$L^{p}$ boundedness of the bilinear Hilbert transform in the lower range $1/2<p\leq 2/3$

The bilinear Hilbert transform $\mathrm{BHT}(f,g)(x)=\mathrm{p.v.}\int_{\mathbb{R}} f(x-t)g(x+t)\,dt/t$ is known since Lacey and Thiele to map $L^{p_1}\times L^{p_2}$ boundedly into $L^{p}$, $1/p_1+1/p_2=1/p$, whenever $2/3<p<\infty$ and $1<p_1,p_2\leq\infty$. Whether the estimates extend to the remaining scaling range $1/2<p\leq 2/3$ — the lower boundary $1/2$ being forced by the relation $1/p=1/p_1+1/p_2$ with $p_1,p_2\leq\infty$ — has remained open since 1997. No proof is known for any exponent tuple in this range, and no counterexample with $L^{p}$ inputs is known either: all known negative results concern larger operator classes or modified input spaces. The problem asks for a definitive resolution in either direction, including the endpoint $p=2/3$, where only distributional estimates with double-logarithmic losses are available.

Background

The bilinear Hilbert transform is the bilinear singular integral

$$\mathrm{BHT}(f_1,f_2)(x):=\mathrm{p.v.}\int_{\mathbb{R}} f_1(x-t)\,f_2(x+t)\,\frac{dt}{t},$$

where p.v. denotes the Cauchy principal value. In Fourier coordinates it is the bilinear multiplier operator with symbol $\operatorname{sgn}(\xi_1-\xi_2)$:

$$\mathrm{BHT}(f_1,f_2)(x)=\int_{\mathbb{R}^2}\operatorname{sgn}(\xi_1-\xi_2)\,\hat f_1(\xi_1)\hat f_2(\xi_2)\,e^{2\pi i x(\xi_1+\xi_2)}\,d\xi_1 d\xi_2.$$

The operator was introduced by A. P. Calderón in his study of the Cauchy integral along Lipschitz curves: the relevant commutators can be rewritten as averages of bilinear Hilbert transforms, and Calderón conjectured in the mid-1960s that $\mathrm{BHT}$ maps $L^2\times L^2$ into $L^1$. The operator is modulation invariant — multiplying an input by $e^{2\pi i \xi x}$ translates the output's Fourier support — which places it outside the classical Calderón–Zygmund theory of singular integrals (the framework that handles the ordinary Hilbert transform) and is the source of its difficulty. In two celebrated papers, Lacey and Thiele proved that $\mathrm{BHT}$ maps $L^{p_1}\times L^{p_2}$ boundedly into $L^p$ whenever $1<p_1,p_2\leq\infty$, $1/p_1+1/p_2=1/p$, and $2/3<p<\infty$ (the case $p>2$ in 1997, the general range in 1999), confirming Calderón's conjecture. Their proof created time-frequency analysis: inputs are decomposed into wave packets (functions localized simultaneously in time and frequency), and the sum is organized into trees (collections of wave packets whose frequency supports all lie inside that of one distinguished packet) whose contributions are controlled by orthogonality and combinatorial selection. The scaling relation $1/p=1/p_1+1/p_2$ with $p_1,p_2\leq\infty$ forces $p>1/2$, so $2/3$ is not a natural barrier of the operator but a barrier of the current method; the conjectured truth is boundedness for all $1/2<p<\infty$. The region of known boundedness is an open hexagon in the exponent simplex $1/p_1+1/p_2+1/p_3=1$ (where $p_3$ is the dual output exponent), whose six corners all correspond to hypothetical endpoint estimates $L^1\times L^2\to L^{2/3}$ of some dual of $\mathrm{BHT}$. Several adjacent facts delimit what any solution must overcome. First, Lacey (2000) showed that the threshold $2/3$ is sharp, modulo endpoints, for the entire modulation-invariant model class: discrete model sums with arbitrary bounded coefficients — which is what generic tree-selection arguments actually bound — cannot be bounded below $2/3$. Hence a positive result must exploit the specific structure of the symbol $\operatorname{sgn}(\xi_1-\xi_2)$, not merely modulation invariance. Second, at the endpoint $p=2/3$ itself, Bilyk and Grafakos (2006) and Di Plinio and Thiele (2016) proved only distributional-type estimates with, respectively, single- and double-logarithmic losses; whether even the double logarithm can be removed (as it can for the quartile operator (the discrete Walsh-model analogue of $\mathrm{BHT}$, built from dyadic step functions instead of Fourier waves), by Demeter and Di Plinio 2013) is open, and no strong or weak-type $L^{2/3}$ bound is known. Third, no counterexample with $L^p$ inputs exists: the negative results of Dai and Lu (2014) show failure only for Fourier–Lebesgue input spaces $\mathcal{F}L^{p'}$ (functions whose Fourier transform lies in $L^{p'}$) with the other input in $L^p$, and Miyachi and Tomita (2017) showed unboundedness on products of Hardy spaces $H^p$ with $p\leq 1$ (the Hardy scale is the standard $L^p$ substitute below exponent $1$, built from maximal averages rather than absolute values). Fourth, Dai and Lu proved that multipliers with symbols exponentially damped transversely to the singularity line $\{\xi_1=\xi_2\}$, at any fixed damping strength $\delta>0$, are bounded in the full range $1/2<p<\infty$, but with constants that blow up as $\delta\to 0$ whenever $1/2<p\leq 2/3$; passing to the undamped limit that recovers the true $\mathrm{BHT}$ symbol is exactly the open question. Subsequent work has confirmed rather than eroded the $2/3$ barrier: Uraltsev and Warchalski (2022) proved uniform-in-parameter bounds for the family $\mathrm{BHT}_\beta$ exactly in $p>2/3$ (the full range where uniform bounds can hold), and Fraccaroli, Saari, and Thiele (2024) extended uniform bounds to quasiconformally transformed symbols still only in the strict local $L^2$ range. For $p<1$ the target $L^p$ is quasi-Banach, so duality, convexity, and standard interpolation tools are unavailable — a structural source of difficulty for the lower part of the range.

Problem Statement

Determine whether the bilinear Hilbert transform

$$\mathrm{BHT}(f_1,f_2)(x)=\mathrm{p.v.}\int_{\mathbb{R}} f_1(x-t)\,f_2(x+t)\,\frac{dt}{t}$$

extends to a bounded bilinear operator from $L^{p_1}(\mathbb{R})\times L^{p_2}(\mathbb{R})$ into $L^{p}(\mathbb{R})$, where $1<p_1,p_2\leq\infty$ and $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$, for exponents in the remaining open range $\frac{1}{2}<p\leq\frac{2}{3}$. A complete solution decides the dichotomy: either (i) a positive answer — a proof of boundedness for all exponent tuples in the stated range $\frac{1}{2}<p\leq\frac{2}{3}$, including a strong- or weak-type bound at the endpoint $p=\frac{2}{3}$; or (ii) a negative answer — an explicit construction of input functions in $L^{p_1}\times L^{p_2}$ demonstrating that the operator norm is infinite for some tuple with $\frac{1}{2}<p\leq\frac{2}{3}$. Partial information such as estimates for damped approximating symbols (whose constants blow up as the damping tends to zero) or for Fourier–Lebesgue/Hardy input spaces does not resolve the question, which concerns the Lebesgue spaces themselves.

References

1. Wei Dai and Guozhen Lu, "$L^{p}$ estimates for the bilinear Hilbert transform for $1/2<p\leq2/3$: A counterexample and generalizations to non-smooth symbols", arXiv:1409.3875 (2014), https://arxiv.org/abs/1409.3875
2. Michael Lacey and Christoph Thiele, "$L^{p}$ estimates on the bilinear Hilbert transform for $2<p<\infty$", Annals of Mathematics (2) 146 (1997), 693–724, DOI: 10.2307/2952458, https://doi.org/10.2307/2952458
3. Michael Lacey and Christoph Thiele, "On Calderón's conjecture", Annals of Mathematics (2) 149 (1999), 475–496, DOI: 10.2307/120971, https://doi.org/10.2307/120971
4. Michael T. Lacey, "The bilinear maximal functions map into $L^{p}$ for $2/3<p\leq 1$", Annals of Mathematics (2) 151 (2000), 35–57, DOI: 10.2307/121111, https://doi.org/10.2307/121111
5. Dmitriy Bilyk and Loukas Grafakos, "Distributional estimates for the bilinear Hilbert transform", Journal of Geometric Analysis 16 (2006), 563–584, DOI: 10.1007/BF02922131, https://doi.org/10.1007/BF02922131
6. Ciprian Demeter and Francesco Di Plinio, "Endpoint bounds for the quartile operator", Journal of Fourier Analysis and Applications 19 (2013), 836–856, DOI: 10.1007/s00041-013-9275-4, https://doi.org/10.1007/s00041-013-9275-4
7. Francesco Di Plinio and Christoph Thiele, "Endpoint bounds for the bilinear Hilbert transform", Transactions of the American Mathematical Society 368 (2016), 3931–3972, DOI: 10.1090/tran/6548, https://doi.org/10.1090/tran/6548
8. Akihiko Miyachi and Naohito Tomita, "A remark on the unboundedness of the bilinear Hilbert transform on Hardy spaces", RIMS Kôkyûroku Bessatsu B65 (2017), 153–157, Zbl: 1390.42021, https://zbmath.org/6870707
9. Gennady Uraltsev and Michał Warchalski, "The full range of uniform bounds for the bilinear Hilbert transform", arXiv:2205.09851 (2022), https://arxiv.org/abs/2205.09851
10. Marco Fraccaroli, Olli Saari, and Christoph Thiele, "Uniform bounds for bilinear symbols with linear K-quasiconformally embedded singularity", Analysis & PDE 18 (2025), 2293–2323, DOI: 10.2140/apde.2025.18.2293 (preprint arXiv:2402.11661), https://doi.org/10.2140/apde.2025.18.2293
11. Lars Becker and Polona Durcik, "The shifted bilinear Hilbert transform", arXiv:2603.20173 (2026), https://arxiv.org/abs/2603.20173