Bilinear Bochner–Riesz means: the $\delta=0$ bilinear disc multiplier in one dimension and the $L^2\times L^\infty\to L^2$ smoothness threshold in higher dimensions

The bilinear Bochner–Riesz means $S^\delta_R$ — the bilinear Fourier multiplier on $\mathbb{R}^n\times\mathbb{R}^n$ with symbol $(1-(|\xi|^2+|\eta|^2)/R^2)^\delta_+$ — govern the spherical summability of products of Fourier series, and their $L^{p_1}\times L^{p_2}\to L^p$ boundedness in the scaling-critical Hölder range $1/p=1/p_1+1/p_2$ is a recognized open problem in multilinear harmonic analysis. The open questions left by Bernicot, Grafakos, Song and Yan (J. d'Analyse Math. 2015) are: (a) in dimension $n=1$, whether the $\delta=0$ means — the bilinear disc multiplier — are bounded in the part of the Banach triangle where exactly one of $p_1,p_2,p'$ lies in $(1,2)$; (b) in dimensions $n\ge2$, whether the means are bounded from $L^2\times L^\infty$ to $L^2$ for the intermediate smoothness $0<\delta\le(n-1)/2$; and (c) whether the known sufficient $\delta$-thresholds for generic exponents are sharp.

Background

For a Schwartz function $f$ on $\mathbb{R}^n$ (i.e. a smooth function all of whose derivatives decay faster than any power), the (linear) Bochner–Riesz means of order $\delta\ge0$ are the Fourier multiplier operators with symbol $(1-|\xi|^2/R^2)_+^{\delta}$, where $(t)_+=\max(t,0)$ and $R>0$; they model summation of Fourier integrals and series over balls with a smoothing weight that fades at the boundary. The classical Bochner–Riesz problem asks for the smallest $\delta$ such that these means are bounded on $L^p(\mathbb{R}^n)$ uniformly in $R$; at $\delta=0$ the symbol is the characteristic function of the ball, and Fefferman's celebrated counterexample shows this ball multiplier is unbounded on $L^p(\mathbb{R}^n)$ for every $p\ne2$ whenever $n\ge2$.

The bilinear analogue is motivated by the spherical summability of products of Fourier series: the norm convergence, as $R\to\infty$, of $\sum_{|m|^2+|k|^2\le R^2}\hat f(m)\hat g(k)e^{2\pi i(m+k)\cdot x}$ to $f(x)g(x)$ reduces to uniform-in-$R$ bounds for the bilinear Bochner–Riesz means

$$S^\delta_R(f,g)(x)=\iint e^{2\pi ix\cdot(\xi+\eta)}\Big(1-\frac{|\xi|^2+|\eta|^2}{R^2}\Big)^{\delta}_+\hat f(\xi)\hat g(\eta)\,d\xi\,d\eta,$$

i.e. the bilinear multiplier with symbol $m^\delta(\xi,\eta)=(1-|\xi|^2-|\eta|^2)_+^\delta$, which is singular along the sphere $\{|\xi|^2+|\eta|^2=1\}$, a hypersurface of dimension $2n-1$ inside $\mathbb{R}^{2n}$. The central question is $L^{p_1}\times L^{p_2}\to L^p$ boundedness under the Hölder relation $1/p=1/p_1+1/p_2$, which is the scaling-critical case and allows $p$ as small as $1/2$. The Banach triangle is the part of this exponent simplex with $p_1,p_2,p\ge1$ (equivalently $p_1,p_2,p'\ge1$, where $p'=p/(p-1)$ is the Hölder conjugate of $p$), and the local $L^2$ triangle is the subregion $p_1,p_2,p'\ge2$ — the exponent range familiar from the Lacey–Thiele theory of the bilinear Hilbert transform, the prototypical bilinear singular integral whose symbol is singular along a line rather than a curve.

The main landmarks are as follows. Grafakos and Li (2006) settled the $n=1$, $\delta=0$ case on the local $L^2$ range: the characteristic function of the disc is a bilinear multiplier for $2\le p_1,p_2,p'<\infty$, and it fails at the three vertices $(2,2,1)$, $(2,\infty,2)$, $(\infty,2,2)$. Diestel and Grafakos (2007) proved a Fefferman-type negative result in dimensions $n\ge2$: at $\delta=0$, boundedness fails whenever exactly one of $p_1,p_2,p'$ is less than $2$. Bernicot, Grafakos, Song and Yan (2015) then gave an essentially complete picture for $n=1$ and $\delta>0$ inside the Banach triangle (bounded in the strict local $L^2$ case already for $\delta\ge0$, at the endpoints and on the triangle boundary for $\delta>0$), and for $n\ge2$ proved: boundedness at $(2,2,1)$ for every $\delta>0$ (sharp); boundedness $L^2\times L^\infty\to L^2$ for $\delta>(n-1)/2$; boundedness $L^1\times L^\infty\to L^1$ for $\delta>n/2$; and sufficient conditions $\delta>n\alpha(p_1,p_2)-1$ for $1\le p_1,p_2<2n/(n+1)$. They also observed that $S^0$ fails at $L^2\times L^\infty\to L^2$ and that even smooth bilinear symbols need not give bounded operators, so the linear $L^2$ theory has no bilinear counterpart. Jeong, Lee and Vargas (2018) introduced a decomposition into products of linear Bochner–Riesz square functions (a square function here being the $\ell^2$-valued sum of the frequency-localized pieces of a function) and improved the thresholds for $p_1,p_2\ge2$, sharply at $(2,2,1)$ and, when $n=2$, on the square $[2,4]^2$; but at the point $(2,\infty)$ their threshold is still $(n-1)/2$. Work on the maximal operators $\sup_{R>0}|S^\delta_R|$ (Jeong–Lee 2020; Kaur–Shrivastava 2022) and on weighted estimates at the critical index $\delta=n-1/2$ requires even more smoothness and leaves the non-maximal cores untouched. What remains open at $\delta=0$ in $n=1$, at the endpoint $(2,\infty,2)$ in $n\ge2$, and for generic exponents in $n\ge2$ is the subject of the problem statement.

Problem Statement

Let $S^\delta_R$ be the bilinear Bochner–Riesz means on $\mathbb{R}^n$ and let $\delta_c(p_1,p_2,p;n)=\inf\{\delta\ge0:\sup_{R>0}\|S^\delta_R\|_{L^{p_1}\times L^{p_2}\to L^p}<\infty\}$ be the sharp smoothness threshold under the Hölder relation $1/p=1/p_1+1/p_2$ with $1\le p_1,p_2\le\infty$. Determine the boundedness region of the means, i.e. compute $\delta_c$ and the behavior at $\delta=\delta_c$, for the following loci left open by Bernicot–Grafakos–Song–Yan (2015) and still open in the literature:

(a) Dimension $n=1$, endpoint $\delta=0$: for every triplet with $1<p_1,p_2,p'<\infty$ such that exactly one of $p_1,p_2,p'$ lies in $(1,2)$ (the interior of the Banach triangle minus the local $L^2$ triangle), decide whether $\sup_{R>0}\|S^0_R\|_{L^{p_1}(\mathbb{R})\times L^{p_2}(\mathbb{R})\to L^p(\mathbb{R})}<\infty$ — equivalently, whether the characteristic function of the disc is a bilinear multiplier on that triplet.

(b) Dimensions $n\ge2$, the endpoint $(2,\infty,2)$ (and by symmetry $(\infty,2,2)$): determine $\delta_c(2,\infty,2;n)$. Known: $S^0$ is unbounded and $S^\delta$ is bounded for $\delta>(n-1)/2$; decide boundedness for every $0<\delta\le(n-1)/2$.

(c) Dimensions $n\ge2$, generic exponents: determine whether the known sufficient conditions — the Jeong–Lee–Vargas thresholds for $p_1,p_2\ge2$ (sharp only near $(2,2,1)$ and, for $n=2$, on $[2,4]^2$) and the Bernicot–Grafakos–Song–Yan condition $\delta>n\alpha(p_1,p_2)-1$ for $1\le p_1,p_2<2n/(n+1)$ — are sharp, by proving matching necessary conditions or improving the sufficient $\delta$-range for each Hölder triplet. Here, writing $a_n=\frac{n+1}{2n}$ and $b_n=\frac{n+1}{2n}+\frac{n-1}{n^2+n}$, the exponent $\alpha(p_1,p_2)$ is the piecewise function

$$\alpha(p_1,p_2)=\begin{cases}\tfrac{4}{n+1}, & (1/p_1,1/p_2)\in(a_n,b_n)\times(a_n,b_n),\\ \tfrac{2}{n+1}-\tfrac{n-1}{2n}+\tfrac{1}{p_2}, & (1/p_1,1/p_2)\in(a_n,b_n)\times[b_n,1],\\ \tfrac{2}{n+1}-\tfrac{n-1}{2n}+\tfrac{1}{p_1}, & (1/p_1,1/p_2)\in[b_n,1]\times(a_n,b_n),\\ \tfrac{1}{p_1}+\tfrac{1}{p_2}-\tfrac{n-1}{n}, & (1/p_1,1/p_2)\in[b_n,1]\times[b_n,1],\end{cases}$$

defined in the source paper (equation (3.10) there, with the infinitesimal parameter $\epsilon$ set to $0$).

At each locus, boundedness holds for $\delta>\delta_c$ and unboundedness for $\delta<\delta_c$, and the behavior at $\delta=\delta_c$ itself is left to be decided; the common objective is the sharp $L^{p_1}\times L^{p_2}\to L^p$ boundedness region of $S^\delta$, of which (a)–(c) are the concrete unfinished parts in the Banach range.

References

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