Eugenia Malinnikova

Stanford University | SU · Department of Mathematics

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 Phragm\'en-Lindel\"of theorem. In this note we first describe the connection of the Hardy uncertainty to the Schr\"odinger eq...

Let $\varphi_{\lambda}$ be an eigenfunction of the Laplace-Beltrami operator on a smooth compact Riemannian manifold $(M,g)$, i.e., $\Delta_g \varphi_{\lambda} + \lambda \varphi_{\lambda}=0$. We show that $\varphi_{\lambda}$ satisfies a local Bernstein inequality, namely for any geodesic ball $B_g(x,r)$ in $M$ there holds: $\sup_{B_g(x,r)}|\nabla\v...

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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...

Let $$u_{k}$$ u k be a solution of the Helmholtz equation with the wave number k , $$\varDelta u_{k}+k^{2} u_{k}=0$$ Δ u k + k 2 u k = 0 , on (a small ball in) either $${\mathbb {R}}^{n}$$ R n , $${\mathbb {S}}^{n}$$ S n , or $${\mathbb {H}}^{n}$$ H n . For a fixed point p , we define $$M_{u_{k}}(r)=\max _{d(x,p)\le r}|u_{k}(x)|.$$ M u k ( r ) = ma...

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $C^{1}$ boundary and let $u_\lambda$ be a Dirichlet Laplace eigenfunction in $\Omega$ with eigenvalue $\lambda$. We show that the $(n-1)$-dimensional Hausdorff measure of the zero set of $u_\lambda$ does not exceed $C(\Omega)\sqrt{\lambda}$. This result is new even for the case of domains with...

Let $u_k$ be a solution of the Helmholtz equation with the wave number $k$, $\Delta u_k+k^2 u_k=0$, on (a small ball in) either $\mathbb{R}^n$, $\mathbb{S}^n$, or $\mathbb{H}^n$. For a fixed point $p$, we define \[M_{u_k}(r)=\max_{d(x,p)\le r}|u_k(x)|.\] It is known that the following three ball inequality \[M_{u_k}(2r)\le C(k,r,\alpha)M_{u_k}(r)^{...

Let (M n , g) be a closed n-dimensional Riemannian mani-fold, where g = (gij) is C 1-smooth metric. Consider the sequence of eigenfunctions u k of the Laplace operator on M. Let B be a ball on M. We prove that the number of nodal domains of u k that intersect B is not greater than C1 Volumeg(B) Volumeg(M) k + C2k n−1 n , where C1, C2 depend on M. T...

Let $(M, g)$ be a closed Riemannian manifold, where g is $C^1$-smooth metric. Consider the sequence of eigenfunctions $u_k$ of the Laplace operator on M. Let $B$ be a ball on $M$. We prove a sharp estimate of the number of nodal domains of $u_k$ that intersect $B$. The problem of local bounds for the volume and for the number of nodal domains was r...

Consider a solution $u$ to $\Delta u +Vu=0$ on $\mathbb{R}^2$, where $V$ is real-valued, measurable and $|V|\leq 1$. If $|u(x)| \leq \exp(-C |x| \log^{1/2}|x|)$, $|x|>2$, where $C$ is a sufficiently large absolute constant, then $u\equiv 0$.

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 th...

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 exp...

We discuss generalizations of Rubio de Francia’s inequality for Triebel–Lizorkin and Besov spaces, continuing the research from Osipov (Sb Math 205(7): 1004–1023, 2014) and answering Havin’s question to one of the authors. Two versions of Rubio de Francia’s operator are discussed: it is shown that exponential factors are needed for the boundedness...

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...

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 t...

Let ΔM be the Laplace operator on a compact n-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions u : ΔMu+λu = 0. In dimension n = 2 we refine the Donnelly–Fefferman estimate by showing that H1({u = 0}) ≤Cλ3/4−β for some β∈ (0, 1/4). The proof employs the Donnelly–Fefferman estimate and a combinatorial arg...

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 $Q$...

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^\gamma \text{ on } \f...

The purpose of this short note is to provide a new and very short proof of a result by Sudakov, offering an important improvement of the classical result by Kolmogorov-Riesz on compact subsets of Lebesgue spaces.

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.

We extend Strichartz’s uncertainty principle (Strichartz, J Funct Anal 84:97–114, 1989) from the setting of the Sobolev space \(W^{1,2}({\mathbb {R}})\) to more general Besov spaces \(B^{1/p}_{p,1}({\mathbb {R}})\). The main result gives an estimate from below of the trace of a function from the Besov space on a uniformly distributed discrete subse...

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 sh...

Let $\Delta_M$ be the Laplace operator on a compact $n$-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions $u:\Delta u + \lambda u =0$. In dimension $n=2$ we refine the Donnelly-Fefferman estimate by showing that $H^1(\{u=0 \})\le C\lambda^{3/4-\beta}$, $\beta \in (0,1/4)$. The proof employs the Donnelli-...

Let \({h_g^\infty}\) be the space of harmonic functions in the unit ball that are bounded by some increasing radial function that tends to infinity as r goes to one; these spaces are called growth spaces. We describe functions in growth spaces by the Cesàro means of their expansions in harmonic polynomials and apply this characterization to study c...

We prove that if a solution of the discrete time-dependent Schr\"odinger equation with bounded real potential decays fast at two distinct times then the solution is trivial. For the free Shr\"odinger operator and for operators with compactly supported time-independent potentials a sharp analog of the Hardy uncertainty principle is obtained, using a...

We prove that if a solution of the discrete time-dependent Schrödinger equation with bounded time-independent real potential decays fast at two distinct times then the solution is trivial. For the free Shrödinger operator or operators with compactly supported potential a sharp analog of the Hardy uncertainty principle is obtained. The argument is b...

We give an elementary argument to prove the Three Balls Theorem for continuous harmonic functions in \({\mathbb {R}}^n\) which can be adapted to the case of discrete harmonic functions on the lattice. The discrete analog of the Three Balls Theorem that we obtain contains an additional term that depends on the mesh size of the lattice and goes to ze...

We consider harmonic functions in the unit ball of $\mathbb{R}^{n+1}$ that are unbounded near the boundary but can be estimated from above by some (rapidly increasing) radial weight $w$. Our main result gives some conditions on $w$ that guarantee the estimate from below on the harmonic function by a multiple of this weight. In dimension two this re...

Spaces of harmonic functions in upper half-space with controlled growth near the boundary are described in terms of multiresolution approximations. The results are applied to prove the law of the iterated logarithm for the oscillation of harmonic functions along vertical lines.

Let $h_g^\infty$ be the space of harmonic functions in the unit ball that are bounded by some increasing radial function $g(r)$ with $\lim_{r\rightarrow 1} g(r)=+\infty$; these spaces are called growth spaces. We describe functions in growth spaces by the Ces\`aro means of their expansions in harmonic polynomials and apply this characterization to...

Let $u$ and $v$ be harmonic in $\Omega \subset \R^n$ functions with the same zero set $Z$. We show that the ratio $f$ of such functions is always well-defined and is real analytic. Moreover it satisfies the maximum and minimum principles. For $n=3$ we also prove the Harnack inequality and the gradient estimate for the ratios of harmonic functions,...

Let $K_\theta$ be a model space generated by an inner function $\theta$. We study the Schatten class membership of embeddings $I : K_\theta \to L^2(\mu)$, $\mu$ a positive measure, and of composition operators $C_\phi:K_\theta\to H^2(\mathbb D)$ with a holomprphic function $\phi:\mathbb D\rightarrow \mathbb D$. In the case of one-component inner fu...

We study the determining sets for discrete harmonic functions on the square lattices. The stability and regularization of the reconstruction of harmonic functions from its values on a part of a domain are discussed. For some specific configurations, we use the logarithmic convexity estimates to obtain error bounds and propose an optimal choice of t...

We prove a strong uncertainty principle for Riesz bases in L^2(R^d) and show that the orthonormal basis constructed by Bourgain possesses the optimal phase-space localization.

We study compactness property of composition operator acting from a model space generated by an inner function to the Hardy space.

We prove that there does not exist an orthonormal basis {b n } for L 2(R) such that the sequences {μ(b n )}, \(\{\mu(\widehat{b_{n}})\}\) , and \(\{\Delta(b_{n})\Delta(\widehat{b_{n}})\}\) are bounded. A higher dimensional version of this result that involves generalized dispersions is also obtained. The main tool is a time-frequency localization i...

We study harmonic functions which admit a certain majorant in the unit ball in $\R^m $. We prove that when the majorant fulfills a doubling condition, the extremal growth or decay may occur only along small sets of radii, and we give precise estimates of these exceptional sets.

We study radial behavior of harmonic functions in the unit disk belonging to the Korenblum class. We prove that functions which admit two-sided Korenblum estimate either oscillate or have slow growth along almost all radii.

We use a Carleman type inequality of Koch and Tataru to obtain quantitative estimates of unique continuation for solutions of second order elliptic equations with singular lower order terms. First we prove a three sphere inequality and then describe two methods of propagation of smallness from sets of positive measure.

We study radial behavior of analytic and harmonic functions, which admit a certain majorant in the unit disk. We prove that extremal growth or decay may occur only along small sets of radii and give precise estimates of these exceptional sets.

Let $u$ be a solution of a generalized Cauchy–Riemann system in $\mathbb{R}^n$. Suppose that $|u|\le1$ in the unit ball and $|u|\le\varepsilon$ on some closed set $E$. Classical results say that if $E$ is a set of positive Lebesgue measure, then $|u|\le C\varepsilon^\alpha$ on any compact subset of the unit ball. In the present work the same estima...

We give a description of measures on the unit sphere in R n that are orthogonal to the gradients of harmonic function. If n ≥ 3 such measure can be singular with respect to the Lebesgue measure. The work is inspired by the recent results of A. Cialdea and B. Gustafsson and D. Khavinson.

In certain classes of subharmonic functions u on C distinguished in terms of lower bounds for the Riesz measure of u, a sharp estimate is obtained for the rate of approximation by functions of the form log |f(z)|, where f is an entire function. The results complement and generalize those recently obtained by Yu. Lyubarskii and Eu. Malinnikova.

Generalizations of the Runge theorem and the Hartogs–Rosenthal theorem are given in the paper. We consider Riemannian manifold instead of the complex plane and harmonic differential forms instead of analytic complex functions. The role of rational functions is played by the elementary harmonic forms, which are introduced in the paper.

We study a generalization of the Hadamard theorem on three circles to harmonic differential forms. An inequality for the L 2-norms of a harmonic form over concentric spheres is proved. Also, we obtain an estimate for the L ∞-norms.

We say that a pair of closed subsets of R n admits separation of singularities of p-harmonic forms if each p-form that is harmonic off the union of the sets can be written as the sum of two forms each harmonic off one of the sets. The problem comes from the classical result of Aronszajn on separation of singularities for analytic functions that cor...