Endpoint bounds and sharpness of the boundedness region for the bilinear fractional integral along curves

For a plane curve $(t,\gamma(t))$ satisfying the regularity hypothesis (H.) of Li, Yu and Zhao — monotone, with power-type growth at $0$ and $\infty$ quantified by four log-asymptotic exponents $\omega_{0,1},\omega_{0,2},\omega_{\infty,1},\omega_{\infty,2}$ — the bilinear fractional integral $H_{\alpha,\gamma}(f,g)(x)=\int_0^\infty f(x-t)g(x-\gamma(t))\,dt/t^{1-\alpha}$ ($0<\alpha<1$) and its linear companion $I_{\alpha,\gamma}$ were shown by those authors to be bounded $L^p\times L^q\to L^r$ whenever the exponent triple $(1/p,1/q,1/r)$ lies in the interior of two explicit polyhedral regions, together with restricted weak-type estimates at the $p=\infty$ and $q=\infty$ vertices and an almost sharp Hardy–Littlewood–Sobolev inequality for $I_{\alpha,\gamma}$: the $L^p\to L^q$ bound holds for $1/p-1/q=\alpha/M$ with $M\in(\omega_{0,2},\omega_{\infty,1})$ and can hold only for $M\in[\omega_{0,1},\omega_{\infty,2}]$. Three linked gaps remain open: the weak-type estimates near the $p=1$ and $q=1$ vertices carry an explicit $\varepsilon$-deficit in the target exponent, and it is unknown whether any estimate survives at the limiting vertices; no boundary theory exists for the two regions beyond the restricted weak-type estimates at the $p=\infty$ and $q=\infty$ vertices, so it is unknown whether boundedness extends to their closures and in which sense; and no necessity result is known for $H_{\alpha,\gamma}$, nor a closure of the sufficiency–necessity gap for $I_{\alpha,\gamma}$. Determine the exact boundedness region of these two operators under (H.) — endpoints included, necessity included.

Background

The classical bilinear fractional integral on $\mathbb{R}^n$ is

$$H_\alpha(f,g)(x):=\int_{\mathbb{R}^n}f(x-t)\,g(x+t)\,\frac{dt}{|t|^{n-\alpha}},\qquad 0<\alpha<n,$$

a bilinear analogue of the fractional integration behind the Hardy–Littlewood–Sobolev inequality, i.e. the statement that $f\mapsto\int_{\mathbb{R}^n}f(\cdot-y)|y|^{\alpha-n}dy$ maps $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ when $1/q=1/p-\alpha/n$. Kenig–Stein, Grafakos, and Grafakos–Kalton proved that $H_\alpha$ maps $L^p(\mathbb{R}^n)\times L^q(\mathbb{R}^n)$ to $L^r(\mathbb{R}^n)$ when $1/r=1/p+1/q-\alpha/n$ and the triple $(1/p,1/q,1/r)$ lies in the interior of an explicit pentagon, and that a restricted weak-type estimate holds at each vertex of that pentagon. Here a restricted weak-type $(p,q,r)$ estimate means the bound $\|T(\chi_A,\chi_B)\|_{L^{r,\infty}}\lesssim|A|^{1/p}|B|^{1/q}$ for characteristic functions of finite-measure sets, where $L^{r,\infty}$ is the weak $L^r$ (Lorentz) space with quasinorm $\|h\|_{L^{r,\infty}}=\sup_{\lambda>0}\lambda\,|\{x:|h(x)|>\lambda\}|^{1/r}$; exponent triples are written as points $(1/p,1/q,1/r)$ and boundedness regions as convex hulls of such points.

Replacing the fractional kernel by the principal-value kernel $dt/t$ gives the bilinear Hilbert transform, bounded $L^p\times L^q\to L^r$ for $1/p+1/q=1/r$ with $p,q>1$ and $r>2/3$ by Lacey and Thiele; extending this to $1/2<r\le 2/3$ is a famous open problem. Curved variants, in which the two arguments of the input functions are displaced along a plane curve $(t,\gamma(t))$ rather than by $\pm t$, form an active branch of time-frequency and oscillatory-integral analysis: Li proved the monomial-curve case $L^2\times L^2\to L^1$; Li–Xiao handled polynomial curves $P(t)$ with no constant or linear term, obtaining $r>(d-1)/d$ for degree $d$ (sharp up to the endpoint) and $r>1/2$ for homogeneous curves $t^d$; Lie treated the class of non-flat curves (NFC), later extended to the Banach-triangle range $r\ge 1$; Li–Yu established $r>1/2$ for a class of convex curves. Fractional integration along curves itself originated with Ricci–Stein's study of fractional integration along manifolds on nilpotent groups.

The source paper studies the fractional curved operator. For $0<\alpha<1$ set

$$H_{\alpha,\gamma}(f,g)(x):=\int_0^\infty f(x-t)\,g(x-\gamma(t))\,\frac{dt}{t^{1-\alpha}},\qquad I_{\alpha,\gamma}f(x):=\int_0^\infty|f(x-\gamma(t))|\,\frac{dt}{t^{1-\alpha}}.$$

The curve is required to satisfy Hypothesis (H.): $\gamma\in C^2(\mathbb{R}_+)$ is monotone with $\gamma(t)\to0$ and $\gamma'(t)\to0$ as $t\to0^+$, the log-asymptotic power exponents

$$\omega_{0,1}:=\liminf_{t\to0^+}\frac{\ln|\gamma(t)|}{\ln t},\quad\omega_{0,2}:=\limsup_{t\to0^+}\frac{\ln|\gamma(t)|}{\ln t},\quad\omega_{\infty,1}:=\liminf_{t\to\infty}\frac{\ln|\gamma(t)|}{\ln t},\quad\omega_{\infty,2}:=\limsup_{t\to\infty}\frac{\ln|\gamma(t)|}{\ln t}$$

satisfy $\omega_{\infty,1}\ge\omega_{0,2}$ and $\omega_{0,1}>1$, and there are positive constants $C_2^{(1)},C_1^{(1)},C_1^{(2)}$ such that $|t\gamma'(t)/\gamma(t)|\le C_2^{(1)}$ and $|t^j\gamma^{(j)}(t)/\gamma(t)|\ge C_1^{(j)}$ for $j=1,2$. Hypothesis (H.) is essentially satisfied by any polynomial with no constant or linear term, e.g. $\gamma(t)=t^\beta$ with $\beta>1$, in which case all four exponents collapse to $\beta$.

Li–Yu–Zhao prove that $H_{\alpha,\gamma}$ is bounded $L^p(\mathbb{R})\times L^q(\mathbb{R})\to L^r(\mathbb{R})$ whenever $(1/p,1/q,1/r)$ lies in the open convex hull $\mathcal{R}_1$ of eight explicit points whenever $\frac{\omega_{\infty,2}-\omega_{\infty,1}}{\omega_{\infty,2}-1}<\frac{\alpha}{1-\alpha}<\frac{\omega_{\infty,2}}{\omega_{\infty,2}-\omega_{\infty,1}+1}$, and in a second region $\mathcal{R}_2$ whenever $\frac{\alpha}{1-\alpha}\ge\omega_{\infty,1}$; the vertex sets are listed in the problem statement. The proof interpolates between restricted weak-type estimates at the $p=\infty$ and $q=\infty$ vertices (Proposition 2.4, with the Hardy–Littlewood–Sobolev parameter restricted to the open range $M\in(\omega_{0,2},\omega_{\infty,1})$) and weak-type estimates near the $p=1$ and $q=1$ vertices that carry an explicit $\varepsilon$-deficit in the target exponent: for example (Proposition 4.1), $\|H_{\alpha,\gamma}(f,g)\|_{L^{r(\varepsilon),\infty}}\le C\|f\|_{L^1}\|g\|_{L^{2r(\varepsilon)}}$ with $1/r(\varepsilon)=\frac{2(\omega_{\infty,2}-\alpha)}{2\omega_{\infty,2}-\omega_{\infty,1}}-\varepsilon$, the limiting value being exactly the vertex exponent; Propositions 5.1, 4.3 and 5.3 are the analogous statements at the other $p=1$/$q=1$ vertices. For the linear operator they prove an almost sharp Hardy–Littlewood–Sobolev inequality (Lemma 2.2): $\|I_{\alpha,\gamma}f\|_{L^q}\le C\|f\|_{L^p}$ holds if $1/p-1/q=\alpha/M$ with $M\in(\omega_{0,2},\omega_{\infty,1})$, while any such estimate forces $1/p-1/q=\alpha/N$ with $N\in[\omega_{0,1},\omega_{\infty,2}]$; for curves with pure power asymptotics at both ends the two ranges coincide and the condition is sharp (Remark 2.3). A technical novelty is the stationary point $t_0$ with $\gamma'(t_0)=1$, at which the change of variables $(x,t)\mapsto(x-t,x-\gamma(t))$ degenerates; under (H.) at most one such point exists and the critical dyadic scales (shells $t\approx 2^j$ containing it) require separate treatment.

What the paper leaves open is exactly the sharp picture: apart from the restricted weak-type estimates already available at the $p=\infty$ and $q=\infty$ vertices, no estimate is proved at any other boundary point of the two regions, the $\varepsilon$-deficits at the $p=1$/$q=1$ vertices are not removed, no necessity (unboundedness outside the regions, sharpness of the $\alpha$-ranges) is proved for $H_{\alpha,\gamma}$, and the sufficiency–necessity gap for $I_{\alpha,\gamma}$ is open for curves whose log-asymptotic exponents are not attained.

Problem Statement

Fix $0<\alpha<1$ and a curve $\gamma$ satisfying Hypothesis (H.) of Li-Yu-Zhao (as stated in the Background), and consider

$$H_{\alpha,\gamma}(f,g)(x):=\int_0^\infty f(x-t)\,g(x-\gamma(t))\,\frac{dt}{t^{1-\alpha}},\qquad I_{\alpha,\gamma}f(x):=\int_0^\infty|f(x-\gamma(t))|\,\frac{dt}{t^{1-\alpha}}.$$

Write points as exponent triples $(1/p,1/q,1/r)$. Let $\mathcal{R}_1$ be the open convex hull of the eight points $A(\alpha,0,0)$, $B(1,0,1-\alpha)$, $C_1(0,\alpha/\omega_{\infty,1},0)$, $C_2(0,\alpha/\omega_{0,2},0)$, $D_1(0,1,1-\alpha/\omega_{\infty,1})$, $D_2(0,1,1-\alpha/\omega_{0,2})$, $E\bigl(1,\frac{\omega_{\infty,2}-\alpha}{2\omega_{\infty,2}-\omega_{\infty,1}},\frac{2(\omega_{\infty,2}-\alpha)}{2\omega_{\infty,2}-\omega_{\infty,1}}\bigr)$ and $F\bigl(\frac{\omega_{\infty,2}-\alpha}{2\omega_{\infty,2}-\omega_{\infty,1}},1,\frac{2(\omega_{\infty,2}-\alpha)}{2\omega_{\infty,2}-\omega_{\infty,1}}\bigr)$. Let $\mathcal{R}_2$ be the open convex hull of $A,B,C_1,C_2,D_1,D_2$ together with $H\bigl(\frac{\omega_{\infty,1}-\alpha}{\omega_{\infty,1}},1,\frac{2(\omega_{\infty,1}-\alpha)}{\omega_{\infty,1}}\bigr)$ and, if $\omega_{\infty,1}\le2\alpha$, the point $G_1\bigl(1,\frac{1-\alpha}{2-\omega_{\infty,1}},\frac{2(1-\alpha)}{2-\omega_{\infty,1}}\bigr)$, or, if $2\alpha<\omega_{\infty,1}$, the two points $G_1'\bigl(1,\frac{\alpha}{\omega_{\infty,1}},1\bigr)$ and $G_2'\bigl(1,\frac{1}{2},1\bigr)$.

Determine the exact boundedness region of these two operators under Hypothesis (H.), completing the program of the source paper within its own setting:

(1) Endpoints and removal of the epsilon-losses (the highlighted boundary cases of part (2)). In the range $\frac{\omega_{\infty,2}-\omega_{\infty,1}}{\omega_{\infty,2}-1}<\frac{\alpha}{1-\alpha}<\frac{\omega_{\infty,2}}{\omega_{\infty,2}-\omega_{\infty,1}+1}$ (where strong type on $\mathcal{R}_1$ is known): does $H_{\alpha,\gamma}$ admit any estimate at vertex $E$, i.e. an $L^1\times L^q\to L^{r,\infty}$ or restricted weak-type bound with $1/q=\frac{\omega_{\infty,2}-\alpha}{2\omega_{\infty,2}-\omega_{\infty,1}}$ and $1/r=\frac{2(\omega_{\infty,2}-\alpha)}{2\omega_{\infty,2}-\omega_{\infty,1}}$ exactly (the epsilon-deficit removed), and symmetrically at vertex $F$? Analogously, in the range $\frac{\alpha}{1-\alpha}\ge\omega_{\infty,1}$ (region $\mathcal{R}_2$): what estimates, if any, hold at the limiting exponents of the $G$ and $H$ vertices?

(2) Boundary behavior of the regions. For each point of the boundaries of $\overline{\mathcal{R}_1}$ and $\overline{\mathcal{R}_2}$, determine whether $H_{\alpha,\gamma}:L^p\times L^q\to L^r$ is bounded and in which sense (strong type, weak type $L^{r,\infty}$, or restricted weak type); in particular, does strong-type boundedness extend from the open hulls to the closed convex hulls, or to which faces and vertices?

(3) Necessity and sharpness. For which triples outside $\overline{\mathcal{R}_1}$ (respectively $\overline{\mathcal{R}_2}$, in the respective alpha-regimes) does the strong-type bound fail? Are the restrictions on $\alpha$ relative to the exponents $\omega$ in the two regimes treated by the source theorems sharp? And close the gap in the Hardy-Littlewood-Sobolev inequality for $I_{\alpha,\gamma}$: for exactly which $M$ does $\|I_{\alpha,\gamma}f\|_{L^q}\lesssim\|f\|_{L^p}$ hold for $1<p<q<\infty$ with $1/p-1/q=\alpha/M$, given that sufficiency is known for $M\in(\omega_{0,2},\omega_{\infty,1})$ and necessity for $M\in[\omega_{0,1},\omega_{\infty,2}]$?

These three parts form one objective, the definitive boundedness picture for the two operators: the problem is fully answered when, for each operator, each of the two alpha-regimes above, and each exponent triple, either a boundedness estimate with its optimal type is proved or unboundedness is demonstrated. Constants may depend on the curve, on $\alpha$, and on the exponents, but not on the input functions.

References

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