Ana Vargas
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Ana Vargas has a total of 19 co-authors. Here are the top co-authors by number of shared publications:
- Detlef Müller
Christian-Albrechts-Universität zu Kiel
8 shared publications - Stefan Buschenhenke
Mathematisches Seminar, C.A.-Universität Kiel, Kiel, Germany
8 shared publications - Anthony Carbery
University of Edinburgh
3 shared publications - Fernando Soria
Universidad Autónoma de Madrid
3 shared publications - Terence Tao
Department of Mathematics, University of California, Los Angeles, USA
3 shared publications - Jonathan Bennett
School of Mathematics, The University of Birmingham, Birmingham, UK
2 shared publications
Latest Publications
- Open Access
- 27 August 2022
- Published by Springer Nature in Mathematische Annalen
- Vol. 387 (1-2) , 17-56
Fourier restriction for smooth hyperbolic 2-surfaces
We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in $$\mathbb {R}^3.$$ 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.- 27 August 2022
- Published by Springer Nature in Mathematische Annalen
- Vol. 387 (1-2) , 17-56
Open AccessFourier restriction for smooth hyperbolic 2-surfaces
We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in $$\mathbb {R}^3.$$ 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. - Open Access
- 7 February 2022
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 301 (2) , 1913-1938
A Fourier restriction theorem for a perturbed hyperbolic paraboloid: polynomial partitioning
We consider a surface with negative curvature in $${{\mathbb {R}}}^3,$$ 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 one-variate perturbations, and we hope that the present paper will further help to pave the way for the study of general perturbations of the saddle by means of the polynomial partitioning method.- 7 February 2022
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 301 (2) , 1913-1938
Open AccessA Fourier restriction theorem for a perturbed hyperbolic paraboloid: polynomial partitioning
We consider a surface with negative curvature in $${{\mathbb {R}}}^3,$$ 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 one-variate perturbations, and we hope that the present paper will further help to pave the way for the study of general perturbations of the saddle by means of the polynomial partitioning method. - 1 July 2021
- Published by Springer Nature in The Journal of Geometric Analysis
- Vol. 31 (7) , 6941-6986
Partitions of Flat One-Variate Functions and a Fourier Restriction Theorem for Related Perturbations of the Hyperbolic Paraboloid
- 1 July 2021
- Published by Springer Nature in The Journal of Geometric Analysis
- Vol. 31 (7) , 6941-6986
Partitions of Flat One-Variate Functions and a Fourier Restriction Theorem for Related Perturbations of the Hyperbolic Paraboloid
- 24 February 2021
- Published by Springer Nature
- p. 193-222
On Fourier Restriction for Finite-Type Perturbations of the Hyperbolic Paraboloid
- 24 February 2021
- Published by Springer Nature
- p. 193-222
On Fourier Restriction for Finite-Type Perturbations of the Hyperbolic Paraboloid
- Open Access
Fourier restriction for smooth hyperbolic 2-surfaces
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.
Open AccessFourier restriction for smooth hyperbolic 2-surfaces
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.
- Open Access
A Fourier restriction theorem for a perturbed hyperbolic paraboloid: polynomial partitioning
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.Open AccessA Fourier restriction theorem for a perturbed hyperbolic paraboloid: polynomial partitioning
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. - 5 August 2019
- Published by Wiley in Proceedings of the London Mathematical Society
- Vol. 120 (1) , 124-154
A Fourier restriction theorem for a perturbed hyperbolic paraboloid
- 5 August 2019
- Published by Wiley in Proceedings of the London Mathematical Society
- Vol. 120 (1) , 124-154
A Fourier restriction theorem for a perturbed hyperbolic paraboloid
- Open Access
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 $h(y).$Open AccessOn 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 $h(y).$ - Open Access
A Fourier restriction theorem for a perturbed hyperbolic paraboloid
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.
Open AccessA Fourier restriction theorem for a perturbed hyperbolic paraboloid
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.
- Open Access
Improved bound for the bilinear Bochner-Riesz operator
We study $L^p\times L^q\to L^r$ bounds for the bilinear Bochner-Riesz operator $\mathcal{B}^\alpha$, $\alpha>0$ in $\mathbb{R}^d,$ $d\ge2$, which is defined by \[ {\mathcal B}^{\alpha}(f,g)=\iint_{\mathbb{R}^d\times\mathbb{R}^d} e^{2\pi i x\cdot(\xi+\eta)} (1-|\xi|^2-|\eta|^2 )^{\alpha}_+ ~ \widehat{f}(\xi)\,\widehat{g}(\eta)\,d\xi d\eta.\] We make use of a decomposition which relates the estimates for $\mathcal{B}^\alpha$ to those of the square function estimates for the classical Bochner-Riesz operators. In consequence, we significantly improve the previously known bounds.Open AccessImproved bound for the bilinear Bochner-Riesz operator
We study $L^p\times L^q\to L^r$ bounds for the bilinear Bochner-Riesz operator $\mathcal{B}^\alpha$, $\alpha>0$ in $\mathbb{R}^d,$ $d\ge2$, which is defined by \[ {\mathcal B}^{\alpha}(f,g)=\iint_{\mathbb{R}^d\times\mathbb{R}^d} e^{2\pi i x\cdot(\xi+\eta)} (1-|\xi|^2-|\eta|^2 )^{\alpha}_+ ~ \widehat{f}(\xi)\,\widehat{g}(\eta)\,d\xi d\eta.\] We make use of a decomposition which relates the estimates for $\mathcal{B}^\alpha$ to those of the square function estimates for the classical Bochner-Riesz operators. In consequence, we significantly improve the previously known bounds.