$L^p$ and $H^1$ Boundedness of Oscillatory Singular Integral Operators with Hölder Class Kernels

Abstract

For oscillatory singular integrals with polynomial phases and Hölder class kernels, we establish their uniform boundedness on $L^p$ spaces as well as a sharp logarithmic bound on the Hardy space $H^1$. These results improve the ones in (Pan in Forum Math 31: 535–542, 2019) by removing the restriction that the phase polynomials be quadratic.

1 Introduction

Let $n, d \in \mathbb {N}$, $x=(x_1, \ldots , x_n) \in \mathbb {R}^n$. Let

$$\begin{aligned} P(x) =\sum _{|\alpha |\le d} a_\alpha x^\alpha \end{aligned}$$

(1)

where $\alpha = (\alpha _1, \ldots , \alpha _n)$, $|\alpha | = \alpha _1 + \cdots + \alpha _n$, $x^\alpha =x_{1}^{\alpha _1}\cdots x_{n}^{\alpha _n}$ and $a_\alpha \in \mathbb {R}$. For each nonconstant polynomial P(x) we let

$$\begin{aligned} \Vert P\Vert _o = \frac{\sum _{|\alpha | = 1} |a_\alpha |}{ \sum _{2\le |\alpha | \le d} |a_\alpha |^{1/|\alpha |}}. \end{aligned}$$

(2)

When $\deg (P)=1$, the value of $\Vert P\Vert _o$ shall be interpreted as $\infty$.

For a Calderón–Zygmund type singular kernel K(x), let $T_{P, K}$ be the oscillatory singular integral operator defined by

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

(3)

In [7] the author proved the following:

Theorem 1.1

Suppose that $\deg (P) = 2$ and there exist $q > 2$ and $\delta >0$ such that

$$\begin{aligned} {{\varvec{CZ}}}(q,\delta ) \quad \left\{ \begin{array}{ll} (a) &{} \bigg (\displaystyle {} \int _{s \le |x| \le 2s}|K(x)|^q dx\bigg )^{1/q} \le A s^{-n/q^\prime } \quad \text{ for } s > 0; \\ (b) &{} \displaystyle {|K(x - y) - K(x)| \le \frac{A |y|^\delta }{|x|^{n+\delta }}} \quad \text{ for } |x| \ge 2|y|; \\ (c) &{} \displaystyle {\bigg | \int _{s_1 \le |x| \le s_2} K(x) dx\bigg | \le A } \text{ for } 0< s_1 < s_2. \end{array} \right. \end{aligned}$$

Then,

(i) For $1< p < \infty$, there exists a positive constant $C_p$ such that

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

(4)

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

(ii) There exists a positive constant C such that

$$\begin{aligned} \Vert T_{P, K} f\Vert _{H^1(\mathbb {R}^n)} \le C (1 + \log ^+\Vert P\Vert _o) \Vert f\Vert _{H^1(\mathbb {R}^n)} \end{aligned}$$

(5)

for all $f \in H^1(\mathbb {R}^n)$. The constant C may depend on n, $\delta$, q and A, but is otherwise independent of K and the coefficients of P. The bound given in (5) is the best possible in the sense that the logarithmic function cannot be replaced by any function with a slower rate of growth.

The above conditions (a)–(c) are commonly referred to as the size, smoothness and cancellation conditions for singular kernels, respectively. In classical Calderón–Zygmund theory of singular integrals, one assumes that condition (a) holds for $q= \infty$ together with the $C^1$ condition $|\nabla K(x)| \le C|x|^{-n-1}$ instead of the weaker Hölder continuity condition (b), as well as (c).

The restriction that P(x) be quadratic (i.e. $\deg (P)=2$) in the above theorem is clearly a severe one. For the operator $T_{P,K}$ with a $C^1$ kernel K, both the $L^p$ bound in (1) and the $H^1$ bound in (2) have been known to be true when the phase polynomial P is of arbitrary degree (for $L^p$ see [8]; for $H^1$ see [1]). The main purpose of this paper is to show that, for $T_{P,K}$ with a kernel K in the Hölder class, the same $L^p$ and $H^1$ bounds are true when the degree of the phase polynomial P is arbitrary. We have the following:

Theorem 1.2

Let P(x) be a real-valued polynomial of any positive degree. Suppose that K(x) satisfies CZ($q,\delta$)(a)–(c) for some $q > 1$ and $\delta >0$. Then,

(i) For $1< p < \infty$, there exists a positive constant $C_p$ 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 n, p, $\delta$, q, $\deg (P)$ and A, but is otherwise independent of K and the coefficients of P;

(ii) There exists a positive constant C such that

$$\begin{aligned} \Vert T_{P, K} f\Vert _{H^1(\mathbb {R}^n)} \le C (1 + \log ^+\Vert P\Vert _o) \Vert f\Vert _{H^1(\mathbb {R}^n)} \end{aligned}$$

(7)

for all $f \in H^1(\mathbb {R}^n)$. The constant C may depend on n, $\delta$, q, $\deg (P)$ and A, but is otherwise independent of K and the coefficients of P. The bound given in (7) is the best possible in the sense that the logarithmic function cannot be replaced by any function with a slower rate of growth.

We point out that, aside from lifting the restriction that $\deg (P)=2$ from Theorem 1.1, Theorem 1.2 also improves the range of q from $q > 2$ to the more natural range $q >1$.

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. A subscript may be added to the symbol $\lesssim$ to indicate a particular dependence as appropriate.

2 $L^p$ Boundedness

In this section we will establish part (i) of Theorem 1.2. For $u \in \mathbb {R}^n$ and $r > 0$ we let B(u, r) denote the ball $\{x \in \mathbb {R}^n:|x - u| \le r\}$. An important tool will be the following lemma from [6].

Lemma 2.1

Let P(x) be given as in (1). Let $R > 0$ and let $\psi : \mathbb {R}^n \rightarrow \mathbb {C}$ be an integrable function supported in B(0, R/2). Then

$$\begin{aligned} \bigg |\int _{\mathbb {R}^n}e^{iP(x)}\psi (x) dx\bigg | \lesssim _{d,n} \sup _{v\in B(0, R\Lambda ^{-1/d})} \int _{\mathbb {R}^n}|\psi (x) - \psi (x - v)|dx, \end{aligned}$$

(8)

where $\Lambda := \sum _{1 \le |\alpha | \le d} |a_\alpha | R^{|\alpha |}$.

For $h > 0$, we let $T_{P, K, h}$ denote the truncation of $T_{P, K}$ given by

$$\begin{aligned} T_{P, K, h} f(x) = \text{ p.v. } \int _{|x-y| \le h} e^{i P(x - y)} K(x - y) f(y) dy. \end{aligned}$$

(9)

We will establish the following uniform $L^p$ boundedness theorem:

Theorem 2.1

Let P(x) be a real-valued polynomial of any degree and $h > 0$. Suppose that K(x) satisfies CZ($q,\delta$)(a)–(c) for some $q > 1$ and $\delta >0$. Then, for $1< p < \infty$, there exists a positive constant $C_p$ such that

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

(10)

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

Proof

Without loss of generality we may assume that P(x) is nonconstant and $P(0)=0$. In order to prove (10) we shall use induction on $\deg (P)$. When $\deg (P)=1$, by $P(x-y) = P(x)-P(y)$, (10) follows from the $L^p$ boundedness of singular integrals ([4], page 300). Suppose that for a $d \ge 2$, (10) holds for all P with $\deg (P) \le d-1$.

We now assume that $\deg (P) = d$, i.e.

$$\begin{aligned}P(x) =\sum _{|\alpha |\le d} a_\alpha x^\alpha \end{aligned}$$

with $\sum _{|\alpha |= d} |a_\alpha | \ne 0$.

It is easy to see that for $t > 0$, $t^{n}K(t x)$ satisfies conditions CZ($q,\delta$)(a)–(c) with the same q, $\delta$ and A. Thus, by rescaling if necessary we may assume that $\sum _{|\alpha |= d} |a_\alpha | =1$.

Let

$$\begin{aligned} R(x) = P(x) - \sum _{|\alpha |= d} a_\alpha x^\alpha . \end{aligned}$$

(11)

Then $\deg (R) \le d-1$ and for $0 < h \le 8$

$$\begin{aligned} |T_{P, K, h} f(x) - T_{R, K, h} f(x)| \lesssim \int _{|x-y| \le 8} |x-y|^d |K(x-y)| |f(y)| dy. \end{aligned}$$

(12)

By (12), Hölder’s inequality and CZ($q,\delta$)(a),

$$\begin{aligned} \Vert T_{P, K, h} f - T_{R, K, h} f\Vert _{L^p(\mathbb {R}^n)}\lesssim & {} \bigg (\int _{|x| \le 8} |x|^d |K(x)| dx\bigg ) \Vert f\Vert _{L^p(\mathbb {R}^n)}\\\lesssim & {} \big (\Vert f\Vert _{L^p(\mathbb {R}^n)}\big ) \sum _{j=-\infty }^2 \bigg (\int _{2^j \le |x| \le 2^{j+1}} |x|^{dq^\prime }dx\bigg )^{1/q^\prime }\\&\times \bigg (\int _{2^j \le |x| \le 2^{j+1}}|K(x)|^q dx\bigg )^{1/q}\\\lesssim & {} \big (\Vert f\Vert _{L^p(\mathbb {R}^n)}\big ) \sum _{j=-\infty }^2 \big (2^{jdq^\prime } 2^{jn}\big )^{{1/q^\prime }}2^{-jn/q^{\prime }}\\\lesssim & {} \Vert f\Vert _{L^p(\mathbb {R}^n)}. \end{aligned}$$

Thus, for $0 < h \le 8$

$$\begin{aligned} \Vert T_{P, K, h} f \Vert _{L^p(\mathbb {R}^n)} \lesssim \Vert T_{R, K, h} f\Vert _{L^p(\mathbb {R}^n)} + \Vert f\Vert _{L^p(\mathbb {R}^n)} \lesssim \Vert f\Vert _{L^p(\mathbb {R}^n)}. \end{aligned}$$

(13)

For the rest of this proof we assume that $h > 8$. For $j \le 2$ let $I_j = [2^{j}, 2^{j+1}]$,

$$\begin{aligned} \psi _j(x) = \chi _{I_j}(|x|) K(x) \end{aligned}$$

and let $S_j$ denote the following operator

$$\begin{aligned} S_j f(x) = \int _{\mathbb {R}^n} e^{i P(x - y)} \psi _j(x - y) f(y) dy. \end{aligned}$$

(14)

Then

$$\begin{aligned} \widehat{S_j f}(\xi ) = m_j(\xi ) \hat{f}(\xi ) \end{aligned}$$

(15)

where

$$\begin{aligned} m_j(\xi ) = \int _{\mathbb {R}^n}e^{i(P(x)- 2\pi \xi \cdot x)} \psi _j(x) dx. \end{aligned}$$

(16)

For each j, we will now apply Lemma 2.1 to the estimate of $\Vert m_j\Vert _\infty$ with $R = 2^{j+2}$. By

$$\begin{aligned} P(x) - 2 \pi \xi \cdot x = \sum _{k=1}^n \bigg (\frac{\partial P}{\partial x_k}(0) - 2\pi \xi _k\bigg )x_k + \sum _{2 \le |\alpha | \le d} a_{\alpha } x^\alpha , \end{aligned}$$

and

$$\begin{aligned} \Lambda := \sum _{k=1}^n \bigg |\frac{\partial P}{\partial x_k}(0) - 2\pi \xi _k\bigg | R + \sum _{2 \le |\alpha | \le d} |a_{\alpha }| R^{|\alpha |} \ge \bigg (\sum _{|\alpha |= d} |a_{\alpha }|\bigg ) R^{d} = R^d, \end{aligned}$$

we have $R \Lambda ^{-1/d} \le 1$. Thus, for any $\xi \in \mathbb {R}^n$,

$$\begin{aligned} |m_j(\xi )| \lesssim \sup _{v \in B(0, 1)} \int _{\mathbb {R}^n}|\psi _j(x) - \psi _j(x-v)| dx. \end{aligned}$$

(17)

For any $v \in B(0, 1)$, $j \ge 2$ and $x \in B(0, 2^{j+1})\backslash B(0, 2^j)$ we have $|x| \ge 2|v|$ and

$$\begin{aligned} |B(0, 2^j)\Delta B(v, 2^j)| + |B(0, 2^{j+1})\Delta B(v, 2^{j+1})| \lesssim 2^{j(n-1)}. \end{aligned}$$

(18)

Thus, by CZ($q,\delta$)(b),

$$\begin{aligned}&|\psi _j(x) - \psi _j(x-v)| \le \chi _{I_j}(|x|) |K(x) - K(x-v)| \\&\quad + | \chi _{I_j}(|x|) - \chi _{I_j}(|x-v|)| |K(x-v)| \\&\lesssim \chi _{I_j}(|x|) \bigg (\frac{|v|^\delta }{|x|^{n+\delta }}\bigg ) \end{aligned}$$

$$\begin{aligned} + (\chi _{B(0, 2^j)\Delta B(v, 2^j)}(x) + \chi _{B(0, 2^{j+1})\Delta B(v, 2^{j+1})}(x))|K(x-v)|. \end{aligned}$$

(19)

Since $(B(0, 2^j)\Delta B(v, 2^j))\cup (B(0, 2^{j+1})\Delta B(v, 2^{j+1})) \subseteq B(v, 2^{j+2})\backslash B(v, 2^{j-1})$, by (17)–(19) and CZ($q,\delta$)(a) we have

$$\begin{aligned}&|m_j(\xi )| \lesssim \int _{2^j \le |x| \le 2^{j+1}} \frac{dx}{|x|^{n+\delta }} \\&\quad + \int _{\mathbb {R}^n} (\chi _{B(0, 2^j)\Delta B(v, 2^j)}(x) + \chi _{B(0, 2^{j+1})\Delta B(v, 2^{j+1})}(x))|K(x-v)|dx \\&\lesssim 2^{-j(n+\delta )} 2^{jn} + (|B(0, 2^j)\Delta B(v, 2^j)| + |B(0, 2^{j+1})\Delta B(v, 2^{j+1})|)^{1/q^\prime }\\&\quad \times \bigg (\int _{B(v, 2^{j+2})\backslash B(v, 2^{j-1})}|K(x-v)|^q dx\bigg )^{1/q} \\&\lesssim 2^{-j\delta } + 2^{j(n-1)/q^\prime }2^{-jn/q^\prime } \\&\lesssim 2^{-j \mu } \end{aligned}$$

where $\mu = \min \{\delta , 1/q^\prime \}$. It follows from Plancherel’s theorem that

$$\begin{aligned} \Vert S_j\Vert _{L^2(\mathbb {R}^n)\rightarrow L^2(\mathbb {R}^n)} \lesssim 2^{-j \mu }. \end{aligned}$$

(20)

By CZ($q,\delta$)(a), for any $t_1, t_2$ satisfying $0< t_1 < t_2$ and $t_2/t_1 \lesssim 1$,

$$\begin{aligned} \Vert \chi _{B(0, t_2)\backslash B(0, t_1)} |K|\Vert _{L^1(\mathbb {R}^n)} \lesssim 1. \end{aligned}$$

(21)

Thus,

$$\begin{aligned}&\Vert S_j\Vert _{L^1(\mathbb {R}^n)\rightarrow L^1(\mathbb {R}^n)} + \Vert S_j\Vert _{L^\infty (\mathbb {R}^n)\rightarrow L^\infty (\mathbb {R}^n)}\nonumber \\&\quad \lesssim \Vert \chi _{B(0, 2^{j+1})\backslash B(0, 2^j)} K\Vert _{L^1(\mathbb {R}^n)} \lesssim 1. \end{aligned}$$

(22)

By the Riesz–Thorin interpolation theorem, for $1< p < \infty$,

$$\begin{aligned} \Vert S_j\Vert _{L^p(\mathbb {R}^n)\rightarrow L^p(\mathbb {R}^n)} \lesssim 2^{-j \mu _p}, \end{aligned}$$

(23)

where

$$\begin{aligned} \mu _p = \mu (1 - |1 - 2/p|) > 0 .\end{aligned}$$

Let $m= [\log _2 h]$. Then

$$\begin{aligned} |T_{P, K, h} f | \le |T_{P, K, 4} f | + \sum _{j=2}^{m-1} |S_j f| + |(\chi _{B(0, h)\backslash B(0, 2^m)} |K|)*|f|. \end{aligned}$$

(24)

It follows from (13), (21) and (23) that

$$\begin{aligned}&\Vert T_{P, K, h} f\Vert _{L^p(\mathbb {R}^n)} \lesssim \bigg (1 + \sum _{j=2}^{m-1} 2^{-j \mu _p} + \Vert \chi _{B(0, h)\backslash B(0, 2^m)} |K|\Vert _{L^1(\mathbb {R}^n)}\bigg )\Vert f\Vert _{L^p(\mathbb {R}^n)} \\&\quad \lesssim \Vert f\Vert _{L^p(\mathbb {R}^n)}. \end{aligned}$$

The proof of Theorem 2.1 is now complete. $\square$

By using

$$\begin{aligned} T_{P, K} f = \lim _{h\rightarrow \infty } T_{P, K, h}f \end{aligned}$$

interpreted in the distributional sense, we obtain (6) for all test functions f. Part (i) of Theorem 1.2 then follows by standard arguments.

3 $H^1 \rightarrow H^1$ Estimates

As for the $L^p$ boundeness, the $H^1$ arguments in [7] relied both on the phase being quadratic as well as the condition CZ($q,\delta$) with a $q > 2$. To prove part (ii) of Theorem 1.2, we will let the degree of the phase polynomial be any positive integer while assuming that K satisfies CZ($q,\delta$) with a $q > 1$.

Lemma 3.1

Let $d \ge 2$, $P(x) =\sum _{|\alpha |\le d} a_\alpha x^\alpha$ and K(x) satisfy

$$\begin{aligned} \sup _{s > 0} s^{n/q^\prime } \Vert \chi _{B(0, 2s)\backslash B(0, s)} K \Vert _{L^q(\mathbb {R}^n)} \lesssim 1 \end{aligned}$$

for some $q > 1$. Then,

(i) for any $0< a < b$ and $\nu \ge 1$,

$$\begin{aligned}&\int _{a \le |x| \le b} |x|^\nu |K(x)| dx \lesssim (b-a)^{1/q^\prime }b^{\nu - 1/q^\prime }; \end{aligned}$$

(25)

$$\begin{aligned}&\int _{a \le |x| \le b} |K(x)| dx \lesssim 1+\ln (b/a); \end{aligned}$$

(26)

(ii) for any $\lambda \ge 1$,

$$\begin{aligned} \begin{aligned} \int _{|x| \ge \lambda } \bigg |K(x) \int _{B(0,1)} e^{i P(x-y)} f(y) dy\bigg | dx \\ \lesssim \bigg [\lambda \bigg (\sum _{2 \le |\alpha | \le d} |a_{\alpha }|^{1/|\alpha |}\bigg )^2\bigg ]^{-1/(2\gamma ^\prime d)} \Vert f\Vert _{L^{\gamma ^\prime }(B(0,1))} \end{aligned} \end{aligned}$$

(27)

where $\gamma = \min \{q, 2\}$.

Proof

(i) Let $N = [\log _2(b/a)]$. Then

$$\begin{aligned}&\int _{a \le |x| \le b} |x|^\nu |K(x)| dx \lesssim \bigg (\int _{a \le |x| \le b} |x|^{1-n}dx\bigg )^{1/q^\prime } \\&\quad \times \bigg (\int _{a \le |x| \le 2^{N+1}a} |x|^{(\nu +(n-1)/q^\prime )q} |K(x)|^q dx\bigg )^{1/q} \\&\lesssim (b-a)^{1/q^\prime } \bigg (\sum _{j=0}^N (2^j a)^{(\nu +(n-1)/q^\prime )q} \Vert \chi _{B(0, 2^{j+1}a)\backslash B(0, 2^ja)} K \Vert ^q_{L^q(\mathbb {R}^n)}\bigg )^{1/q} \\&\lesssim (b-a)^{1/q^\prime } \bigg (\sum _{j=0}^N(2^j a)^{(\nu -1/q^\prime )q+ n q/q^\prime }((2^ja)^{-n/q^\prime })^q\bigg )^{1/q} \\&\lesssim (b-a)^{1/q^\prime } (2^Na)^{\nu -1/q^\prime } \lesssim (b-a)^{1/q^\prime } b^{\nu -1/q^\prime }, \end{aligned}$$

which proves (25). The proof of (26) is simpler and will be omitted.

(ii) Since $1 < \gamma \le q$, we have for any $s > 0$,

$$\begin{aligned}&\Vert \chi _{B(0, 2s)\backslash B(0, s)} K \Vert _{L^\gamma (\mathbb {R}^n)} \lesssim \Vert \chi _{B(0, 2s)\backslash B(0, s)} K \Vert _{L^q(\mathbb {R}^n)} |B(0, 2s)|^{1/\gamma - 1/q}\\&\quad \lesssim s^{-n/q^\prime + n(1/\gamma - 1/q)} = s^{-n/\gamma ^\prime }.\end{aligned}$$

For any $\lambda \ge 1$, by Hölder’s inequality and applying Lemma 2.3 in [1] (taking p to be $\gamma ^\prime \ge 2$),

$$\begin{aligned}&\int _{|x| \ge \lambda } \bigg |K(x) \int _{B(0,1)} e^{i P(x-y)} f(y) dy\bigg | dx \\&\lesssim \sum _{j=0}^\infty \Vert \chi _{B(0, 2^{j+1}\lambda )\backslash B(0, 2^j \lambda )} K \Vert _{L^\gamma (\mathbb {R}^n)} \\&\quad \times \bigg (\int _{B(0, 2^{j+1}\lambda )}\bigg |\int _{B(0,1)} e^{i P(x-y)} f(y) dy\bigg |^{\gamma ^\prime } dx\bigg )^{1/\gamma ^\prime } \\&\lesssim \bigg (\sum _{j=0}^\infty (2^j \lambda )^{-n/\gamma ^\prime }(2^j \lambda )^{(2nd-1)/(2\gamma ^\prime d)}\bigg ) \\&\quad \times \bigg (\sum _{2 \le |\alpha | \le d} |a_{\alpha }|^{1/|\alpha |}\bigg )^{-1/(\gamma ^\prime d)} \Vert f\Vert _{L^{\gamma ^\prime }(B(0,1))} \\&\lesssim \bigg (\sum _{j=0}^\infty 2^{-j/(2\gamma ^\prime d)}\bigg ) \bigg [\lambda \bigg (\sum _{2 \le |\alpha | \le d} |a_{\alpha }|^{1/|\alpha |}\bigg )^2\bigg ]^{-1/(2\gamma ^\prime d)} \Vert f\Vert _{L^{\gamma ^\prime }(B(0,1))} \\&\lesssim \bigg [\lambda \bigg (\sum _{2 \le |\alpha | \le d} |a_{\alpha }|^{1/|\alpha |}\bigg )^2\bigg ]^{-1/(2\gamma ^\prime d)} \Vert f\Vert _{L^{\gamma ^\prime }(B(0,1))} . \end{aligned}$$

$\square$

Lemma 3.2

Let K(x) be given as in Lemma 3.1 and Q(x) be a polynomial satisfying $\nabla Q(0) = 0$. Let f be a Lebesgue measurable function satisfying

$$\begin{aligned}&\text{ supp }(f) \subseteq B(0, 1); \end{aligned}$$

(28)

$$\begin{aligned}&\Vert f\Vert _\infty \le 1; \end{aligned}$$

(29)

$$\begin{aligned}&\int _{B(0,1)} f(y) dy = 0. \end{aligned}$$

(30)

Then, there exists a $C > 0$ such that

$$\begin{aligned} \int _{|x|\ge 2}\bigg |K(x)\int _{B(0,1)}e^{iQ(x-y)} f(y) dy\bigg | dx \le C. \end{aligned}$$

(31)

The constant C may depend on $\deg (Q)$ but is otherwise independent of the coefficients of Q(x).

Proof

When $\deg (Q) \le 1$, by $\nabla Q(0) = 0$, (31) follows from (30) trivially.

Suppose that $d \ge 2$ and (31) holds for all Q(x) satisfying $\deg (Q) \le d-1$ and $\nabla Q(0) = 0$.

Assume that $\deg (Q) = d$ and $\nabla Q(0) = 0$. Then

$$\begin{aligned} Q(x) = \sum _{|\alpha |=d} q_\alpha x^\alpha + R(x) \end{aligned}$$

where $\deg (R) \le d-1$ and $\nabla R(0) = 0$. Thus,

$$\begin{aligned} \int _{|x|\ge 2}\bigg |K(x)\int _{B(0,1)}e^{iR(x-y)} f(y) dy\bigg | dx \lesssim 1. \end{aligned}$$

Let

$$\begin{aligned} \beta = \max \bigg \{2, \bigg (\sum _{|\alpha |=d}|q_\alpha |\bigg )^{-1/(d-1)}\bigg \}.\end{aligned}$$

Then

$$\begin{aligned}&\int _{2 \le |x|\le \beta }\bigg |K(x)\int _{B(0,1)}e^{iQ(x-y)} f(y) dy\bigg | dx \nonumber \\&\quad \lesssim \int _{2 \le |x|\le \beta }\bigg |K(x)\int _{B(0,1)}\bigg (e^{iQ(x-y)} -e^{i(\sum _{|\alpha |=d}q_\alpha x^\alpha + R(x-y))}\bigg ) f(y) dy\bigg |\nonumber \\&dx + 1 \nonumber \\&\quad \lesssim \bigg (\sum _{|\alpha |=d}|q_\alpha |\bigg )\int _{2 \le |x| \le \beta }|x|^{d-1}|K(x)| dx + 1 \nonumber \\&\quad \lesssim \bigg (\sum _{|\alpha |=d}|q_\alpha |\bigg ) (\beta -2)^{1/q^\prime } \beta ^{d-1 - 1/q^\prime } +1 \lesssim 1. \end{aligned}$$

(32)

Let $\gamma = \min \{q, 2\}$. By Hölder’s inequality, (29) and Lemma 4.3 in [5] (after interpolating between the $L^2 \rightarrow L^2$ bound there and a trivial $L^1 \rightarrow L^\infty$ bound), we have

$$\begin{aligned}&\int _{|x|\ge \beta }\bigg |K(x)\int _{B(0,1)}e^{iQ(x-y)} f(y) dy\bigg | dx \lesssim \sum _{j=0}^\infty \bigg (\int _{2^j \beta \le |x| \le 2^{j+1} \beta } |K(x)|^{\gamma } dx\bigg )^{1/\gamma }\nonumber \\&\quad \times \bigg (\int _{|x| \le 2^{j+1}\beta }\bigg |\int _{B(0,1)}e^{iQ(x-y)} f(y)dy|\bigg |^{\gamma ^\prime }dx\bigg )^{1/\gamma ^\prime }\nonumber \\&\quad \lesssim \sum _{j=0}^\infty (2^j \beta )^{-n/\gamma ^\prime } \bigg [(2^j \beta )^{n/2}\bigg ((2^j \beta )^{d-1} \sum _{|\alpha |=d}|q_\alpha |\bigg )^{-1/(4(d-1))}\bigg ]^{2/\gamma ^\prime }\\&\times \Vert f\Vert _{L^\gamma (B(0,1))}\nonumber \\&\quad \lesssim \bigg [\beta \bigg (\sum _{|\alpha |=d}|q_\alpha |\bigg )^{1/(d-1)}\bigg ]^{-1/2 \gamma ^\prime } \lesssim 1.\nonumber \end{aligned}$$

(33)

Now (31) follows from (32) and (33). $\square$

We will now prove part (ii) of Theorem 1.2.

Proof

Let P(x) be a real-valued polynomial of any positive degree. Suppose that K(x) satisfies CZ($q,\delta$)(a)–(c) for some $q > 1$ and $\delta >0$. Let $\gamma = \min \{q, 2\}$. For P with $\deg (P) = 1$, we have $\Vert P\Vert _o = \infty$ in which case (7) holds trivially. Thus we may assume that $d = \deg (P) \ge 2$.

Since $T_{P, K}$ is translation invariant, by the standard atomic theory of Hardy spaces, it suffices to prove that

$$\begin{aligned} \Vert T_{P, K} f \Vert _{L^{1}(\mathbb {R}^n)} \lesssim 1 + \log ^+ \Vert P\Vert _o \end{aligned}$$

(34)

holds for every $H^1(\mathbb {R}^n)$ atom $f(\cdot )$ which is supported in a ball centered at the origin (see [2, 3, 9, 10]). Additionally, due to the invariance of the CZ($q,\delta$) conditions under $K(x) \rightarrow t^n K(tx)$ and the invariance of $\Vert \Vert _o$ under $P(x) \rightarrow P(tx)$, we may assume that f satisfies (28)–(30).

First we will prove that

$$\begin{aligned} \int _{|x|\ge 2}\bigg |K(x)\int _{B(0,1)}e^{iP(x-y)} f(y) dy\bigg | dx \lesssim 1 + \log ^+ \Vert P\Vert _o. \end{aligned}$$

(35)

Let

$$\begin{aligned} a = \max \bigg \{2, \bigg (\sum _{|\alpha |=1}|a_\alpha |\bigg )^{-1}\bigg \}, b = \max \bigg \{a, \bigg (\sum _{2 \le |\alpha |\le d}|a_\alpha |^{1/|\alpha |}\bigg )^{-2}\bigg \}.\end{aligned}$$

By Lemma 3.1(ii) and (29),

$$\begin{aligned} \begin{aligned} \int _{|x| \ge b} \bigg |K(x) \int _{B(0,1)} e^{i P(x-y)} f(y) dy\bigg | dx \\ \lesssim \bigg [b \bigg (\sum _{2 \le |\alpha | \le d} |a_{\alpha }|^{1/|\alpha |}\bigg )^2\bigg ]^{-1/(2\gamma ^\prime d)} \Vert f\Vert _{L^{\gamma ^\prime }(B(0,1))} \lesssim 1. \end{aligned} \end{aligned}$$

(36)

Let

$$\begin{aligned} Q(x) = P(0) + \sum _{2 \le |\alpha | \le d} a_\alpha x^\alpha .\end{aligned}$$

By Lemmas 3.2 and 3.1(i),

$$\begin{aligned}&\int _{2 \le |x| \le a} \bigg |K(x) \int _{B(0,1)} e^{i P(x-y)} f(y) dy\bigg | dx \nonumber \\&\quad \lesssim \int _{2 \le |x| \le a} \bigg |K(x) \int _{B(0,1)} \bigg (e^{i P(x-y)} - e^{i Q(x-y)}\bigg ) f(y) dy\bigg | dx + 1\nonumber \\&\quad \lesssim \bigg (\sum _{|\alpha |=1}|a_\alpha |\bigg ) \int _{2 \le |x| \le a}|x||K(x)| dx \nonumber \\&\quad \lesssim \bigg (\sum _{|\alpha |=1}|a_\alpha |\bigg ) (a-2)^{1/q^\prime } a^{1-1/q^\prime } \lesssim 1. \end{aligned}$$

(37)

Thus, by (36) and (37), (35) would follow if we can prove that

$$\begin{aligned} \int _{a \le |x|\le b}\bigg |K(x)\int _{B(0,1)}e^{iP(x-y)} f(y) dy\bigg | dx \lesssim 1 + \log ^+ \Vert P\Vert _o. \end{aligned}$$

(38)

Since (38) holds trivially when $a = b$, we may assume that $a < b$. Then

$$\begin{aligned} \bigg (\sum _{|\alpha |=1}|a_\alpha |\bigg )^{-1} \le a < b = \bigg (\sum _{2 \le |\alpha |\le d}|a_\alpha |^{1/|\alpha |}\bigg )^{-2}. \end{aligned}$$

(39)

For $y \in B(0,1)$,

$$\begin{aligned} \bigg | e^{i P(x-y)}- e^{i (\sum _{|\alpha |=1} a_\alpha x^\alpha + Q(x-y))} \bigg | \le \min \bigg \{2, \sum _{|\alpha |=1}|a_\alpha |\bigg \} \lesssim a^{-1}. \end{aligned}$$

(40)

By (40), (29) ,(26) and (39),

$$\begin{aligned}&\int _{a \le |x| \le b} \bigg |K(x) \int _{B(0,1)} e^{i P(x-y)} f(y) dy\bigg | dx\\&\quad \lesssim \int _{a \le |x| \le b} \bigg |K(x) \int _{B(0,1)} \\&\qquad \times \bigg (e^{i P(x-y)} - e^{i (\sum _{|\alpha |=1} a_\alpha x^\alpha + Q(x-y))}\bigg ) f(y) dy\bigg | dx + 1\\&\quad \lesssim a^{-1} \int _{a \le |x| \le b}|K(x)| dx + 1 \lesssim a^{-1}\ln (b/a) + 1 \\&\quad \lesssim a^{-1}\ln (b/a^2) + \sup _{t\ge 2}(t^{-1}\ln t) \lesssim 1 + \log ^+ \Vert P\Vert _o. \end{aligned}$$

This proves (38) and, in turn, (35).

By part (i) of Theorem 1.2, CZ($q,\delta$)(b), (35) and (29),

$$\begin{aligned}&\Vert T_{P, K} f \Vert _{L^{1}(\mathbb {R}^n)} \lesssim \int _{|x| \le 2} |T_{P, K} f (x)| dx\\&\quad + \int _{|x| \ge 2} \bigg |\int _{B(0,1)} e^{iP(x-y)} (K(x-y) -K(x)) f(y) dy \bigg | dx \\&\quad + \int _{|x|\ge 2}\bigg |K(x)\int _{B(0,1)}e^{iP(x-y)} f(y) dy\bigg | dx \\&\lesssim \Vert T_{P, K} f\Vert _{L^2(\mathbb {R}^n)} + \int _{|x| \ge 2} \int _{B(0,1)} \frac{|y|^\delta |f(y)|}{|x|^{n+\delta }} dy dx + (1 + \log ^+ \Vert P\Vert _o)\\&\lesssim \Vert f\Vert _{L^2(\mathbb {R}^n)} + \Vert f\Vert _{L^1(\mathbb {R}^n)} + (1 + \log ^+ \Vert P\Vert _o) \lesssim + (1 + \log ^+ \Vert P\Vert _o). \end{aligned}$$

The proof of part (ii) of Theorem 1.2 is now complete. $\square$