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João P. G. Ramos

Publications and source records attributed to João P. G. Ramos.

At least 37 records · Page 2Linked to original sources
arXiv2023

Sharp Gaussian decay for the one-dimensional harmonic oscillator

We prove a conjecture by Vemuri by proving sharp bounds on $\ell^κ$ sums of Hermite functions multiplied by an exponentially decaying factor. More explicitly, we prove that, for each $y>0,$ we have \[ \sum_{n \ge 1} |h_n(x)|^κ \frac{e^{-κn y}}{n^β} \ll_y x^{\frac{1}{2} - 2β} e^{-κx^2 \tanh(y)/2}, \] for all $x \in \mathbb{R}$ sufficiently large. Our proof involves the classical Plancherel-Rotach asymptotic formula for Hermite polynomials and a careful local analysis near the maximum point of such a bound.

math.CA↗
arXiv2023

A Faber-Krahn type inequality for log-subharmonic functions in the hyperbolic ball

Assume that $Δ_h$ is the hyperbolic Laplacian in the unit ball $\mathbb{B}$ and assume that $Φ_n$ is the unique radial solution of Poisson equation $Δ_h \log Φ_n =-4 (n-1)^2$ satisfying the condition $Φ_n(0)=1$ and $Φ_n(ζ)=0$ for $ζ\in \partial\mathbb{B}$. We explicitly solve the question of maximizing $$ R_n(f,Ω)= \frac{\int_Ω|f(x)|^2 Φ_n^α(|x|) \, dτ(x)}{\|f\|^2_{\mathbf{B}^2_α}}, $$ over all $f \in\mathbf{B}^2_α$ and $Ω\subset \mathbb{B}$ with $τ(Ω) = s,$ where $dτ$ denotes the invariant measure on $\mathbb{B},$ and $\|f\|_{{B}^2_α}^2 = \int_\mathbb{B} |f(x)|^2 Φ_n^α(|x|) dτ(x) < \infty.$ This result extends the main result of Tilli and the second author \cite{ramostilli} to a higher-dimensional context. Our proof relies on a version of the techniques used for the two-dimensional case, with several additional technical difficulties arising from the definition of the weights $Φ_n$ through hypergeometric functions. Additionally, we show that an immediate relationship between a concentration result for log-sunharmonic functions and one for the Wavelet transform is only available in dimension one.

math.AP↗
arXiv2022

Perturbed Fourier uniqueness and interpolation results in higher dimensions

We obtain new Fourier interpolation and -uniqueness results in all dimensions, extending methods and results by the first author and M. Sousa, and by the second author. We show that the only Schwartz function which, together with its Fourier transform, vanishes on surfaces close to the origin-centered spheres whose radius are square roots of integers, is the zero function. In the radial case, these surfaces are spheres with perturbed radii, while in the non-radial case, they can be graphs of continuous functions over the sphere. As an application, we translate our perturbed Fourier uniqueness results to perturbed Heisenberg uniqueness for the hyperbola, using the interrelation between these fields introduced and studied by Bakan, Hedenmalm, Montes-Rodriguez, Radchenko and Viazovska.

math.CA↗
arXiv2022

On Gaussian decay rates of harmonic oscillators and equivalences of related Fourier uncertainty principles

We make progress on a question by Vemuri on the optimal Gaussian decay of harmonic oscillators, proving the original conjecture up to an arithmetic progression of times. The techniques used are a suitable translation of the problem at hand in terms of the free Schrödinger equation, the machinery developed in the work of Cowling, Escauriaza, Kenig, Ponce and Vega , and a lemma which relates decay on average to pointwise decay. Such a lemma produces many more consequences in terms of equivalences of uncertainty principles. Complementing such results, we provide endpoint results in particular classes induced by certain Laplace transforms, both to the decay Lemma and to the remaining cases of Vemuri's conjecture, shedding light on the full endpoint question.

math.CA↗
arXiv2022

A Faber-Krahn inequality for wavelet transforms

For some special window functions $ψ_β \in H^2(\mathbb{C}^+),$ we prove that, over all sets $Δ\subset \mathbb{C}^+$ of fixed hyperbolic measure $ν(Δ),$ the ones over which the Wavelet transform $W_{\overline{ψ_β}}$ with window $\overline{ψ_β}$ concentrates optimally are exactly the discs with respect to the pseudohyperbolic metric of the upper half space. This answers a question raised by Abreu and Dörfler. Our techniques make use of a framework recently developed in a previous work by F. Nicola and the second author, but in the hyperbolic context induced by the dilation symmetry of the Wavelet transform. This leads us naturally to use a hyperbolic rearrangement function, as well as the hyperbolic isoperimetric inequality, in our analysis.

math.FA↗
arXiv2022

The Multilinear Spherical Maximal Function in one dimension

In dimension $n=1$ we obtain $L^{p_1}(\mathbb R) \times\dots\times L^{p_m}(\mathbb R)$ to $L^p(\mathbb R)$ boundedness for the multilinear spherical maximal function in the largest possible open set of indices and we provide counterexamples that indicate the optimality of our results.

math.CA↗
arXiv2022

Uniqueness of solutions to nonlinear Schrödinger equations from their zeros

We show novel types of uniqueness and rigidity results for Schrödinger equations in either the nonlinear case or in the presence of a complex-valued potential. As our main result we obtain that the trivial solution $u=0$ is the only solution for which the assumptions $u(t=0)\vert_{D}=0, u(t=T)\vert_{D}=0$ hold, where $D\subset \mathbb{R}^d$ are certain subsets of codimension one. In particular, $D$ is discrete for dimension $d=1$. Our main theorem can be seen as a nonlinear analogue of discrete Fourier uniqueness pairs such as the celebrated Radchenko--Viazovska formula, and the uniqueness result of the second author and M. Sousa for powers of integers. As an additional application, we deduce rigidity results for solutions to some semilinear elliptic equations from their zeros.

math.AP↗
arXiv2020

Uniform bounds for oscillatory and polynomial Carleson operators

We prove that a variety of oscillatory and polynomial Carleson operators are uniformly bounded on the family of parameters under considerations. As a particular application of our techniques, we prove uniform bounds for oscillatory Carleson operators near a single scale version of the quadratic Carleson operator.

math.CA↗
arXiv2020

On optimal autocorrelation inequalities on the real line

We study autocorrelation inequalities, in the spirit of Barnard and Steinerberger's work. In particular, we obtain improvements on the sharp constants in some of the inequalities previously considered by these authors, and also prove existence of extremizers to these inequalities in certain specific settings. Our methods consist of relating the inequalities in question to other classical sharp inequalities in Fourier analysis, such as the sharp Hausdorff--Young inequality, and employing functional analysis as well as measure theory tools in connection to a suitable dual version of the problem to identify and impose conditions on extremizers.

math.CA↗
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.

math.CA↗
arXiv2020

The Cauchy problem for the $L^2-$critical generalized Zakharov-Kuznetsov equation in dimension 3

We prove local well-posedness for the $L^2$ critical generalized Zakharov-Kuznetsov equation in $H^s, \, s \in (3/4,1).$ We also prove that the equation is "almost well-posedness" for initial data $u_0 \in H^s, \, s \in [1,2),$ in the sense that the solution belongs to a certain intersection $C([0,T] : H^s(\mathbb{R}^3)) \cap X^s_T$ and is unique within that class, where we can ensure continuity of the data-to-solution map in an only slightly larger space. We also prove that solutions satisfy the expected conservation of $L^2-$mass for the whole $s \in (3/4,2)$ range, and energy for $s \in (1,2).$ By a limiting argument, this implies, in particular, global existence for small initial data in $H^1.$ Finally, we study the question of almost everywhere (a.e.) convergence of solutions of the initial value problem to initial data.

math.AP↗
arXiv2020

Maximal function estimates and local well-posedness for the generalized Zakharov--Kuznetsov equation

We prove a high-dimensional version of the Strichartz estimates for the unitary group associated to the free Zakharov--Kuznetsov equation. As a by--product, we deduce maximal estimates which allow us to prove local well-posedness for the generalized Zakharov--Kuznetsov equation in the whole subcritical case whenever $d \ge 4, k \ge 4,$ complementing the recent results of Kinoshita and Herr--Kinoshita. Finally, we use some of those maximal estimates in order to prove pointwise convergence results for the flow of the generalized Zakharov--Kuznetsov equation in any dimension, in the same spirit of a recent manuscript by Compaan, Lucà and Staffilani.

math.AP↗
arXiv2020

Perturbed interpolation formulae and applications

We employ functional analysis techniques in order to deduce that some classical and recent interpolation results in Fourier analysis can be suitably perturbed. As an application of our techniques, we obtain generalizations of Kadec's 1/4-theorem for interpolation formulae in the Paley-Wiener space both in the real and complex case, as well as a perturbation result on the recent Radchenko-Viazovska interpolation result and the Cohn-Kumar-Miller-Radchenko-Viazovska result for Fourier interpolation with derivatives in dimensions 8 and 24. We also provide several applications of the main results and techniques, all relating to recent contributions in interpolation formulae and uniqueness sets for the Fourier transform.

math.CA↗
arXiv2020

On the endpoint behaviour of oscillatory maximal function

Inspired by a question of Lie, we study boundedness in subspaces of $L^1(\mathbb{R})$ of oscillatory maximal functions. In particular, we construct functions in $L^1(\mathbb{R})$ which are never integrable under action of our class of maximal functions. On the other hand, we prove that these maximal functions map certain classes of spaces resembling Sobolev spaces into $L^1(\mathbb{R})$ continuously under mild curvature assumptions on the phase $γ$.

math.CA↗
arXiv2020

Weak differentiability for fractional maximal functions of general $L^{p}$ functions on domains

Let $Ω\subset \mathbb{R}^{n}$ be bounded a domain. We prove under certain structural assumptions that the fractional maximal operator relative to $Ω$ maps $L^{p}(Ω) \to W^{1,p}(Ω)$ for all $p > 1$, when the smoothness index $α\geq 1$. In particular, the results are valid in the range $p \in (1, n/(n-1)]$ that was previously unknown. As an application, we prove an endpoint regularity result in the domain setting.

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