Boundedness of the bilinear Hilbert transform and (sub)bilinear maximal function along variable curves with x-measurable profile

For a fixed smooth non-flat curve profile $\gamma(t)$, the bilinear Hilbert transform $H_\Gamma$ along $\Gamma=(t,-\gamma(t))$ and its (sub)bilinear maximal companion $M_\Gamma$ are bounded on the full Hölder range of Lebesgue exponents (Lie 2015–2018; Gaitan–Lie 2020, sharp for the maximal function). Gaitan and Lie frame as the open follow-up problem the extension to variable curves $\Gamma_x=(t,-\gamma(x,t))$ whose profile is merely measurable in the base point $x$ while non-flat and doubling in $t$. The linear variable-curve theory is settled (Lie, Adv. Math. 2024), but no boundedness result for any bilinear operator along an x-dependent curve exists. The recorded problem asks for a proof or disproof of Hölder-range $L^p\times L^q\to L^r$ boundedness of $H^B_\Gamma$ and $M^B_\Gamma$ for non-flat profiles measurable in $x$, including polynomial profiles $\sum_j a_j(x)t^{\alpha_j}$ with measurable coefficients.

Background

The bilinear Hilbert transform is the singular integral $\mathrm{BHT}(f,g)(x)=\mathrm{p.v.}\int_{\mathbb{R}}f(x-t)\,g(x+t)\,\frac{dt}{t}$, introduced by A. Calderón in the 1960s in connection with the Cauchy integral on Lipschitz curves. Its boundedness on products of Lebesgue spaces (Calderón's conjecture) was resolved by Lacey and Thiele: $\mathrm{BHT}$ maps $L^p\times L^q$ continuously into $L^r$ whenever $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ (Hölder indices) with $1<p,q\le\infty$ and $\frac{2}{3}<r<\infty$, and Lacey proved the analogous result for the flat bilinear maximal function. Whether this range is optimal — it is conjectured to extend down to $r>\frac{1}{2}$ — remains a well-known open problem in time-frequency analysis, the branch of harmonic analysis that decomposes inputs into spatially and spectrally localized wave packets (for example Gabor frames) and exploits orthogonality between them.

A natural curved generalization replaces the straight trajectory $t\mapsto t$ in the arguments of $f$ and $g$ by a planar curve. For $\Gamma=(t,-\gamma(t))$ one studies $H_\Gamma(f,g)(x)=\mathrm{p.v.}\int_{\mathbb{R}}f(x-t)\,g(x+\gamma(t))\,\frac{dt}{t}$ and its maximal companion $M_\Gamma(f,g)(x)=\sup_{\epsilon>0}\frac{1}{2\epsilon}\int_{-\epsilon}^{\epsilon}|f(x-t)\,g(x+\gamma(t))|\,dt$; the maximal function governs the almost-everywhere convergence and differentiation of bilinear averages along the curve, a continuous analogue of the norm-convergence problem for non-conventional bilinear ergodic averages. Li proved the first curved result, $L^2\times L^2\to L^1$ for $\gamma(t)=t^d$; Lie then established, for the class $\mathcal{NF}$ of smooth “non-flat” curves near the origin and near infinity — curves such as generalized polynomials $\sum_j a_j t^{\alpha_j}$ with $\alpha_j\notin\{0,1\}$, whose curvature-type degeneracy is excluded at both ends — that $H_\Gamma$ is bounded on the Banach range $1<p<\infty$, $1<q\le\infty$, $1\le r<\infty$, sharp up to endpoints. Li and Xiao complemented this with uniform-in-coefficients estimates for polynomial curves and counterexamples showing that $M_\Gamma$ fails for $r<1$; Li and Yu later handled convex curves for $r>\frac{1}{2}$, and Bényi–Hu–Lie recently reached the maximal range $r>\frac{1}{2}$ for monomial curves through a bilinear Hilbert–Carleson operator. The source paper of the present problem (Gaitan–Lie) completed the fixed-curve picture for the maximal function: $M_\Gamma$ is bounded exactly on the Hölder range $1<p,q\le\infty$, $1\le r\le\infty$, and this range is sharp for $\gamma\in\mathcal{NF}$.

All of these results fix one curve, independent of the base point $x$ at which the operator is evaluated. The variable-curve setting lets the curve depend on the base point: $\Gamma_x=(t,-\gamma(x,t))$. This is the curved analogue of Zygmund's classical conjecture on differentiation of integrals along variable vector fields, and even the linear theory is delicate: for the linear Hilbert transform and maximal function along variable homogeneous curves, Guo–Hickman–Lie–Roos needed the coefficient to be Lipschitz (maximal case) or measurable but constant in the second variable (singular case), and Lie's “unified approach” program eventually settled the linear themes — $L^p$ bounds, $1<p<\infty$, for the linear Hilbert transform, its maximal function, and Carleson-type operators along variable curves $\gamma(x,t)$ that are merely measurable in $x$ and non-flat in $t$ (published in Advances in Mathematics, 2024). For bilinear operators, by contrast, no boundedness result along an $x$-dependent curve is known: measurable $x$-dependence destroys the bilinear Fourier-multiplier structure on which every existing proof rests, because the oscillatory phase of the multiplier itself varies with the base point.

The source paper explicitly frames this as its continuation: it announces “a unified approach for the boundedness of $H_\Gamma$ and $M_\Gamma$ in the case in which $\Gamma=(t,-\gamma(x,t))$ … where $\gamma$ is a suitable non-degenerate curve that is smooth and doubling in $t$ but only measurable in $x$”, calling that study “significantly more complex” than the fixed-curve theory treated there. The announced sequel has not appeared, so the bilinear variable-curve boundedness problem is open.

Problem Statement

Let $\gamma:\mathbb{R}^2\to\mathbb{R}$ belong to the variable-curve class of Lie's unified program: $\gamma(\cdot,t)$ is measurable for every $t\neq 0$; for almost every $x$, $\gamma(x,\cdot)$ is finitely piecewise $C^{2+\delta}$ (for some $\delta>0$) and non-flat — with non-vanishing curvature-type non-degeneracy — near both the origin and infinity, the non-degeneracy and doubling behavior in $t$ holding uniformly in $x$ up to a controlled exceptional region (the class $\mathcal{M}^x_t\mathcal{NF}$; as a model, $\gamma(x,t)=\sum_{j=1}^{d}a_j(x)\,t^{\alpha_j}$ with arbitrary measurable functions $a_j$ and $\alpha_j\in\mathbb{R}\setminus\{0,1\}$). For the family of curves $\Gamma_x=(t,-\gamma(x,t))$ define the bilinear Hilbert transform along the variable curve

$H^B_\Gamma(f,g)(x)=\mathrm{p.v.}\int_{\mathbb{R}}f(x-t)\,g(x+\gamma(x,t))\,\frac{dt}{t}$

and the (sub)bilinear maximal function along the variable curve

$M^B_\Gamma(f,g)(x)=\sup_{\epsilon>0}\frac{1}{2\epsilon}\int_{-\epsilon}^{\epsilon}\left|f(x-t)\,g(x+\gamma(x,t))\right|\,dt.$

Determine whether $H^B_\Gamma$ and $M^B_\Gamma$ extend to bounded operators $L^p(\mathbb{R})\times L^q(\mathbb{R})\to L^r(\mathbb{R})$ on Hölder indices $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ in this variable setting. The announced milestone is the diagonal bounds $L^p\times L^p\to L^{p/2}$ for $2\le p<\infty$, an intermediate goal rather than the resolution; the task is settled by proving or disproving the natural full target, the range known for $x$-independent curves, namely $1<p,q\le\infty$ and $1\le r\le\infty$ for $M^B_\Gamma$ and $1<p<\infty$, $1<q\le\infty$, $1\le r<\infty$ for $H^B_\Gamma$.

References

1. Alejandra Gaitan and Victor Lie, The Boundedness of the (Sub)Bilinear Maximal Function Along “Non-flat” Smooth Curves, Journal of Fourier Analysis and Applications 26 (2020), article 69. DOI: 10.1007/s00041-020-09770-6, https://doi.org/10.1007/s00041-020-09770-6 (source paper; preprint arXiv:1903.11002)
2. Victor Lie, On the boundedness of the bilinear Hilbert transform along “non-flat” smooth curves, American Journal of Mathematics 137 (2015), 313–363. DOI: 10.1353/ajm.2015.0013, https://doi.org/10.1353/ajm.2015.0013
3. Victor Lie, On the boundedness of the bilinear Hilbert transform along “non-flat” smooth curves. The Banach triangle case (L^r, 1 ≤ r < ∞), Revista Matemática Iberoamericana 34 (2018), 331–353. DOI: 10.4171/rmi/987, https://doi.org/10.4171/rmi/987
4. Victor Lie, A unified approach to three themes in harmonic analysis (I & II), Advances in Mathematics 437 (2024), article 109385. DOI: 10.1016/j.aim.2023.109385, https://doi.org/10.1016/j.aim.2023.109385 (preprint arXiv:1902.03807)
5. Alejandra Gaitan and Victor Lie, Non-zero to zero curvature transition: Operators along hybrid curves with no quadratic (quasi-)resonances, Advances in Mathematics 478 (2025), article 110356. DOI: 10.1016/j.aim.2025.110356, https://doi.org/10.1016/j.aim.2025.110356 (preprint arXiv:2402.03451)
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