Bilinear Hilbert Transforms Associated with Plane Curves

Abstract

We prove that the bilinear Hilbert transforms and maximal functions along certain general plane curves are bounded from $L^2({{\mathbb {R}}})\times L^2({{\mathbb {R}}})$ to $L^1({{\mathbb {R}}})$.

1 Introduction

Since the initial breakthroughs for singular integrals along curves and surfaces by Nagel, Rivière, Stein, Wainger, et al., in the 1970s (see for example [14, 15] and [17] for some of their works on Hilbert transforms along curves), extensive research in this area of harmonic analysis has been done and a great many fascinating and important results have been established, which culminate in a general theory of singular Radon transforms (see for instance Christ et al. [2]).

Another attractive area, parallel to the above one, is the bilinear extension of the classical Hilbert transform. The boundedness of such bilinear transforms was conjectured by Calderón and motivated by the study of the Cauchy integral on Lipschitz curves. In the 1990s, this conjecture was verified by Lacey and Thiele in a breakthrough pair of papers [8, 9]. In their works, a systematic and delicate method was developed, inspired by the famous works of Carleson [1] and Fefferman [3], which is nowadays referred as the method of time-frequency analysis. Over the past two decades, this method has emerged as a powerful analytic tool to handle problems that are related to multilinear analysis.

We are interested in the study of bilinear/multilinear singular integrals along curves and surfaces—a problem that is closely related to the two areas above. (We refer the readers to Li [11] for connections of this problem with ergodic theory and multilinear oscillatory integrals.) To begin with, we consider a model case—the truncated bilinear Hilbert transforms along plane curves. One formulation of the problem is as follows.

Let $\Gamma (t)=(t, \gamma (t)):(-1, 1)\rightarrow {{\mathbb {R}}}^2$ be a curve in ${{\mathbb {R}}}^2$. With $\Gamma$ we associate the truncated bilinear Hilbert transform operator $H_{\Gamma }$ given by the principal value integral

$$\begin{aligned} H_{\Gamma }(f, g)(x)=\int _{-1}^{1} \! f(x-t)g(x-\gamma (t) )t^{-1}{\mathrm{d}}t \qquad (x\in {{\mathbb {R}}}), \end{aligned}$$

(1.1)

where $f$ and $g$ are Schwartz functions on ${{\mathbb {R}}}$. When the function $\gamma$ has certain curvature (or “non-flat”, i.e., not “resembling” a line) conditions, the boundedness properties of this operator (e.g., whether it is bounded from $L^{p_1}({{\mathbb {R}}})\times L^{p_2}({{\mathbb {R}}})$ to $L^r({{\mathbb {R}}})$ for certain $p_1$, $p_2$, and $r$) are of great interest to us.

Li [11] studied such an operator (the integral defining $H_{\Gamma }(f, g)(x)$ in [11] is over ${{\mathbb {R}}}$) with the curve being defined by a monomial (i.e., $\gamma (t)=t^d$, $d\in {{\mathbb {N}}}$, $d\ge 2$) and proved that it is bounded from $L^2({{\mathbb {R}}})\times L^2({{\mathbb {R}}})$ to $L^1({{\mathbb {R}}})$. In his proof, he combined results and tools from both time-frequency analysis and oscillatory integral theory and used ingeniously a uniformity concept (the so-called $\sigma$-uniformity; see [11, Section 6]). Lie [13] improved Li’s results both qualitatively, by extending monomials to more general curves (certain “slow-varying” curves with extra curvature assumptions), and quantitatively, by improving the estimates. Instead of using Li’s method of $\sigma$-uniformity, Lie used a Gabor frame decomposition to discretize certain operators in a smart way and then worked with the discretized operators which have variables separated on the frequency side and preserve certain main characteristics (see the appendix of [13] for a detailed comparison between their methods).

Another interesting aspect of this problem was considered by Li and the second author [12], in which they studied the case when the curve is defined by a polynomial with different emphasis of getting bounds uniform in coefficients of the polynomial and the full range of indices $(p_1,p_2,r)$. They provided, among other results, complete answers (except to the endpoint case) for $H_\Gamma$ when the polynomial is “non-flat” near the origin (i.e., without a linear term). When the polynomial has a linear term, however, the full range of indices for the corresponding uniform estimates is extremely difficult to find and remains open.

In this paper we consider a family of general “non-flat” curves and provide an easy-to-check criterion for a curve whose associated bilinear Hilbert transform is bounded from $L^2({{\mathbb {R}}})\times L^2({{\mathbb {R}}})$ to $L^1({{\mathbb {R}}})$ (for the precise statement of our results, see Sect. 2). Our goal is to extend Li [11]’s method and results to general plane curves, (in some sense) recover Lie [13]’s results without using the wave packets to discretize the operators, and also prove the boundedness of corresponding maximal functions.

Our criterion, motivated by results in Lie [13] and Nagel et al. [16], mainly asks one to check whether certain bounds of various expressions involving derivatives of a quotient are satisfied. In [16] a simple necessary and sufficient condition is provided (among other results) for the $L^2$-boundedness of the Hilbert transform along the curve $\Gamma$ with $\gamma$ odd, that is, one needs to check whether an auxiliary function $h(t)=t\gamma '(t)-\gamma (t)$ has bounded doubling time. Both Lie’s and our results indicate that an appropriate replacement for $h$ in the bilinear setting might be in the form of a quotient (see the $Q_{\epsilon }(t)$ defined in Sect. 2). We still do not know whether our criterion is a necessary condition for certain “non-flat” curves.

In our main estimates in Sect. 4, we apply the $TT^*$ method both in frequency space (with an extra size restriction $|\gamma '(2^{-j})|>2^{-m}$) and in time space (with an extra restriction on the function space), then we combine both results to get the fast decay needed in proving the boundedness of the desired operator. Since we are considering general curves, certain uniformity of estimates is important, hence we formulate carefully the assumptions on curves and pay special attention to the dependence on parameters of all bounds, especially when we apply a quantitative version of the method of stationary phase.

We also establish analogous results for the bilinear maximal function along $\Gamma$ (defined below) by using the arguments of [12, Section 7] and our main estimates in Sect. 4.

$$\begin{aligned} M_\Gamma (f,g)(x) =\sup _{0<\epsilon <1}\epsilon ^{-1} \int _{0}^\epsilon \left| f(x-t) g(x-\gamma (t))\right| {\mathrm{d}}t \qquad (x\in {{\mathbb {R}}}). \end{aligned}$$

(1.2)

We note that such an operator along a “non-flat” polynomial was already carefully studied in [12]. Much deeper and more elegant results for a linear curve can be found in Lacey [7].

Notations The Fourier transform of $f$ is $\widehat{f}(\xi )={{\mathcal {F}}}[f](\xi )=\int _{{{\mathbb {R}}}} \! f(x) e^{-2\pi i\xi x} \,\mathrm d x$ and its inverse Fourier transform is ${{\mathcal {F}}}^{-1}[g](x)=\int _{{{\mathbb {R}}}} \! g(\xi ) e^{2\pi i\xi x} \,\mathrm d \xi$. Let $\mathbf 1 _{a, n}$ be the indicator function of interval $a\cdot [n, n+1)$ for $a, n\in {{\mathbb {R}}}$ and $1_{I}$ the indicator function of interval I. The indices $(p_1,p_2,r)$ are always assumed to satisfy $1/p_1+1/p_2=1/r$, $p_1>1$, $p_2>1$, and $r>1/2$. We use $C$ to denote an absolute constant which may be depending on the curve and different from line to line.

2 Statement of Theorems

For any $a\in {{\mathbb {R}}}$, we say that a curve $\Gamma (t)=(t, \gamma (t)+a):(-1, 1)\rightarrow {{\mathbb {R}}}^2$ ¹ belongs to a family of curves, $\mathbf F (-1, 1)$, if the function $\gamma$ satisfies the following conditions (2.1)–(2.4). There exists a constant $0<A_1<1/2$ such that on $(-A_1, A_1)\setminus \{0\}$ the function $\gamma$ is of class $C^N$ ($N\ge 5$) and $\gamma '\ne 0$. Let $Q_{\epsilon }(t)=\gamma (\epsilon t)/\epsilon \gamma '(\epsilon )$. For $0<|\epsilon |<c_0<A_1/4$ and $1/4\le |t|\le 4$, we have

$$\begin{aligned}&\left| D^j Q_{\epsilon }(t)\right| \le C_1, \quad 0\le j\le N,\end{aligned}$$

(2.1)

$$\begin{aligned}&\left| D^2 Q_{\epsilon }(t)\right| \ge c_1, \end{aligned}$$

(2.2)

² and

$$\begin{aligned} \big |(D^2Q_{\epsilon })^2(t)-D^1Q_{\epsilon }(t)D^3Q_{\epsilon }(t)\big |\ge c_2, \quad \mathrm{if}\, \big |\gamma '(\epsilon )\big |\le K_1|\epsilon |^{c_1}, \end{aligned}$$

(2.3)

or

$$\begin{aligned} \big |2(D^2Q_{\epsilon })^2(t)-D^1Q_{\epsilon }(t)D^3Q_{\epsilon }(t)\big |\ge c_3, \quad \mathrm{if}\, \big |\gamma '(\epsilon )\big |\ge K_2|\epsilon |^{-c_1}. \end{aligned}$$

(2.3′)

Let $\Delta _j=\big |2^{-j}\gamma '(2^{-j})\big |^{-1}$. If $\gamma ''(\epsilon )\gamma '(\epsilon )<0$ for $0<\epsilon <c_0$, then there exist $K_3\in {{\mathbb {Z}}}$ and $K_4\in {{\mathbb {N}}}$ such that

$$\begin{aligned} \Delta _{j+K_3} \ge 2 \Delta _j, \quad \mathrm{if}\,j\ge K_4. \end{aligned}$$

(2.4)

Theorem 2.1

If $\Gamma \in \mathbf F (-1, 1)$, then $H_{\Gamma }(f, g)$ can be extended to a bounded operator from $L^2({{\mathbb {R}}})\times L^2({{\mathbb {R}}})$ to $L^1({{\mathbb {R}}})$.

The analogous version for bilinear maximal functions is as follows.

Theorem 2.2

If $\Gamma \in \mathbf F (-1, 1)$, then $M_\Gamma (f,g)$ is a bounded operator from $L^2({{\mathbb {R}}})\times L^2({{\mathbb {R}}})$ to $L^1({{\mathbb {R}}})$.

Remark 2.3

By combining the results in this paper with the time-frequency analysis arguments in [12], the boundedness of $H_\Gamma$ and $M_\Gamma$ from $L^{p_1}({{\mathbb {R}}})\times L^{p_2}({{\mathbb {R}}})$ to $L^{r}({{\mathbb {R}}})$ may be obtained for $r<1$. We do not carry the details out in this paper. The lower bound of such $r$, as indicated in [12, Theorem 4], is closely related to the decay rate of the size of the sublevel set

$$\begin{aligned} \big \{|t|<1: |\gamma '(t)-1| <h\big \}, \end{aligned}$$

(2.5)

as $h\rightarrow 0^+$. In particular, if the size of (2.5) is bounded by $c_\nu h^{\nu }$ for some $\nu >0$ and $c_\nu >0$, then $H_\Gamma$ and $M_\Gamma$ are expected to be bounded from $L^{p_1}({{\mathbb {R}}})\times L^{p_2}({{\mathbb {R}}})$ to $L^{r}({{\mathbb {R}}})$ given $r>\max \{1/(1+\nu ),1/2\}$; see [12, Theorem 4] when $\gamma$ is a polynomial.

Remark 2.4

• (1) We use $A_1$, $c_0$, $c_1$, $c_2$, $c_3$, $C_1$, $K_1$, $K_2$, $K_3$, and $K_4$ throughout this paper.

• (2) The condition (2.1) with $j=1$ implies that

  $$\begin{aligned} \big |D^1 Q_{\epsilon }(t)\big |\ge 1/C_1, \quad \mathrm{for}\, 0<|\epsilon |<c_0/4,\, 1/4\le |t|\le 4. \end{aligned}$$

• (3) If $\gamma ''(\epsilon )\gamma '(\epsilon )>0$ for $0<\epsilon <c_0$, then (2.4) always holds with $K_3=1$.

• (4) We now compare our assumptions (2.1)–(2.4) with Lie’s assumptions (1)–(5) in [13, P. 3]. The (2.4) implies Lie’s (1). The (2.1) and (2.2) correspond to Lie’s (2) and (4) (the $Q''$ part) while the (2.3) essentially corresponds to Lie’s (5).

• (5) Note that the curves considered here are not necessarily differentiable at the origin (they can even have a pole). One explanation for this phenomenon is that the bilinear Hilbert transform possesses certain symmetry between its two functions $f$ and $g$ (as well as its two variables $\xi$ and $\eta$ on the frequency side) that we can take advantage of to somehow transfer the case with a pole to the case without a pole (see the two expressions of $B_{j, m}^{\varphi }(f, g)$ at the beginning of Sect. 4).

Remark 2.5

Here are some curves $\Gamma (t)=(t, \gamma (t))$ that belong to $\mathbf F (-1, 1)$:

• (1) Those smooth curves that have contact with $t$-axis at the origin of finite order $\ge 2$ (namely, $\gamma (0)=\gamma '(0)=\cdots =\gamma ^{(d-1)}(0)=0$, but $\gamma ^{(d)}(0)\ne 0$ for some natural number $d\ge 2$), for example, $\gamma (t)=t^d$ or $e^{t^d}-1$ if $d\ge 2$;

• (2) The function $\gamma$ has a pole at the origin of finite order $\ge 1$ (namely, $\gamma (t)=t^{-n}h(t)$ for some natural number $n\ge 1$ and some smooth function $h$ with $h(0)\ne 0$);

• (3) $\gamma (t)=$ a linear combination of finitely many terms of the form $|t|^{\alpha }|\log |t||^{\beta }$ for $\alpha , \beta \in {{\mathbb {R}}}$ and $\alpha \ne 0, 1$;

• (4) $\gamma (t)=\mathrm{sgn}(t)|t|^{\alpha }$ or $|t|^{\alpha }|\log |\log |t|||^{\beta }$ for $\alpha , \beta \in {{\mathbb {R}}}$ and $\alpha \ne 0, 1$.

3 Preliminaries

In this section, we first study a special oscillatory integral which occurs in later sections. The results are standard, but we include a proof for completeness and the convenience of the readers.

Let $\rho \in C_0^{\infty }({{\mathbb {R}}})$ be a real-valued function with ${{\mathrm{supp}}}{\rho }\subset [1/2, 2]$, $\xi , \eta \in {{\mathbb {R}}}$, $\eta \ne 0$, $A>1$ a constant, and

$$\begin{aligned} I\big (\lambda ,\epsilon , \xi , \eta \big )=1_{[-A, A]}(\xi /\eta )\int _{0}^{\infty } \! \rho (t)e^{i \lambda \phi _{\epsilon }(t, \xi , \eta )}\,{\mathrm{d}}t, \quad \lambda >1, \end{aligned}$$

where

$$\begin{aligned} \phi _{\epsilon }\big (t, \xi , \eta \big )=Q_{\epsilon }(t)+(\xi /\eta )t. \end{aligned}$$

Lemma 3.1

Assume that $Q_{\epsilon }\in C^N([1/4, 4])$ ($N\ge 5$) is a real-valued function such that $|D^j Q_{\epsilon }|\le C_1$ for $0\le j\le N$ and $|D^2 Q_{\epsilon }|\ge c_1$ for constants $C_1$ and $c_1$. If $\chi \in C_0^{\infty }({{\mathbb {R}}})$ has its support contained in an interval of length $c_1/12$, then either one of the following two statements holds.

• (1) We have

  $$\begin{aligned} \chi \big (-\xi /\eta \big )I\big (\lambda ,\epsilon , \xi , \eta \big )=O\big (\lambda ^{-(N-1)}\big ). \end{aligned}$$

  (3.1)

• (2) For each pair $(\xi , \eta )$ with $-\xi /\eta \in {{\mathrm{supp}}}\chi$, there exists a unique $t=t(\xi , \eta )\in [1/3, 13/6]$ such that $t(\xi , \eta )=(Q_{\epsilon }')^{-1}(-\xi /\eta )$ is $(N-1)$-times differentiable and satisfies

  $$\begin{aligned} D_{t}^1 \phi _{\epsilon }\big (t(\xi , \eta ), \xi , \eta \big )=0 \end{aligned}$$

  (3.2)

  and

  $$\begin{aligned} \begin{aligned} \chi \big (-\xi /\eta \big )I\big (\lambda ,\epsilon , \xi , \eta \big )&=C\chi \big (-\xi /\eta \big )1_{[-A, A]}\big (\xi /\eta \big )\rho \big (t(\xi , \eta )\big )\\&\quad \cdot \big |D^2_{t}\phi _{\epsilon }(t(\xi , \eta ), \xi , \eta )\big |^{-1/2}e^{i \lambda \phi _{\epsilon }(t(\xi , \eta ), \xi , \eta )}\lambda ^{-1/2}\\&\quad \quad +O\big (\lambda ^{-3/2}\big ) \end{aligned} \end{aligned}$$

  (3.3)

  with $C$ being an absolute constant.

Furthermore, the implicit constants in (3.1) and (3.3) are independent of $\lambda$, $\epsilon$, $\xi$, and $\eta$.

Proof

Due to (2.2), we observe that $Q_{\epsilon }'$ is monotone on $[1/4, 4]$ and that, for any $t\in [1/3, 13/6]$ and $r\in (0, 1/12]$, $Q_{\epsilon }'$ is a bijection from $B(t, r)$ ³ to an interval which contains $B(Q_{\epsilon }'(t), c_1 r)$.

Assume that there exist $a\in [1/2, 2]$ and $(\xi _0, \eta _0)$ with $-\xi _0/\eta _0\in {{\mathrm{supp}}}\chi$ such that $|D_{t}^1 \phi _{\epsilon }(a, \xi _0, \eta _0)|<c_1/12$, otherwise we get (3.1) by integration by parts.

Since $D_{t}^1 \phi _{\epsilon }(t, \xi , \eta )=Q_{\epsilon }'(t)+\xi /\eta$, we have that $-\xi _0/\eta _0\in B(Q_{\epsilon }'(a), c_1/12)$. It follows from the observation above that there exists a unique $a_0\in [1/4, 4]$ such that $a_0\in B(a, 1/12)$ and $Q_{\epsilon }'(a_0)=-\xi _0/\eta _0$. Thus ${{\mathrm{supp}}}\chi \subset B(Q_{\epsilon }'(a_0), c_1/12)$. The observation above then implies that, for each pair $(\xi , \eta )$ with $-\xi /\eta \in {{\mathrm{supp}}}\chi$, there exists a unique $t(\xi , \eta )\in B(a_0, 1/12)$ such that $Q_{\epsilon }'(t(\xi , \eta ))=-\xi /\eta$, which is (3.2). In particular, $t(\xi , \eta )=(Q_{\epsilon }')^{-1}(-\xi /\eta )$, whose differentiability is a consequence of the inverse function theorem.

Note that $B(t(\xi , \eta ), 1/12)\subset [1/4, 9/4]$ and we also have

$$\begin{aligned} \big |D^1_{t} \phi _{\epsilon }(t, \xi , \eta )\big |=\big |D^1_{t} \phi _{\epsilon }(t, \xi , \eta )-D^1_{t} \phi _{\epsilon }(t(\xi , \eta ), \xi , \eta )\big |\ge c_1\big |t-t(\xi , \eta )\big |. \end{aligned}$$

Applying to $I(\lambda ,\epsilon , \xi , \eta )$ the method of stationary phase on $B(t(\xi , \eta ), 1/12)$ and integration by parts outside $B(t(\xi , \eta ), 1/24)$ yields (3.3).$\square$

Remark 3.2

A similar argument in high dimensions can be found in the proof of [4, Proposition 2.4]. For the method of stationary phase, the reader can check [6, Section 7.7].

We quote below Li’s [11, Theorem 6.2] with a small modification in the statement for the sake of our later application, however its proof remains the same. Let $\sigma \in (0, 1]$, $\mathbf I \subset {{\mathbb {R}}}$ be a fixed bounded interval, and $U(\mathbf I )$ a nontrivial subset of $L^2(\mathbf I )$ such that the $L^2$-norm of every element of $U(\mathbf I )$ is uniformly bounded by a constant. We say that a function $f\in L^2(\mathbf I )$ is $\sigma$ -uniform in $U(\mathbf I )$ if

$$\begin{aligned} \left| \int _\mathbf{I } \! f(x) \overline{u(x)} \,\mathrm d x \right| \le \sigma \Vert f\Vert _{L^2(\mathbf I )} \quad \mathrm{for \, all}\, u\in U(\mathbf I ). \end{aligned}$$

Lemma 3.3

Let ${{\fancyscript{L}}}$ be a bounded sublinear functional from $L^2(\mathbf I )$ to ${{\mathbb {C}}}$, $S_\sigma$ the set of all functions that are $\sigma$-uniform in $U(\mathbf I )$,

$$\begin{aligned} A_\sigma = \sup \big \{|{{\fancyscript{L}}}(f)|/\Vert f\Vert _{L^2(\mathbf I )} : f\in S_\sigma , f\ne 0\big \}, \end{aligned}$$

and

$$\begin{aligned} M=\sup _{u\in U(\mathbf I )}|{{\fancyscript{L}}}(u)|. \end{aligned}$$

Then

$$\begin{aligned} \Vert {{\fancyscript{L}}}\Vert \le \max \big \{A_\sigma , 2\sigma ^{-1} M\big \}. \end{aligned}$$

We also need the following theorem to handle the minor part in Sect. 6. This theorem is a variant of the results in [10, Theorem 2.1] concerning estimates for certain paraproducts. The only change is that the standard dyadic sequence $\{2^{\alpha j}\}_{j\in {{\mathbb {Z}}}}$ with $\alpha \in {{\mathbb {N}}}\backslash \{0\}$ (in [10]) is replaced by a dyadic-like sequence $\{\Delta _j\}$ here, while the proof remains the same; see [10, Sections 3 and 4].

Theorem 3.4

Let $L\in {{\mathbb {Z}}}$ and let $\{\Delta _j\}_{j>L}$ be a sequence of positive numbers which is dyadic-like, i.e., there is a $K\in {{\mathbb {Z}}}$ such that for all $j>L$ and $j+K>L$ the following holds

$$\begin{aligned} {\Delta _{j+K} } \ge 2 {\Delta _j}. \end{aligned}$$

(3.4)

Let $\Phi _1$ and $\Phi _2$ be Schwartz functions on ${{\mathbb {R}}}$ whose Fourier transforms are standard bump functions supported on $[-2,-1/2]\cup [1/2,2]$ and $[-1,1]$ respectively, and $\widehat{\Phi _2}(0)=1$. For $(n_1,n_2)\in {{\mathbb {Z}}}^2$ and $l =1$ or $2$, set

$$\begin{aligned} {{\mathcal {M}}}_{l,n_1,n_2}\big (\xi ,\eta \big ) = \sum _{j>L}\widehat{\Phi _l}\Big (\frac{\xi }{2^j}\Big ) e^{ 2\pi i n_1 \frac{\xi }{2^j} } \widehat{\Phi _{3-l}}\Big (\frac{\eta }{\Delta _j} \Big ) e^{ 2\pi i n_2 \frac{\eta }{\Delta _j} }. \end{aligned}$$

Then for $l=1$ and $2$, for any $p_1$, $p_2>1$ with $1/r = 1/p_1+1/p_2$, there is a constant $C$ independent of $(n_1,n_2)$ such that for all $f_1\in L^{p_1}({{\mathbb {R}}})$, $f_2\in L^{p_2}({{\mathbb {R}}})$, the following holds

$$\begin{aligned} \left\| \Pi _{l,n_1,n_2} (f_1,f_2) \right\| _r \le C\big (1+n_1^2\big )^{10}\big (1+n_2^2\big )^{10} \Vert f_1\Vert _{p_1}\Vert f_2\Vert _{p_2}, \end{aligned}$$

where

$$\begin{aligned} \Pi _{l,n_1,n_2}\big (f_1,f_2\big )(x) =\iint \! {{\mathcal {M}}}_{l,n_1,n_2}\big (\xi ,\eta \big ) \widehat{f_1}(\xi )\widehat{f_2}(\eta ) e^{2\pi i (\xi +\eta )x } \,{\mathrm{d}}\xi \,{\mathrm{d}}\eta . \end{aligned}$$

4 The Main Estimates

Let $\widehat{\varphi }\in C_{0}^{\infty }({{\mathbb {R}}})$ such that $\widehat{\varphi }=1$ on $\{t\in {{\mathbb {R}}} : 3/8\le |t|\le 17/8\}$ and ${{\mathrm{supp}}}\widehat{\varphi }\subset \{t\in {{\mathbb {R}}} : 1/4\le |t|\le 9/4\}$. For $j, m\in {{\mathbb {N}}}$ denote $\epsilon _j=2^{-j}$ and

$$\begin{aligned} K_{j, m}(\xi , \eta )=\int _{0}^{\infty } \! \rho (t)e^{-2\pi i 2^m\eta \phi _{\epsilon _j}(t, \xi , \eta )}\,{\mathrm{d}}t, \end{aligned}$$

where $\rho$ and $\phi _{\epsilon _j}$ are as defined at the beginning of Sect. 3.

For $f, g\in L^2({{\mathbb {R}}})$ denote, when $|\gamma '(\epsilon _j)|\le K_1|\epsilon _j|^{c_1}$,

$$\begin{aligned} B_{j, m}^{\varphi }(f, g)(x)=\big |\gamma '(\epsilon _j)\big |^{1/2}\!\iint \! \widehat{f}(\xi )\widehat{\varphi }(\xi ) \widehat{g}(\eta )\widehat{\varphi }(\eta ) e^{2\pi i \big (\gamma '(\epsilon _j)\xi +\eta \big )x}K_{j, m}(\xi , \eta ) \,{\mathrm{d}}\xi \,{\mathrm{d}}\eta \!, \end{aligned}$$

and, when $|\gamma '(\epsilon _j)|\ge K_2|\epsilon _j|^{-c_1}$,

$$\begin{aligned} B_{j, m}^{\varphi }(f, g)(x)=\big |\gamma '(\epsilon _j)\big |^{-1/2}\iint \! \widehat{f}(\xi )\widehat{\varphi }(\xi ) \widehat{g}(\eta )\widehat{\varphi }(\eta ) e^{2\pi i \big (\xi +\gamma '(\epsilon _j)^{-1}\eta \big )x}\\ K_{j, m}(\xi , \eta ) \,{\mathrm{d}}\xi \,{\mathrm{d}}\eta . \end{aligned}$$

Proposition 4.1

Assume that $\Gamma (t)=(t, \gamma (t))\in \mathbf F (-1, 1)$.⁴ For any $\beta <1$, there exist an $L\in {{\mathbb {N}}}$ and a constant $C_{\beta }$ such that whenever $j\ge L$, $m\in {{\mathbb {N}}}$, $n\in {{\mathbb {Z}}}$, and $f, g\in L^2({{\mathbb {R}}})$, we have

• (1) if $|\gamma '(\epsilon _j)|\le K_1|\epsilon _j|^{c_1}$, then

  $$\begin{aligned} \big \Vert B_{j, m}^{\varphi }(f, g)\mathbf 1 _{2^m\gamma '(\epsilon _j)^{-1}, n}\big \Vert _{1}\le C_{\beta } C_{j, m}\Vert f\Vert _{2} \Vert g\Vert _{2}, \end{aligned}$$

  where

  $$\begin{aligned} C_{j, m}=\left\{ \begin{array}{ll} 2^{-m/16} &{} \quad \mathrm{if}\; |\gamma '(\epsilon _j)|> 2^{-m},\\ 2^{-\beta m/4} &{} \quad \mathrm{if}\; |\gamma '(\epsilon _j)|\le 2^{-m}; \end{array}\right. \end{aligned}$$

  (4.1)

• (2) if $|\gamma '(\epsilon _j)|\ge K_2|\epsilon _j|^{-c_1}$, then

  $$\begin{aligned} \big \Vert B_{j, m}^{\varphi }(f, g)\mathbf 1 _{2^m\gamma '(\epsilon _j), n}\big \Vert _{1}\le C_{\beta } C_{j, m}'\Vert f\Vert _{2} \Vert g\Vert _{2}, \end{aligned}$$

  where

  $$\begin{aligned} C_{j, m}'=\left\{ \begin{array}{ll} 2^{-m/16} &{} \quad \mathrm{if}\; |\gamma '(\epsilon _j)|<2^{m},\\ 2^{-\beta m/4} &{} \quad \mathrm{if}\; |\gamma '(\epsilon _j)|\ge 2^{m}. \end{array}\right. \end{aligned}$$

  (4.2)

The rest of this section is devoted to the proof of Proposition 4.1.

We first observe that there is actually a symmetry between the case $|\gamma '(\epsilon _j)|\le K_1|\epsilon _j|^{c_1}$ and the case $|\gamma '(\epsilon _j)|\ge K_2|\epsilon _j|^{-c_1}$, hence we only prove the former case while the other one can be handled similarly. We can also simplify the domain of integration of $B_{j, m}^{\varphi }(f, g)(x)$ by using a decomposition $\widehat{\varphi }=\widehat{\varphi }1_{(0, \infty )}+\widehat{\varphi }1_{(-\infty , 0]}$, which allows us to restrict the domain to one of the cubes $(\pm [1/4, 9/4])\times (\pm [1/4, 9/4])$. We still use $\widehat{\varphi }$ below but with its support contained in either $[1/4, 9/4]$ or $[-9/4, -1/4]$ (and this won’t cause any problem).

The proof is split into three parts. In the first part, we apply the $TT^*$ method to estimate $\Vert B_{j, m}^{\varphi }(f, g)\Vert _{1}$, during which procedure we need a standard result from the oscillatory integral theory and a necessary condition $|\gamma '(\epsilon _j)|>2^{-m}$. The bound we get (see (4.4) below) is efficient when $|\gamma '(\epsilon _j)|$ is large but inefficient when $|\gamma '(\epsilon _j)|$ is close to $2^{-m}$. In the second part, with the help of Lemma 3.3 (the method of $\sigma$-uniformity introduced in Li [11]), we can put certain restrictions on the function $f$ (or $g$) and reduce the estimate of $\Vert B_{j, m}^{\varphi }(f, g)\Vert _{1}$ to a restricted version, to which the $TT^*$ method can be applied without extra assumptions on the size of $|\gamma '(\epsilon _j)|$. The bound we get in this part (see (4.23) and (4.24) below) is efficient when $|\gamma '(\epsilon _j)|$ is small (even when $|\gamma '(\epsilon _j)|$ is close to $2^{-m}$) but inefficient when $|\gamma '(\epsilon _j)|$ is large (see also Lie [13, p. 18]). In the last part, we take advantage of both results and prove the desired estimate.

4.1 Part 1: $j\ge L$, $m\in {{\mathbb {N}}}$ such that $|\gamma '(\epsilon _j)|>2^{-m}$

⁵ We first prove that, for $h\in L^2({{\mathbb {R}}})$,

$$\begin{aligned} \left| \int \! B_{j, m}^{\varphi }(f, g)(x)h(x) \, {\mathrm{d}}x\right| \le C\big (2^m|\gamma '(\epsilon _j)|\big )^{-1/6}\Vert f\Vert _2\Vert g\Vert _2\cdot \big (2^{-m}|\gamma '(\epsilon _j)|\big )^{1/2}\Vert h\Vert _2, \end{aligned}$$

(4.3)

which trivially leads to the estimate

$$\begin{aligned} \big \Vert B_{j, m}^{\varphi }(f, g)\mathbf 1 _{2^m\gamma '(\epsilon _j)^{-1}, n}\big \Vert _1 \le C\big (2^m|\gamma '(\epsilon _j)|\big )^{-1/6}\Vert f\Vert _2\Vert g\Vert _2. \end{aligned}$$

(4.4)

We can find a finite open cover of the interval $[-10, 10]$ by using open intervals of length $c_1/12$, associated with which we can construct a partition of unity. By inserting this partition of unity we reduce the estimate of $\int \! B_{j, m}^{\varphi }(f, g)(x)h(x) \, {\mathrm{d}}x$ to

$$\begin{aligned}&\int \! \widetilde{B}_{j, m}^{\varphi }(f, g)(x)h(x) \, {\mathrm{d}}x \\&=\big |\gamma '(\epsilon _j)\big |^{1/2} \iint \! \widehat{f}\widehat{\varphi }(\xi ) \widehat{g}\widehat{\varphi }(\eta ) {{\mathcal {F}}}^{-1}[h]\big (\gamma '(\epsilon _j)\xi +\eta \big ) \chi \big (-\xi /\eta \big )K_{j, m}\big (\xi , \eta \big ) \,{\mathrm{d}}\xi \,{\mathrm{d}}\eta , \end{aligned}$$

where $\chi$ is smooth and supported in an interval of length $c_1/12$.

We can then apply Lemma 3.1 to $\chi (-\xi /\eta )K_{j, m}(\xi , \eta )$. If (3.1) holds, then an application of Hölder’s inequality yields

$$\begin{aligned} \left| \int \! \widetilde{B}_{j, m}^{\varphi }(f, g)(x)h(x) \, {\mathrm{d}}x\right| \le C 2^{-m/2}\Vert f\Vert _2\Vert g\Vert _2\cdot \big (2^{-m}|\gamma '(\epsilon _j)|\big )^{1/2}\Vert h\Vert _2. \end{aligned}$$

(4.5)

This estimate immediately leads to (4.3).

Below we assume that the second statement in Lemma 3.1 holds. Applying (3.3) yields

$$\begin{aligned} \int \! \widetilde{B}_{j, m}^{\varphi }(f, g)(x)h(x) {\mathrm{d}}x&=C\big (2^{-m}|\gamma '(\epsilon _j)|\big )^{1/2} \nonumber \\&\quad \cdot \iint \! \widehat{f}\widehat{\varphi }(\xi ) \widehat{g}\widehat{\varphi }(\eta ) {{\mathcal {F}}}^{-1}[h]\big (\gamma '(\epsilon _j)\xi +\eta \big )\nonumber \\&\quad \quad a\big (\xi , \eta \big ) e^{-2\pi i 2^m\eta \phi _{\epsilon _j}\big (t(\xi , \eta ), \xi , \eta \big )} \,{\mathrm{d}}\xi {\mathrm{d}}\eta , \end{aligned}$$

(4.6)

where we have omitted the error term in (3.3) (since it leads to the same bound as in (4.5)), and $a(\xi , \eta )$ is defined as

$$\begin{aligned} a\big (\xi , \eta \big )=\chi \big (-\xi /\eta \big )\rho \big (t(\xi , \eta )\big )|\eta |^{-1/2}\big |D^2_{t}\phi _{\epsilon _j}(t(\xi , \eta ), \xi , \eta )\big |^{-1/2}. \end{aligned}$$

Applying to the double integral in (4.6) a change of variables $\gamma '(\epsilon _j)\xi +\eta \rightarrow \xi$, $\eta /\gamma '(\epsilon _j)\rightarrow \eta$ and then Hölder’s inequality yields

$$\begin{aligned} \left| \int \! \widetilde{B}_{j, m}^{\varphi }(f, g)(x)h(x) \, {\mathrm{d}}x\right| \le C\Vert T_{j, m}(f, g)\Vert _2\cdot \big (2^{-m}|\gamma '(\epsilon _j)|\big )^{1/2}\Vert h\Vert _2, \end{aligned}$$

(4.7)

where

$$\begin{aligned} T_{j, m}(f, g)(\xi )&=\int \! \widehat{f}\widehat{\varphi }\big (\gamma '(\epsilon _j)^{-1}\xi -\eta \big ) \widehat{g}\widehat{\varphi }\big (\gamma '(\epsilon _j)\eta \big ) a\big (\gamma '(\epsilon _j)^{-1}\xi -\eta , \gamma '(\epsilon _j)\eta \big ) \\&\quad \quad \cdot e^{-2\pi i 2^m\big (\gamma '(\epsilon _j)\eta \big )\phi _{\epsilon _j}\Big (t\big (\gamma '(\epsilon _j)^{-1}\xi -\eta , \gamma '(\epsilon _j)\eta \big ), \gamma '(\epsilon _j)^{-1}\xi -\eta , \gamma '(\epsilon _j)\eta \Big )} \,{\mathrm{d}}\eta . \end{aligned}$$

We then have, after a change of variables,

$$\begin{aligned} \big \Vert T_{j, m}(f, g)\big \Vert _2^2&=\int \! T_{j, m}(f, g)(\xi )\overline{T_{j, m}(f, g)(\xi )} \,{\mathrm{d}}\xi \nonumber \\&=\int {\mathrm{d}}\tau \iint \! F_{\tau }(x)G_{\tau }(y)A_{\tau }(x, y)e^{-2\pi i 2^m P_{\tau }(x, y)}\,{\mathrm{d}}x \,{\mathrm{d}}y, \end{aligned}$$

(4.8)

where

$$\begin{aligned} F_{\tau }(x)&= \widehat{f}\widehat{\varphi }(x-\tau )\overline{\widehat{f}\widehat{\varphi }(x)},\\ G_{\tau }(y)&= \widehat{g}\widehat{\varphi }(y+\gamma '(\epsilon _j)\tau )\overline{\widehat{g}\widehat{\varphi }(y)},\\ A_{\tau }(x, y)&= a(x-\tau , y+\gamma '(\epsilon _j)\tau )\overline{a(x, y)}, \end{aligned}$$

and

$$\begin{aligned} P_{\tau }(x, y)=P_1\big (x-\tau , y+\gamma '(\epsilon _j)\tau \big )-P_1(x, y) \end{aligned}$$

with

$$\begin{aligned} P_1(x, y)=y\phi _{\epsilon _j}\big (t(x, y), x, y\big ). \end{aligned}$$

In order to estimate the inner double integral in (4.8), we first show that there exists an $L\in {{\mathbb {N}}}$ such that if $j\ge L$ then

$$\begin{aligned} \left| \frac{\partial ^2 P_{\tau }}{\partial y \partial x}(x, y)\right| \asymp |\tau |. \end{aligned}$$

(4.9)

Recall that $t(x, y)$ satisfies (3.2) (with $\xi$, $\eta$, and $\epsilon$ replaced by $x$, $y$, and $\epsilon _j$ respectively). By implicit differentiation, we get

$$\begin{aligned} \frac{\partial t}{\partial x}(x, y)=-\frac{1}{yQ_{\epsilon _j}''(t(x, y))} \quad \mathrm{and} \quad \frac{\partial t}{\partial y}(x, y)=-\frac{Q_{\epsilon _j}'(t(x, y))}{yQ_{\epsilon _j}''(t(x, y))}. \end{aligned}$$

By (2.1), (2.2), and (2.3), we then have

$$\begin{aligned} \frac{\partial ^2 t}{\partial x \partial y}(x, y)=\frac{1}{y^2}\frac{\big (Q_{\epsilon _j}''\big )^2-Q_{\epsilon _j}'Q_{\epsilon _j}'''}{(Q_{\epsilon _j}'')^3}\big (t(x, y)\big )\asymp 1 \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^2 t}{\partial y^2}(x, y)=\frac{1}{y^2}\frac{Q_{\epsilon _j}'\big (2(Q_{\epsilon _j}'')^2-Q_{\epsilon _j}'Q_{\epsilon _j}'''\big )}{(Q_{\epsilon _j}'')^3}\big (t(x, y)\big )\lesssim 1. \end{aligned}$$

By using (3.2) we also get

$$\begin{aligned} \frac{\partial ^2 P_1}{\partial y \partial x}(x, y)=\frac{\partial t}{\partial y}(x, y). \end{aligned}$$

Noticing that $|\gamma '(\epsilon _j)|$ is small if $L$ is large, by the mean value theorem we get (4.9).

Let $\tau _0=(2^m|\gamma '(\epsilon _j)|)^{-1/3}$. We have the following splitting of (4.8):

$$\begin{aligned} \Vert T_{j, m}(f, g)\Vert _2^2&= \left( \int _{|\tau |<\tau _0}+\int _{\tau _0\le |\tau |\le 10}{\mathrm{d}}\tau \right) \\&\quad \cdot \iint F_{\tau }(x)G_{\tau }(y)A_{\tau }(x, y)e^{-2\pi i 2^m P_{\tau }(x, y)}\,{\mathrm{d}}x \,{\mathrm{d}}y. \end{aligned}$$

Applying the trivial estimate and Hörmander’s [5, Theorem 1.1] to the two parts above (and also Hölder’s inequality) yields

$$\begin{aligned} \big \Vert T_{j, m}(f, g)\big \Vert _2^2&\le C\tau _0\Vert f\Vert _2^2\Vert g\Vert _2^2+C\int _{\tau _0\le |\tau |\le 10} \! \big (2^m|\tau |\big )^{-1/2} \Vert F_{\tau }\Vert _2\Vert G_{\tau }\Vert _2 \,{\mathrm{d}}\tau \\&\le C\big (\tau _0+(2^m|\gamma '(\epsilon _j)|\tau _0)^{-1/2}\big )\Vert f\Vert _2^2\Vert g\Vert _2^2\\&\le C\big (2^m|\gamma '(\epsilon _j)|\big )^{-1/3}\Vert f\Vert _2^2\Vert g\Vert _2^2. \end{aligned}$$

To conclude, the desired estimate (4.3) follows from (4.5), (4.7), and the estimate above of $\Vert T_{j, m}(f, g)\Vert _2$.

4.2 Part 2: $j\ge L$, $m\in {{\mathbb {N}}}$

⁶ We can find a finite open cover of the interval $[-36C_1, 36C_1]$ by using open intervals of length $c_1/24$, associated with which we can construct a partition of unity $\{\chi _s : 1\le s\le \Theta \}$ such that $\sum _s \chi _s\equiv 1$ in $[-36C_1, 36C_1]$ and each $\chi _s$ is smooth and supported in an interval that belongs to the finite open cover above.

Lemma 3.1 will be applied to $\chi _s(-\xi /\eta )K_{j, m}(\xi , \eta )$ (below). Here we denote $S$ to be the collection of all $1\le s\le \Theta$ for which the second statement in Lemma 3.1 holds. Let $\mathbf I$ be either $[1/4, 9/4]$ or $[-9/4, -1/4]$, and

$$\begin{aligned} U(\mathbf I ):=\big \{u_{s, r, \eta }(\xi )\in L^2(\mathbf I ) : s\in S, r\in {{\mathbb {R}}}, 1/16C_1\le |\eta |\le 9C_1\big \}, \end{aligned}$$

where

$$\begin{aligned} u_{s, r, \eta }(\xi )=\chi _s\big (-\xi /\eta \big )e^{2\pi i\big ( 2^m\eta \phi _{\epsilon _j}(t(\xi , \eta ), \xi , \eta )+r\xi \big )}. \end{aligned}$$

According to Lemma 3.3, we finish this part in three steps.

Step 1: Let $\widehat{f}|_\mathbf{I }$, the restriction of $\widehat{f}$ to $\mathbf I$, be an arbitrary function in $L^2(\mathbf I )$ that is $\sigma$-uniform in $U(\mathbf I )$.

We first note that $B_{j, m}^{\varphi }(f, g)(x)$ in the time space can be expressed as

$$\begin{aligned} B_{j, m}^{\varphi }(f, g)(x)=\big |\gamma '(\epsilon _j)\big |^{1/2} \int _{0}^{\infty } \! f*\varphi \big (\gamma '(\epsilon _j)x-2^m t\big )g*\varphi \big (x-2^m Q_{\epsilon _j}(t)\big )\rho (t) \,{\mathrm{d}}t, \end{aligned}$$

(4.10)

which leads to, for $h\in L^2({{\mathbb {R}}})$,

where $g_{j, m, l}=1_{I_{j, m, l}}\cdot g*\varphi$ with $I_{j, m, l}=[\alpha _{j, l}-C_1 2^m, \alpha _{j, l+1}+C_1 2^m]$ and $\alpha _{j, l}=|\gamma '(\epsilon _j)|^{-1}l$. In the frequency space we then have

Let $\widehat{\varphi _1}\in C_{0}^{\infty }({{\mathbb {R}}})$ such that $\widehat{\varphi _1}(\eta )=1$ if $|\eta |\in [1/8C_1, 9C_1/2]$ and ${{\mathrm{supp}}}\widehat{\varphi _1}\subset \{x\in {{\mathbb {R}}} : 1/16C_1\le |x|\le 9C_1\}$. By using $1=\widehat{\varphi _1}(\eta )+(1-\widehat{\varphi _1}(\eta ))$ and the power series of $e^{2\pi i \gamma '(\epsilon _j)\xi (x-\alpha _{j, l})}$, we get

$$\begin{aligned} \int \! B_{j, m}^{\varphi }(f, g)(x)h(x) \, {\mathrm{d}}x=I+II, \end{aligned}$$

where

$$\begin{aligned} I&=\big |\gamma '(\epsilon _j)\big |^{1/2}\sum _{l\in {{\mathbb {Z}}}}\sum _{p=0}^{\infty }\frac{(2\pi i)^p}{p!}\iint \! \widehat{f}\widehat{\varphi }(\xi )\xi ^p e^{2\pi i \gamma '(\epsilon _j)\alpha _{j, l}\xi }\\&\quad \cdot \widehat{g_{j, m, l}}(\eta )\widehat{\varphi _1}(\eta ) K_{j, m}(\xi , \eta ){{\mathcal {F}}}^{-1}\Big [\big (\gamma '(\epsilon _j)(\cdot -\alpha _{j, l})\big )^p\big (\mathbf 1 _{|\gamma '(\epsilon _j)|^{-1}, l}h\big )(\cdot )\Big ](\eta ) \,{\mathrm{d}}\xi \,{\mathrm{d}}\eta \end{aligned}$$

and

$$\begin{aligned} II&=\big |\gamma '(\epsilon _j)\big |^{1/2}\sum _{l\in {{\mathbb {Z}}}}\sum _{p=0}^{\infty }\frac{(2\pi i)^p}{p!}\iint \! \widehat{f}\widehat{\varphi }(\xi )\xi ^p e^{2\pi i \gamma '(\epsilon _j)\alpha _{j, l}\xi }\\&\quad \cdot \widehat{g_{j, m, l}}(\eta )\big (1-\widehat{\varphi _1}(\eta )\big ) K_{j, m}(\xi , \eta ){{\mathcal {F}}}^{-1}\Big [\big (\gamma '(\epsilon _j)(\cdot -\alpha _{j, l})\big )^p\big (\mathbf 1 _{|\gamma '(\epsilon _j)|^{-1}, l}h\big )(\cdot )\Big ]\\&\quad \quad (\eta ) \,{\mathrm{d}}\xi \,{\mathrm{d}}\eta . \end{aligned}$$

We first estimate Sum II. When $1-\widehat{\varphi _1}(\eta )\ne 0$, Remark 2.4 (2) implies that the gradient of the phase function of $K_{j, m}(\xi , \eta )$ has a uniform lower bound, which leads to the bound $K_{j, m}(\xi , \eta )=O(2^{-m})$. Then by Hölder’s inequality we get

$$\begin{aligned} |II|\le C2^{-m}\big |\gamma '(\epsilon _j)\big |^{1/2}\Vert 1_\mathbf{I }\widehat{f}\Vert _2\sum _{l\in {{\mathbb {Z}}}}\Vert g_{j, m, l}\Vert _2\Vert \mathbf 1 _{|\gamma '(\epsilon _j)|^{-1}, l}h\Vert _2. \end{aligned}$$

Applying the Cauchy–Schwarz inequality yields

$$\begin{aligned} |II|\le \left\{ \begin{array}{ll} C 2^{-m/2}\big \Vert 1_\mathbf{I }\widehat{f}\big \Vert _2\Vert g\Vert _2\cdot \big (2^{-m}|\gamma '(\epsilon _j)|\big )^{1/2}\Vert h\Vert _2, &{} \quad \mathrm{if}\, |\gamma '(\epsilon _j)|\le 2^{-m},\\ C\big |\gamma '(\epsilon _j)\big |^{1/2}\big \Vert 1_\mathbf{I }\widehat{f}\big \Vert _2\Vert g\Vert _2\cdot \big (2^{-m}|\gamma '(\epsilon _j)|\big )^{1/2}\Vert h\Vert _2, &{} \quad \mathrm{if}\, |\gamma '(\epsilon _j)|> 2^{-m}. \end{array}\right. \end{aligned}$$

(4.11)

The estimate of Sum I, by using the partition of unity we have constructed at the beginning of this subsection, can be reduced to

$$\begin{aligned} I_s&=\big |\gamma '(\epsilon _j)\big |^{1/2}\sum _{l\in {{\mathbb {Z}}}}\sum _{p=0}^{\infty }\frac{(2\pi i)^p}{p!}\iint \! \widehat{f}\widehat{\varphi }(\xi )\xi ^p e^{2\pi i \gamma '(\epsilon _j)\alpha _{j, l}\xi }\widehat{g_{j, m, l}}(\eta )\widehat{\varphi _1}(\eta )\\&\quad \cdot \chi _s\big (-\xi /\eta \big ) K_{j, m}(\xi , \eta ){{\mathcal {F}}}^{-1}\Big [\big (\gamma '(\epsilon _j)(\cdot -\alpha _{j, l})\big )^p\big (\mathbf 1 _{|\gamma '(\epsilon _j)|^{-1}, l}h\big )(\cdot )\Big ](\eta ) \,{\mathrm{d}}\xi \,{\mathrm{d}}\eta \end{aligned}$$

for any $1\le s\le \Theta$. We apply Lemma 3.1 to $\chi _s(-\xi /\eta )K_{j, m}(\xi , \eta )$. If (3.1) holds, then $I_s$ is bounded by (4.11) too. Hence we may assume that the second statement in Lemma 3.1 holds. Applying (3.3) yields

$$\begin{aligned} I_s&=C\big (2^{-m}|\gamma '(\epsilon _j)|\big )^{1/2}\sum _{l\in {{\mathbb {Z}}}}\sum _{p=0}^{\infty }\frac{(2\pi i)^p}{p!} \\&\quad \cdot \int \! {{\mathfrak {M}}}(\eta ) \widehat{g_{j, m, l}}(\eta ) {{\mathcal {F}}}^{-1}\Big [\big (\gamma '(\epsilon _j)(\cdot -\alpha _{j, l})\big )^p\big (\mathbf 1 _{|\gamma '(\epsilon _j)|^{-1}, l}h\big )(\cdot )\Big ](\eta )\,{\mathrm{d}}\eta , \end{aligned}$$

where we have omitted the error term in (3.3) (since it leads to the same bound as in (4.11)), and ${{\mathfrak {M}}}(\eta )$ is defined as

$$\begin{aligned} {{\mathfrak {M}}}(\eta ):=\int _\mathbf{I } \! b(\xi , \eta )\widehat{f}(\xi )\chi _s\big (-\xi /\eta \big )e^{-2\pi i \big ( 2^m\eta \phi _{\epsilon _j}(t(\xi , \eta ), \xi , \eta )-\gamma '(\epsilon _j)\alpha _{j, l}\xi \big )} \,{\mathrm{d}}\xi \end{aligned}$$

with

$$\begin{aligned} b(\xi , \eta )=\widehat{\varphi }(\xi )\widehat{\varphi _1}(\eta )\xi ^p\rho \big (t(\xi , \eta )\big )|\eta |^{-1/2}\big |D^2_{t}\phi _{\epsilon _j}(t(\xi , \eta ), \xi , \eta )\big |^{-1/2}. \end{aligned}$$

Using the Fourier series of $b(\xi , \eta )$ and the assumption that $\widehat{f}|_\mathbf{I }$ is $\sigma$-uniform in $U(\mathbf I )$, we have

$$\begin{aligned} \big |{{\mathfrak {M}}}(\eta )\big |\le C9^{p}\sigma \big \Vert 1_\mathbf{I }\widehat{f}\big \Vert _2. \end{aligned}$$

Hence by using Hölder’s and the Cauchy–Schwarz inequalities we get

$$\begin{aligned} |I_s|\!\le \! \left\{ \begin{array}{ll} \!\!C\sigma \big \Vert 1_\mathbf{I }\widehat{f}\big \Vert _2\Vert g\Vert _2\cdot \big (2^{-m}|\gamma '(\epsilon _j)|\big )^{1/2}\Vert h\Vert _2, &{} \quad \! \mathrm{if}\, |\gamma '(\epsilon _j)|\le 2^{-m},\\ \!\!C\big (2^m|\gamma '(\epsilon _j)|\big )^{1/2}\sigma \big \Vert 1_\mathbf{I }\widehat{f}\big \Vert _2\Vert g\Vert _2\cdot \big (2^{-m}|\gamma '(\epsilon _j)|\big )^{1/2}\Vert h\Vert _2, &{}\! \quad \mathrm{if}\, |\gamma '(\epsilon _j)|> 2^{-m}. \end{array}\right. \end{aligned}$$

To conclude Step 1, if $\sigma >2^{-m/2}$, then the bound above of $I_s$ and (4.11) lead to, for $h\in L^{\infty }({{\mathbb {R}}})$,

$$\begin{aligned}&\left| \int \! B_{j, m}^{\varphi }(f, g)(x)\mathbf 1 _{2^m\gamma '(\epsilon _j)^{-1}, n}(x)h(x) \, {\mathrm{d}}x\right| \nonumber \\&\quad \le \left\{ \begin{array}{ll} C\sigma \big \Vert 1_\mathbf{I }\widehat{f}\big \Vert _2\Vert g\Vert _2\Vert h\Vert _{\infty }, &{} \quad \mathrm{if}\, |\gamma '(\epsilon _j)|\le 2^{-m},\\ C\big (2^m|\gamma '(\epsilon _j)|\big )^{1/2}\sigma \big \Vert 1_\mathbf{I }\widehat{f}\big \Vert _2\Vert g\Vert _2\Vert h\Vert _{\infty }, &{} \quad \mathrm{if}\, |\gamma '(\epsilon _j)|> 2^{-m}. \end{array}\right. \end{aligned}$$

(4.12)

Step 2: We now assume that $\widehat{f}|_\mathbf{I }\in U(\mathbf I )$.

By using (4.10), a change of variables $x\rightarrow 2^m \gamma '(\epsilon _j)^{-1}(x+\gamma '(\epsilon _j)Q_{\epsilon _j}(t))$, and Hölder’s inequality, we have, for $h\in L^{\infty }({{\mathbb {R}}})$,

$$\begin{aligned} \left| \int \! B_{j, m}^{\varphi }(f, g)(x)h(x) \, {\mathrm{d}}x\right|&=2^m\big |\gamma '(\epsilon _j)\big |^{-1/2}\bigg |\iint _{0}^{\infty } \! f*\varphi \big (2^m(x\!+\!\gamma '(\epsilon _j)Q_{\epsilon _j}(t)\!-\!t)\big ) \nonumber \\&\quad \cdot g*\varphi \big (2^m \gamma '(\epsilon _j)^{-1}x\big )h_{j, m}\big (x\!+\!\gamma '(\epsilon _j)Q_{\epsilon _j}(t)\big )\rho (t) \,{\mathrm{d}}t\,{\mathrm{d}}x\bigg |\nonumber \\&\le C\Vert g\Vert _2\Vert T_1(h)\Vert _2, \end{aligned}$$

(4.13)

where $h_{j, m}(x)=h(2^m \gamma '(\epsilon _j)^{-1}x)$ and

$$\begin{aligned} T_1(h)(x)=2^{m/2}\int _{0}^{\infty } \! f*\varphi \big (2^m(x+\gamma '(\epsilon _j)Q_{\epsilon _j}(t)-t)\big )h_{j, m}\big (x+\gamma '(\epsilon _j)Q_{\epsilon _j}(t)\big )\rho (t) \,{\mathrm{d}}t. \end{aligned}$$

Let $\widehat{f}|_\mathbf{I }=u_{s, r, \eta }(\xi )$ for arbitrarily fixed $s\in S$, $r\in {{\mathbb {R}}}$, and $1/16C_1\le |\eta |\le 9C_1$. By applying the Fourier inversion formula to $f*\varphi$ and changing variables, we get

$$\begin{aligned} \big \Vert T_1(h)\big \Vert _2^2=2^m\int \! \left| \int _0^{\infty }K_{1}(x, t) h_{j, m}\big (x-2^{-m}r+\gamma '(\epsilon _j)Q_{\epsilon _j}(t)\big )\rho (t) \,{\mathrm{d}}t\right| ^2\,{\mathrm{d}}x, \end{aligned}$$

(4.14)

where

$$\begin{aligned} K_{1}(x, t)=\int \! \widehat{\varphi }(\xi )\chi _s\big (-\xi /\eta \big )e^{2\pi i 2^m\eta \big [\phi _{\epsilon _j}(t(\xi , \eta ), \xi , \eta )+y(x, t)(\xi /\eta )\big ]}\, \mathrm d \xi \end{aligned}$$

(4.15)

with

$$\begin{aligned} y=y(x, t)=x+\gamma '(\epsilon _j)Q_{\epsilon _j}(t)-t. \end{aligned}$$

Let $\chi _M$ be a smooth cut-off function supported in $[-M, M]$, which equals $1$ in $[-M/2, M/2]$. We decompose the right-hand side of (4.14) into two parts by using the decomposition $1=(1-\chi _M(x))+\chi _M(x)$ to restrict the integration domain of $x$ to $\{x\in {{\mathbb {R}}} : |x|\ge M/2\}$ and $\{x\in {{\mathbb {R}}} : |x|<M\}$ respectively for a sufficiently large constant $M$. The former part is bounded by

$$\begin{aligned} C2^{-m}\Vert h\Vert _{\infty }^2, \end{aligned}$$

(4.16)

since integration by parts yields $K_{1}(x, t)=O(2^{-m}|x|^{-1})$.

We next consider the latter part with $|x|<M$. After inserting a partition of unity, we may replace $K_{1}(x, t)$ by $\widetilde{\chi }(-y(x, t))K_{1}(x, t)$ with a smooth cut-off function $\widetilde{\chi }$ supported in an interval of sufficiently small length. Then by repeating the argument in the proof of Lemma 3.1, we have that either $\widetilde{\chi }(-y(x, t))K_{1}(x, t)=O(2^{-m})$ (leading to the bound (4.16)) or the phase function in (4.15) has a critical point $\xi (x, t)$ satisfying

$$\begin{aligned} t\big (\xi (x, t), \eta \big )=-y(x, t). \end{aligned}$$

This equation, together with $\partial _{t}\phi _{\epsilon _j}(t(\xi , \eta ), \xi , \eta )=0$ (namely, Eq. (3.2) satisfied by $t(\xi , \eta )$), yields

$$\begin{aligned} \xi (x, t)=-\eta Q_{\epsilon _j}'\big (-y(x, t)\big ). \end{aligned}$$

By using the method of stationary phase in a neighborhood of $\xi (x, t)$ and integration by parts away from it, we get the following asymptotic formula.

$$\begin{aligned} \begin{aligned} \widetilde{\chi }\big (-y(x, t)\big )K_{1}(x, t)&=C\widetilde{\chi }\big (-y(x, t)\big )\chi _s\big (-\xi (x, t)/\eta \big )\widehat{\varphi }\big (\xi (x, t)\big )\\&\quad \quad \Big |\partial _{\xi }t\big (\xi (x, t), \eta \big )\Big |^{-1/2} \\&\quad \cdot e^{2\pi i 2^m\eta Q_{\epsilon _j}\big (-y(x, t)\big )}2^{-m/2}+O\big (2^{-3m/2}\big ). \end{aligned} \end{aligned}$$

By using the leading term above and a change of variables $u=Q_{\epsilon _j}(t)$, we now need to estimate

$$\begin{aligned} \int \! \chi _M(x)\left| \int h_{j, m}\big (x-2^{-m}r+\gamma '(\epsilon _j)u\big )k(x, u)e^{2\pi i 2^m\eta Q_{\epsilon _j}\Big (-y\big (x, Q_{\epsilon _j}^{-1}(u)\big )\Big )} \,{\mathrm{d}}u\right| ^2 \,{\mathrm{d}}x, \end{aligned}$$

(4.17)

where

$$\begin{aligned} \begin{aligned} k(x, u)&=\widetilde{\chi }\Big (-y\big (x, Q_{\epsilon _j}^{-1}(u)\big )\Big )\chi _s\Big (-\xi \big (x, Q_{\epsilon _j}^{-1}(u)\big )/\eta \Big )\widehat{\varphi }\Big (\xi \big (x, Q_{\epsilon _j}^{-1}(u)\big )\Big ) \\&\quad \cdot \Big |\partial _{\xi }t\big (\xi (x, Q_{\epsilon _j}^{-1}(u)\big ), \eta \big )\Big |^{-1/2}\rho \Big (Q_{\epsilon _j}^{-1}(u)\Big )\Big (Q_{\epsilon _j}'\big (Q_{\epsilon _j}^{-1}(u)\big )\Big )^{-1}. \end{aligned} \end{aligned}$$

We use the $TT^*$ method for (4.17). By changing variables $u_1=\upsilon +\tau$, $u_2=\upsilon$, followed by $x\rightarrow x-\gamma '(\epsilon _j)\upsilon$, (4.17) becomes

$$\begin{aligned} \int \mathrm d \tau \int \! H_{\tau }(x) \,{\mathrm{d}}x \int \! K_{\tau , x}(\upsilon )e^{2\pi i 2^m\eta P_{\tau , x}(\upsilon )} \,{\mathrm{d}}\upsilon , \end{aligned}$$

(4.18)

where all three integrals are over some finite intervals,

$$\begin{aligned} H_{\tau }(x)&= h_{j, m}\big (x-2^{-m}r+\gamma '(\epsilon _j)\tau \big )\overline{h_{j, m}(x-2^{-m}r)},\\ K_{\tau , x}(\upsilon )&= \chi _M\big (x-\gamma '(\epsilon _j)\upsilon \big )k\big (x-\gamma '(\epsilon _j)\upsilon , \upsilon +\tau \big )\overline{k(x-\gamma '(\epsilon _j)\upsilon , \upsilon )}, \end{aligned}$$

and

$$\begin{aligned} P_{\tau , x}(\upsilon )=P_{2}\big (x+\gamma '(\epsilon _j)\tau , \upsilon +\tau \big )-P_{2}(x, \upsilon ) \end{aligned}$$

with

$$\begin{aligned} P_{2}(x, \upsilon )=Q_{\epsilon _j}\Big (-\big [x-Q_{\epsilon _j}^{-1}(\upsilon )\big ]\Big ). \end{aligned}$$

Before applying integration by parts to the innermost integral in (4.18) we first estimate its phase function $P_{\tau , x}(\upsilon )$. Actually we have that if $|\gamma '(\epsilon _j)|/|x|$ is sufficiently small then

$$\begin{aligned} \big |D_{\upsilon }P_{\tau , x}(\upsilon )\big |\asymp |x||\tau | \end{aligned}$$

(4.19)

and

$$\begin{aligned} \big |D^2_{\upsilon }P_{\tau , x}(\upsilon )\big |\lesssim |x||\tau |. \end{aligned}$$

(4.20)

The (4.19) follows from the mean value theorem and the following estimates

$$\begin{aligned} \frac{\partial ^2 P_{2}}{\partial x\partial \upsilon }(x, \upsilon )=-\frac{Q_{\epsilon _j}^{''}\big (-[x-Q_{\epsilon _j}^{-1}(\upsilon )]\big )}{Q_{\epsilon _j}^{'}\big (Q_{\epsilon _j}^{-1}(\upsilon )\big )} \asymp 1 \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^2 P_{2}}{\partial \upsilon ^2}(x, \upsilon )=x\cdot \frac{Q_{\epsilon _j}^{'}\big (-[x-Q_{\epsilon _j}^{-1}(\upsilon )]\big )}{\big (Q_{\epsilon _j}^{'}\big (Q_{\epsilon _j}^{-1}(\upsilon )\big )\big )^2}\cdot \frac{\big (Q_{\epsilon _j}''\big )^2-Q_{\epsilon _j}'Q_{\epsilon _j}'''}{\big (Q_{\epsilon _j}'\big )^2}(c)\asymp |x|, \end{aligned}$$

where $c$ is between $-[x-Q_{\epsilon _j}^{-1}(\upsilon )]$ and $Q_{\epsilon _j}^{-1}(\upsilon )$. The (4.20) can be proved similarly.

Therefore, if $|\gamma '(\epsilon _j)|/|x|$ is sufficiently small, for any $\beta <1$ we have

$$\begin{aligned} \left| \int \! K_{\tau , x}(\upsilon )e^{2\pi i 2^m\eta P_{\tau , x}(\upsilon )} \,{\mathrm{d}}\upsilon \right| \le C\min \big \{1, (2^m|x||\tau |)^{-1}\big \}\le C \big (2^m|x||\tau |\big )^{-\beta }. \end{aligned}$$

We now estimate (4.18) by splitting it into two parts (depending on the size of $|\gamma '(\epsilon _j)|/|x|$) and using the trivial estimate and the bound above respectively. Then it is bounded by

$$\begin{aligned} C\big (|\gamma '(\epsilon _j)|+2^{-\beta m} \big )\Vert h\Vert _{\infty }^2. \end{aligned}$$

(4.21)

To conclude Step 2, by (4.13), (4.14), (4.16), and (4.21), we get, for $h\in L^{\infty }({{\mathbb {R}}})$,

$$\begin{aligned}&\left| \int \! B_{j, m}^{\varphi }(f, g)(x)\mathbf 1 _{2^m\gamma '(\epsilon _j)^{-1}, n}(x)h(x) \, {\mathrm{d}}x\right| \nonumber \\&\quad \le \left\{ \begin{array}{ll} C2^{-\beta m/2}\Vert g\Vert _2\Vert h\Vert _{\infty }, &{} \quad \mathrm{if}\, |\gamma '(\epsilon _j)|\le 2^{-m},\\ C\big (\max \{|\gamma '(\epsilon _j)|, 2^{-\beta m}\}\big )^{1/2} \Vert g\Vert _2\Vert h\Vert _{\infty }, &{} \quad \mathrm{if}\, |\gamma '(\epsilon _j)|> 2^{-m}. \end{array}\right. \end{aligned}$$

(4.22)

Step 3: To conclude this subsection (namely, Part 2), by using Lemma 3.3 and the estimates (4.12) and (4.22), we get that for any $\beta <1$

$$\begin{aligned} \big \Vert B_{j, m}^{\varphi }(f, g)\mathbf 1 _{2^m\gamma '(\epsilon _j)^{-1}, n}\big \Vert _{1}\le C2^{-\beta m/4}\Vert f\Vert _{2} \Vert g\Vert _{2}, \quad \mathrm{if}\, \big |\gamma '(\epsilon _j)\big |\le 2^{-\beta m},\quad \end{aligned}$$

(4.23)

and

$$\begin{aligned} \big \Vert B_{j, m}^{\varphi }(f, g)\mathbf 1 _{2^m\gamma '(\epsilon _j)^{-1}, n}\big \Vert _{1}\le C2^{m/4}\big |\gamma '(\epsilon _j)\big |^{1/2}\Vert f\Vert _{2} \Vert g\Vert _{2}, \quad \mathrm{if}\, |\gamma '(\epsilon _j)|\ge 2^{-\beta m}. \end{aligned}$$

(4.24)

4.3 Part 3: Conclusion

If $|\gamma '(\epsilon _j)|\ge 2^{-\beta m}$, balancing (4.4) with (4.24) yields (see also Lie [13, p. 20])

$$\begin{aligned} \big \Vert B_{j, m}^{\varphi }(f, g)\mathbf 1 _{2^m\gamma '(\epsilon _j)^{-1}, n}\big \Vert _{1}&\le C\min \Big \{\big (2^m|\gamma '(\epsilon _j)|\big )^{-1/6}, 2^{m/4}|\gamma '(\epsilon _j)|^{1/2}\Big \}\Vert f\Vert _{2} \Vert g\Vert _{2}\\&\le C 2^{-m/16}\Vert f\Vert _{2} \Vert g\Vert _{2}. \end{aligned}$$

If $|\gamma '(\epsilon _j)|\le 2^{-\beta m}$, the (4.23) is already good enough. This finishes the proof of Proposition 4.1.

5 Estimate of $\Vert B_{j, m}^{\Phi }(f, g)\Vert _{1}$

Let $\widehat{\Phi }\in C_{0}^{\infty }({{\mathbb {R}}})$ be supported in $\{\xi \in {{\mathbb {R}}} : 1/2\le |\xi |\le 2\}$ and $B_{j, m}^{\Phi }(f, g)$ be as defined at the beginning of Sect. 4 (with $\varphi$ there replaced by $\Phi$).

Proposition 5.1

Assume that $\Gamma (t)=(t, \gamma (t))\in \mathbf F (-1, 1)$.⁷ For any $\beta <1$, there exist an $L\in {{\mathbb {N}}}$ and a constant $C'_{\beta }$ such that whenever $j\ge L$, $m\in {{\mathbb {N}}}$, and $f, g\in L^2({{\mathbb {R}}})$ we have

$$\begin{aligned} \Vert B_{j, m}^{\Phi }(f, g)\Vert _{1}\le C'_{\beta }A_{j, m}^{\beta } \Vert f\Vert _{2} \Vert g\Vert _{2}, \end{aligned}$$

(5.1)

where $A_{j, m}$ equals $C_{j, m}$ if $|\gamma '(\epsilon _j)|\le K_1|\epsilon _j|^{c_1}$ and $C_{j, m}'$ if $|\gamma '(\epsilon _j)|\ge K_2|\epsilon _j|^{-c_1}$ (with $C_{j, m}$ and $C_{j, m}'$ defined as in Proposition 4.1).

Remark 5.2

This proposition is a consequence of Proposition 4.1. It is essentially the Lemma 5.1 contained in the arXiv preprint (arXiv:0805.0107) (which was later published as Li [11]).

Proof of Proposition 5.1

We only prove the case when $|\gamma '(\epsilon _j)|\le K_1|\epsilon _j|^{c_1}$ while the other case can be handled similarly. Let $\phi$ be a Schwartz function on ${{\mathbb {R}}}$ such that $\int \!\phi =1$ and ${{\mathrm{supp}}}\widehat{\phi }\subset [-1/100, 1/100]$. Denote $\phi _K(x)=K^{-1}\phi (K^{-1}x)$. We have

$$\begin{aligned} B_{j, m}^{\Phi }(f, g)(x)&=\big |\gamma '(\epsilon _j)\big |^{1/2} \sum _{n\in {{\mathbb {Z}}}}\sum _{k_1, k_2\in {{\mathbb {Z}}}} \int _{0}^{\infty } \! \big (\mathbf 1 _{2^m, n\!+\!k_1}*\phi _{2^m} \cdot f*\Phi \big ) \big (\gamma '(\epsilon _j)x\!-\!2^m t\big )\\&\quad \cdot \Big (\mathbf 1 _{2^m\gamma '(\epsilon _j)^{-1}, n+k_2}*\phi _{2^m\gamma '(\epsilon _j)^{-1}} \cdot g*\Phi \Big )\\&\quad \,\,\big (x-2^m Q_{\epsilon _j}(t)\big )\rho (t) \,{\mathrm{d}}t \cdot \mathbf 1 _{2^m\gamma '(\epsilon _j)^{-1}, n}. \end{aligned}$$

We then make the decomposition $B_{j, m}^{\Phi }(f, g)(x):=\mathrm{I}+\mathrm{II}$ by splitting the inner summation for $k_1, k_2$ into two parts such that the first part, denoted by I, sums over $\{k_1, k_2\in {{\mathbb {Z}}} : \max \{|k_1|, |k_2|\}\ge A\}$ and the second one, denoted by II, over $\{k_1, k_2\in {{\mathbb {Z}}} : \max \{|k_1|, |k_2|\}< A\}$ with $A=C_{j, m}^{-(1-\beta )/2}>1$.

Using the fast decay of $\mathbf 1 _{2^m, n+k_1}*\phi _{2^m}$ and $\mathbf 1 _{2^m\gamma '(\epsilon _j)^{-1}, n+k_2}*\phi _{2^m\gamma '(\epsilon _j)^{-1}}$ yields that

$$\begin{aligned} |\mathrm I |&\le C\big |\gamma '(\epsilon _j)\big |^{1/2}\sum _{\begin{array}{c} k_1, k_2\in {{\mathbb {Z}}}\\ \max \{|k_1|, |k_2|\}\ge A \end{array}}\\&\quad \int _{0}^{\infty } \! \frac{\big |f*\Phi \big (\gamma '(\epsilon _j)x-2^m t\big ) g*\Phi \big (x-2^m Q_{\epsilon _j}(t)\big )\rho (t)\big |}{\big (1+|t+k_1|\big )^{N_1}\big (1+|\gamma '(\epsilon _j)Q_{\epsilon _j}(t)+k_2|\big )^{N_2}} \,{\mathrm{d}}t\\&\le C\big (A^{1-N_1}+A^{1-N_2}\big )\big |\gamma '(\epsilon _j)\big |^{1/2}\cdot \\&\quad \int _{0}^{\infty } \! \big |f*\Phi \big (\gamma '(\epsilon _j)x-2^m t\big ) g*\Phi \big (x-2^m Q_{\epsilon _j}(t)\big )\rho (t)\big | \,{\mathrm{d}}t \end{aligned}$$

for any $N_1, N_2\in {{\mathbb {N}}}$. By Hölder’s and Young’s inequalities, we get

$$\begin{aligned} \Vert \mathrm I \Vert _1\le C\big (A^{1-N_1}+A^{1-N_2}\big )\Vert f\Vert _2\Vert g\Vert _2\le C''_{\beta }C_{j, m}^{\beta }\Vert f\Vert _2\Vert g\Vert _2, \end{aligned}$$

(5.2)

where the second inequality holds whenever $N_1, N_2\ge (1+\beta )/(1-\beta )$.

On the other hand, since

$$\begin{aligned} {{\mathrm{supp}}}\big ({{\mathcal {F}}}[\mathbf 1 _{2^m, n+k_1}*\phi _{2^m} \cdot f*\Phi ]\big )\subset \big \{\xi \in {{\mathbb {R}}} : 3/8\le |\xi |\le 17/8\big \} \end{aligned}$$

and

$$\begin{aligned} {{\mathrm{supp}}}\big ({{\mathcal {F}}}[\mathbf 1 _{2^m\gamma '(\epsilon _j)^{-1}, n+k_2}*\phi _{2^m\gamma '(\epsilon _j)^{-1}} \cdot g*\Phi ]\big )\subset \big \{\xi \in {{\mathbb {R}}} : 3/8\le |\xi |\le 17/8\big \}, \end{aligned}$$

we then have

$$\begin{aligned} \mathrm{II}&=\sum _{n\in {{\mathbb {Z}}}}\sum _{\begin{array}{c} k_1, k_2\in {{\mathbb {Z}}}\\ \max \{|k_1|, |k_2|\}< A \end{array}}B_{j, m}^{\varphi }\big (\mathbf 1 _{2^m, n+k_1}*\phi _{2^m} \cdot f*\Phi ,\\&\qquad \mathbf 1 _{2^m\gamma '(\epsilon _j)^{-1}, n+k_2}*\phi _{2^m\gamma '(\epsilon _j)^{-1}} \cdot g*\Phi \big )(x)\mathbf 1 _{2^m\gamma '(\epsilon _j)^{-1}, n}(x). \end{aligned}$$

Using Proposition 4.1 and the Cauchy–Schwarz inequality, we have

$$\begin{aligned} \Vert \mathrm{II}\Vert _1\le C_{\beta }C_{j, m}A^2\Vert f\Vert _2\Vert g\Vert _2=C_{\beta }C_{j, m}^{\beta }\Vert f\Vert _2\Vert g\Vert _2. \end{aligned}$$

(5.3)

The desired inequality (5.1) follows from (5.2) and (5.3).$\square$

6 Proof of Theorem 2.1

We prove Theorem 2.1 in this section. Let $\rho \in C_{0}^{\infty }({{\mathbb {R}}})$ be an odd function supported in $\{t\in {{\mathbb {R}}} : 1/2\le |t| \le 2 \}$ and $\rho _j(t) =2^j\rho (2^j t)$ such that

$$\begin{aligned} 1/t = \sum _{j\in {{\mathbb {Z}}}}\rho _j(t), \quad \mathrm{if}\, t\ne 0. \end{aligned}$$

Then

$$\begin{aligned} H_{\Gamma }(f, g)(x)=\sum _{j\ge 0} \int _{-1}^{1} \! f(x-t)g\big (x-\gamma (t)\big )\rho _j(t) \,{\mathrm{d}}t. \end{aligned}$$

Let $L\in {{\mathbb {N}}}$. If $0\le j\le L$, we can trivially estimate the $L^1$-norm of each summand above by Hölder’s inequality and get a bound in the form of $C\Vert f\Vert _2\Vert g\Vert _2$. Hence we may assume $j> L$ below. By the Fourier inversion formula we need to estimate

$$\begin{aligned} \widetilde{H}_{\Gamma }(f, g)(x)= \sum _{j> L} \iint \! \widehat{f}(\xi ) \widehat{g}(\eta ) {{\mathfrak {m}}}_j(\xi , \eta ) e^{2\pi i (\xi +\eta )x} \,{\mathrm{d}}\xi \,{\mathrm{d}}\eta , \end{aligned}$$

where

$$\begin{aligned} {{\mathfrak {m}}}_j(\xi , \eta )=\int _{{{\mathbb {R}}}} \! \rho (t)e^{-2\pi i\big (2^{-j}\xi t+\eta \gamma (2^{-j}t)\big )}\,{\mathrm{d}}t. \end{aligned}$$

(6.1)

Let $\widehat{\Phi }\in C_{0}^{\infty }({{\mathbb {R}}})$ be an even nonnegative function supported in $\{\xi \in {{\mathbb {R}}} : 1/2\le |\xi |\le 2\}$ such that

$$\begin{aligned} \sum _{m\in {{\mathbb {Z}}}}\widehat{\Phi }\left( \frac{\xi }{2^m}\right) =1, \quad \mathrm{if}\, \xi \ne 0. \end{aligned}$$

Let $m, m'\in {{\mathbb {Z}}}$. Set

$$\begin{aligned} {{\mathfrak {m}}}_{j,m,m'}(\xi ,\eta )= \widehat{\Phi }\left( \frac{\xi }{2^{j+m}}\right) \widehat{\Phi }\left( \frac{\eta }{2^{m'}\Delta _j}\right) {{\mathfrak {m}}}_j(\xi ,\eta ) \end{aligned}$$

with $\Delta _j$ defined as in Sect. 2. Then ${{\mathfrak {m}}}_j(\xi ,\eta )$ can be decomposed as the sum of

$$\begin{aligned} {{\mathfrak {m}}}_{j,+,+}(\xi ,\eta )&= \sum _{\begin{array}{c} \max \{m,m'\} \ge 0\\ |m'-m|< C \end{array}} {{\mathfrak {m}}}_{j,m,m'}(\xi ,\eta ), \\ {{\mathfrak {m}}}_{j,-,-}(\xi ,\eta )&= \sum _{m< 0} \!\quad \sum _{m'< 0} \,\,\,\, {{\mathfrak {m}}}_{j,m,m'}(\xi ,\eta ), \\ {{\mathfrak {m}}}_{j,-,+}(\xi ,\eta )&= \sum _{m' \ge 0}\sum _{m\le m'-C} {{\mathfrak {m}}}_{j,m,m'}(\xi ,\eta ), \end{aligned}$$

and

$$\begin{aligned} {{\mathfrak {m}}}_{j,+,-}(\xi ,\eta ) = \sum _{m \ge 0}\sum _{m'\le m-C} {{\mathfrak {m}}}_{j,m,m'}(\xi ,\eta ), \end{aligned}$$

where $C$ is a large constant (to be determined later; see (6.5) below). Then

$$\begin{aligned} \widetilde{H}_{\Gamma }(f, g)(x)&= \sum _{(*,**)\in {{\mathcal {A}}}} \sum _{j> L}\iint \! \widehat{f}(\xi )\widehat{g}(\eta ) {{\mathfrak {m}}}_{j,*,**}(\xi ,\eta ) e^{2\pi i(\xi +\eta )x} \,{\mathrm{d}}\xi \,{\mathrm{d}}\eta \\&{=:}\sum _{(*,**)\in {{\mathcal {A}}}} \widetilde{H}_{(*,**)}(f, g)(x), \end{aligned}$$

where the index set ${{\mathcal {A}}}$ is given by

$$\begin{aligned} {{\mathcal {A}}} = \big \{(+,+), (-,-), (-,+), (+,-)\big \}. \end{aligned}$$

(6.2)

We split $\widetilde{H}_{\Gamma }(f, g)(x)$ into two parts:

Major part: $(*,**)= (+,+)$;

Minor part: $(*,**) =(-,-)$, $(-,+)$, and $(+,-)$.

The essential difficulty in the proof of Theorem 2.1 lies in the estimates of the major part. All our preparations in Sects. 3–5 are done for it. The minor part can be reduced to classical paraproducts by using the Taylor and Fourier series expansions, and then handled by Theorem 3.4.

The following proposition completes the proof of Theorem 2.1.

Proposition 6.1

Using the notations above, if $L$ is sufficiently large we have

• (i) For the major part, if $(*,**)=(+,+)$ then

  $$\begin{aligned} \big \Vert \widetilde{H}_{(*,**)}(f, g)\big \Vert _1 \le C\Vert f\Vert _2\Vert g\Vert _2. \end{aligned}$$

• (ii) For the minor part, if $(*,**)\ne (+,+)$ then

  $$\begin{aligned} \big \Vert \widetilde{H}_{(*,**)}(f, g)\big \Vert _{r} \le C\Vert f\Vert _{p_1}\Vert g\Vert _{p_2}, \end{aligned}$$

  for all $p_1>1$ and $p_2>1$ such that $1/r= 1/p_1+1/p_2$.

The rest of this section is devoted to the proof of this proposition.

6.1 Estimates of the Major Part

We consider the case $\max \{m, m'\}\ge 0$ and $|m-m'|<C$ in this subsection. Actually it suffices to prove the special case $m'=m\in {{\mathbb {N}}}$ to which we can easily reduce the case $m'=m+b$ (for each integer $b$ with $1\le |b|< C$) simply by replacing $\gamma$ by a constant multiple of $\gamma$, namely, $2^{-b}\gamma$. We also notice that there is a symmetry between the two cases: $t\ge 0$ and $t\le 0$ and they can be handled similarly.

With these simplifications it suffices to prove

$$\begin{aligned} \bigg \Vert \sum _{j> L} \sum _{m\in {{\mathbb {N}}}} T_{j, m}(f, g)\bigg \Vert _1\le C\Vert f\Vert _2\Vert g\Vert _2, \end{aligned}$$

(6.3)

where

$$\begin{aligned} T_{j, m}(f, g)(x)=\iint \! \widehat{f}(\xi ) \widehat{g}(\eta ) \widehat{\Phi }\left( \frac{2^{-j}\xi }{2^m}\right) \widehat{\Phi }\left( \frac{2^{-j}\gamma '(2^{-j})\eta }{2^{m}}\right) \\ {{\mathfrak {m}}}^{+}_j(\xi , \eta )e^{2\pi i (\xi +\eta )x} \,{\mathrm{d}}\xi \,{\mathrm{d}}\eta \end{aligned}$$

with

$$\begin{aligned} {{\mathfrak {m}}}^{+}_j(\xi , \eta )=\int _{0}^{\infty } \! \rho (t)e^{-2\pi i\big (2^{-j}\xi t+\eta \gamma (2^{-j}t)\big )}\,{\mathrm{d}}t. \end{aligned}$$

To prove (6.3) we first apply a change of variables and get

$$\begin{aligned} T_{j, m}(f, g)(x)&=2^{2(j+m)}\big |\gamma '(2^{-j})\big |^{-1} \\&\quad \cdot \iint \! \widehat{f_{j, m}}(\xi )\widehat{\Phi }(\xi ) \widehat{g_{j, m}}(\eta )\widehat{\Phi }(\eta ) e^{2\pi i \big (2^{j+m}/\gamma '\left( 2^{-j}\right) \big )\big (\gamma '\left( 2^{-j}\right) \xi +\eta \big )x}\\&\quad \quad K_{j, m}(\xi , \eta ) \,{\mathrm{d}}\xi \,{\mathrm{d}}\eta , \end{aligned}$$

where

$$\begin{aligned} f_{j, m}(x)&= 2^{-j-m}f\big (2^{-j-m}x\big ),\\ g_{j, m}(x)&= 2^{-j-m}\gamma '\big (2^{-j}\big )g\Big (2^{-j-m}\gamma '\big (2^{-j}\big )x\Big ), \end{aligned}$$

and

$$\begin{aligned} K_{j, m}(\xi , \eta )={{\mathfrak {m}}}^{+}_j\Big (2^{j+m}\xi , 2^{j+m}\eta /\gamma '\big (2^{-j}\big )\Big )=\int _{0}^{\infty } \! \rho (t)e^{-2\pi i 2^m\big (\xi t+\eta Q_{2^{-j}}(t)\big )}\,{\mathrm{d}}t \end{aligned}$$

with

$$\begin{aligned} Q_{2^{-j}}(t)=\gamma \Big (2^{-j}t\Big )/\Big (2^{-j}\gamma '\big (2^{-j}\big )\Big ). \end{aligned}$$

Let $\mathbf 1 _0$ be the indicator function of $\{x\in {{\mathbb {R}}} : 1/2\le |x|\le 2\}$. Then

$$\begin{aligned} \big \Vert T_{j, m}(f, g)\big \Vert _1&=2^{j+m}\big |\gamma '(2^{-j})\big |^{-1/2}\Big \Vert B_{j, m}^{\Phi }\big ({{\mathcal {F}}}^{-1}[\widehat{f_{j, m}}\mathbf 1 _0], {{\mathcal {F}}}^{-1}[\widehat{g_{j, m}}\mathbf 1 _0]\big )\Big \Vert _1,\\&\le C2^{-m/32}2^{j+m}\Big |\gamma '\big (2^{-j}\big )\Big |^{-1/2}\big \Vert \widehat{f_{j, m}}\mathbf 1 _0\big \Vert _2\big \Vert \widehat{g_{j, m}}\mathbf 1 _0\big \Vert _2,\\&\le C2^{-m/32}\big \Vert \widehat{f}(\cdot )\mathbf 1 _0\big (2^{-j-m}\cdot \big )\big \Vert _2\big \Vert \widehat{g}(\cdot )\mathbf 1 _0\big (2^{-j-m}\gamma '\big (2^{-j}\big )\cdot \big )\big \Vert _2, \end{aligned}$$

where we have applied Propositions 4.1 and 5.1 if $L$ is sufficiently large. Thus,

$$\begin{aligned} \bigg \Vert \sum _{j> L}&\sum _{m\in {{\mathbb {N}}}} T_{j, m}(f, g)\bigg \Vert _1\\&\le C\sum _{m\in {{\mathbb {N}}}}2^{-m/32}\sum _{j> L}\big \Vert \widehat{f}(\cdot )\mathbf 1 _0\big (2^{-j-m}\cdot \big )\big \Vert _2\big \Vert \widehat{g}(\cdot )\mathbf 1 _0\Big (2^{-j-m}\gamma '\big (2^{-j}\big )\cdot \Big )\big \Vert _2\\&\le C \Vert f\Vert _2\Vert g\Vert _2. \end{aligned}$$

In the last inequality, we have used the Cauchy–Schwarz inequality and the bound

$$\begin{aligned} \sum _{j> L}\mathbf 1 _0\Big (2^{-j-m}\gamma '\big (2^{-j}\big )\eta \Big )\le C, \end{aligned}$$

which follows from the condition (2.4) (and also Remark 2.4 (3)). This finishes the estimates of the major part.

6.2 Estimates of the Minor Part

For the minor part, we begin with $(*,**) =(-,-)$. Notice

$$\begin{aligned} {{\mathfrak {m}}}_{j,-,-}(\xi ,\eta )&=\sum _{m'=-\infty }^{-1}\left( \sum _{m=-\infty }^{m'}{{\mathfrak {m}}}_{j,m,m'}(\xi ,\eta )\right) + \sum _{m=-\infty }^{-1} \left( \sum _{m'=-\infty }^{m-1}{{\mathfrak {m}}}_{j,m,m'}(\xi ,\eta )\right) \\&{=:} \sum _{m'=-\infty }^{-1} {{\mathfrak {m}}}_{j,-,m'}(\xi ,\eta )+\sum _{m=-\infty }^{-1} {{\mathfrak {m}}}_{j,m,-}(\xi ,\eta ). \end{aligned}$$

The treatments of ${{\mathfrak {m}}}_{j,-,m'}(\xi ,\eta )$ and ${{\mathfrak {m}}}_{j,m,-}(\xi ,\eta )$ are similar. We show how to handle ${{\mathfrak {m}}}_{j,-,m'}(\xi ,\eta )$. Set

$$\begin{aligned} \widehat{\Psi }(\xi )=\sum \limits _{m\le 0}\widehat{\Phi }\left( \frac{\xi }{2^m}\right) . \end{aligned}$$

Then, by applying the Taylor expansion to ${{\mathfrak {m}}}_j(\xi , \eta )$, we have

$$\begin{aligned} {{\mathfrak {m}}}_{j,-,m'}(\xi ,\eta )=\sum _{p,q=0}^\infty \frac{c_{j,p,q}}{p!q!} 2^{m'(p+q)} {{\mathcal {N}}}_{j,p,q}(\xi ,\eta ), \end{aligned}$$

where

$$\begin{aligned} c_{j,p,q} = \int \! \rho (t)(-2\pi i t)^{p} \Big (-2\pi i \gamma \big (2^{-j}t\big )\Delta _j) \Big )^q \,{\mathrm{d}}t \end{aligned}$$

(6.4)

and

$$\begin{aligned} {\mathcal {N}}_{j,p,q}(\xi ,\eta )=\widehat{\Psi }\left( \frac{\xi }{2^{j+m'}}\right) \left( \frac{\xi }{2^{j+m'}}\right) ^p \widehat{\Phi }\left( \frac{\eta }{ 2^{m'}\Delta _j}\right) \left( \frac{\eta }{ 2^{m'}\Delta _j}\right) ^q. \end{aligned}$$

Since $\rho$ is an odd function, $c_{j,0,0}=0$ for all $j\in {{\mathbb {Z}}}$ and thus we do not need to consider ${{\mathcal {N}}}_{j,0,0}(\xi ,\eta )$. This yields a decay factor as follows

$$\begin{aligned} 2^{m'(p+q)} \le 2^{m'} \quad \mathrm{if}\quad (p,q) \ne (0,0), \end{aligned}$$

which allows us to sum over $m'< 0$ later.

The condition (2.1) gives

$$\begin{aligned} |c_{j,p,q}|\le \Vert \rho \Vert _1(4\pi )^p(2\pi C_1)^q, \end{aligned}$$

which leads to

$$\begin{aligned} \sum _{p,q \ge 0 }\frac{|c_{j,p,q}|}{p!q!} < C<\infty \end{aligned}$$

for some constant $C$ independent of $j$.

Set

$$\begin{aligned} {{\mathcal {N}}}_{p,q}(\xi ,\eta ) =\sum _{j >L} {{\mathcal {N}}}_{j,p,q}(\xi ,\eta ) = \sum _{j>L}\widehat{\Psi }\left( \frac{\xi }{2^{j+m'}}\right) \left( \frac{\xi }{2^{j+m'}}\right) ^p\\ \widehat{\Phi }\left( \frac{\eta }{ 2^{m'}\Delta _j }\right) \left( \frac{\eta }{ 2^{m'}\Delta _j }\right) ^q. \end{aligned}$$

It suffices to show that ${{\mathcal {N}}}_{p,q}$, as a bilinear multiplier, maps $L^{p_1}\times L^{p_2}$ to $L^r$ with a bound independent of ${m'}$. Indeed, the dependence of ${m'}$ can be removed easily via the following claim:

Claim 6.2

Assume ${{\mathcal {M}}}(\xi ,\eta )$, as a symbol for a bilinear multiplier, maps $L^{p_1}\times L^{p_2}$ to $L^r$ with a bound $A$. Here $p_1>1$, $p_2>1$, and $1/p_1+1/p_2=1/r.$

Let $R>0$ be any constant. Then ${{\mathcal {M}}}_R(\xi ,\eta )={{\mathcal {M}}}(R\xi ,R\eta )$ is also a bounded bilinear multiplier which maps $L^{p_1}\times L^{p_2}$ to $L^r$ with the same bound $A$.

Claim 6.2 can be proved by a standard rescaling argument and we omit the details here.

Applying the same arguments to ${{\mathfrak {m}}}_{j,m,-}(\xi ,\eta )$, we obtain the corresponding multiplier

$$\begin{aligned} \tilde{{{\mathcal {N}}}}_{p,q}(\xi ,\eta ) = \sum _{j>L} \widehat{\Phi }\left( \frac{\xi }{2^{j+m-1}}\right) \left( \frac{\xi }{2^{j+m-1}}\right) ^p \widehat{\Psi }\left( \frac{\eta }{2^{m-1}\Delta _j }\right) \left( \frac{\eta }{2^{m-1}\Delta _j }\right) ^q. \end{aligned}$$

Again, Claim 6.2 allows us to dispose the factor $2^{m-1}$ on the right-hand side.

To sum up, the case $(*,**)=(-,-)$ is reduced to establishing the boundedness of the bilinear multipliers whose symbols are given by

$$\begin{aligned} \sum _{j>L}\widehat{\Psi }\left( \frac{\xi }{2^{j}}\right) \left( \frac{\xi }{2^{j}}\right) ^p\widehat{\Phi }\left( \frac{\eta }{ \Delta _j }\right) \left( \frac{\eta }{ \Delta _j }\right) ^q \end{aligned}$$

and

$$\begin{aligned} \sum _{j>L }\widehat{\Phi }\left( \frac{\xi }{2^{j}}\right) \left( \frac{\xi }{2^{j}}\right) ^p\widehat{\Psi }\left( \frac{\eta }{ \Delta _j }\right) \left( \frac{\eta }{ \Delta _j }\right) ^q, \end{aligned}$$

which is ensured by Theorem 3.4 (with $(n_1,n_2)=(0,0)$ there).

Now we turn to the case $(*,**)=(-,+)$. Notice

$$\begin{aligned} {{\mathfrak {m}}}_{j,-,+} (\xi ,\eta )=\sum _{m'\ge 0} \widetilde{\mathfrak {m}}_{j,-,m'}(\xi ,\eta ), \end{aligned}$$

where

$$\begin{aligned} \widetilde{\mathfrak {m}}_{j,-,m'}(\xi ,\eta ) := \sum _{m=-\infty }^{m'-C}{{\mathfrak {m}}}_{j,m,m'}(\xi ,\eta ). \end{aligned}$$

Applying the Fourier series of ${{\mathfrak {m}}}_{j}(\xi ,\eta )$ yields

$$\begin{aligned} \widetilde{\mathfrak {m}}_{j,-,m'}(\xi ,\eta )=&\widehat{\Psi }\left( \frac{\xi }{2^{j+{m'}-C}}\right) \widehat{\Phi }\left( \frac{\eta }{2^{m'} \Delta _j}\right) {{\mathfrak {m}}}_j(\xi , \eta ) \\ =&\widehat{\Psi }\left( \frac{\xi }{2^{j+{m'}-C}}\right) \widehat{\Phi }\left( \frac{\eta }{2^{m'} \Delta _j}\right) \sum _{n_1,n_2\in {{\mathbb {Z}}}} a_{n_1,n_2} e^{2\pi i \left( n_1 \frac{\xi }{2^{j+{m'}-C}}+n_2 \frac{\eta }{2^{m'}\Delta _j}\right) }. \end{aligned}$$

Since

$$\begin{aligned} \left| \partial _t\left( \eta \gamma (2^{-j}t)\right) \right| >2^9 |\partial _t(\xi 2^{-j}t)| \end{aligned}$$

(6.5)

given $C$ sufficiently large, integration by parts gives the following fast decay

$$\begin{aligned} |a_{n_1,n_2}| \le C_N \big (1+n_1^2+n_2^2\big )^{-N} 2^{-{m'}N}\quad \mathrm{for \, any}\quad N\in {{\mathbb {N}}}. \end{aligned}$$

Consequently, we only need to handle the following multiplier, for a fixed ${m'}$ and a fixed pair $(n_1,n_2)$,

$$\begin{aligned} {{\mathcal {N}}}_{n_1,n_2} (\xi ,\eta ) = \sum _{j>L} \widehat{\Psi }\left( \frac{\xi }{2^{j+{m'}-C}}\right) e^{2\pi i n_1 \frac{\xi }{2^{j+{m'}-C}}} \widehat{\Phi }\left( \frac{\eta }{2^{m'} \Delta _j}\right) e^{2\pi in_2 \frac{\eta }{2^{m'}\Delta _j}}.\quad \end{aligned}$$

(6.6)

Similarly, in the case $(*,**)=(+,-)$ we need to handle

$$\begin{aligned} \tilde{{{\mathcal {N}}}}_{n_1,n_2} (\xi ,\eta )= \sum _{j>L} \widehat{\Phi }\left( \frac{\xi }{2^{j+m}}\right) e^{2\pi i n_1 \frac{\xi }{2^{j+m}}} \widehat{\Psi }\left( \frac{\eta }{2^{m-C} \Delta _j}\right) e^{2\pi in_2 \frac{\eta }{2^{m-C}\Delta _j}}. \end{aligned}$$

(6.7)

By Claim 6.2, the factor $2^{m'}$ in (6.6) and the factor $2^m$ in (6.7) are disposable. Thus the problem is reduced to establishing the boundedness of the paraproducts whose symbols are given by

$$\begin{aligned} \sum _{j>L} \widehat{\Psi }\left( \frac{\xi }{2^{j-C}}\right) e^{2\pi i n_1 \frac{\xi }{2^{j-C}}} \widehat{\Phi }\left( \frac{\eta }{ \Delta _j}\right) e^{2\pi in_2 \frac{\eta }{\Delta _j}} \end{aligned}$$

(6.8)

and

$$\begin{aligned} \sum _{j>L} \widehat{\Phi }\left( \frac{\xi }{2^{j}}\right) e^{2\pi i n_1 \frac{\xi }{2^{j}}} \widehat{\Psi }\left( \frac{\eta }{2^{-C} \Delta _j}\right) e^{2\pi in_2 \frac{\eta }{2^{-C}\Delta _j}}, \end{aligned}$$

(6.9)

with bounds growing no faster than a polynomial of $(1+n_1^2+n_2^2)$, which are ensured by Theorem 3.4. Indeed, Theorem 3.4 is applicable to the multiplier (6.9), since the sequence $\{2^{-C}\Delta _j\}_{j>L}$ satisfies the condition (3.4). For the multiplier (6.8), one can perform a rescaling $(\xi ,\eta )\rightarrow (2^{-C}\xi , 2^{-C}\eta )$ and then apply Theorem 3.4 again since the sequence $\{2^C\Delta _j\}_{j>L}$ also satisfies the condition (3.4). This finishes the estimates of the minor part.

7 The Bilinear Maximal Functions

This section is devoted to the proof of Theorem 2.2. The arguments we use here are essentially those from [12, Section 7]. Recall that

$$\begin{aligned} M_\Gamma (f,g)(x) =\sup _{0<\epsilon <1} \epsilon ^{-1} \int _{0 }^\epsilon \! f(x-t)g(x-\gamma (t)) \,{\mathrm{d}}t , \end{aligned}$$

(7.1)

where we have assumed that $f$ and $g$ are both nonnegative. We want to show

$$\begin{aligned} \big \Vert M_\Gamma (f,g) \big \Vert _1\le C\Vert f\Vert _2\Vert g\Vert _2. \end{aligned}$$

(7.2)

The proof of (7.2) is almost identical to the proof of Theorem 2.1 in Sect. 6. One noticeable difference is that the minor part of (7.1) can be controlled pointwisely by the Hardy–Littlewood maximal function.

Let $\rho \in C^{\infty }_{0}([1/4, 1])$ be nonnegative with $\rho (1/2) =1$. Set $\rho _j(t) = 2^{j}\rho (2^jt)$. It suffices to establish the boundedness of the following maximal function

$$\begin{aligned} M^*(f,g) (x)= \sup _{j >L} \int \! f(x-t)g(x-\gamma (t)) \rho _j(t)\,{\mathrm{d}}t{=:} \sup _{j >L} M_j(f,g)(x), \end{aligned}$$

where $L\in {{\mathbb {N}}}$. Like what we did in Sect. 6, we have

$$\begin{aligned} M_j(f,g)(x)&= \iint \! \widehat{f}(\xi )\widehat{g}(\eta ) {{\mathfrak {m}}}_{j}(\xi , \eta ) e^{2\pi i (\xi +\eta )x}\,{\mathrm{d}}\xi \,{\mathrm{d}}\eta \\&= \sum _{(*,**)\in {{\mathcal {A}}}} \iint \! \widehat{f}(\xi )\widehat{g}(\eta ) {{\mathfrak {m}}}_{j,*,**}(\xi ,\eta ) e^{2\pi i (\xi +\eta )x}\,{\mathrm{d}}\xi \,{\mathrm{d}}\eta \\&= :\sum _{(*,**)\in {{\mathcal {A}}}} M_{j,*,**}(f,g)(x), \end{aligned}$$

where ${{\mathfrak {m}}}_j(\xi ,\eta )$ and ${{\mathcal {A}}}$ are as defined in (6.1) and (6.2) respectively. It suffices to prove

$$\begin{aligned} \left\| \sup _{j}|M_{j,*,**}(f,g)|\right\| _1 \le C\Vert f\Vert _2\Vert g\Vert _2 \end{aligned}$$

(7.3)

for each pair $(*,**)\in {{\mathcal {A}}}$.

Lemma 7.1

(Minor part) Let $M(f)$ denote the Hardy–Littlewood maximal function of $f$. If $L$ is sufficiently large and $(*,**)\ne (+,+)$, then there is a constant $C>0$ such that

$$\begin{aligned} \sup _{j>L} |M_{j,*,**}(f,g)(x)| \le CM(f)(x)M(g)(x). \end{aligned}$$

As a consequence of this lemma and the boundedness of the Hardy–Littlewood maximal function, we have

$$\begin{aligned} \left\| \sup _{j>L} |M_{j,*,**}(f,g)|\right\| _{1} \le C\Vert M(f)\Vert _{2}\Vert M(g)\Vert _{2}\le C \Vert f\Vert _{2}\Vert g\Vert _{2}. \end{aligned}$$

This proves (7.3) when $(*,**)\ne (+,+)$.

Proposition 7.2

(Major part) If $L$ is sufficiently large we have

$$\begin{aligned} \left\| \sup _{j>L} |M_{j,+,+}(f,g)|\right\| _1 \le C \Vert f\Vert _2\Vert g\Vert _2. \end{aligned}$$

This proposition is essentially the result obtained in Sect. 6.1. Indeed, we have the following pointwise estimate

$$\begin{aligned} \sup _{j>L} \big |M_{j,+,+}(f,g)(x)\big | \le \sum _{j>L} \big |M_{j,+,+}(f,g)(x)\big |. \end{aligned}$$

Then (6.3) implies

$$\begin{aligned} \left\| \sup _{j>L} |M_{j,+,+}(f,g)| \right\| _1\le \bigg \Vert \sum _{j>L} |M_{j,+,+}(f,g)|\bigg \Vert _1\le C\Vert f\Vert _2\Vert g\Vert _2. \end{aligned}$$

It remains to verify Lemma 7.1. We first consider the case $(*,**) =(-,-)$. Most of the calculation in Sect. 6.2 remains valid. In particular, we have

$$\begin{aligned} {{\mathfrak {m}}}_{j,-,-}(\xi ,\eta ):&= \sum _{p,q\in {{\mathbb {N}}}} \frac{c_{j,p,q}}{p!q!} \widehat{\Psi }\left( {\frac{\xi }{2^{j}}} \right) \left( \frac{\xi }{2^j}\right) ^p\widehat{\Psi }\left( \frac{\eta }{\Delta _j }\right) \left( \frac{\eta }{\Delta _j}\right) ^q. \end{aligned}$$

Notice

$$\begin{aligned} \sup _{j>L}\left| \int \!\widehat{\Psi }\left( {\frac{\xi }{2^{j}}} \right) \left( \frac{\xi }{2^j}\right) ^p \widehat{f}(\xi ) e^{2\pi i \xi x}\,{\mathrm{d}}\xi \right| \le C_1' M(f)(x) \end{aligned}$$

and

$$\begin{aligned} \sup _{j>L}\left| \int \! \widehat{\Psi }\left( \frac{\eta }{\Delta _j }\right) \left( \frac{\eta }{\Delta _j}\right) ^q \widehat{g} (\eta )e^{2\pi i \eta x}\,{\mathrm{d}}\eta \right| \le C_2' M(g)(x), \end{aligned}$$

where $C_1'$ and $C_2'$ depend at most exponentially on $p$ and $q$. Thus

$$\begin{aligned} \sup _{j>L} \left| \iint \! \widehat{f}(\xi ) \widehat{g}(\eta ) {{\mathfrak {m}}}_{j,-,-} (\xi ,\eta ) e^{2\pi i (\xi +\eta )x} \,{\mathrm{d}}\xi \,{\mathrm{d}}\eta \right| \le CM(f)(x)M(g)(x), \end{aligned}$$

which proves Lemma 7.1 when $(*,**)=(-,-)$.

The cases $(-,+)$ and $(+,-)$ are similar, hence we only show how to handle the former one. Using the same notations as in Sect. 6.2, we have

$$\begin{aligned} {{\mathfrak {m}}}_{j,-,+}(\xi ,\eta ) =\sum _{m'\ge 0} \widetilde{\mathfrak {m}}_{j,-,m'}(\xi ,\eta ) \end{aligned}$$

(7.4)

and

$$\begin{aligned} \widetilde{\mathfrak {m}}_{j,-,m'}(\xi ,\eta ) = \widehat{\Psi }\left( \frac{\xi }{2^{j+{m'}-C}}\right) \widehat{\Phi }\left( \frac{\eta }{2^{m'} \Delta _j}\right) \! \sum _{n_1,n_2\in {{\mathbb {Z}}}} a_{n_1,n_2} e^{2\pi i \left( n_1 \frac{\xi }{2^{j+{m'}-C}}+n_2 \frac{\eta }{2^{m'}\Delta _j}\right) }, \end{aligned}$$

(7.5)

where

$$\begin{aligned} |a_{n_1,n_2}| \le C_N \big (1+n_1^2+n_2^2\big )^{-N} 2^{-{m'}N} \quad \mathrm{for \, any} \quad N\in {{\mathbb {N}}}. \end{aligned}$$

(7.6)

Then

$$\begin{aligned} \sup _{j>L} \left| \int \! \widehat{\Psi }\left( \frac{\xi }{2^{j+{m'}-C}}\right) e^{2\pi i n_1 \frac{\xi }{2^{j+{m'}-C}}} \widehat{f}(\xi )e^{2\pi i \xi x}\,{\mathrm{d}}\xi \right| \le C\big (1+n_1^2\big ) M(f)(x)\quad \end{aligned}$$

(7.7)

and

$$\begin{aligned} \sup _{j>L} \left| \int \! \widehat{\Phi }\left( \frac{\eta }{2^{m'} \Delta _j}\right) e^{ 2\pi i {n_2 \frac{\eta }{2^{m'}\Delta _j}}} \widehat{g}(\eta )e^{2\pi i \eta x} \,{\mathrm{d}}\eta \right| \le C\big (1+n_2^2\big ) M(g)(x).\quad \end{aligned}$$

(7.8)

To get (7.7) and (7.8), we have applied the following fact

$$\begin{aligned} \sup _{t>0}\Big |f*\Omega _t\left( x-\frac{n}{t}\right) \Big | \le C_{\Omega }\big (1+n^2\big ) M(f)(x),\quad \end{aligned}$$

where

$$\begin{aligned} \Omega _t(x) = t\Omega (tx). \end{aligned}$$

Then (7.4), (7.5), (7.6), (7.7) and (7.8) yield

$$\begin{aligned} \sup _{j>L} \big |M_{j,-,+}(f,g)(x)\big | \le C M(f)(x)M(g)(x), \end{aligned}$$

as desired.