Boundary cases of the Tao–Wright $L^p$-improving characterization for averaging operators along curves

Tao and Wright (J. Amer. Math. Soc. 16 (2003), 605–638) determined the local $L^p$-improving mapping region for averaging operators associated with a smooth double fibration: under the Hörmander bracket condition on the two fiber-tangent vector fields, strong-type $(L^{p_1}, L^{p_2'})$ boundedness holds when a derived exponent pair $(c_1, c_2)$ lies in the interior of the Newton polytope $P$ of the bracket data, and fails outside $P$. The boundary of $P$ — the endpoint exponents — was left open: the proof loses an $\varepsilon$ in the exponents when refining sets of times to central sets of fixed width and requires increasing regularity near the boundary, and the authors write that they 'do not know how to get around this loss'. This problem asks for the completion of the characterization at the boundary: restricted weak-type and strong-type estimates at boundary points of $P$ for general smooth double fibrations, or an explicit smooth counterexample. The endpoint question has been resolved only in special cases — model curves via Fourier-analytic tools, and polynomial-like data with generalized affine-arclength weights by Christ–Dendrinos–Stovall–Street, who explicitly state that the boundary of the polytope was left open by Tao–Wright and answered by them only in special cases — so the general unweighted endpoint case remains open.

Background

An averaging operator, or generalized Radon transform, along a family of curves takes a function on one manifold to its integrals over a smoothly varying family of curves. The prototypical example is convolution with a smooth compactly supported measure on a finite-type curve — a curve at every point of which some finite-order derivative of the parametrization is nonzero, so that the curve deviates from any straight line to finite order — such as the parabola $(t, t^2)$ in $\mathbb{R}^2$; another is the X-ray transform restricted to lines whose directions lie along a curve, which integrates a function over each line of such a family. Tao and Wright studied such operators in the double fibration framework from integral geometry: one fixes a sextuple $(\Sigma, M_1, M_2, \pi_1, \pi_2, 0)$ where $\Sigma$ is an $n$-dimensional smooth manifold, $M_1, M_2$ are $(n-1)$-dimensional, and $\pi_j : \Sigma \to M_j$ are submersions (differentials of maximal rank) mapping $0$ to $0$; the averaging operator $R$ is defined by duality through $\int_\Sigma f_1(\pi_1(x)) f_2(\pi_2(x)) a(x)\,dx$ with $a$ a smooth cutoff near $0$. The $L^p$-improving problem asks for the set of exponent pairs $(p_1, p_2)$ with $1 \le p_1 < p_2' \le \infty$ (where $p'$ is the dual exponent) for which $R$ maps $L^{p_1}(M_1)$ locally into $L^{p_2'}(M_2)$; the regime $p_2' > p_1$ is called improving because the output is more integrable than the input. It is convenient to work with the restricted weak-type variant, in which $f_1, f_2$ are characteristic functions of sets, and Tao–Wright reformulated it geometrically: the sextuple is of restricted weak-type $(p_1, p_2')$ if and only if $|\Omega| \lesssim |\pi_1(\Omega)|^{1/p_1} |\pi_2(\Omega)|^{1/p_2}$ for all sets $\Omega$ near $0$ — a purely geometric comparison between the volume of a set and the volumes of its two projections.

The relevant structure is encoded by two smooth nonvanishing vector fields $X_1, X_2$ tangent to the fibers of $\pi_1, \pi_2$ (defined up to multiplication by nonvanishing scalar functions). The Hörmander condition — that finitely many iterated Lie brackets $X_w$ of $X_1$ and $X_2$, indexed by words $w$ (finite strings in the letters 1 and 2), span the tangent space $T_0 \Sigma$ — is necessary for any nontrivial estimate. The degree of a word is the pair counting its 1s and 2s, and for each $n$-tuple $I$ of words one sets $\lambda_I = \det(X_{w_1}, \dots, X_{w_n})$. The Newton polytope $P$ is the closed convex hull of all degrees $\deg(I)$ with $\lambda_I(0) \neq 0$, closed upward in the coordinatewise order. For a candidate exponent pair define $c_1 = p_2/(p_1 + p_2 - p_1 p_2)$ and $c_2 = p_1/(p_1 + p_2 - p_1 p_2)$. Tao–Wright's main theorem proves their Conjecture 1.9 away from endpoints: strong type $(p_1, p_2')$ holds when $(c_1, c_2)$ lies in the interior of $P$, and when $(c_1, c_2)$ lies outside $P$ even restricted weak type fails. For convolution with arclength measure on a maximally curved curve in $\mathbb{R}^{n-1}$ the admissible region is the interior of an explicit trapezoid; for the restricted X-ray transform, an explicit triangle. The proof is geometric-combinatorial: sets of times are refined to central sets (sets whose diameter is comparable to a parameter called their width and which are non-concentrated, in the sense that no interval much shorter than the width captures a large fraction of the set), flowed along the vector fields, and compared with two-parameter Carnot–Carathéodory balls $B(x; \delta_1, \delta_2)$, the sets reachable from $x$ by flowing time $\delta_1$ along $X_1$ and $\delta_2$ along $X_2$; the single-scale estimate takes the form of a two-parameter isoperimetric inequality $|\Omega| \approx |B(0; \alpha_1, \alpha_2)|$, where $\alpha_j = |\Omega|/|\pi_j(\Omega)|$ are the thicknesses of $\Omega$.

Two obstructions prevent the argument from reaching the boundary of $P$: refining sets of times to central sets of fixed width costs an $\varepsilon$ in the exponents ('We do not know how to get around this loss'), and the argument demands more and more regularity of the data as $(c_1, c_2)$ approaches $\partial P$. Even in model cases the endpoint is out of reach for these methods: for convolution with arclength measure on the parabola the endpoint is $L^{3/2} \to L^3$, while the combinatorial argument yields only the Lorentz-space endpoint $L^{3/2,1} \to L^{3,\infty}$ (Lorentz spaces $L^{p,r}$ refine the Lebesgue space $L^p$ with a second, fine index $r$); the strong endpoint is nevertheless true in that model case via complex interpolation and $L^2$ smoothing — Fourier-analytic tools that do not extend to the general double-fibration setting. Subsequent work built a parallel endpoint theory in a modified setting: replacing arclength by affine arclength (for a polynomial curve $\gamma$ in $\mathbb{R}^d$, the measure $|\det(\gamma', \dots, \gamma^{(d)})|^{1/(d(d+1))}\,dt$), Stovall proved endpoint and uniform $L^p$-improving bounds for polynomial curves, and Christ–Dendrinos–Stovall–Street proved full endpoint Lebesgue estimates for polynomial-like (nilpotent) double fibrations — data for which the Lie algebra generated by the fiber-tangent fields is nilpotent, so that the iterated flow maps are polynomial — with essentially optimal generalized affine-arclength weights, explicitly noting that the behavior on the boundary of the polytope in Tao–Wright's original (unweighted) setting was left open and answered by them only in special cases. The unweighted, general $C^\infty$ endpoint case — the gap between Tao–Wright's interior theorem and their conjecture — is the subject of this problem.

Problem Statement

Resolve the endpoint cases of Tao–Wright's characterization of local $L^p$-improving for averaging operators along curves. Precisely: let $(\Sigma, M_1, M_2, \pi_1, \pi_2, 0)$ be a smooth double fibration with $\dim \Sigma = n$, $\dim M_j = n-1$, each $\pi_j$ a submersion near $0$ fixing $0$; let $X_1, X_2$ be smooth nonvanishing vector fields tangent to the fibers of $\pi_1, \pi_2$; assume the Hörmander condition at $0$ (iterated brackets $X_w$, indexed by words $w$ in the letters 1 and 2, span $T_0 \Sigma$). Let $P$ be the Newton polytope, the closed convex hull of $\{x \in \mathbb{R}^2_{\ge 0} : x \ge \deg(I), \ \lambda_I(0) \neq 0\}$ over $n$-tuples $I$ of words, where $\lambda_I = \det(X_{w_1}, \dots, X_{w_n})$. For $1 \le p_1 < p_2' \le \infty$ set $c_1 = p_2/(p_1 + p_2 - p_1 p_2)$ and $c_2 = p_1/(p_1 + p_2 - p_1 p_2)$, and define strong type and restricted weak type as in Tao–Wright's Definitions 1.1–1.2 (local bilinear estimates $\int_\Sigma f_1(\pi_1(x)) f_2(\pi_2(x)) a(x)\,dx \lesssim \|f_1\|_{L^{p_1}} \|f_2\|_{L^{p_2}}$ and its characteristic-function version, near $0$). Tao–Wright proved strong type when $(c_1, c_2) \in \operatorname{int} P$ and failure of even restricted weak type when $(c_1, c_2) \notin P$. The open problem is the boundary case: for $(c_1, c_2) \in \partial P$, determine whether the sextuple is of restricted weak-type $(p_1, p_2')$ — geometrically, whether the one-sided strengthening of the two-parameter isoperimetric-type comparison $|\Omega| \approx |B(0; \alpha_1, \alpha_2)|$ of Tao–Wright's (63), with thicknesses $\alpha_j = |\Omega|/|\pi_j(\Omega)|$, holds in the direction that Tao–Wright note would give restricted-type boundedness on the entire boundary of $P$, removing the $\varepsilon$-loss in the refinement of sets of times to central sets of fixed width — and whether it is of strong type $(p_1, p_2')$. The question is posed at every boundary point for all smooth ($C^\infty$) data satisfying the Hörmander condition, and a negative resolution at a boundary point is an explicit smooth double fibration satisfying the Hörmander condition together with a boundary point of its Newton polytope at which the restricted weak-type or strong-type estimate fails.

References

1. Terence Tao, James Wright, 'L^p improving bounds for averages along curves', Journal of the American Mathematical Society 16 (2003), no. 3, 605–638. DOI: 10.1090/s0894-0347-03-00420-x, URL: https://doi.org/10.1090/s0894-0347-03-00420-x (author final version: arXiv math/0108137, https://arxiv.org/abs/math/0108137).
2. Philip T. Gressman, 'L^p-improving properties of averages on polynomial curves and related integral estimates', Mathematical Research Letters 16 (2009), no. 6, 971–989. DOI: 10.4310/mrl.2009.v16.n6.a5, URL: https://doi.org/10.4310/mrl.2009.v16.n6.a5.
3. Spyridon Dendrinos, Norberto Laghi, James Wright, 'Universal L^p improving for averages along polynomial curves in low dimensions', Journal of Functional Analysis 257 (2009), no. 5, 1355–1378. DOI: 10.1016/j.jfa.2009.05.011, URL: https://doi.org/10.1016/j.jfa.2009.05.011.
4. Betsy Stovall, 'Endpoint L^p → L^q bounds for integration along certain polynomial curves', Journal of Functional Analysis 259 (2010), no. 12, 3205–3229. DOI: 10.1016/j.jfa.2010.08.008, URL: https://doi.org/10.1016/j.jfa.2010.08.008 (preprint: arXiv 1710.07668, https://arxiv.org/abs/1710.07668).
5. Spyridon Dendrinos, Betsy Stovall, 'Uniform bounds for convolution and restricted X-ray transforms along degenerate curves', Journal of Functional Analysis 268 (2015), no. 3, 585–633. DOI: 10.1016/j.jfa.2014.10.012, URL: https://doi.org/10.1016/j.jfa.2014.10.012.
6. Betsy Stovall, 'Uniform L^p-improving for weighted averages on curves', Analysis & PDE 7 (2014), no. 5, 1109–1136. DOI: 10.2140/apde.2014.7.1109, URL: https://doi.org/10.2140/apde.2014.7.1109 (preprint: arXiv 1710.07677, https://arxiv.org/abs/1710.07677).
7. Michael Christ, Spyridon Dendrinos, Betsy Stovall, Brian Street, 'Endpoint Lebesgue estimates for weighted averages on polynomial curves', American Journal of Mathematics 142 (2020), no. 6, 1661–1731. DOI: 10.1353/ajm.2020.0042, URL: https://doi.org/10.1353/ajm.2020.0042 (preprint: arXiv 1710.07688, https://arxiv.org/abs/1710.07688).
8. Jonathan Hickman, 'Uniform L^p_x–L^q_{x,r} improving for dilated averages over polynomial curves', Journal of Functional Analysis 270 (2016), no. 2, 560–608. DOI: 10.1016/j.jfa.2015.10.011, URL: https://doi.org/10.1016/j.jfa.2015.10.011 (preprint: arXiv 1507.02919, https://arxiv.org/abs/1507.02919).
9. Hyerim Ko, Sanghyuk Lee, Sewook Oh, 'Sharp smoothing properties of averages over curves', Forum of Mathematics, Pi 11 (2023), article e4. DOI: 10.1017/fmp.2023.2, URL: https://doi.org/10.1017/fmp.2023.2 (preprint: arXiv 2105.01628, https://arxiv.org/abs/2105.01628).
10. Junfeng Li, Naijia Liu, Zengjian Lou, Haixia Yu, 'L^p_x → L^q_{x,u} estimates for dilated averages over planar curves', Mathematische Zeitschrift 311 (2025), no. 2, article 31. DOI: 10.1007/s00209-025-03826-3, URL: https://doi.org/10.1007/s00209-025-03826-3 (preprint: arXiv 2401.16040, https://arxiv.org/abs/2401.16040).
11. Junfeng Li, Haixia Yu, Minqun Zhao, 'The boundedness of the bilinear fractional integrals along curves', Canadian Journal of Mathematics, published online 22 August 2025, pp. 1–31. DOI: 10.4153/S0008414X25101193, URL: https://doi.org/10.4153/S0008414X25101193 (preprint: arXiv 2411.14830, https://arxiv.org/abs/2411.14830).
12. Ye Zhang, 'Loomis–Whitney inequalities on corank 1 Carnot groups', Annales Fennici Mathematici 49 (2024), no. 2, 437–459. DOI: 10.54330/afm.146800, URL: https://doi.org/10.54330/afm.146800.
13. Kaiyi Huang, Betsy Stovall, Sarah Tammen, 'Incidences among flows', arXiv 2509.08937 (2025), URL: https://arxiv.org/abs/2509.08937.