arXiv subjects

Javier Minguillón

Publications and source records attributed to Javier Minguillón.

3 recordsLinked to original sources

arXiv2026

A higher-dimensional Schrödinger maximal estimate along families of bi-Lipschitz curves

Let $Θ\subset\mathbb{R}^n$, $n\geq 1$ be compact with upper Minkowski dimension $β$, and let $γ(x,t,θ)$ be a family of curves which is bi-Lipschitz in the first variable and Lipschitz in the last two. We prove the maximal estimate $$\left\|\sup_{(τ,θ)\in[0,1)\timesΘ}\left|e^{iτΔ}f(γ(\,\cdot\,,τ,θ))\right|\right\|_{L^2(B^n_1)}\lesssim_s\|f\|_{H^s(\mathbb{R}^n)},$$ for every $s>\tfrac{n(β+1)}{2(n+1)}$, in the range $0\leβ\le n-1$. In the range $n-1<β\leq n$ we prove the threshold $s>\min\big\lbrace\fracβ2 + \frac 1{2(n+1)},\frac n2\big\rbrace$. The proof relies on a fractal $L^2$ estimate of Du--Zhang. We also use the temporal localization of Lee, the locally constant property, and a counting argument allowed by the Lipschitz properties of the curve.

math.CA↗

arXiv2026

On the Pointwise Convergence of Solutions to the Schrödinger Equation Along Certain Highly Tangential Curves

We investigate the Sobolev regularity required for almost everywhere convergence to the initial datum of solutions to the linear Schrödinger equation along certain tangential curves. In the regime $α<\tfrac12$, we analyze maximal estimates for expressions of the form $e^{itΔ}f(x+γ(t))$ over specific $α$-Hölder curves $γ$ and initial data $f\in H^s(\mathbb{R}^n)$. For the model family $γ(t)=(t^{α_1},\ldots,t^{α_n})$, where $α=\min_j α_j$, we show that the critical regularity is $s=\max\left\{\frac{1-2α}{2},\frac{n}{2(n+1)}\right\}.$

math.CA↗

arXiv2024

A Note on Almost Everywhere Convergence Along Tangential Curves to the Schrödinger Equation Initial Datum

In this short note, we give an easy proof of the following result: for $ n\geq 2, $ $\underset{t\to0}{\lim} \,e^{itΔ}f\left(x+γ(t)\right) = f(x) $ almost everywhere whenever $ γ$ is an $ α- $Hölder curve with $ \frac12\leq α\leq 1 $ and $ f\in H^s(\mathbb{R}^n) $, with $ s > \frac{n}{2(n+1)} $. This is the optimal range of regularity up to the endpoint.

math.CA↗