$

which together imply that

$$\begin{aligned} |I_2(x,y)| \lesssim \frac{\chi _{[2\lambda ^{-\rho }, \, 10\lambda ^{-\rho }]}(|x - y|)}{|x - y|^{n}}. \end{aligned}$$

(38)

By (37)–(38),

$$\begin{aligned} \Vert T_2\Vert _{L^p(\mathbb {R}^n)\rightarrow L^p(\mathbb {R}^n)} \lesssim \lambda ^{-\rho \delta } \int _{|x| \ge 4\lambda ^{-\rho }} \frac{dx}{|x|^{n+\delta }} +\int _{2\lambda ^{-\rho } \le |x| \le 10\lambda ^{-\rho }} \frac{dx}{|x|^n} \lesssim 1. \end{aligned}$$

(39)

Now $T_3f$ is the only term left to be treated. For any $h \in \mathbb {R}^n$, let $Q_h = h + (\lambda ^{-\rho } I)^n$ and $Q_h^*= h + (9\lambda ^{-\rho } I)^n$ where $I = (-1/2, 1/2]$. Let $\phi _\beta (x, u) = D^\beta _2\Phi (x, x, u)$ for $\beta \in (\mathbb {N}\cup \{0\})^n$ and define the polynomial $P_{h, u}(x, y)$ by

$$\begin{aligned} P_{h, u}(x, y) = \sum _{1 \le |\beta | \le N_0 -1}\bigg (\sum _{|\alpha | \le N_0 - |\beta | -1} \frac{1}{\alpha !\beta !}D_1^\alpha \phi _{\beta }(h, u)(x-h)^\alpha (y-x)^{\beta }\bigg ). \end{aligned}$$

Thus, for any $h \in \mathbb {R}^n$, $x \in Q_h^*$, $y \in Q_h$ and $|u| < r_0$,

$$\begin{aligned} |\Phi (x, y, u) - (\Phi (x, x, u) + P_{h, u}(x, y))| |\varphi (x,y)| \lesssim \sum _{j=1}^{N_0}|x-y|^j|x-h|^{N_0-j}. \end{aligned}$$

For any $f \in L^p(\mathbb {R}^n)$ and any $h \in \mathbb {R}^n$, we have $\text{ supp }(T_3(\chi _{Q_h}f)) \subseteq Q_h^*$ and thus,

$$\begin{aligned} \bigg |T_3(\chi _{Q_h}f)(x) - e^{i\lambda \Phi (x, x, u)} T_{\lambda P_{h, u}, \, {\tilde{K}}}(\chi _{Q_h}f)(x)\bigg | \lesssim \sum _{j=1}^{N_0} \lambda ^{1-(N_0 - j)\rho } \int _{Q_h} \frac{|f(y)|dy}{|x-y|^{n-j}}\nonumber \\ \end{aligned}$$

(40)

where ${\tilde{K}}(x, y) = K(x,y)(1- \theta (\lambda ^\rho (x-y)))\varphi (x,y)$. It is easy to verify that (2), (7) and (4) are all satisfied by ${\tilde{K}}(\, \cdot \, , \cdot \, )$ uniformly in $\lambda$. By (40) and Theorem 1.3,

$$\begin{aligned}{} & {} \Vert T_3(\chi _{Q_h}f)\Vert _{L^p(\mathbb {R}^n)} \lesssim \bigg (1 + \sum _{j=1}^{N_0}\lambda ^{1-(N_0 - j)\rho } \int _{|x| \le 10 \lambda ^{-\rho }} \frac{dx}{|x|^{n-j}}\bigg )\times \Vert \chi _{Q_h}f\Vert _{L^p(\mathbb {R}^n)} \nonumber \\{} & {} \quad \lesssim \Vert \chi _{Q_h}f\Vert _{L^p(\mathbb {R}^n)}. \end{aligned}$$

(41)

By

$$\begin{aligned}{} & {} |T_3f|^p = \bigg |\sum _{h \in (\lambda ^{-\rho })\mathbb {Z}^n} \chi _{Q_h^*}T_3(\chi _{Q_h}f)\bigg |^p \\{} & {} \quad \le \bigg |\sum _{h \in (\lambda ^{-\rho })\mathbb {Z}^n} \chi _{Q_h^*}\bigg |^{p/p^\prime } \bigg (\sum _{h \in (\lambda ^{-\rho })\mathbb {Z}^n}|T_3(\chi _{Q_h}f)|^p\bigg ) \lesssim \sum _{h \in (\lambda ^{-\rho })\mathbb {Z}^n}|T_3(\chi _{Q_h}f)|^p \end{aligned}$$

and (41), we get

$$\begin{aligned} \Vert T_3\Vert _{L^p(\mathbb {R}^n)\rightarrow L^p(\mathbb {R}^n)} \lesssim 1 \end{aligned}$$

(42)

for $1< p < \infty$. It follows from (27), (36), (39) and (42) that

$$\begin{aligned} \Vert T_{\lambda \Phi , \, \varphi K}\Vert _{L^p(\mathbb {R}^n)\rightarrow L^p(\mathbb {R}^n)} \lesssim 1\end{aligned}$$

for $1< p < \infty$. $\square$

4 Extension to $L^p$ spaces with $A_p$ weights

As pointed earlier, the conclusions of Theorem 1.4 continue to hold when the spaces $L^p(\mathbb {R}^n, dx)$ is replaced by the weighted spaces $L^p(\mathbb {R}^n, wdx)$ as long as w is in the class $A_p$ [7] whose definition is given below:

Definition 4.1

Let $p \in (1, \, \infty )$. A nonnegative, locally integrable function $w(\cdot )$ on $\mathbb {R}^n$ is said to be in the Muckenhoupt weight class $A_p(\mathbb {R}^n)$ if there exists a constant $C > 0$ such that

$$\begin{aligned} \bigg (\frac{1}{|Q|} \int _Q w(y) dy\bigg ) \bigg (\frac{1}{|Q|} \int _Q w(y)^{-1/(p-1)} dy\bigg ) ^{p-1} \le C \end{aligned}$$

(43)

holds for all cubes Q in $\mathbb {R}^n$. The smallest such constant C in (43) is the corresponding $A_p$ constant of w.

Let

$$\begin{aligned} \Vert f\Vert _{p, w} = \bigg (\int _{\mathbb {R}^n}|f(x)|^p w(x) dx\bigg )^{1/p},\end{aligned}$$

and

$$\begin{aligned} \displaystyle { L^p(\mathbb {R}^n, wdx) = \{f: \,\Vert f\Vert _{p, w} < \infty \}}. \end{aligned}$$

We shall need the following result due to Coifman and Fefferman:

Theorem 4.1

([3]) For each $p \in (1, \infty )$ and each $w \in A_p(\mathbb {R}^n)$, there exists a $\nu \in (0,1)$ such that $w^{1+\nu } \in A_p(\mathbb {R}^n)$. Both $\nu$ and the $A_p$ constant of $w^{1+\nu }$ depend on n, p and the $A_p$ constant of w only.

We shall now state the weighted version of Theorem 1.4 and give a brief sketch of its proof while leaving out most of the technical details.

Theorem 4.2

Let U be an open set in $\mathbb {R}^m$ and G be a compact subset of U. Let $\Phi (x, y, u) \in C^\infty (\mathbb {R}^n\times \mathbb {R}^n\times U)$ and $\varphi (x,y) \in C^\infty _0(\mathbb {R}^n\times \mathbb {R}^n)$ be such that, for every $u \in U$, $\Phi (\,\cdot \,, \, \cdot \,, u)$ is of finite type at every point in $(supp (\varphi ))\cap \Delta$. Let K(x, y) be a Hölder class Calderón-Zygmund kernel, i.e. there exist $\delta , A > 0$ such that K(x, y) satisfies (2), (7) and (4). Let $p \in (1, \infty )$ and $w \in A_p(\mathbb {R}^n)$. Then there exists a positive constant $C_{p,w}$ such that

$$\begin{aligned} \Vert T _{\lambda \Phi ,\, \varphi K}f\Vert _{p, w} \le C_{p, w} \Vert f\Vert _{p, w} \end{aligned}$$

(44)

for all $f \in L^p(\mathbb {R}^n, wdx)$, $\lambda \in \mathbb {R}$ and $u \in G$. The constant $C_{p,w}$ may depend on $p, n, m, \delta , A, \varphi$, G and $A_p$ the constant of w, but is independent of $\lambda$ and u.

Proof

By (27), it suffices to prove $\Vert T_j f\Vert _{p, w} \lesssim \Vert f\Vert _{p, w}$ for $j=1, 2, 3$ and $\lambda > 2$.

For $T_1$, by (25),

$$\begin{aligned} |T_1f| \lesssim (\ln (\lambda ))\mathcal Mf, \end{aligned}$$

where $\mathcal M$ is the Hardy-Littlewood maximal operator. By Theorem 4.1 and the weighted $L^p$ boundedness of $\mathcal M$,

$$\begin{aligned} \Vert T_1f\Vert _{p, w^{1+\nu }} \lesssim (\ln (\lambda )) \Vert f\Vert _{p, w^{1+\nu }} \end{aligned}$$

(45)

for a certain $\nu > 0$ (see [5]). By (36) and (45) and a result of Stein and Weiss in [17], we obtain

$$\begin{aligned} \Vert T_1f\Vert _{p, w} \lesssim (\ln (\lambda ))^{1/(1+\nu )+|1-2/p|}\lambda ^{-(1-|1-2/p|)\nu /(12(1+\nu )k_0l_0)} \Vert f\Vert _{p, w}\\ \lesssim \Vert f\Vert _{p, w} \end{aligned}$$

For $T_2$, one can use (37)–(38) to get $|T_2f| \lesssim \mathcal Mf$ and thus

$$\begin{aligned} \Vert T_2f\Vert _{p, w} \lesssim \Vert f\Vert _{p, w}.\end{aligned}$$

Finally, for the treament of $T_3f$, one uses Theorem 3.2 in [2] instead of Theorem 1.3 but otherwise follows the steps in the proof of Theorem 1.4 to arrive at

$$\begin{aligned} \Vert T_3f\Vert _{p, w} \lesssim \Vert f\Vert _{p, w}. \square \end{aligned}$$

5 Real analytic phases

In this section we will show how one can use Theorem 1.4 (and Theorem 4.2) to obtain the uniform $L^p$ boundedness of oscillatory singular integral operators with Hölder class kernels and real-analytic phase functions $\lambda \Phi (x, y, u)$ when the parameter u is in a compact subset of $\mathbb {R}$.

Theorem 5.1

Let U be an open set in $\mathbb {R}$ and G be a compact subset of U. Let $\varphi (x,y) \in C^\infty _0(\mathbb {R}^n\times \mathbb {R}^n)$ and $\Phi (x, y, u)$ be real-analytic for (x, y) in an open neighborhood of $\text{ supp }(\varphi )$ and $u \in U$. Let K(x, y) be a Hölder class Calderón-Zygmund kernel, i.e. there exist $\delta , A > 0$ such that K(x, y) satisfies (2), (7) and (4). Let $p \in (1, \infty )$ and $w \in A_p(\mathbb {R}^n)$. Then there exists a positive constant $C_{p,w}$ such that

$$\begin{aligned} \Vert T _{\lambda \Phi ,\, \varphi K}f\Vert _{p, w} \le C_{p, w} \Vert f\Vert _{p, w} \end{aligned}$$

(46)

for all $f \in L^p(\mathbb {R}^n, wdx)$, $\lambda \in \mathbb {R}$ and $u \in G$. The constant $C_{p,w}$ may depend on $p, n, \delta , A, \varphi$, G and the $A_p$ constant of w, but is independent of $\lambda$ and u.

Proof

Without loss of generality we may assume that $\text{ supp }(\varphi ) = \overline{B(0, r_0)}$, $U = (-2r_0, \, 2r_0)$ and $G = [-r_0,\, r_0]$ for some $r_0 > 0$. Let

$$\begin{aligned} E = \{u \in [-r_0,\, r_0]: \, \Phi (\, \cdot \, , \, \cdot \,, u) \, \, \text{ fails } \text{ to } \text{ have } \text{ finite } \text{ type } \text{ at } \text{ some } \text{ point }\}. \end{aligned}$$

In the case where $E = \emptyset$, (46) follows from Theorem 4.2.

Suppose that $E \ne \emptyset$. For each $u_0 \in E$ and $1 \le j, k \le n$, there exists a $(x_0, y_0)$ such that all partial derivatives

$$\begin{aligned}\bigg \{D^\alpha _1D^\beta _2 \bigg (\frac{\partial ^2 \Phi (x, y, u_0)}{\partial x_j \partial y_k}\bigg ): \, \, \alpha , \beta \in (\mathbb {N}\cup \{0\})^n\bigg \} \end{aligned}$$

vanish at $(x_0, y_0)$ which, by real-analyticity, implies that

$$\begin{aligned} \frac{\partial ^2 \Phi (x, y, u_0)}{\partial x_j \partial y_k} = 0 \end{aligned}$$

for all $(x, y) \in \overline{B(0, r_0)}$ and $1 \le j, k \le n$.

If E has a limit point p, then there exists a sequence $\{u_l\}_{l=1}^\infty$ in $E\backslash \{p\}$ such that

$$\begin{aligned} \lim _{l\rightarrow \infty } u_l = p. \end{aligned}$$

Thus,

$$\begin{aligned} \frac{\partial ^2 \Phi (x, y, u_l)}{\partial x_j \partial y_k} = 0 \end{aligned}$$

for all $(x, y) \in \overline{B(0, r_0)}$, $l \in \mathbb {N}$ and $1 \le j, k \le n$. Again by real-analyticity,

$$\begin{aligned} \frac{\partial ^2 \Phi (x, y, u)}{\partial x_j \partial y_k} = 0 \end{aligned}$$

for all $(x, y) \in \overline{B(0, r_0)}$, $u \in (-2r_0, \, 2r_0)$ and $1 \le j, k \le n$. Thus, $\Phi (x, y, u)$ can be written as $\phi (x, u) + \psi (y, u)$ and (46) follows trivially.

Thus we may now assume that $E \, (\ne \emptyset )$ has no limit points. By using a translation and shrinking $r_0$ if necessary, we may further assume that $E = \{0\}$ and

$$\begin{aligned} \Phi (x, y, u) = \sum _{k=0}^\infty \bigg (\frac{u^k}{k!}\bigg ) \frac{\partial ^k\Phi (x, y, 0)}{\partial u^k}. \end{aligned}$$

Since $\Phi (\, \cdot \, , \, \cdot \,, 0)$ fails to be of finite type at least at one point while for every $u \ne 0$, $\Phi (\, \cdot \, , \, \cdot \,, u)$ has finite type at every point, there exists a $k \in \mathbb {N}$ such that $\displaystyle {\frac{\partial ^k\Phi (x, y, 0)}{\partial u^k}}$ has finite type at (0, 0). Let $k_0$ be the smallest such k. Then each $\displaystyle {\frac{1}{j!}\frac{\partial ^j\Phi (x, y, 0)}{\partial u^j}}$ can be written as $\phi _j(x) + \psi _j(y)$ for $0 \le j \le k_0-1$ and

$$\begin{aligned} \lambda \Phi (x, y, u) = \lambda \sum _{j=0}^{k_0-1}\big (\phi _j(x) + \psi _j(y)\big ) + (\lambda u^{k_0}) \Psi (x, y, u) \end{aligned}$$

(47)

where

$$\begin{aligned} \Psi (x, y, u) = \frac{1}{k_0!}\frac{\partial ^{k_0}\Phi (x, y, 0)}{\partial u^{k_0}} + \sum _{j=k_0 + 1}^\infty \bigg (\frac{u^{j-k_0}}{j!}\bigg ) \frac{\partial ^j\Phi (x, y, 0)}{\partial u^j}. \end{aligned}$$

(48)

Since $\displaystyle {\frac{\partial ^{k_0}\Phi (\, \cdot \, , \, \cdot \,, 0)}{\partial u^{k_0}}}$ has finite type at (0, 0), by continuity, for ${\tilde{r}}_0 > 0$ sufficiently small and $|u| \le {\tilde{r}}_0$, $\Psi (\, \cdot \, , \, \cdot \,, u)$ also has finite type at every point of $\overline{B_{2n}({\tilde{r}}_0)}$. Let

$$\begin{aligned} \displaystyle {{\tilde{f}}(y) = e^{i\lambda \big (\sum _{j=0}^{k_0-1}\psi _j(y)\big )} f(y)}.\end{aligned}$$

By Theorem 4.2 (after shrinking $\text{ supp }(\varphi )$ if necessary) and (47)–(48),

$$\begin{aligned} \Vert T _{\lambda \Phi ,\, \varphi K}f\Vert _{p, w} = \Vert T _{(\lambda u^{k_0}) \Psi ,\, \varphi K}{\tilde{f}}\Vert _{p, w} \le C_p \Vert f\Vert _{p, w}.\end{aligned}$$

$\square$