A unified geometric perspective on Zygmund’s conjecture for maximal functions associated with vector fields
May 06, 2026
Abstract
The Zygmund vector field maximal function conjecture is a long-standing open problem. This paper establishes a new boundedness criterion that significantly weakens the existing conditions in the literature. Specifically, the required decay condition is relaxed from the power-type decay of Bourgain for Zygmund’s conjecture and the exponential-logarithmic decay of Lacey and Li for Stein’s conjecture, to a logarithmic polynomial decay. Unlike the traditional framework that separates finite-type and non-finite-type operators, this paper offers a unified geometric view of both settings. The new criterion forms a natural continuation of a long-standing research line in harmonic analysis: it situates several pivotal conditions from earlier foundational works within a single developmental trajectory. Additionally, motivated by Lacey and Li’s work, a non-centered rectangular maximal operator tailored to the underlying geometry is shown to satisfy weak \((1,1)\) boundedness. This operator serves as a novel tool for the subsequent study of the Zygmund and Stein conjectures.
1 Introduction↩︎
The study of maximal and singular integral operators along lower-dimensional manifolds is a central theme in modern harmonic analysis, with connections to complex analysis, partial differential equations, and ergodic theory. In particular, Zygmund’s conjecture addresses maximal functions associated with vector fields.
This paper focuses on advancing Zygmund’s conjecture. Let \(\Omega\subset \mathbb{R}^2\) be an open bounded set, and let \(v\) be a \(C^1\) vector field defined on a neighborhood of \(\overline{\Omega}\). For a sufficiently small parameter \(\epsilon_0>0\), we study \[M_vf(x) = \sup_{\epsilon< \epsilon_0} \left|A_\epsilon f(x)\right| = \sup_{\epsilon<\epsilon_0} \left|\frac{1}{\epsilon}\int_{-\epsilon}^\epsilon f(x+tv(x))\,dt \right| \cdot \chi_\Omega(x),\] where \(\chi_\Omega\) denotes the characteristic function supported on \(\Omega\). By refining and extending Bourgain’s classical argument ([1]) for maximal functions associated with planar vector fields, this paper establishes a new boundedness criterion that significantly weakens the existing conditions in the literature. As a consequence, our result strengthens Remark 3.35 in Lacey and Li ([2]), where a comparison is made between their condition and that arising from Bourgain’s argument. We note that Lacey and Li [2] consider Hilbert transforms associated with planar vector fields, a related but more delicate operator.
Since \(\overline{\Omega}\) is compact and \(v\) is \(C^1\) on a neighborhood of \(\overline{\Omega}\), we may fix a constant \(B > 0\) such that \(\|v\|_{C^1} \le B\) on this neighborhood. Throughout this paper, we choose the parameter \(\epsilon_0 > 0\) sufficiently small depending only on \(\Omega\) and \(B\). This choice ensures both that the line segments \(\left\{x + tv(x) : |t| \le \epsilon_0\right\}\) remain within the domain of \(v\) for all \(x \in \Omega\), and that \(\epsilon_0\) satisfies the necessary smallness conditions.
For \(x \in \Omega\) and \(t \in [-\epsilon_0, \epsilon_0]\), we define the function \[w_x(t) = \left|\det \left[ v(x+tv(x)), v(x)\right]\right|.\] The following is Bourgain’s original theorem:
Theorem 1 ([1]). If \(v\) is \(C^1\) and satisfies \[\label{bourgain} \left|\left\{ t\in [-\epsilon, \epsilon]: w_x(t) <\tau \sup_{[-\epsilon, \epsilon]} w_x(t)\right\}\right| \leq C\tau^{c_0}\epsilon, \quad \forall 0<\tau<1, \forall 0<\epsilon<\epsilon_0, \forall x\in \Omega,\qquad{(1)}\] for some constants \(0<c_0, C<\infty\). Then \(M_v\) is bounded on \(L^2\).
Boundedness can be obtained under a weaker condition in Remark 3.35 in Lacey and Li [2]:
Theorem 2. If \(v\) is \(C^1\) and satisfies \[\label{li} \left|\left\{ t\in [-\epsilon, \epsilon]: w_x(t) <\tau \sup_{[-\epsilon, \epsilon]} w_x(t)\right\}\right| \leq C e^{-\sigma(\log \frac{1}{\tau})^{c_1}} \epsilon, \quad \forall 0<\tau<1, \forall 0<\epsilon<\epsilon_0, \forall x\in \Omega,\qquad{(2)}\] for some constants \(0<c_1, \sigma, C<\infty\). (This is of interest when \(c_1<1\).) Then \(M_v\) is bounded on \(L^2\).
The following is the main result, which requires an even weaker condition:
Theorem 3. If \(v\) is \(C^1\) and satisfies \[\label{zhang} \left|\left\{ t\in [-\epsilon, \epsilon]: w_x(t) <\tau \sup_{[-\epsilon, \epsilon]} w_x(t)\right\}\right| \leq C \left(\log \frac{1}{\tau}\right)^{-p}\epsilon, \quad \forall 0<\tau<1, \forall 0<\epsilon<\epsilon_0, \forall x\in \Omega,\qquad{(3)}\] for some constants \(0<C<\infty\) and \(1<p<\infty\). Then \(M_v\) is bounded on \(L^2\).
Remark 4. By a straightforward modification of the proof, the decay rate \(\left(\log \frac{1}{\tau}\right)^{-p}\) in ?? can be replaced by iterated logarithmic variants, such as \(\left(\log \frac{1}{\tau}\right) \left(\log \log \frac{1}{\tau}\right)^{-p}\) or \(\left(\log \frac{1}{\tau}\right) \left(\log \log \frac{1}{\tau}\right) \left(\log \log \log \frac{1}{\tau}\right)^{-p}\). We do not pursue these routine refinements here.
Unlike traditional frameworks that treat finite-type and non-finite-type operators separately, Section 1.1 presents a unified geometric perspective, situating several pivotal conditions from earlier foundational works within a single developmental trajectory. In Section 2, we modify Bourgain’s arguments to prove Theorems 2 and 3. We retain his unified technical framework for maximal functions along vector fields while substantially weakening the decay condition. In Section 3, motivated by Lacey and Li’s work, and to offer a novel tool for studying the Zygmund and Stein conjectures, a non-centered rectangular maximal operator tailored to the underlying geometry is introduced and shown to satisfy weak \((1,1)\) boundedness.
1.1 A unified geometric view for boundedness criteria↩︎
The foundational insight of Stein and Wainger in late 1970s explicitly introduced several central \(L^p\) boundedness problems concerning the role of curvature. In this direction, Christ, Nagel, Stein, and Wainger [3] showed that \(L^p\) boundedness of singular and maximal Radon transforms follows from a finite type condition: Hörmander’s condition. In particular, Hörmander’s condition is shown to be equivalent to a Jacobian nondegeneracy condition, which in turn implies the integrability condition \[\label{J} \int_B |J(\tau)|^{-\sigma}\,d\tau <\infty\tag{1}\] for sufficiently small \(\sigma>0\), where \(J(\tau)\) denotes an associated Jacobian.
In Bourgain [1], the verification that any analytic \(v\) satisfies condition ?? is carried out via an equivalent condition: there exists \(\sigma>0\) such that \[\label{J39} \frac{1}{\epsilon}\int_{-\epsilon}^\epsilon \left(\frac{w_x(t)}{\sup_{[-\epsilon, \epsilon]}w_x(t)}\right)^{-\sigma}\,dt\lesssim 1,\tag{2}\] uniformly in \(x\) and \(\epsilon\); see (2.3) in [1]. This condition is of a similar form to 1 . The main difference is that no Hörmander condition is assumed, so uniformity in \(\epsilon\) is imposed explicitly across scales, rather than following automatically.
Finite type conditions, which involve curvature, have been extended to various contexts, including discrete operators, multi-parameter settings, and non-translation invariant cases. In the real analytic setting—where a certain finite-type condition holds automatically—Stein, Street, and Zhang [4], [5] obtained a full characterization of bounded multi-parameter singular Radon transforms. In the non-analytic setting, such a finite-type condition does not automatically hold, yet it has long been understood that boundedness does not necessarily require finite-type conditions. For example, the operator \[f\mapsto \int_{-1}^1 f\left(x+t, y+e^{-\frac{1}{|t|}}\,\text{sgn}\,t\right)\,\frac{dt}{t}\] remains \(L^p\)-bounded despite the failure of finite type conditions (the curvature of the underlying curve at the origin vanishes); see Carbery, Christ, Vance, Wainger, and Watson [6]. See also Zhang [7] for a result in the non-translation-invariant non-finite-type setting, of the same strength and form as [6]. The approach naturally degenerates to that of [6] in translation-invariant cases. Moreover, a condition of the form 2 crystallizes within the proof (Proposition 4.12 of [7]). Therefore conditions of the form 2 also govern some non-finite-type settings.
In contrast, the present paper employs weaker conditions ?? and ?? , under which condition 2 no longer holds. For the reader’s convenience, we record here the corresponding integrability conditions to facilitate direct comparison. Condition ?? is equivalent to \[\label{239} \frac{1}{\epsilon}\int_{-\epsilon}^\epsilon \exp\!\left(\sigma\left(-\log \frac{w_x(t)}{\sup_{[-\epsilon, \epsilon]}w_x(t)}\right)^{c_1}\right) \,dt\lesssim 1,\tag{3}\] uniformly in \(x\) and \(\epsilon\), for some \(\sigma>0\). Condition ?? is equivalent to \[\label{339} \frac{1}{\epsilon}\int_{-\epsilon}^\epsilon \left(-\log \frac{w_x(t)}{\sup_{[-\epsilon, \epsilon]}w_x(t)}\right)^q \,dt \lesssim 1,\tag{4}\] uniformly in \(x\) and \(\epsilon\), for some \(1<q<\infty\). In light of the relationships among 2 , 3 , and 4 , the conditions introduced in this paper may be viewed as a natural continuation of a long-standing line of inquiry in the subject.
The hierarchy of decay conditions established above provides a glimpse into the underlying geometric structure, where \(w_x(t)\) quantifies the angular variation of \(v\). Crucially, less rigidity is now imposed on this angular variation. For instance, the modulus of \(w_x(t)\) can fluctuate between being of order \(\sim 1\) and vanishing as rapidly as \(e^{-1/|t|^\gamma}\) for \(0 < \gamma < 1\).
To delve deeper into the underlying geometry, it is instructive to see its direct influence on operator boundedness and examine how it functions within the proof. For simplicity, consider the \(\mathbb{R}^2\) case. With the Lebesgue measure, the role of a suitable metric geometry in securing boundedness is fundamental: in finite-type settings, a standard homogeneous—though not necessarily isotropic—metric suffices (Figure 1). In non-finite-type settings, however, such as those in [6], [7], a dynamic “rotation” behavior is required to counteract the highly singular nature of the operator (Figure 2). Both [1] and the present paper implicitly incorporate this rotational behavior; specifically, \(\sup_{[-\epsilon, \epsilon]} w_x(t)\) is derived from the smallest rectangle containing all the parallelograms spanned by the fixed vector \(\epsilon v(x)\) and the rotating vector \(\epsilon v(x+t v(x))\) with \(t\) varying in \([-\epsilon, \epsilon]\) (Figure 3). With the Lebesgue measure and the metrics from [1], [6], [7], the spaces are non-homogeneous; however, the analytical challenges of this setting were successfully navigated in [6], [7], suggesting that \(L^p\) boundedness for the operators in [1] follow from analogous non-homogeneous Calderón–Zygmund and Littlewood–Paley theories.
![Figure 3: This figure displays Bourgain’s rectangle R_{x,\epsilon} of center x, length 2\epsilon|v(x)|, and width 2 \epsilon \sup_{[-\epsilon, \epsilon]} w_x(t)/|v(x)|.](/article/2605.04394/_pdflatex/mpuiozdq.png "fig:")
Such underlying geometry is typically uncovered through distinct technical frameworks: a duality argument combined with Schur’s test ([1]), the \(TT^\ast\) method ([3], [4], [7]), and time-frequency analysis ([2]).
2 Proof of the main result↩︎
In this section, we modify Bourgain’s arguments to prove Theorems 2 and 3. We retain his unified technical framework for maximal functions along vector fields while substantially weakening the decay condition. This extends the theory to encompass finite-type, intermediate degenerate, and highly non-finite-type behaviors within a single framework, removing long-standing geometric limitations inherent in other approaches.
Proof of Theorems 2 and 3. The first instance in which condition ?? is used in [1] appears in equation (3.20), where it is shown that there exists \(0<C<\infty\), \[\label{doubling} \sup_{[-2\epsilon, 2\epsilon]} w_x(t) \leq C \sup_{[-\epsilon, \epsilon]} w_x(t), \quad \forall x\in \Omega, \forall 0<\epsilon<\epsilon_0.\tag{5}\] If we assume instead condition ?? , then since there exists \(0<\tau_0<1\) (independent of \(x\) and \(\epsilon\)) such that \[C \left(\log \frac{1}{\tau_0}\right)^{-p} <1,\] we have \[\begin{align} \left|\left\{t\in [-2\epsilon, 2\epsilon]: w_x(t) < \tau_0 \sup_{[-2\epsilon, 2\epsilon]} w_x(t)\right\}\right| < 2\epsilon, \quad \forall x\in \Omega, \forall 0<\epsilon<\epsilon_0. \end{align}\] Hence \[\begin{align} \sup_{[-\epsilon, \epsilon]} w_x(t) \geq \tau_0 \sup_{[-2\epsilon, 2\epsilon]} w_x(t). \end{align}\] Therefore 5 is obtained.
If we assume instead condition ?? , 5 follows similarly.
Lemma 3.21 of [1] and its arguments remain the same under conditions ?? and ?? , respectively. The second place that requires modification is the last line of equation (3.27) in [1]: \[\begin{align} &\quad \int_{-\epsilon}^\epsilon \left( 1+\epsilon T |v(x_0)|^{-1}w_x(t)\right)^{-2}\,dt\\ &\leq \left|\left\{ t\in [-\epsilon, \epsilon]: w_x(t)<\tau \sup_{[-\epsilon, \epsilon]} w_x(t)\right\}\right| + \epsilon \left( \epsilon T |v(x_0)|^{-1} \tau \sup_{[-\epsilon, \epsilon]} w_x(t)\right)^{-2}\\ &\leq C\tau^{c_0} \epsilon + C\epsilon (\tau T\delta)^{-2}, \quad \forall 0<\tau<1. \end{align}\] Recall in [1], Bourgain constructs the rectangle \(R = R_{x,\epsilon}\) centered at \(x\) with scale \(\epsilon\), as a rectangle with length \(2\epsilon|v(x)|\) along the direction \(v(x)\) and with width \(\delta = \delta(R_{x,\epsilon}):= \epsilon \sup_{[-\epsilon, \epsilon]} w_x(t)/|v(x)|\). The following estimates 6 –9 are implemented for \(f\) whose Fourier transform is supported on \(B(0,2T)\backslash B(0,T)\), and the parameter \(T\) in 6 –9 satisfies \(T\delta >1\).
If we assume instead condition ?? , we have \[\label{23939} \begin{align} &\quad \int_{-\epsilon}^\epsilon \left( 1+\epsilon T |v(x_0)|^{-1}w_x(t)\right)^{-2}\,dt\\ &\leq \left|\left\{ t\in [-\epsilon, \epsilon]: w_x(t)<\tau \sup_{[-\epsilon, \epsilon]} w_x(t)\right\}\right| + \epsilon \left( \epsilon T |v(x_0)|^{-1} \tau \sup_{[-\epsilon, \epsilon]} w_x(t)\right)^{-2}\\ &\leq C e^{-\sigma \left(\log \frac{1}{\tau}\right)^{c_1}} \epsilon + C\epsilon (\tau T\delta)^{-2}, \quad \forall 0<\tau<1. \end{align}\tag{6}\] If we assume instead condition ?? , we have \[\label{33939} \begin{align} &\quad \int_{-\epsilon}^\epsilon \left( 1+\epsilon T |v(x_0)|^{-1}w_x(t)\right)^{-2}\,dt\\ &\leq \left|\left\{ t\in [-\epsilon, \epsilon]: w_x(t)<\tau \sup_{[-\epsilon, \epsilon]} w_x(t)\right\}\right| + \epsilon \left( \epsilon T |v(x_0)|^{-1} \tau \sup_{[-\epsilon, \epsilon]} w_x(t)\right)^{-2}\\ &\leq C\left(\log \frac{1}{\tau}\right)^{-p} \epsilon + C\epsilon (\tau T\delta)^{-2}, \quad \forall 0<\tau<1. \end{align}\tag{7}\]
We therefore need corresponding modifications for (3.29) of [1]: \[\begin{align} \left\|A_\epsilon f\big|_R\right\|_2 &\lesssim \left( \frac{1}{T\delta}+\frac{1}{\epsilon}\int_{-\epsilon}^\epsilon \left( 1+\epsilon T |v(x_0)|^{-1}w_x(t)\right)^{-2}\,dt\right) \left\|f\left(\chi_{R'}\ast \psi_{\frac{1}{T}} \right)\right\|_2 \\ &\lesssim (T\delta)^{-c} \left\|f\left(\chi_{R'}\ast \psi_{\frac{1}{T}} \right)\right\|_2, \quad \text{for some } c>0, \end{align}\] where \(\psi\) is a fixed Schwarz function whose Fourier transform is supported in the unit disk, \(\psi_{\frac{1}{T}} (x)= T^2 \psi(Tx)\), and \(R'\) denotes the doubled rectangle of \(R\). Recall condition ?? is of interest only when \(c_1 <1\). By taking \(0<\tau<1\) in 6 to be \(\tau_1\) such that \[e^{-\sigma \left(\log \frac{1}{\tau_1}\right)^{c_1}} = (\tau_1 T\delta)^{-2},\] we have \[\left(2\log \, (T\delta)\right)^{c_1} = \left(2\log \, \frac{1}{\tau_1} + \sigma \left(\log \frac{1}{\tau_1}\right)^{c_1}\right)^{c_1} \leq \left(2\log \, \frac{1}{\tau_1}\right)^{c_1} + \left(\sigma \left(\log \frac{1}{\tau_1}\right)^{c_1}\right)^{c_1} \lesssim \left(\log \, \frac{1}{\tau_1}\right)^{c_1},\] and thus \[\label{11} \begin{align} \big\|A_\epsilon f\big|_R\big\|_2 &\lesssim \left( \frac{1}{T\delta} + e^{-\sigma \left(\log \frac{1}{\tau_1}\right)^{c_1}}\right) \left\|f\left(\chi_{R'}\ast \psi_{\frac{1}{T}} \right)\right\|_2 \\ &\lesssim e^{-\sigma' \left(\log T\delta \right)^{c_1}}\left\|f\left(\chi_{R'}\ast \psi_{\frac{1}{T}} \right)\right\|_2, \end{align}\tag{8}\] for some \(\sigma'>0\).
By taking \(0<\tau<1\) in 7 to be \(\tau_2\) such that \[\left(\log \frac{1}{\tau_2} \right)^{-p} = (\tau_2T\delta)^{-2},\] we have for any \(\eta>0\), \[\tau_2^{2+\eta} =\frac{\tau_2^2}{\big(\frac{1}{\tau_2}\big)^\eta} \lesssim \frac{\tau_2^2}{\big(\log \frac{1}{\tau_2}\big)^p} = (T\delta)^{-2},\] and thus, by fixing a sufficiently small \(\eta>0\), \[\label{12} \begin{align} \big\|A_\epsilon f\big|_R\big\|_2 &\lesssim \left( \frac{1}{T\delta} + \left(\log \frac{1}{\tau_2}\right)^{-p}\right) \left\|f\left(\chi_{R'}\ast \psi_{\frac{1}{T}} \right)\right\|_2 \\ &\lesssim \left( \log \left((CT\delta)^{\frac{2}{2+\eta}}\right)\right)^{-p} \left\|f\left(\chi_{R'}\ast \psi_{\frac{1}{T}} \right)\right\|_2 \\ &\lesssim \left(\log CT\delta \right)^{-p} \left\|f\left(\chi_{R'}\ast \psi_{\frac{1}{T}} \right)\right\|_2, \end{align}\tag{9}\] for some \(C>0\).
Section 4 in [1] remains the same.
Now let \(f\) be an arbitrary \(L^2\) function (no longer assume it is supported in an annulus of radius \(\sim T\)). Let \(f= \sum_{T \,\text{dyadic}} f_T\) be a Littlewood-Paley decomposition with \(\text{supp}\, \hat{f}_T \subset B(0,2T)\backslash B(0,T)\). In [1], \[\Omega_{\epsilon,s}:= \left\{x\in \Omega : 2^{-s-1} \leq \epsilon |v(x)|^{-1} \sup_{[-\epsilon, \epsilon]} w_x(t) = \delta(R_{x,\epsilon}) <2^{-s}\right\},\] and \[\Omega_{\epsilon, s}':=\bigcup_{x\in \Omega_{\epsilon,s}} R_{x,\epsilon}',\] where \(R_{x,\epsilon}'\) denotes the doubled rectangle of \(R_{x,\epsilon}\).
The last place that requires modification is the last several equation displays in Page 20 in [1]: cover \(\Omega_{\epsilon,s}\) with rectangles \(R_l\) of length \(\sim \epsilon\) and width \(\delta \sim 2^{-s}\) (\(s\in \mathbb{N}\)), then \[\left\|A_\epsilon\left(f_{2^{s+j}}\right)\chi_{\Omega_{\epsilon,s}} \right\|_2^2 \leq \sum_l \left\|A_\epsilon\left(f_{2^{s+j}}\right)\chi_{R_l}\right\|_2^2 \lesssim 2^{-cj} \sum_l \left\|f_{2^{s+j}} \left(\chi_{R_l'}\ast \psi_{\frac{1}{2^{s+j}}}\right)\right\|_2^2,\] where \(R_l'\) denotes a doubled rectangle of \(R_l\), and thus \[\|(5.4)\|_2^2 \lesssim \sum_j 2^{-cj} \sum_{\epsilon, s} \left\|f_{2^{s+j}} \left(\chi_{\Omega_{\epsilon,s}'} \ast \psi_{2^{-s-j}}\right)\right\|_2^2 \lesssim \sum_j 2^{-cj} \sum_s \|f_{2^{s+j}}\|_2^2 \lesssim \|f\|_2^2,\] where \(0<\epsilon<\epsilon_0\) takes dyadic values.
With 8 , we have instead \[\left\|A_\epsilon\left(f_{2^{s+j}}\right)\chi_{\Omega_{\epsilon,s}} \right\|_2^2 \leq \sum_l \left\|A_\epsilon\left(f_{2^{s+j}}\right)\chi_{R_l}\right\|_2^2 \lesssim e^{-\sigma''j^{c_1}} \sum_l \left\|f_{2^{s+j}} \left(\chi_{R_l'}\ast \psi_{\frac{1}{2^{s+j}}}\right)\right\|_2^2, \text{ for some } \sigma''>0,\] and thus for some \(\sigma'''>0\), \[\|(5.4)\|_2^2 \lesssim \sum_j e^{-\sigma'''j^{c_1}} \sum_{\epsilon, s} \left\|f_{2^{s+j}} \left(\chi_{\Omega_{\epsilon,s}'} \ast \psi_{2^{-s-j}}\right)\right\|_2^2 \lesssim \sum_j e^{-\sigma'''j^{c_1}} \sum_s \|f_{2^{s+j}}\|_2^2 \lesssim \|f\|_2^2.\]
With 9 , we have instead
\[\left\|A_\epsilon\left(f_{2^{s+j}}\right)\chi_{\Omega_{\epsilon,s}} \right\|_2^2 \leq \sum_l \left\|A_\epsilon\left(f_{2^{s+j}}\right)\chi_{R_l}\right\|_2^2 \lesssim \frac{1}{j^{2p}} \sum_l \left\|f_{2^{s+j}} \left(\chi_{R_l'}\ast \psi_{\frac{1}{2^{s+j}}}\right)\right\|_2^2,\] and thus \[\|(5.4)\|_2^2 \lesssim \sum_j e^{-\sigma'''j^{c_1}} \sum_{\epsilon, s} \left\|f_{2^{s+j}} \left(\chi_{\Omega_{\epsilon,s}'} \ast \psi_{2^{-s-j}}\right)\right\|_2^2 \lesssim \sum_j \frac{1}{j^p} \sum_s \|f_{2^{s+j}}\|_2^2 \lesssim \|f\|_2^2. \qedhere\] ◻
3 A non-centered rectangular maximal operator↩︎
In this section, motivated by Lacey and Li’s work, we introduce a non-centered rectangular maximal operator tailored to the underlying geometry and show its weak \((1,1)\) boundedness. This offers a novel tool for the subsequent study of the Zygmund and Stein conjectures.
Instead of the centered maximal function associated with rectangles \(R_{x,\epsilon}\) as in [1], Lacey and Li [2] introduced an uncentered maximal function. The uncentered maximal function is arguably more suitable, as operators associated with vector fields—particularly in the non-analytic setting—do not necessarily satisfy conditions such as ?? , ?? , and ?? , meaning they lack inherent rigidity.
Let \(v\) be a \(C^1\) vector field on \(\mathbb{R}^2\). Assume \(|\nabla v|<B\). For any arbitrary rectangle \(R= I\times J\) in the plane, assuming \(|I|\geq |J|\), denote the length \(L(R)=|I|\) and width \(W(R) =|J|\). Lacey and Li consider the set of rectangles \(R\) with \(L(R)< \frac{1}{100B}\) and \(W(R)<\frac{B}{100}\). Let \[V(R) = \left\{x\in R: \left(\text{the angle between } v(x) \text{ and the long axis of } R\right) < W(R)/(2L(R))\right\}.\] Note \(V(R)\) is an open subset of \(R\). For any \(0<w<\frac{B}{100}\) and \(0<\delta<1\) fixed, Lacey and Li’s maximal functions are defined as \[M_{v,\delta, w} f(x) = \sup_{\substack{|V(R)|\geq \delta |R|\\ w\leq W(R)<2w}} \frac{\chi_R(x)}{|R|} \int_R |f(y)|\,dy.\] They proved that if \(v\) is \(C^{1+\eta}\) for some \(\eta>0\), and if \(M_{v,\delta, w}\) is bounded \(L^p \to L^{p, \infty}\) for some \(1<p<2\) with norm \(\lesssim \delta^{-N}\) for all \(w, \delta\), then the local Hilbert transform along \(v\) is bounded on \(L^2\).
However, such boundedness for \(M_{v,\delta, w}\) has not been established. From our perspective, the geometry of a simply connected rectangular domain may still not be well adapted to the irregular behavior of operators associated with vector fields. In fact, we prove boundedness for the following maximal functions associated with \(V(R)\):
Theorem 5. For any \(0<\theta<\frac{1}{100}\) and \(0<\delta<1\), \[\widetilde{M}_{v,\delta, \theta} f(x) = \sup_{\substack{|V(R)|\geq \delta |R|\\ \theta\leq W(R)/L(R)<2\theta}} \frac{\chi_{V(R)}(x)}{|V(R)|} \int_{V(R)} |f(y)|\,dy\] is of weak type \((1,1)\), and hence bounded on \(L^p\) for \(1<p\leq\infty\); the operator norms are \(\lesssim \delta^{-1}\).
Our goal is not to establish boundedness of the Hilbert transform associated with vector fields, but rather to isolate a structural shift at the level of non-centered rectangular maximal functions. Further extensions of this geometric framework—both for such maximal functions and for other components of the argument in [2]—would likely be required to resolve Stein’s conjecture.
Proof of Theorem 5. Fix arbitrary \(f\in L^1(\mathbb{R}^2)\). For any \(\lambda>0\), consider the open set \(\left\{\widetilde{M}_{v, \delta, \theta}f>\lambda\right\}\). Fix an arbitrary compact subset \(K\). \(K\) is covered by finitely many \(V(R_1), \ldots, V(R_N)\) such that \(|V(R_i)|\geq \delta |R_i|\), \(\theta \leq W(R_i)/L(R_i) <2\theta\), and \[\frac{1}{|V(R_i)|}\int_{V(R_i)} |f| >\lambda.\] Find a maximal disjoint subcollection of them, ordered by the lengths of the rectangles from largest to smallest, denoted by \(V(R_1), \ldots, V(R_k)\). Let \(R_1', \ldots, R_k'\) be the \(10\)-fold dilations of the rectangles \(R_1, \ldots, R_k\), respectively, preserving their respective centers and eccentricities. Then \[K\subseteq \bigcup_{i=1}^k R_i'.\] Indeed, if this were not the case, for a point \(x_0\in K\) not covered by \(R_1', \ldots, R_k'\), we have \(x_0\in V(R_0)\) for some \(R_0\). By how we choose the maximal disjoint collection, \(V(R_0)\) must intersect \(V(R_i)\) for some \(1\leq i \leq k\) with \(L(R_0)\leq L(R_i)\). Choose any point \(z_0\in V(R_0) \cap V(R_i)\). The angle \(\phi_1\) between \(v(z_0)\) and the long axis of \(R_0\) is less than \(W(R_0)/(2L(R_0))\), and the angle \(\phi_2\) between \(v(z_0)\) and the long axis of \(R_i\) is less than \(W(R_i)/(2L(R_i))\). Hence the angle \(\phi_0\) between long axes of \(R_0\) and \(R_i\) satisfies \[\phi_0 \leq \phi_1 + \phi_2 < \frac{W(R_0)}{2L(R_0)} + \frac{W(R_i)}{2L(R_i)}<2\theta \leq 2\frac{W(R_i)}{L(R_i)}.\] Therefore for arbitrary \(z\in R_0, y\in R_i\), the absolute value of the projection of \(y-z\) along the short axis of \(R_i\) is \[\leq W(R_0)\cos \phi_0 + L(R_0) \sin \phi_0 \leq W(R_0)+ L(R_i) \phi_0 \leq 2W(R_i) + 2W(R_i) = 4W(R_i),\] and the absolute value of the projection of \(y-z\) along the long axis of \(R_i\) is \[\leq W(R_0)\sin \phi _0 + L(R_0) \cos \phi_0 \leq L(R_0) + L(R_0) =2L(R_0).\] Thus \(R_0\subseteq R_i'\), and in particular, \(x_0\in R_i'\); contradiction. Hence \(K\subseteq \bigcup R_i'\).
Therefore we have \[\begin{align} |K| \leq \sum_{i=1}^k |R_i'| \leq 100 \sum_{i=1}^k |R_i| \leq \frac{100}{\delta} \sum_{i=1}^k |V(R_i)| \leq \frac{100}{\delta \lambda} \sum_{i=1}^k \int_{V(R_i)} |f| \leq \frac{100}{\delta \lambda} \|f\|_1. \end{align}\] Hence \(\widetilde{M}_{v, \delta, \theta}\) is of weak type \((1,1)\) with norm \(\lesssim \delta^{-1}\). ◻
Lingxiao Zhang
Department of Mathematics
University of Connecticut
341 Mansfield Road, Storrs, CT 06269, USA
Email: lingxiao.zhang@uconn.edu
MSC: Primary 42B25, 42B20