For non-degenerate modulation parameters, the known LpL^p bounds of BHT are equivalent to a family of generalized restricted weak-type (GRWT) estimates on an open hexagon H\mathcal H of exponent tuples; at the three short open boundary segments S\mathcal S, which correspond to weak-L2/3L^{2/3} targets, the strongest known estimate (Di Plinio–Thiele, Trans. AMS 2016) carries a double-logarithmic correction, whereas the discrete Walsh-model (quartile) analogue satisfies the corresponding logarithm-free bounds. The problem is Conjecture 1 of Di Plinio–Thiele: prove or refute that the GRWT estimate holds at every tuple of S\mathcal S for every non-degenerate BHT — equivalently, that the double-logarithmic correction can be removed — together with its explicitly identified technical companion: whether the continuous multi-frequency Calderón–Zygmund decomposition can avoid its starred correction terms.

Background

The bilinear Hilbert transforms are the operators

BHTb⃗(f1,f2)(x)=p.v.∫Rf1(x−b1t) f2(x−b2t) dtt,\mathrm{BHT}_{\vec b}(f_1,f_2)(x)=\mathrm{p.v.}\int_{\mathbb R} f_1(x-b_1t)\,f_2(x-b_2t)\,\frac{dt}{t},

where b⃗=(b1,b2)\vec b=(b_1,b_2) with b1≠b2b_1\ne b_2 (the non-degenerate case; when b1=b2b_1=b_2 the operator degenerates to a pointwise product of a Hilbert transform and a function). They arose in Calderón's program on commutators of the Hilbert transform and are the prototype of modulation-invariant singular integrals: their Fourier symbols are singular along a line in frequency space and are invariant under simultaneous modulation of the input and output frequencies, a feature that places them beyond the classical Coifman–Meyer theory of multilinear singular integrals. Lacey and Thiele proved that BHTb⃗\mathrm{BHT}_{\vec b} maps Lp1×Lp2L^{p_1}\times L^{p_2} boundedly into LpL^p whenever 1<p1,p2≤∞1<p_1,p_2\le\infty, 1p1+1p2=1p\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}, and p>23p>\frac23, with a constant that may depend on the separation b1−b2b_1-b_2. It is convenient to dualize. For a non-degenerate vector β⃗∈R3\vec\beta\in\mathbb R^3 (no two components equal; by translation and scaling invariance one may take β⃗\vec\beta of unit length orthogonal to (1,1,1)(1,1,1)), define the trilinear form

Λβ⃗(f1,f2,f3)=∫Rp.v.∫Rf1(x−β1t) f2(x−β2t) f3(x−β3t) dtt dx,\Lambda_{\vec\beta}(f_1,f_2,f_3)=\int_{\mathbb R}\mathrm{p.v.}\int_{\mathbb R} f_1(x-\beta_1t)\,f_2(x-\beta_2t)\,f_3(x-\beta_3t)\,\frac{dt}{t}\,dx,

so that Λβ⃗(f1,f2,f3)=⟨BHTb⃗(f1,f2),f3⟩\Lambda_{\vec\beta}(f_1,f_2,f_3)=\langle\mathrm{BHT}_{\vec b}(f_1,f_2),f_3\rangle for a suitable b⃗\vec b. The sharp distributional formulation of the Lacey–Thiele theorem is a family of generalized restricted weak-type (GRWT) estimates, indexed by tuples α⃗=(α1,α2,α3)\vec\alpha=(\alpha_1,\alpha_2,\alpha_3) with α1+α2+α3=1\alpha_1+\alpha_2+\alpha_3=1. For α⃗\vec\alpha in the open hexagon

H={α⃗:α1+α2+α3=1, max⁡jαj≤1, min⁡jαj≥−12},\mathcal H=\{\vec\alpha:\alpha_1+\alpha_2+\alpha_3=1,\ \max_j\alpha_j\le 1,\ \min_j\alpha_j\ge-\tfrac12\},

the GRWT estimate at α⃗\vec\alpha asserts that for all sets Fj⊂RF_j\subset\mathbb R of finite measure and all functions ∣fj∣≤1Fj|f_j|\le\mathbf 1_{F_j},

Λβ⃗(f1,f2,f3)≤Cβ⃗ Cα⃗∏j=13∣Fj∣αj,\Lambda_{\vec\beta}(f_1,f_2,f_3)\le C_{\vec\beta}\,C_{\vec\alpha}\prod_{j=1}^3|F_j|^{\alpha_j},

with one extra convention: if j∗j^* is an index attaining min⁡jαj\min_j\alpha_j (the unique index with αj≤0\alpha_j\le 0 in the region of interest), the function fj∗f_{j^*} is only required to be supported on a major subset Fj∗′⊂Fj∗F'_{j^*}\subset F_{j^*} with ∣Fj∗′∣≥12∣Fj∗∣|F'_{j^*}|\ge\frac12|F_{j^*}|, which may depend on the triple (F1,F2,F3)(F_1,F_2,F_3). This passage to a major subset is the natural bilinear analogue of weak-type estimates and is indispensable when some αj≤0\alpha_j\le 0; for instance, when j∗=3j^*=3 the GRWT estimate at α⃗\vec\alpha is equivalent to a weak-type bound ∥BHTb⃗(f1,f2)∥L1/(1−α3),∞≤C∏j=1,2∣Fj∣αj\|\mathrm{BHT}_{\vec b}(f_1,f_2)\|_{L^{1/(1-\alpha_3),\infty}}\le C\prod_{j=1,2}|F_j|^{\alpha_j} for ∣fj∣≤1Fj|f_j|\le\mathbf 1_{F_j}. The six vertices of H\mathcal H all correspond to hypothetical L1×L2→L2/3L^1\times L^2\to L^{2/3} estimates for some dual of the bilinear Hilbert transform, and the three short open sides

S=⋃j=13Sj,Sj={α⃗:α1+α2+α3=1, αj=−12, max⁡k≠jαk<1}\mathcal S=\bigcup_{j=1}^3\mathcal S_j,\qquad \mathcal S_j=\{\vec\alpha:\alpha_1+\alpha_2+\alpha_3=1,\ \alpha_j=-\tfrac12,\ \max_{k\ne j}\alpha_k<1\}

correspond to endpoint estimates at target exponent p=23p=\frac23. This threshold is structural for modulation-invariant operators: Lacey proved that the bilinear maximal function — and with it the entire modulation-invariant class — cannot map into LpL^p for any p≤23p\le\frac23; whether the bilinear Hilbert transform itself admits any bounds past this threshold is a separate and still-open question. The endpoint theory on S\mathcal S is only partially understood. Bilyk and Grafakos obtained the GRWT estimates on S\mathcal S with a single logarithmic correction; Di Plinio and Thiele improved this to a double logarithm. Their Corollary 4 states that for all α⃗∈Sj\vec\alpha\in\mathcal S_j,

Λβ⃗(f1,f2,f3)≤Cβ⃗∏k=13∣Fk∣αk max⁡{(min⁡k≠j(1−αk))−1, log⁡log⁡(ee+∣Fj∣min⁡k≠j∣Fk∣)},\Lambda_{\vec\beta}(f_1,f_2,f_3)\le C_{\vec\beta}\prod_{k=1}^3|F_k|^{\alpha_k}\,\max\Big\{\Big(\min_{k\ne j}(1-\alpha_k)\Big)^{-1},\ \log\log\Big(\mathrm e^{\mathrm e}+\frac{|F_j|}{\min_{k\ne j}|F_k|}\Big)\Big\},

still in the GRWT sense (major subset allowed). The interior of the hexagon carries no logarithmic losses, so the question whether the double logarithm is an artifact of the current technology is sharp. For the quartile operator — the discrete Walsh-model analogue of BHT used in time-frequency analysis, whose Walsh wave packets are compactly localized in both time and frequency — Demeter and Di Plinio proved the log-free analogue, which is a stronger form of the present conjecture in the discrete model. The proofs of Di Plinio–Thiele rest on a sharpened multi-frequency Calderón–Zygmund decomposition (MFCZ): a decomposition of an LpL^p input, 1≤p<21\le p<2, into a good part amenable to Hilbert-space methods and a bad part localized on spatial intervals and orthogonal to the NN relevant bad frequencies, originally due to Nazarov, Oberlin, and Thiele. In the continuous setting the bad part must be prepared to have mean zero against approximately log⁡N\log N frequencies and must be tested against Ingham-type wave packets (wave packets with almost exponential spatial decay, following a classical construction of Ingham), which lack compact time-frequency support; the price is a starred correction factor (t)∗=(1+t)(log⁡(e+t))3(t)_*=(1+t)(\log(\mathrm e+t))^3 appearing in the edge estimates (Theorems 2 and 3 of Di Plinio–Thiele, governing the short side S3\mathcal S_3 and the bisecting line from the vertex A=(1,12,−12)A=(1,\frac12,-\frac12) where α2=12\alpha_2=\frac12), and ultimately the double logarithm of Corollary 4. No such loss occurs in the discrete quartile model, where the offending interactions vanish identically.

Problem Statement

Prove or refute Conjecture 1 of Di Plinio–Thiele (Trans. AMS 368 (2016), 3931–3972; arXiv:1403.5978). Here, for a non-degenerate β⃗=(β1,β2,β3)∈R3\vec\beta=(\beta_1,\beta_2,\beta_3)\in\mathbb R^3 with no two components equal (by translation and scaling invariance one may take β⃗\vec\beta of unit length orthogonal to (1,1,1)(1,1,1)),

Λβ⃗(f1,f2,f3)=∫Rp.v.∫Rf1(x−β1t) f2(x−β2t) f3(x−β3t) dtt dx\Lambda_{\vec\beta}(f_1,f_2,f_3)=\int_{\mathbb R}\mathrm{p.v.}\int_{\mathbb R} f_1(x-\beta_1t)\,f_2(x-\beta_2t)\,f_3(x-\beta_3t)\,\frac{dt}{t}\,dx

is the trilinear form of the bilinear Hilbert transform (Λβ⃗(f1,f2,f3)=⟨BHTb⃗(f1,f2),f3⟩\Lambda_{\vec\beta}(f_1,f_2,f_3)=\langle\mathrm{BHT}_{\vec b}(f_1,f_2),f_3\rangle for a suitable b⃗\vec b with b1≠b2b_1\ne b_2). The conjecture asserts that the generalized restricted weak-type estimate

Λβ⃗(f1,f2,f3)≤Cβ⃗ Cα⃗∏j=13∣Fj∣αj\Lambda_{\vec\beta}(f_1,f_2,f_3)\le C_{\vec\beta}\,C_{\vec\alpha}\prod_{j=1}^3|F_j|^{\alpha_j}

holds for all tuples α⃗\vec\alpha in

S=⋃j=13Sj,Sj={α⃗:α1+α2+α3=1, αj=−12, max⁡k≠jαk<1},\mathcal S=\bigcup_{j=1}^3\mathcal S_j,\qquad \mathcal S_j=\{\vec\alpha:\alpha_1+\alpha_2+\alpha_3=1,\ \alpha_j=-\tfrac12,\ \max_{k\ne j}\alpha_k<1\},

in the GRWT sense: for all sets Fj⊂RF_j\subset\mathbb R of finite measure and all ∣fj∣≤1Fj|f_j|\le\mathbf 1_{F_j}, where the input at an index j∗j^* attaining min⁡jαj\min_j\alpha_j is required only to be supported on a major subset Fj∗′⊂Fj∗F'_{j^*}\subset F_{j^*} with ∣Fj∗′∣≥12∣Fj∗∣|F'_{j^*}|\ge\frac12|F_{j^*}| that may depend on (F1,F2,F3)(F_1,F_2,F_3). Constants may depend on the fixed tuple α⃗∈S\vec\alpha\in\mathcal S and on the non-degenerate β⃗\vec\beta, but on no other parameter. A positive answer must in particular remove the double-logarithmic correction of Corollary 4 of the source on all of S\mathcal S and would yield the logarithm-free interior blow-up rates and Lorentz–Orlicz refinements near the L2/3L^{2/3} threshold. A negative answer must exhibit a non-degenerate β⃗\vec\beta, a tuple α⃗∈S\vec\alpha\in\mathcal S, and finite-measure sets with subindicator inputs such that no choice of major subset can make the estimate hold — i.e. a distributional lower bound showing that some logarithmic correction is necessary on the short sides. The source identifies as the central technical obstruction — and this problem includes as an explicit companion question — whether the continuous multi-frequency Calderón–Zygmund decomposition can be sharpened to avoid the starred correction terms (⋅)∗(\cdot)_* (the price of preparing bad parts to have mean zero against approximately log⁡N\log N frequencies and of testing against Ingham-type almost-exponentially decaying wave packets, which lack the compact time-frequency support that makes the discrete Walsh/quartile model sharp). Establishing a continuous decomposition with the discrete model's sharpness, or log-free GRWT bounds on any nontrivial portion of S\mathcal S in the continuous setting, constitutes partial progress toward this problem; the full target remains the complete settle (proof or refutation) of Conjecture 1 on all of S\mathcal S.

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