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Diogo Oliveira e Silva

Publications and source records attributed to Diogo Oliveira e Silva.

At least 37 records · Page 2Linked to original sources
arXiv2021

Global maximizers for adjoint Fourier restriction inequalities on low dimensional spheres

We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler-Lagrange equation being smooth, a fact of independent interest which we establish in a companion paper. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{iξ\cdotω}$, for some $ξ$, thereby extending previous work of Christ & Shao to arbitrary dimensions $d\geq 2$ and general even exponents.

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arXiv2020

Smoothness of solutions of a convolution equation of restricted-type on the sphere

Let $\mathbb{S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb{R}^d$, $d\geq 2$, equipped with surface measure $σ_{d-1}$. An instance of our main result concerns the regularity of solutions of the convolution equation \[ a\cdot(fσ_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}}=f,\text{ a.e. on }\mathbb{S}^{d-1}, \] where $a\in C^\infty(\mathbb{S}^{d-1})$, $q\geq 2(d+1)/(d-1)$ is an integer, and the only a priori assumption is $f\in L^2(\mathbb{S}^{d-1})$. We prove that any such solution belongs to the class $C^\infty(\mathbb{S}^{d-1})$. In particular, we show that all critical points associated to the sharp form of the corresponding adjoint Fourier restriction inequality on $\mathbb{S}^{d-1}$ are $C^\infty$-smooth. This extends previous work of Christ & Shao to arbitrary dimensions and general even exponents, and plays a key role in a companion paper.

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arXiv2020

On regularity and mass concentration phenomena for the sign uncertainty principle

The sign uncertainty principle of Bourgain, Clozel & Kahane asserts that if a function $f:\mathbb{R}^d\to \mathbb{R}$ and its Fourier transform $\widehat{f}$ are nonpositive at the origin and not identically zero, then they cannot both be nonnegative outside an arbitrarily small neighborhood of the origin. In this article, we establish some equivalent formulations of the sign uncertainty principle, and in particular prove that minimizing sequences exist within the Schwartz class when $d=1$. We further address a complementary sign uncertainty principle, and show that corresponding near-minimizers concentrate a universal proportion of their positive mass near the origin in all dimensions.

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arXiv2020

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 $q > \frac{10}{3}$.

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arXiv2019

A Universality Law For Sign Correlations of Eigenfunctions of Differential Operators

We establish a universality law for sequences of functions $\{w_n\}_{n \in \mathbb{N}}$ satisfying a form of WKB approximation on compact intervals. This includes eigenfunctions of generic Schrödinger operators, as well as Laguerre and Chebyshev polynomials. Given two distinct points $x, y \in \mathbb{R}$, we ask how often do $w_n(x)$ and $w_n(y)$ have the same sign. Asymptotically, one would expect this to be true half the time, but this turns out to not always be the case. Under certain natural assumptions, we prove that, for all $x \neq y$, $$ \frac{1}{3} \leq \lim_{N \to \infty} \frac{1}{N} \# \left\{0 \leq n < N: \mathrm{sgn}(w_n(x)) = \mathrm{sgn}(w_n(y)) \right\} \leq \frac{2}{3}, $$ and that these bounds are optimal, and can be attained. Our methods extend to other questions of similar flavor and we also discuss a number of open problems.

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arXiv2018

Extremizers for adjoint Fourier restriction on hyperboloids: the higher dimensional case

We prove that in dimensions $d \geq 3$, the non-endpoint, Lorentz-invariant $L^2 \to L^p$ adjoint Fourier restriction inequality on the $d$-dimensional hyperboloid $\mathbb{H}^d \subseteq \mathbb{R}^{d+1}$ possesses maximizers. The analogous result had been previously established in dimensions $d=1,2$ using the convolution structure of the inequality at the lower endpoint (an even integer); we obtain the generalization by using tools from bilinear restriction theory.

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arXiv2018

On extremizers for Strichartz estimates for higher order Schrödinger equations

For an appropriate class of convex functions $ϕ$, we study the Fourier extension operator on the surface $\{(y, |y|^2+ϕ(y)):y\in\mathbb{R}^2\}$ equipped with projection measure. For the corresponding extension inequality, we compute optimal constants and prove that extremizers do not exist. The main tool is a new comparison principle for convolutions of certain singular measures that holds in all dimensions. Using tools of concentration-compactness flavor, we further investigate the behavior of general extremizing sequences. Our work is directly related to the study of extremizers and optimal constants for Strichartz estimates of certain higher order Schrödinger equations. In particular, we resolve a dichotomy from the recent literature concerning the existence of extremizers for a family of fourth order Schrödinger equations, and compute the corresponding operator norms exactly where only lower bounds were previously known.

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arXiv2018

Sharp Strichartz inequalities for fractional and higher order Schrödinger equations

We investigate a class of sharp Fourier extension inequalities on the planar curves $s=|y|^p$, $p>1$. We identify the mechanism responsible for the possible loss of compactness of nonnegative extremizing sequences, and prove that extremizers exist if $1 4$. In particular, this resolves the dichotomy of Jiang, Pausader & Shao concerning the existence of extremizers for the Strichartz inequality for the fourth order Schrödinger equation in one spatial dimension. One of our tools is a geometric comparison principle for $n$-fold convolutions of certain singular measures in $\mathbb{R}^d$, developed in a companion paper. We further show that any extremizer exhibits fast $L^2$-decay in physical space, and so its Fourier transform can be extended to an entire function on the whole complex plane. Finally, we investigate the extent to which our methods apply to the case of the planar curves $s=y|y|^{p-1}$, $p>1$.

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arXiv2018

A comparison principle for convolution measures with applications

We establish the general form of a geometric comparison principle for $n$-fold convolutions of certain singular measures in $\mathbb{R}^d$ which holds for arbitrary $n$ and $d$. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in a companion paper.

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arXiv2018

Sharp mixed norm spherical restriction

Let $d\geq 2$ be an integer and let $2d/(d-1) < q \leq \infty$. In this paper we investigate the sharp form of the mixed norm Fourier extension inequality \begin{equation*} \big\|\widehat{fσ}\big\|_{L^q_{\rm rad}L^2_{\rm ang}(\mathbb{R}^d)} \leq {\bf C}_{d,q}\, \|f\|_{L^2(\mathbb{S}^{d-1},{\rm d}σ)}, \end{equation*} established by L. Vega in 1988. Letting $\mathcal{A}_d \subset (2d/(d-1), \infty]$ be the set of exponents for which the constant functions on $\mathbb{S}^{d-1}$ are the unique extremizers of this inequality, we show that: (i) $\mathcal{A}_d$ contains the even integers and $\infty$; (ii) $\mathcal{A}_d$ is an open set in the extended topology; (iii) $\mathcal{A}_d$ contains a neighborhood of infinity $(q_0(d), \infty]$ with $q_0(d) \leq \left(\tfrac{1}{2} + o(1)\right) d\log d$. In low dimensions we show that $q_0(2) \leq 6.76\,;\,q_0(3) \leq 5.45 \,;\, q_0(4) \leq 5.53 \,;\, q_0(5) \leq 6.07$. In particular, this breaks for the first time the even exponent barrier in sharp Fourier restriction theory. The crux of the matter in our approach is to establish a hierarchy between certain weighted norms of Bessel functions, a nontrivial question of independent interest within the theory of special functions.

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arXiv2018

A sharp nonlinear Hausdorff-Young inequality for small potentials

The nonlinear Hausdorff-Young inequality follows from the work of Christ and Kiselev. Later Muscalu, Tao, and Thiele asked if the constants can be chosen independently of the exponent. We show that the nonlinear Hausdorff-Young quotient admits an even better upper bound than the linear one, provided that the function is sufficiently small in the $L^1$ norm. The proof combines perturbative techniques with the sharpened version of the linear Hausdorff-Young inequality due to Christ.

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arXiv2017

Extremizers for Fourier restriction on hyperboloids

The $L^2 \to L^p$ adjoint Fourier restriction inequality on the $d$-dimensional hyperboloid $\mathbb{H}^d \subset \mathbb{R}^{d+1}$ holds provided $6 \leq p < \infty$, if $d=1$, and $2(d+2)/d \leq p\leq 2(d+1)/(d-1)$, if $d\geq2$. Quilodrán recently found the values of the optimal constants in the endpoint cases $(d,p)\in\{(2,4),(2,6),(3,4)\}$ and showed that the inequality does not have extremizers in these cases. In this paper we answer two questions posed by Quilodrán, namely: (i) we find the explicit value of the optimal constant in the endpoint case $(d,p) = (1,6)$ (the remaining endpoint for which $p$ is an even integer) and show that there are no extremizers in this case; and (ii) we establish the existence of extremizers in all non-endpoint cases in dimensions $d \in \{1,2\}$. This completes the qualitative description of this problem in low dimensions.

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arXiv2017

A variational nonlinear Hausdorff-Young inequality in the discrete setting

Following the works of Lyons and Oberlin, Seeger, Tao, Thiele and Wright, we relate the variation of certain discrete curves on the Lie group $\text{SU}(1,1)$ to the corresponding variation of their linearized versions on the Lie algebra. Combining this with a discrete variational Menshov-Paley-Zygmund theorem, we establish a variational Hausdorff-Young inequality for a discrete version of the nonlinear Fourier transform on $\text{SU}(1,1)$.

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arXiv2017

Some recent progress on sharp Fourier restriction theory

The purpose of this note is to discuss several results that have been obtained in the last decade in the context of sharp adjoint Fourier restriction/Strichartz inequalities. Rather than aiming at full generality, we focus on several concrete examples of underlying manifolds with large groups of symmetries, which sometimes allows for simple geometric proofs. We mention several open problems along the way, and include an appendix on integration on manifolds using delta calculus.

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arXiv2017

Some Problems in Harmonic Analysis

In May 2016, we organized a conference in harmonic analysis in honor of Professor Michael Christ, on the campus of the University of Wisconsin in Madison. We are happy to present sixteen open problems, almost all of which were contributed by participants of a problem session held in the afternoon of May 19, 2016.

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arXiv2016

Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots

We study a recent result of Bourgain, Clozel and Kahane, a version of which states that a sufficiently nice function $f:\mathbb{R} \rightarrow \mathbb{R}$ that coincides with its Fourier transform and vanishes at the origin has a root in the interval $(c, \infty)$, where the optimal $c$ satisfies $0.41 \leq c \leq 0.64$. A similar result holds in higher dimensions. We improve the one-dimensional result to $0.45 \leq c \leq 0.594$, and the lower bound in higher dimensions. We also prove that extremizers exist, and have infinitely many double roots. With this purpose in mind, we establish a new structure statement about Hermite polynomials which relates their pointwise evaluation to linear flows on the torus, and applies to other families of orthogonal polynomials as well.

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