Elliptic extension conjecture above rectangles

Let $d\geq 1$, let $\ell\in(0,\infty]^d$, and write

$$A^\ell(\xi_1,\ldots,\xi_d):=(l_1\xi_1,\ldots,l_d\xi_d),\qquad \mathbf{1}:=(1,\ldots,1).$$

For a function $g$ elliptic over $Q^\ell$, define the extension operator

$$\mathcal{E}^\ell_g f(t,x):=\int_{Q^\ell}e^{i(t,x)(g(\xi),\xi)}f(\xi)\,d\xi.$$

Here ellipticity means that $g(\xi)=|\xi|^2+h(\xi)$, with $h(0)=0$, $\nabla h(0)=0$, $D^2h(0)=0$, and the stated uniform $C^N$ smallness condition on $(D^2h)\circ A^{\tilde\ell}$.

Elliptic extension conjecture. For $N$ sufficiently large, $\varepsilon_0>0$ sufficiently large, and $1\leq p,q\leq\infty$ satisfying

$$q=\frac{d+2}{d}p'>p,$$

there exists $C_{p,q,d}<\infty$, independent of $\ell$ and $g$, such that

$$\|\mathcal{E}^\ell_g\|_{L^p\to L^q}\leq C_{p,q,d}$$

for every $\ell\in(0,\infty]^d$ and every $g$ elliptic over $Q^\ell$ with parameters $N,\varepsilon_0$.

The conjecture extends the expected restriction estimate for elliptic hypersurfaces to surfaces elliptic over rectangles, with constants uniform in the sidelengths. It is verified when $d=1$ by Fefferman–Stein and Zygmund, and the paper proves it in the bilinear range $q>\frac{2(d+3)}{d+1}$; the full stated range remains open.