1. Introduction
In this study, we analyse a maximal operator defined by a convex function $\gamma|_{[0,\infty)}$ and a measurable function
$m:\mathbb{R}\rightarrow\mathbb{R}$. Specifically, our focus lies on the operator:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU1.png?pub-status=live)
where $\gamma:\mathbb{R}\rightarrow\mathbb{R}$ is an extension of
$\gamma|_{[0,\infty)}$, which is a even or odd function. Recently, Guo, Hickman, Lie and Roos [Reference Guo, Hickman, Lie and Roos13] proved the Lp boundedness of maximal operators
$\mathcal{M}^{m}_{\gamma}$ for the homogeneous curve
$\gamma(t) = t^n$, with
$n \geqslant 2$, assuming that m is measurable. However, the Lp boundedness of
$\mathcal{M}^{m}_{\gamma}$ for the case n = 1 remains an open problem. So, we focus on flat convex curves, including piecewise linear curves. Given a convex extension
$\gamma:\mathbb{R}\rightarrow \mathbb{R}$, we define the bounded doubling property for a derivative
$\gamma'$ as follows:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn1.png?pub-status=live)
Now, we state the main theorem:
Main Theorem 1.
Let $m:\mathbb{R}\rightarrow\mathbb{R}$ be a measurable function such that
$1\leqslant m(x)\leqslant 2$ for all
$x\in \mathbb{R}$. Suppose that an extension γ of a convex function
$\gamma|_{[0,\infty)}$ satisfies the bounded doubling property of
$\gamma'$ in (1.1), with
$\gamma(0)=0$. Then, there exists a constant Cω such that
$\|\mathcal{M}^{m}_{\gamma}\|_{{L^p(\mathbb{R}^2)}\rightarrow L^p(\mathbb{R}^2)}\leqslant C_{\omega,p}$ holds for
$1 \lt p\leqslant \infty$.
• The theorem can be extended to certain types of piecewise linear curves. Refer to Section 7 in [Reference Carbery, Christ, Vance, Wainger and Watson7] or Remark 5 in [Reference Kim14] for more details. Additionally, the condition (1.1) admits flat convex curves, such as
$\gamma(t)=\text{e}^{-\frac{1}{|t|}}$ and
$\text{e}^{-\text{e}^{\frac{1}{|t|}}}$, which are flat at the origin.
• By using the dilation technique, we can extend our results to
$\|\mathcal{M}^{m}_{\gamma}\|_{{L^p}\rightarrow L^p}\leqslant C\log_{2}(\frac{b}{a})$ under the assumption
$0 \lt a\leqslant m(x)\leqslant b$.
In the view of pointwise convergence, we can drop the assumption $1\leqslant m(x_1)\leqslant 2$.
Corollary 1.1. For a measurable function $m:\mathbb{R}\rightarrow\mathbb{R}$ and a convex extension γ on
$\mathbb{R}^1$ passing through the origin with its derivative
$\gamma'$ satisfying property (1.1), we have
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU2.png?pub-status=live)
for $f\in L^p(\mathbb{R}^2)$.
The study of maximal operators along flat convex curves has a rich history in Harmonic analysis by itself. In the 1970s, Stein and Wainger [Reference Stein and Wainger24] asked the general class of curves $(t,\gamma(t))$ for which there are Lp results for
$\mathcal{M}^{1}_{\gamma}$. In the 1980s, Carlsson
$\textit{et al. }$ [Reference Carlsson, Christ, Córdoba, Duoandikoetxea, Rubio de Francia, Vance, Wainger and Weinberg11] proved that
$\mathcal{M}^{1}_{\gamma}$ is bounded on
$L^p(\mathbb{R}^2)$ under the bounded doubling condition (1.1). In the 1990s, the study of maximal operators was extended to the curves with a variable coefficient, as demonstrated in [Reference Bennett.4, Reference Carbery, Wainger and Wright9, Reference Carbery, Wainger and Wright10, Reference Kim15, Reference Seeger and Wainger23]. Carbery, Wainger and Wright [Reference Carbery, Wainger and Wright9] established the Lp boundedness of
$\mathcal{M}^{x_1}_{\gamma}$ along plane curves γ whose derivative satisfies the infinitesimal doubling property. Under the same assumption, Bennett [Reference Bennett.4] extended the L 2 results for
$\mathcal{M}^{P}_{\gamma}$, where P is a polynomial. As a corollary of our main theorem, we derive the Lp boundedness of
$\mathcal{M}^{P}_{\gamma}$ under much weaker assumptions on γ.
Corollary 1.2. For a polynomial $P:\mathbb{R}\rightarrow\mathbb{R}$ with degree d and a convex extension γ on
$\mathbb{R}^1$ passing through the origin with its derivative
$\gamma'$ satisfying property (1.1), there exists a constant
$C_{\omega, d}$ independent of the coefficients of P such that
$\|\mathcal{M}^{P}_{\gamma}\|_{{L^p(\mathbb{R}^2)}\rightarrow L^p(\mathbb{R}^2)}\leqslant C_{\omega,d,p}$ for
$1 \lt p\leqslant \infty$.
Note that the infinitesimal doubling property implies the bounded doubling property. For more details, refer to [Reference Bennett.4].
1.1. Historical background
Zygmund conjecture is a long-standing open problem in harmonic analysis. This question inquires whether the Lipschitz regularity of u is sufficient to guarantee any non-trivial Lp bounds for the maximal operator:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU3.png?pub-status=live)
where $\gamma(t)=t$. Since the discovery of the Besicovitch set in the 1920s, it has been shown that the conjecture is false when the function u is only Hölder continuous C α with α < 1. However, the problem remains open under the Lipschitz assumption for u. In the 1970s, Stein and Wainger [Reference Stein and Wainger24] proposed an analogous conjecture for the Hilbert transform. Regarding the Hilbert transforms along vector fields, Lacey and Li [Reference Lacey and Li18] made a significant progress regarding the regularity of u in 2006, using time–frequency analysis tools. Later, Bateman and Thiele [Reference Bateman and Thiele2] obtained the Lp estimates for the Hilbert transform along a one-variable vector field. Their proof relied on the commutation relation between the Hilbert transform and Littlewood–Paley projection operators, which cannot be directly applied to the maximal operator
$\mathcal{M}_{\gamma}^m$ due to its sub-linearity. Therefore, the problem for maximal operators remains open. For additional discussion on Stein’s conjecture, we recommend references [Reference Bateman1, Reference Bateman and Thiele2, Reference Lacey and Li17]. In the study of maximal operators, Bourgain [Reference Bourgain5] demonstrated the L 2 boundedness of
$\mathcal{M}^{u}_{t}$ for real analytic functions u. In 1999, Carbery, Seeger, Wainger and Wright [Reference Carbery, Seeger, Wainger and Wright8] examined the maximal operators
$\mathcal{M}_{t}^{m}$ along one variable vector field. One of the authors in this paper further extended this result in [Reference Kim16].
Recently, in [Reference Guo, Hickman, Lie and Roos13], Guo et al. investigated the Lp boundedness of $\mathcal{M}_{\gamma}^u$ under the Lipschitz assumption for u and homogeneous curve
$\gamma(t)=t^n$ for n > 1. Later, Liu, Song and Yu [Reference Liu, Song and Yu20] extended the results to more general curves with the condition
$\left|\frac{t\gamma''(t)}{\gamma'(t)}\right|\sim 1$. A crucial tool used in the proofs of both papers was the local smoothing estimate, which was established in [Reference Beltran, Hickman and Sogge3, Reference Mockenhaupt, Seeger and Sogge21]. For more history, we recommend the study [Reference Lie19] by Victor Lie, which presents a unified approach and includes a more general view of this topic as well as problems related to the concept of non-zero curvature.
1.2. Notation
Let $\psi:\mathbb{R}\rightarrow \mathbb{R}$ be a non-negative
$C^{\infty}$ function supported on
$[-2,2]$ such that
$\psi\equiv 1$ on
$[-1,1]$. Define
$\varphi(t)=\psi(t)-\psi(2t)$ and
$\varphi_l(t)=\frac{1}{2^l}\varphi(\frac{t}{2^l})$. Also, define
$\psi^c(t)=1-\psi(t)$. Note that
$\sum_{l\in \mathbb{Z}} \varphi\left(\frac{t}{2^l}\right)=1\ \text{for }t\neq0$ and
$\text{supp}(\varphi)\subset\left\{\frac{1}{2}\leqslant|x|\leqslant2\right\}$. We define the Littlewood–Paley projection
$\mathcal{L}_sf$ as
$\widehat{\mathcal{L}_sf}(\xi):=\hat{f}(\xi){\varphi}\left(\frac{\xi_1}{2^s}\right)$. We shall use the notation
$A\lesssim_d B$ when
$A\leqslant C_dB$ with a constant
$C_d \gt 0$ depending on the parameter d. Moreover, we write
$A\sim_d B$, if
$A\lesssim_d B$ and
$B\lesssim_d A$. Let M HL be the Hardy–Littlewood maximal operator and M str be the strong maximal operator. Let χA be a characteristic function, which is equal to 1 on A and otherwise 0. Denote the dyadic pieces of intervals by
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU4.png?pub-status=live)
and the corresponding strips by $S_i=I_i\times \mathbb{R}$,
$\tilde{S}_i=\tilde{I}_i\times \mathbb{R}$.
2. Reduction
In this section, we present three propositions that have broad applicability. Let $\Gamma:\mathbb{R}^2\rightarrow \mathbb{R}$ be a measurable function and define a general class of operators
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU5.png?pub-status=live)
Proposition 2.1. Define $T_j^{\text{glo}}f(x_1,x_2):=\psi_{j+4}^c(x_1)T_jf(x_1,x_2)$. Under the measurability assumption of Γ, we have
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU6.png?pub-status=live)
for $1 \lt p\leqslant \infty$.
Proof. Denote that $\tilde{\varphi}(\frac{x}{2^j})=\sum_{k=-3}^{4}\varphi(\frac{x}{2^{j+k}})$, which has a localized support
$|x|\sim 2^j$. Let
$T_j^{\text{loc}}$ and
$T_j^{\text{mid}}$ be operator, defined by
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU7.png?pub-status=live)
Then, we can decompose $T_j-T_j^{\text{glo}}$ into
$T_j^{\text{mid}}+T_j^{\text{loc}}$. For the operator
$T_j^{\text{mid}}$, replace the sup as
$\ell^p$ sum. Then, we have
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU8.png?pub-status=live)
Denote $F(x_1)=\|f(x_1,\cdot)\|_{L^p(dx_2)}$. By applying Minkowski’s integral inequality and a change of variables, we get the pointwise inequality:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn2.png?pub-status=live)
where the second inequality follows form the fact that $\Gamma(x_1,t)$ is independent of x 2. By (2.1) and the Lp boundedness of M HL, we obtain
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU9.png?pub-status=live)
which implies the Lp boundedness of $f\mapsto \sup_{j}|T_j^{\text{mid}}f|$ for p > 1. For the operator
$T_j^{\text{loc}}f$, we observe the localization principle:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU10.png?pub-status=live)
By combining this with $\sup_{j\in \mathbb{Z}}\|T_j\|_p \leqslant C$, we get the following estimate:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU11.png?pub-status=live)
Therefore, we prove $\|\sup_{j}|T_j-T_j^{\text{glo}}|\|_p\leqslant C_p$ for
$1 \lt p\leqslant \infty$.
By Proposition 2.1, in order to prove Theorem 1, it suffices to consider the maximal operator defined as
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU12.png?pub-status=live)
Proposition 2.2 (Space Reduction)
Let ${T}_{j}^{\ell}f(x_1,x_2):=\chi_{S_{\ell}}(x_1,x_2){T}_{j}^{\text{glo}}f(x_1,x_2)$. Then, the following inequality holds:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn3.png?pub-status=live)
Proof. One can obtain (2.2) from the localization $T^{\ell}_jf(x_1,x_2)=T_j^{\ell}(\chi_{\tilde{{S_{\ell}}}}f)(x_1,x_2).$
Combining Proposition 2.1 and Proposition 2.2, we may restrict our attention to the maximal operator defined by $
f\mapsto \sup_j|T_j^{\ell}|$, supported on
$|x_1|\sim 2^{\ell}\gg2^j$.
Proposition 2.3 (Frequency Reduction)
Suppose $\Gamma:\mathbb{R}\times [0,\infty)\rightarrow \mathbb{R}$ is measurable on
$\mathbb{R}^2$ with
$\Gamma(x_1,0)=0$ satisfying the following conditions:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU13.png?pub-status=live)
Let $\widehat{\mathcal{L}_j^{\text{low} }f}(\xi_1,\xi_2):=\hat{f}(\xi_1,\xi_2)\psi(2^j\xi_1)$ for
$f\in \mathcal{S}(\mathbb{R}^2)$. Then, there exists a constant C independent of Γ such that
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU14.png?pub-status=live)
where M i is the Hardy–Littlewood maximal operator taken in the ith variable.
Proof. For $g\in \mathcal{S}(\mathbb{R}^1)$ and
$2^{j-1}\leqslant|t|\leqslant 2^{j+1}$, we have
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU15.png?pub-status=live)
where the second inequality follows form the convexity of $t\mapsto \Gamma(x_1,t)$. For more details, we refer to Lemma 2 in [Reference Cho, Hong, Kim and Woo Yang12] and [Reference Córdoba and Rubio de Francia6]. Since
$T_j(\mathcal{L}_j^{\text{low}}f)(x_1,x_2)$ is a composition of the above two functions, we obtain the desired pointwise inequality.
Set $\widehat{\mathcal{L}_j^{\text{high}}f}(\xi_1,\xi_2)= \hat{f}(\xi_1,\xi_2)\psi^c(2^j\xi_1)$. Following Proposition 2.3, it is enough to show the estimate
$\|\sup_j|T_j^{\ell}(\mathcal{L}_j^{\text{high}}f)|\|_p\lesssim \|f\|_p$.
3. Proof of main theorem 1
Following the reduction section, we only consider $\mathcal{T}_{j}^{\ell}(\mathcal{L}_j^{\text{high}}f)$, which is given by
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU16.png?pub-status=live)
supported on $|x_1|\sim 2^{\ell}\gg 2^j$.
3.1. Main difficulty
In a view of pseudo-differential operator, we write
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU17.png?pub-status=live)
with the symbol $b_j(x_1,\xi_1,\xi_2)$ given by
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU18.png?pub-status=live)
When analysing an oscillatory integral with a phase $t\xi_1+m(x_1)\gamma(t)\xi_2$, it is usual to decompose each frequency variable ξ 1 and ξ 2 with dyadic scale. Specifically, in the case of a homogeneous curve, we can even estimate the asymptotic behaviour of oscillatory integral. However, under the flat condition (1.1), this usual approach does not work, as there are no comparablity condition
$\left|\frac{\gamma'(2t)}{\gamma'(t)}\right|\sim 1$ and a finite type assumption for the curve. To overcome this situation, we will perform an angular decomposition in [Reference Carlsson, Christ, Córdoba, Duoandikoetxea, Rubio de Francia, Vance, Wainger and Weinberg11] for a function f and utilize the method in one of the author’s paper [Reference Kim15].
3.2. Angular decomposition
Set
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU19.png?pub-status=live)
and
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU20.png?pub-status=live)
Note that we have the following Littlewood–Paley estimate in [Reference Carlsson, Christ, Córdoba, Duoandikoetxea, Rubio de Francia, Vance, Wainger and Weinberg11]:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU21.png?pub-status=live)
We have $\mathcal{A}_{j}\mathcal{L}_{j}^{\text{high}}f(x)=\mathcal{A}_{j}f(x)-
\mathcal{L}_{j}^{\text{low}}\mathcal{A}_{j}f(x)$. Then, it gives
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU22.png?pub-status=live)
from the pointwise estimate $|\mathcal{L}_{j}^{\text{low}}f(x_1,x_2)|\lesssim M^1f(x_1,x_2)$. By the vector valued estimate for Hardy–Littlewood maximal operator, the following estimate holds:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn4.png?pub-status=live)
We split $\mathcal{T}^{\ell}_{j}(\mathcal{L}_j^{\text{high}}f)$ into two terms:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU23.png?pub-status=live)
Then, we shall prove the following:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn5.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn6.png?pub-status=live)
We can obtain the estimate (3.2) for p = 2 from the following process:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn7.png?pub-status=live)
Furthermore, the range of p can be extended by a bootstrap argument detailed in Section 3.4. In the following proposition, we focus particularly on the term $\mathcal{T}^{\ell}_{j}(\mathcal{A}_j^c\mathcal{L}_{j}^{\text{high}}f)$ and prove the estimate (3.3). Furthermore, the range of p can be extended by a bootstrap argument detailed in Section 3.4. In the following proposition, we focus particularly on the term
$\mathcal{T}^{\ell}_{j}(\mathcal{A}_j^c\mathcal{L}_{j}^{\text{high}}f)$ and prove the estimate (3.3).
Proposition 3.1. Define the Littlewood–Paley projection $\widehat{\mathcal{L}_jf}(\xi_1,\xi_2):=\hat{f}(\xi_1,\xi_2)$
$\varphi(\frac{\xi_1}{2^j})$ so that
$\mathcal{T}^{\ell}_{j}(\mathcal{A}_j^c\mathcal{L}_{j}^{\text{high}}f)=\sum_{n=0}^{\infty}\mathcal{T}^{\ell}_{j}(\mathcal{A}_j^c\mathcal{L}_{n-j}f)$. For
$f\in L^p(\mathbb{R}^2)$, It holds that
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn8.png?pub-status=live)
for $1 \lt p \lt \infty$ and
$n\geqslant0$.
Note that we need the following:
Lemma 3.1 (Reduction to one variable operator)
Consider the two operators $\mathcal{R}_1$ and
$\mathcal{R}^{\lambda}_2$, given by
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU24.png?pub-status=live)
for $f\in \mathcal{S}(\mathbb{R}^2)$ and
$g\in \mathcal{S}(\mathbb{R})$. Then,
$\|\mathcal{R}_1\|_{L^2(\mathbb{R}^2)\rightarrow L^2(\mathbb{R}^2)}\leqslant\sup_{\lambda\in \mathbb{R}}\|\mathcal{R}^{\lambda}_2\|_{L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})}$.
Proof of Lemma 3.1
Consider a function $f\in \mathcal{S}(\mathbb{R}^2)$ with
$\|f\|_{L^2(\mathbb{R}^2)}=1$. Denote
$\mathcal{F}_2f(x_1,\xi_2)=g_{\xi_2}(x_1)$. By Plancheral’s theorem with respect to x 2, we get
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU25.png?pub-status=live)
which yields the desired estimate.
3.3. Proof of Proposition 3.1
We shall prove $\|\mathcal{T}^{\ell}_j\mathcal{L}_{n-j}\mathcal{A}_j^c\|_{L^2(\mathbb{R}^2)\rightarrow L^2(\mathbb{R}^2)}\lesssim 2^{-\frac{n}{2}}$, which implies
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU26.png?pub-status=live)
We write $\mathcal{T}^{\ell}_j\mathcal{L}_{n-j}\mathcal{A}_j^cf$ as
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU27.png?pub-status=live)
with symbol $a_j(x_1,\xi_1,\xi_2)$ given by
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU28.png?pub-status=live)
By Lemma 3.1, to prove (3.5), it suffices to show
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU29.png?pub-status=live)
where c 1 and c 2 are constants independent of j and λ and $\mathcal{R}^{\lambda}_jg(x):=\int \text{e}^{2\pi i x\xi}a_j(x,\xi,\lambda)\hat{g}(\xi)\text{d}\xi$ for
$g\in \mathcal{S}(\mathbb{R})$. Note that
$x\in \mathbb{R}$ and
$\xi\in \mathbb{R}$. Hereafter, we omit j and λ in operators for simplicity. Observe that we write
$\mathcal{R}$ with kernel K
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU30.png?pub-status=live)
where
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU31.png?pub-status=live)
Recall that $|x|\sim 2^{\ell}\gg 2^j$ and denote
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU32.png?pub-status=live)
for each integer k. We define the functions
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU33.png?pub-status=live)
and use them to split the operator $\mathcal{R}$ as
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU34.png?pub-status=live)
Then, we shall prove the following:
Lemma 3.2. There exist constants C 1 and C 2 independent of $j, \ell$ and λ such that
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn9.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn10.png?pub-status=live)
Proof of (3.6)
Recall that
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU35.png?pub-status=live)
We build our proof upon the following observation:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn11.png?pub-status=live)
Proof of (3.8)
Note that $supp(\psi^c)\subset \left\{|x| \gt \frac{1}{2}\right\}$. We utilize the integration by parts twice with respect to ξ. Then, we get
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU36.png?pub-status=live)
Since $|x-2^jt-y| \gt rsim |x-y|$ on
$x\in Q_k$,
$y\in \mathbb{R}\setminus Q_k'$ for
$\frac{1}{2}\leqslant t\leqslant 2$, we get the desired estimate.
We shall deduce the following estimate:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn12.png?pub-status=live)
Proof of (3.9)
By estimate (3.8) and the disjointness of Qks, we have
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU37.png?pub-status=live)
and the second estimate also holds by the similar way.
By Schur’s lemma with the estimate (3.9), we finish the proof of (3.6).
Proof of (3.7)
For the operator $\mathcal{B}_k$, denote
$g_k(y)=\chi_{Q_k'}(y)g(y)$. By the localization principle, we have
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn13.png?pub-status=live)
To estimate $\|\mathcal{B}_kg_k\|_2$, we write it with the symbol expression again, which is
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU38.png?pub-status=live)
where
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU39.png?pub-status=live)
Observe that
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn14.png?pub-status=live)
Proof of (3.11)
From the support of $A^c_j(\xi,\lambda)$, we have
$|\frac{\xi}{\lambda}|\nsim|\gamma'(2^jt)|$ for
$|t|\sim1$. This enables us to apply the integration by parts with respect to variable t. Then, we get
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU40.png?pub-status=live)
Then, we get the desired estimate.
From the observation (3.11), it is easy to check
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU41.png?pub-status=live)
By Schur’s lemma with the above estimate and (3.10), we obtain (3.7) in Lemma 3.2.
3.4. A bootstrap argument for the proof of Theorem 1
In the spirit of Nagel, Stein and Wainger [Reference Nagel, Stein and Wainger22], we claim that
Lemma 3.3. If $\|\sup_{j}|\mathcal{T}_j^{\ell} f|\|_{L^p(\mathbb{R}^2)}\leqslant C_1\|f\|_{L^p(\mathbb{R}^2)}$ and
$\|\mathcal{T}_j^{\ell} f\|_{L^r(\mathbb{R}^2)}\leqslant C_2\|f\|_{L^r(\mathbb{R}^2)}$ for
$1 \lt r \lt \infty$,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn15.png?pub-status=live)
holds for all q with $\frac{1}{q} \lt \frac{1}{2}(1+\frac{1}{p})$.
Proof. Consider vector valued functions $\mathfrak{f}=\{f_j\}$ and
$\mathfrak{Tf}=\{\mathcal{T}_j^{\ell}f_j\}$. Since the operator
$\mathcal{A}_j$ is a positive, it follows that
$\|\mathfrak{Tf}\|_{L^p({\mathbb{R}}^2,l^\infty)}\lesssim\|\mathfrak{f}\|_{L^p({\mathbb{R}}^2,l^\infty)}$ and
$\|\mathfrak{Tf}\|_{L^r({\mathbb{R}}^2,l^r)}$
$\lesssim \|\mathfrak{f}\|_{L^r({\mathbb{R}}^2,l^r)}$ for r near 1. Applying the Riesz–Thorin interpolation for vector-valued function, we get the conclusion.
Combining (3.4), Proposition 2.3 and Proposition 3.1, we obtain the estimate
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn16.png?pub-status=live)
for p = 2. Moreover, we have
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn17.png?pub-status=live)
for r > 1. By using Lemma 3.3 with (3.13) and (3.14), we obtain (3.12) for $\frac{4}{3} \lt p\leqslant 2$. Then, by setting
$\{f_j\}_{j\in \mathbb{Z}}=\{\mathcal{A}_j^c\mathcal{L}_{n-j}f\}_{j\in \mathbb{Z}}$ in (3.12) and applying interpolation with the decay estimate (3.5), we obtain Proposition 3.1 for
$\frac{4}{3} \lt p\leqslant 2$. To treat the bad part in (3.4), set
$\{f_j\}_{j\in \mathbb{Z}}=\{\mathcal{A}_j\mathcal{L}_j^{\text{high}}f\}_{j\in \mathbb{Z}}$. Then, we apply Lemma 3.3 again to get the first inequality of (3.4), which implies (3.13) for
$\frac{4}{3} \lt p\leqslant 2$. We can iteratively apply Lemma 3.3 with a wider range of p until we get (3.13) for all p > 1. With this, we complete the proof of Main Theorem 1.
4. Application
In this section, we shall prove Corollary 1.1 and Corollary 1.2.
4.1. Proof of Corollary 1.1
For a measurable function $m:\mathbb{R}\rightarrow\mathbb{R}$, denote that
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU42.png?pub-status=live)
By Main Theorem 1 and the second part of Remark 1.1, one can easily check that
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn18.png?pub-status=live)
To prove Corollary 1.1, it suffices to show that for each α > 0 and $k\in \mathbb{Z}$, the set
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU43.png?pub-status=live)
has measure zero. Consider a continuous function gɛ of compact support with $\|f-g_{\varepsilon}\|_p \lt ~\varepsilon$. One can see that
$\limsup_{r\rightarrow0}|S_r^mf(x_1,x_2)-f(x_1,x_2)|\leqslant \mathcal{M}_{\gamma}^{m}(f-g_{\varepsilon})(x)+|g_{\varepsilon}(x)-f(x)|.$ For
$F_{\alpha}^k$ and
$G_{\alpha}^k$, defined by
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU44.png?pub-status=live)
we have $m(E_{\alpha}^k)\leqslant m(F_{\alpha}^k)+m(G_{\alpha}^k)$. Applying estimate (4.1), we get
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU45.png?pub-status=live)
As $\varepsilon\rightarrow 0$, we get the conclusion.
4.2. Proof of Corollary 1.2
In order to achieve our goal of removing the dependence of the coefficients of polynomial P on factors other than its degree, we consider the following lemma.
Lemma 4.1. Given a polynomial P with degree d, we can find a partition $\{s_0,s_1,s_2,\dots,s_{n(d)}\}$ such that for each interval
$[s_i,s_{i+1}]$, there exists a pair
$(m_i,s_{j_i})$ with
$1\leqslant m_i\leqslant d$, satisfying
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn19.png?pub-status=live)
Proof of Lemma 4.1
We seek to construct a partition $\mathcal{P} = \{s_1, s_2, ..., s_{n(d)}\}$ of
$(-\infty,\infty)$ such that, for each subinterval
$[s_i, s_{i+1}]$, there exist non-negative integers mi and ji satisfying (4.2). Consider a polynomial P(x) represented by the following expression:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU46.png?pub-status=live)
where αi are distinct real numbers. Let $U_i=\{x\in\mathbb{R}: |x-\alpha_i| \lt |x-\alpha_k| \text{ for all } k=1,\dots,d_1\}$. For each i and k, let
$\mathcal{U}_i^{k}(1)=\{x\in U_i:2|x-\alpha_i|\geqslant |x-\alpha_k| \}$ and
$\mathcal{U}_i^{k}(0)=\{x\in U_i:2|x-\alpha_i| \lt |x-\alpha_k| \}$. Then, for any
$x\in \mathbb{R}$, there exists an index i such that
$x\in U_i$. We define the set-valued function Fi on
$\{0,1\}^{d_1}$ by
$F_i(a)=\bigcap_{k=1}^{d_1}\mathcal{U}_{i}^k(a_k)$ for
$a=(a_k)\in\{0,1\}^{d_1}$. By using the set-valued function F, we can decompose each set Ui into a finite number of disjoint open intervals, that is,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU47.png?pub-status=live)
For each interval $F_i(a)=[s_i,s_{i+1}]$, we take
$m=\sum_{\{k:a_k=1\}}q_k$ and
$s_{j_i}=\alpha_i$. Observe that we have the following inequalities for each fixed i:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU48.png?pub-status=live)
By using these observation, we have (4.2) on $[s_i,s_{i+1}]$.
To handle a general polynomial, we can employ a similar approach. First, we can express the polynomial as
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU49.png?pub-status=live)
To treat this, we give one more criterion comparing between $2|x-\alpha_i|$ and
$\max\{|x-\beta_k|,|\delta_k|\}$ instead of
$|x-\alpha_k|$. Then. the last part can be proved similarly.
Proof of the Corollory 1.2
Given a polynomial P(x), we obtain a partition $\mathcal{P}=\{s_0,s_1,\dots,s_{n(d)}\}$ from Lemma 4.1. We then decompose
$\mathcal{M}_{\gamma}^{P}f(x)$ as
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU50.png?pub-status=live)
where $\mathcal{M}_{i}f(x):=\chi_{[s_i,s_{i+1}]}(x)\mathcal{M}_{\gamma}^{P}f(x)$. To complete the proof, it suffices to demonstrate that
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU51.png?pub-status=live)
By Lemma 4.1, there exists a pair $(m_i,s)$ such that the following holds for
$[s_{i},s_{i+1}]$:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU52.png?pub-status=live)
Denote that $g_s(x_1,x_2):=f(x_1+s ,x_2)$ and consider the estimate
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU53.png?pub-status=live)
By applying Proposition 2.2, we can reduce matters to $|x_1|\sim 2^{\ell}$:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqn20.png?pub-status=live)
where $\mathcal{P}_{j}^{\ell}g_s(x)$ is defined as
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU54.png?pub-status=live)
for $\ell$ such that
$[2^{\ell-1},2^{\ell+1}]\cap [s_i-s,s_{i+1}-s]\neq \emptyset$. To prove (4.3), it is enough to check the hypothesis of Remark 1.1:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241202182032052-0997:S0013091524000555:S0013091524000555_eqnU55.png?pub-status=live)
where $1\leqslant m_i\leqslant d$. This implies the conclusion.
Acknowledgements
J. Kim was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea under grant NRF-2015R1A2A2A01004568. J. Oh was supported by the National Research Foundation of Korea under grant NRF-2020R1F1A1A01048520 and is currently supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (RS-2024-00461749).