arXiv subjects
Publications and source records attributed to Ana Vargas.
We prove $L^p\rightarrow L^4$ Fourier adjoint restriction estimates for three-dimensional quadratic surfaces in $\mathbb{R}^5$. These estimates are sharp up to the endpoint, and improve results by Oberlin and Guo--Oh.
We investigate the Sobolev regularity required for almost everywhere convergence to the initial datum of solutions to the linear Schr\"odinger equation along certain tangential curves. In the regime $\alpha<\tfrac12$, we analyze maximal estimates for expressions of the form $e^{it\Delta}f(x+\gamma(t))$ over specific $\alpha$-H\"older curves $\gamma$ and initial data $f\in H^s(\mathbb{R}^n)$. For the model family $\gamma(t)=(t^{\alpha_1},\ldots,t^{\alpha_n})$, where $\alpha=\min_j \alpha_j$, we show that the critical regularity is $s=\max\left\{\frac{1-2\alpha}{2},\frac{n}{2(n+1)}\right\}.$
We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in threedimensional euclidean space. Our approach exploits in a crucial way the underlying hyperbolic geometry, which leads to a novel notion of strong transversality and corresponding "exceptional" sets. For the division of these exceptional sets we make crucial and perhaps surprising use of a lemma on level sets for sufficiently smooth one-variate functions from a previous article of ours.
We consider a surface with negative curvature in $\Bbb R^3$ which is a cubic perturbation of the saddle. For this surface, we prove a new restriction theorem, analogous to the theorem for paraboloids proved by L. Guth in 2016. This specific perturbation has turned out to be of fundamental importance also to the understanding of more general classes of perturbations.
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 which is flat at the origin. The case of perturbations of finite type had already been handled before, but the flat case imposes several new obstacles. By means of a decomposition into intervals on which $|h'''|$ is of a fixed size $λ,$ we can apply methods devised in preceding papers, but since we loose control on higher order derivatives of $h$ we are forced to rework the bilinear method for wave packets that are only slowly decaying. Another problem lies in the passage from bilinear estimates to linear estimates, for which we need to require some monotonicity of $h'''.$
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 $h(y).$
In contrast to elliptic surfaces, the Fourier restriction problem for hypersurfaces of non-vanishing Gaussian curvature which admit principal curvatures of opposite signs is still hardly understood. In fact, even for 2-surfaces, the only case of a hyperbolic surface for which Fourier restriction estimates could be established that are analogous to the ones known for elliptic surfaces is the hyperbolic paraboloid or "saddle" z = xy. The bilinear method gave here sharp results for p > 10/3 (Lee 05, Vargas 05, Stovall 17), and this result was recently improved to p > 3.25 (Cho-Lee 17, Kim 17). This paper aims to be a first step in extending those results to more general hyperbolic surfaces. We consider a specific cubic perturbation of the saddle and obtain the sharp result, up to the end-point, for p > 10/3. In the application of the bilinear method, we show that the behavior at small scale in our surface is drastically different from the saddle. Indeed, as it turns out, in some regimes the perturbation term assumes a dominant role, which necessitates the introduction of a number of new techniques that should also be useful for the study of more general hyperbolic surfaces.
We study $L^p\times L^q\to L^r$ bounds for the bilinear Bochner-Riesz operator $\mathcal{B}^α$, $α>0$ in $\mathbb{R}^d,$ $d\ge2$, which is defined by \[ {\mathcal B}^α(f,g)=\iint_{\mathbb{R}^d\times\mathbb{R}^d} e^{2πi x\cdot(ξ+η)} (1-|ξ|^2-|η|^2 )^α_+ ~ \widehat{f}(ξ)\,\widehat{g}(η)\,dξdη.\] We make use of a decomposition which relates the estimates for $\mathcal{B}^α$ to those of the square function estimates for the classical Bochner-Riesz operators. In consequence, we significantly improve the previously known bounds.
The problem of $L^p(R^3)\to L^2(S)$ Fourier restriction estimates for smooth hypersurfaces S of finite type in R^3 is by now very well understood for a large class of hypersurfaces, including all analytic ones. In this article, we take up the study of more general $L^p(R^3)\to L^q(S)$ Fourier restriction estimates, by studying a prototypical class of two-dimensional surfaces with strongly varying curvature conditions. Our approach is based on an adaptation of the so-called bilinear method. We discuss several new features arising in the study of this problem.
In this paper, we show that any solution of the nonlinear Schr{ö}dinger equation $iu\_t+Δu\pm|u|^\frac{4}{N}u=0,$ which blows up in finite time, satisfies a mass concentration phenomena near the blow-up time. Our proof is essentially based on the Bourgain's one~\cite{MR99f:35184}, which has established this result in the bidimensional spatial case, and on a generalization of Strichartz's inequality, where the bidimensional spatial case was proved by Moyua, Vargas and Vega~\cite{MR1671214}. We also generalize to higher dimensions the results in Keraani~\cite{MR2216444} and Merle and Vega~\cite{MR1628235}.
Bilinear restriction estimates have been appeared in work of Bourgain, Klainerman, and Machedon. In this paper we develop the theory of these estimates (together with the analogues for Kakeya estimates). As a consequence we improve the $(L^p,L^p)$ spherical restriction theorem of Wolff from $p > 42/11$ to $p > 34/9$, and also obtain a sharp $(L^p,L^q)$ spherical restriction theorem for $q > 4 - 5/27$.