Endpoint bounds for the bilinear Hilbert transform along non-flat smooth curves: the remaining boundary of the Banach triangle

For a curve $\Gamma=(t,-\gamma(t))$, the curved bilinear Hilbert transform is $H_\Gamma(f,g)(x)=\mathrm{p.v.}\int_{\mathbb{R}} f(x-t)\,g(x+\gamma(t))\,\frac{dt}{t}$. Lie (2018) proved that for every $\gamma$ in the class $\mathcal{NF}^C$ of smooth "non-flat" curves near zero and infinity, $H_\Gamma$ maps $L^p\times L^q\to L^r$ boundedly for all Hölder exponents $1/p+1/q=1/r$ with $1<p<\infty$, $1<q\le\infty$, $1\le r<\infty$, and showed this range is optimal up to endpoints: $r\ge 1$ is sharp for the class as a whole, while the boundary points of the Banach triangle were left undecided. The problem is to determine these endpoints for every $\gamma\in\mathcal{NF}^C$: prove or disprove that $H_\Gamma$ maps $L^\infty\times L^q\to L^q$ ($1\le q<\infty$, weak type included where strong type fails) and $L^1\times L^\infty\to L^1$ (strong or weak type), thereby deciding boundedness at every point of the closed Banach triangle. Later literature — sharp ranges for the companion maximal function, quasi-Banach ranges for convex-curve subclasses, variable and hybrid curves, and trilinear analogues — leaves precisely this endpoint boundary of the singular operator untouched.

Background

The classical bilinear Hilbert transform is the bilinear singular integral

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

where $\mathrm{p.v.}$ denotes the Cauchy principal value. In breakthrough work, Lacey and Thiele proved that $\mathrm{BHT}$ is bounded from $L^p(\mathbb{R})\times L^q(\mathbb{R})$ to $L^r(\mathbb{R})$ whenever the exponents are Hölder indices, i.e. $1/p+1/q=1/r$, with $1<p,q\le\infty$ and $2/3<r<\infty$. A natural curved analogue replaces the linear arguments $x-t$ and $x+t$ by a nonlinear path: for a curve $\Gamma=(t,-\gamma(t))$ one sets

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

These operators are motivated by the study of the Cauchy transform along Lipschitz curves, by the $L^r$-norm convergence of non-conventional bilinear polynomial averages in ergodic theory, and by nonlinear extensions of Roth's theorem in additive combinatorics; the curvature of $\gamma$ destroys the modulation symmetry that powers the classical theory and demands genuinely new time-frequency analysis.

The natural class of curves is Lie's $\mathcal{NF}^C$: $\gamma$ is $C^N$-smooth ($N\ge 4$) with no critical points near zero and infinity; under dyadic rescaling it is asymptotically self-similar in the sense that $\gamma(2^{-j}t)/(2^{-j}\gamma'(2^{-j}))$ approaches a fixed profile $Q(t)$ with $Q''$ bounded away from zero (the "non-flatness" — the curve deviates from any straight line at every scale); and a doubling-type variation condition holds. The class contains every real polynomial with no constant and no linear term, every finite real combination of powers $t^\alpha$ with $\alpha\in(0,\infty)\setminus\{1\}$, and combinations $|t|^\alpha|\log|t||^\beta$ with $\alpha\notin\{-1,0,1\}$; $\mathcal{NF}^C$ additionally allows adding a constant.

The mapping theory of $H_\Gamma$ was built in stages: Li proved the $L^2\times L^2\to L^1$ bound for monomial curves and, crucially, constructed for every degree $d\ge 2$ a polynomial curve on which the operator is unbounded from $L^p\times L^q\to L^r$ for all $p,q>1$ whenever $r<(d-1)/d$; Lie extended the $L^2\times L^2\to L^1$ bound to all of $\mathcal{NF}^C$; Li and Xiao obtained uniform-in-coefficient bounds for polynomial curves for $r>(d-1)/d$; and Lie (2018) established the full interior range $1<p<\infty$, $1<q\le\infty$, $1\le r<\infty$. Because the degree-$d$ counterexamples above approach threshold $1$ as $d\to\infty$, the restriction $r\ge 1$ is optimal for the class as a whole — no uniform quasi-Banach extension (output exponent below $1$) is possible over $\mathcal{NF}^C$.

Viewing the associated trilinear form $\Lambda(f,g,h)=\int H_\Gamma(f,g)\,h$, the admissible Banach-range triples form a closed triangle in the coordinates $(1/p,1/q,1/r')$ with $1/p+1/q+1/r'=1$, the "Banach triangle", whose vertices correspond to $(p,q,r)=(1,\infty,1)$, $(\infty,1,1)$ and $(\infty,\infty,\infty)$. Lie's theorem covers the open triangle together with the $q=\infty$ and $r=1$ edges; since $L^\infty\times L^\infty\to L^\infty$ fails (with one input identically $1$ the operator reduces, up to a sign, to the classical Hilbert transform of the other input, which is unbounded from $L^\infty$ to $L^\infty$), the undecided boundary is exactly the $p=\infty$ edge — $H_\Gamma:L^\infty\times L^q\to L^q$ for $1\le q<\infty$ — together with the vertex $H_\Gamma:L^1\times L^\infty\to L^1$. The companion (sub)bilinear maximal function $M_\Gamma(f,g)(x)=\sup_{\varepsilon>0}\frac{1}{2\varepsilon}\int_{-\varepsilon}^{\varepsilon}|f(x-t)\,g(x+\gamma(t))|\,dt$ was subsequently settled completely by Gaitan and Lie: it is bounded on the full range $1<p,q\le\infty$, $1\le r\le\infty$, failing exactly at the corner vertices $(1,\infty,1)$ and $(\infty,1,1)$ — which isolates the singular-integral endpoints as the one remaining gap. That endpoints are genuinely delicate here is underscored by the classical flat case, where the optimality of the Lacey–Thiele range is still open and near-boundary estimates are known to carry at worst double-logarithmic losses whose complete removal is itself open (Di Plinio–Thiele).

Problem Statement

Let $\mathcal{NF}^C$ be the class of smooth non-flat curves near zero and infinity defined by Lie (2018) (containing all real polynomials with no constant and no linear term, finite real combinations of powers $t^\alpha$ with $\alpha\in(0,\infty)\setminus\{1\}$, and combinations $|t|^\alpha|\log|t||^\beta$ with $\alpha\notin\{-1,0,1\}$, up to additive constants), and for $\gamma\in\mathcal{NF}^C$ let

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

It is known that $H_\Gamma$ maps $L^p(\mathbb{R})\times L^q(\mathbb{R})\to L^r(\mathbb{R})$ boundedly for all Hölder exponents $1/p+1/q=1/r$ with $1<p<\infty$, $1<q\le\infty$, $1\le r<\infty$, and that $r\ge 1$ is optimal for the class as a whole; the point $(\infty,\infty,\infty)$ (i.e. $L^\infty\times L^\infty\to L^\infty$) is known to fail. Determine the endpoint behavior on the remaining boundary of the closed Banach triangle, for every $\gamma\in\mathcal{NF}^C$:

1. Prove or disprove that $H_\Gamma$ maps $L^\infty(\mathbb{R})\times L^q(\mathbb{R})\to L^q(\mathbb{R})$ boundedly for each $1<q<\infty$, and decide the endpoint $(p,q,r)=(\infty,1,1)$, i.e. $L^\infty\times L^1\to L^1$, at least in weak type (i.e. boundedness into the weak space $L^{1,\infty}$).
2. Prove or disprove the vertex bound $H_\Gamma:L^1(\mathbb{R})\times L^\infty(\mathbb{R})\to L^1(\mathbb{R})$, in strong or weak type.

A complete answer decides, at every boundary point of the closed Banach triangle $\{(1/p,1/q,1/r'): 1/p+1/q+1/r'=1,\ 1\le p,q,r\le\infty\}$, whether $H_\Gamma$ is bounded for each curve class member, jointly completing the $L^p$-mapping theory of the curved bilinear Hilbert transform on $\mathcal{NF}^C$; the two items above are the two undecided pieces of that single boundary. Bounds may depend on the curve $\gamma$, as in the interior theorem.

References

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