Local extension estimates for the hyperbolic hyperboloid in three dimensions

Restriction inequalities for the hyperbolic hyperboloid

In this article we establish new inequalities, both conditional and unconditional, for the restriction problem associated to the hyperbolic, or one-sheeted, hyperboloid in three dimensions, endowed with a Lorentz-invariant measure. These inequalities are unconditional (and optimal) in the bilinear range $q > \frac{10}{3}$.

On Fourier restriction for finite-type perturbations of the hyperboloid

In this note, we continue our research on Fourier restriction for hyperbolic surfaces, by studying local perturbations of the hyperbolic paraboloid $z=xy,$ which are of the form $z=xy+h(y),$ where $h(y)$ is a smooth function of finite type. Our results build on previous joint work in which we have studied the case $h(y)=y^3/3$ by means of the bilinear method. As it turns out, the understanding of that special case becomes also crucial for the treatment of arbitrary finite type perturbation terms $h(y).$

Fourier decay of fractal measures on hyperboloids

Let $μ$ be an $α$-dimensional probability measure. We prove new upper and lower bounds on the decay rate of hyperbolic averages of the Fourier transform $\widehatμ$. More precisely, if $\mathbb{H}$ is a truncated hyperbolic paraboloid in $\mathbb{R}^d$ we study the optimal $β$ for which $$\int_{\mathbb{H}} |\hatμ(Rξ)|^2 \, d σ(ξ)\leq C(α, μ) R^{-β}$$ for all $R > 1$. Our estimates for $β$ depend on the minimum between the number of positive and negative principal curvatures of $\mathbb{H}$; if this number is as large as possible our estimates are sharp in all dimensions.

Restriction estimates for hyperboloids in higher dimensions via bilinear estimates

Let $\mathbb{H}$ be a $(d-1)$-dimensonal hyperbolic paraboloid in $\mathbb{R}^d$ and let $Ef$ be the Fourier extension operator associated to $\mathbb{H},$ with $f$ supported in $B^{d-1}(0,2)$. We prove that $\|Ef\|_{L^p (B(0,R))} \leq C_εR^ε\|f\|_{L^p}$ for all $p \geq \frac{2(d+2)}{d}$ whenever $ \frac{d}{2} \geq m + 1$, where $m$ is the minimum between the number of positive and negative principal curvatures of $\mathbb{H}$. Bilinear restriction estimates for $\mathbb{H}$ proved by S. Lee and Vargas play an important role in our argument.

Bilinear identities involving the $k$-plane transform and Fourier extension operators

We prove certain $L^2(\mathbb{R}^n)$ bilinear estimates for Fourier extension operators associated to spheres and hyperboloids under the action of the $k$-plane transform. As the estimates are $L^2$-based, they follow from bilinear identities: in particular, these are the analogues of a known identity for paraboloids, and may be seen as higher-dimensional versions of the classical $L^2(\mathbb{R}^2)$-bilinear identity for Fourier extension operators associated to curves in $\mathbb{R}^2$.

三维双曲抛物面的 \(L^p\) 傅里叶扩展最大化器的存在性

Existence of maximizers for $L^p$ Fourier extension from the hyperbolic paraboloid

研究三维双曲抛物面 \(L^p\) 傅里叶扩展最大化器的存在性，证明最大化器存在且最大化序列子序列在模对称下收敛到最大化器。