Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.07909

A Mizohata--Takeuchi-type estimate for the hyperbolic paraboloid

Jongchon Kim·Liang Wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove Mizohata--Takeuchi-type estimates for the Fourier extension operator associated with the truncated hyperbolic paraboloid in $\R^3$. Our main result controls the weighted $L^2$ norm of the extension operator by the $L^2$ mass of the weight over a multiscale family of parallelepipeds arising from $\ell^2$ decoupling for the hyperbolic paraboloid. We deduce this result from a weighted $L^2$ estimate for functions whose Fourier transforms are supported in a small neighborhood of the hyperbolic paraboloid, which we establish using a bilinear refined decoupling estimate.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jongchon Kim, Liang Wang. 2026-10-06. A Mizohata--Takeuchi-type estimate for the hyperbolic paraboloid. https://arxiv.org/abs/2610.07909

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Minimizing measures for the doubling condition

We study those measures whose doubling constant is the least possible among doubling measures on a given metric space. It is shown that such measures exist on every complete metric space supporting at least one doubling measure. In addition, a connection between minimizers for the doubling constant and superharmonic functions is exhibited. This allows us to show that for the particular case of the Euclidean space $\mathbb R^d$, Lebesgue measure is the only minimizer for the doubling constant (up to constant multiples) precisely when $d=1$ or $d=2$, while for $d\geq3$ there are infinitely many independent minimizers when considering the Euclidean metric. Analogously, in the discrete setting, we can show uniqueness of the counting measure as a minimizer for regular graphs where the standard random walk is a recurrent Markov chain. The counting measure is also shown to be a minimizer in every infinite graph where the cardinality of balls depends solely on their radii.

math.CA↗
arXiv2026

Capacitary Trapping and Brownian Escape: A Sharp Hardy Principle for Absorbing Media

We study Brownian motion killed upon exiting arbitrary open sets $Ω\subset\mathbb R^n$, $n\in[3,\infty)\cap\mathbb N$. Let $α\in(0,1]$. The infimal radius $d_α(x)$ at which the absorber near $x$ carries an $α$-fraction of the Newtonian capacity of the ball of the same radius provides a local measure of electrostatic trapping. We prove that this capacitary loading forces an absorption probability of at least $c_nα$ within a dimension-dependent multiple of the diffusive time scale $d_α(x)^2$. Independently, if $p\in (0,1)$ and $τ_p(x)$ is the time by which Brownian motion started at $x$ has been absorbed with probability at least $p$, then \begin{align*} \int_Ω\frac{|u\left(x\right)|^2}{τ_p\left(x\right)}\,\mathrm d x \le\frac{2}{p^2}\int_Ω|\nabla u\left(x\right)|^2\,\mathrm d x \end{align*} for every $u\in C_{\mathrm{c}}^\infty(Ω)$. Together these statements give a capacitary Hardy inequality with constant $C_nα^{-2}$ on every open set, answering a problem of Maz'ya, where $C_n$ is a positive constant and, moreover, the exponent of $α^{-2}$ is sharp. As applications, we derive corresponding estimates for survival probabilities, a corresponding spectral lower bound (including a recovery of the Maz'ya--Shubin lower bound), heat dissipation, Schrödinger forms, and Fisher information.

math.CA↗
arXiv2026

Sharp two-sided bounds for the product of modified Bessel functions and for its logarithmic derivative

We prove that, for all $x > 0$, the product of modified Bessel functions satisfies \[ \frac{1}{2\sqrt{x^2+ν^2+1/5}} -1$ and the upper bound for $ν\ge 1/2$, and that the sum of the logarithmic derivatives satisfies \[ -\frac{x}{x^2+ν^2-1}<\frac{I_ν'(x)}{I_ν(x)}+\frac{K_ν'(x)}{K_ν(x)}<-\frac{x}{x^2+ν^2+7/20}, \] where the upper bound holds for $ν>-1$ and the lower bound for $ν\ge 1$. All four constants are best possible. The four proofs share the same elementary structure: in each case the relevant auxiliary function satisfies a second-order differential inequality, and the inequality follows from the strong maximum principle. Among other applications, we obtain sharp monotonicity properties of the product and improved bounds for the ratios $I_{ν-1}(x)/I_ν(x)$ and $K_{ν+1}(x)/K_ν(x)$.

math.CA↗