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Eugenia Malinnikova

Publications and source records attributed to Eugenia Malinnikova.

At least 19 recordsLinked to original sources
arXiv2026

Doubling inequalities and propagation of smallness for Schrödinger equations with singular potentials

We study quantitative unique continuation for solutions of the Schrö\-din\-ger equation with complex-valued singular potentials \(V\in L^t\), \(t>d/2\). We obtain a family of scale-invariant \(L^p\to L^q\) Carleman inequalities for the Laplacian and use them to prove the doubling inequalities with explicit dependence on the \(L^t\)-norm of the potential. These estimates are combined with multiscale arguments to obtain propagation of smallness from arbitrary measurable sets of positive measure. As an application of the main results, we prove an upper bound for the BMO norm for the logarithms of Dirichlet-Laplace eigenfunctions in simply connected planar Lipschitz domains.

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arXiv2025

Spectral inequalities for Schrödinger equations and quantitative propagation of smallness in the plane

This paper deals with spectral inequalities for one-dimensional Schrödinger operators with potentials bounded between two increasing functions (weights). The spectral inequality allows one to estimate the norm of a function with bounded spectrum by its values on a certain sensor set. We say that a measurable subset of the real line is thick if the measure of the intersection of this set with any interval of fixed length is bounded from below. First, we consider thick sensor sets a large class of pairs of weights. For potentials constrained between two polynomials, spectral inequalities for a broad class of so-called generalized thick sets are analyzed. A quantitative dependence of the constants in the spectral inequalities on the density of the sensor sets, the growth rate of the potentials, and the spectral interval is established. The proofs rely on a new quantitative propagation of smallness (or quantitative Cauchy uniqueness) for elliptic equations in the plane.

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arXiv2025

Hardy's Uncertainty principle for Schrödinger equations with quadratic Hamiltonians

Hardy's uncertainty principle is a classical result in harmonic analysis, stating that a function in $L^2(\mathbb{R}^d)$ and its Fourier transform cannot both decay arbitrarily fast at infinity. In this paper, we extend this principle to the propagators of Schrödinger equations with quadratic Hamiltonians, known in the literature as metaplectic operators. These operators generalize the Fourier transform and have captured significant attention in recent years due to their wide-ranging applications in time-frequency analysis, quantum harmonic analysis, signal processing, and various other fields. However, the involved structure of these operators requires careful analysis, and most results obtained so far concern special propagators that can basically be reduced to rescaled Fourier transforms. The main contributions of this work are threefold: (1) we extend Hardy's uncertainty principle, covering all propagators of Schrödinger equations with quadratic Hamiltonians, (2) we provide concrete examples, such as fractional Fourier transforms, which arise when considering anisotropic harmonic oscillators, (3) we suggest Gaussian decay conditions in certain directions only, which are related to the structure of the corresponding symplectic projection.

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arXiv2023

On a Bernstein inequality for eigenfunctions

Let $φ_λ$ be an eigenfunction of the Laplace-Beltrami operator on a smooth compact Riemannian manifold $(M,g)$, i.e., $Δ_g φ_λ + λφ_λ=0$. We show that $φ_λ$ satisfies a local Bernstein inequality, namely for any geodesic ball $B_g(x,r)$ in $M$ there holds: $\sup_{B_g(x,r)}|\nablaφ_λ|\leq C_δ\max\left\{\frac{\sqrtλ\log^{2+δ}λ}{r},λ\log^{2+δ}λ\right\}\sup_{B_g(x,r)}|φ_λ|$. We also prove analogous inequalities for solutions of elliptic PDEs in terms of the frequency function.

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arXiv2022

Dynamical versions of Hardy's uncertainty principle: A survey

The Hardy uncertainty principle says that no function is better localized together with its Fourier transform than the Gaussian. The textbook proof of the result, as well as one of the original proofs by Hardy, refers to the Phragmén-Lindelöf theorem. In this note we first describe the connection of the Hardy uncertainty to the Schrödinger equation, and give a new proof of Hardy's result which is based on this connection and the Liouville theorem. The proof is related to the second proof of Hardy, which has been underservedly forgotten. Then we survey the recent results on dynamical versions of Hardy's theorem.

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arXiv2021

On the three ball theorem for solutions of the Helmholtz equation

Let $u_k$ be a solution of the Helmholtz equation with the wave number $k$, $Δu_k+k^2 u_k=0$, on a small ball in either $\mathbb{R}^n$, $\mathbb{S}^n$, or $\mathbb{H}^n$. For a fixed point $p$, we define $M_{u_k}(r)=\max_{d(x,p)\le r}|u_k(x)|.$ The following three ball inequality $M_{u_k}(2r)\le C(k,r,α)M_{u_k}(r)^αM_{u_k}(4r)^{1-α}$ is well known, it holds for some $α\in (0,1)$ and $C(k,r,α)>0$ independent of $u_k$. We show that the constant $C(k,r,α)$ grows exponentially in $k$ (when $r$ is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds.

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arXiv2019

Review of Yau's conjecture on zero sets of Laplace eigenfunctions

This is a review of old and new results and methods related to the Yau conjecture on the zero set of Laplace eigenfunctions. The review accompanies two lectures given at the conference CDM 2018. We discuss the works of Donnelly and Fefferman including their solution of the conjecture in the case of real-analytic Riemannian manifolds. The review exposes the new results for Yau's conjecture in the smooth setting. We try to avoid technical details and emphasize the main ideas of the proof of Nadirashvili's conjecture. We also discuss two-dimensional methods to study zero sets.

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arXiv2019

Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three

Let $Δ_M$ be the Laplace operator on a compact $n$-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions $u:Δu + λu =0$. In dimension $n=2$ we refine the Donnelly-Fefferman estimate by showing that $H^1(\{u=0 \})\le Cλ^{3/4-β}$, $β\in (0,1/4)$. The proof employs the Donnelli-Fefferman estimate and a combinatorial argument, which also gives a lower (non-sharp) bound in dimension $n=3$: $H^2(\{u=0\})\ge cλ^α$, $α\in (0,1/2)$. The positive constants $c,C$ depend on the manifold, $α$ and $β$ are universal.

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arXiv2019

Lecture notes on quantitative unique continuation for solutions of second order elliptic equations

In these lectures we present some useful techniques to study quantitative properties of solutions of elliptic PDEs. Our aim is to outline a proof of a recent result on propagation of smallness. The ideas are also useful in the study of the zero sets of eigenfunctions of Laplace-Beltrami operator and we discuss the connection. Some basic facts about second order elliptic PDEs in divergent form are collected in the Appendix at the end of the notes.

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arXiv2018

Zeros of the Wigner Distribution and the Short-Time Fourier Transform

We study the question under which conditions the zero set of a (cross-) Wigner distribution W (f, g) or a short-time Fourier transform is empty. This is the case when both f and g are generalized Gaussians, but we will construct less obvious examples consisting of exponential functions and their convolutions. The results require elements from the theory of totally positive functions, Bessel functions, and Hurwitz polynomials. The question of zero-free Wigner distributions is also related to Hudson's theorem for the positivity of the Wigner distribution and to Hardy's uncertainty principle. We then construct a class of step functions S so that the Wigner distribution W (f, 1 (0,1)) always possesses a zero f $\in$ S $\cap$ L p for p < $\infty$, but may be zero-free for f $\in$ S $\cap$ L $\infty$. The examples show that the question of zeros of the Wigner distribution may be quite subtle and relate to several branches of analysis.

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arXiv2017

A discrete harmonic function bounded on a large portion of $\mathbb{Z}^2$ is constant

An improvement of the Liouville theorem for discrete harmonic functions on $\mathbb{Z}^2$ is obtained. More precisely, we prove that there exists a positive constant $\varepsilon$ such that if $u$ is discrete harmonic on $\mathbb{Z}^2$ and for each sufficiently large square $Q$ centered at the origin $|u|\le 1$ on a $(1-\varepsilon)$ portion of $Q$ then $u$ is constant.

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arXiv2017

Quantitative propagation of smallness for solutions of elliptic equations

Let $u$ be a solution to an elliptic equation $\text{div}(A\nabla u)=0$ with Lipschitz coefficients in $\mathbb{R}^n$. Assume $|u|$ is bounded by $1$ in the ball $B=\{|x|\leq 1\}$. We show that if $|u| < \varepsilon$ on a set $ E \subset \frac{1}{2} B$ with positive $n$-dimensional Hausdorf measure, then $$|u|\leq C\varepsilon^γ\text{ on } \frac{1}{2}B,$$ where $C>0, γ\in (0,1)$ do not depend on $u$ and depend only on $A$ and the measure of $E$. We specify the dependence on the measure of $E$ in the form of the Remez type inequality. Similar estimate holds for sets $E$ with Hausdorff dimension bigger than $n-1$. For the gradients of the solutions we show that a similar propagation of smallness holds for sets of Hausdorff dimension bigger than $n-1-c$, where $c>0$ is a small numerical constant depending on the dimension only.

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arXiv2017

Two types of Rubio de Francia operators on Triebel--Lizorkin and Besov spaces

We discuss generalizations of Rubio de Francia's inequality for Triebel--Lizorkin and Besov spaces, continuing the research from [5]. Two versions of Rubio de Francia's operator are discussed: it is shown that a rotation factor is needed for the boundedness of the operator in some smooth spaces while it is not essential in other spaces. We study the operators on some "end" spaces of the Triebel--Lizorkin scale and then use usual interpolation methods.

math.FA↗
arXiv2017

Sharp Uniqueness Results for Discrete Evolutions

We prove sharp uniqueness results for a wide class of one-dimensional discrete evolutions. The proof is based on a construction from the theory of complex Jacobi matrices combined with growth estimates of entire functions.

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arXiv2016

Ratios of harmonic functions with the same zero set

We study the ratio of harmonic functions $u,v$, which have the same zero set $Z$ in the unit ball $B\subset \mathbb{R}^n$. The ratio $f=u/v$ can be extended to a real analytic nowhere vanishing function in $B$. We prove the Harnack inequality and the gradient estimate for such ratios in any dimension: for a given compact set $K\subset B$ we show that $\sup_K|f|\le C_1\inf_K|f|$ and $\sup_K\left|\nabla f\right|\le C_2 \inf_K|f|$, where $C_1$ and $C_2$ depend on $K$ and $Z$ only. In dimension two we specify the dependence of the constants on $Z$ in these inequalities by showing that only the number of nodal domains of $u$, i.e. the number of connected components of $B\setminus Z$, plays a role.

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arXiv2015

An uncertainty principle and sampling inequalities in Besov spaces

We extend Strichartz's uncertainty principle [18] from the setting of the Sobolov space W 1,2 (R) to more general Besov spaces B 1/p p,1 (R). The main result gives an estimate from below of the trace of a function from the Besov space on a uniformly distributed discrete subset. We also prove the corresponding result in the multivariate case and discuss some applications to irregular approximate sampling in critical Besov spaces.

math.CA↗
arXiv2015

Uniqueness for discrete Schrodinger evolutions

We prove that if a solution of the discrete time-dependent Schrödinger equation with bounded real potential decays fast at two distinct times then the solution is trivial. For the free Shrödinger operator and for operators with compactly supported time-independent potentials a sharp analog of the Hardy uncertainty principle is obtained, using an argument based on the theory of entire functions. Logarithmic convexity of weighted norms is employed in the case of general real-valued time-dependent bounded potentials. In the latter case the result is not optimal.

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