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Gian Maria Dall'Ara

Publications and source records attributed to Gian Maria Dall'Ara.

26 records · Page 2Linked to original sources
arXiv2019

A robust approach to sharp multiplier theorems for Grushin operators

We prove a multiplier theorem of Mihlin-Hörmander type for operators of the form $-Δ_x - V(x) Δ_y$ on $\mathbb{R}^{d_1}_x \times \mathbb{R}^{d_2}_y$, where $V(x) = \sum_{j=1}^{d_1} V_j(x_j)$, the $V_j$ are perturbations of the power law $t \mapsto |t|^{2σ}$, and $σ\in (1/2,\infty)$. The result is sharp whenever $d_1 \geq σd_2$. The main novelty of the result resides in its robustness: this appears to be the first sharp multiplier theorem for nonelliptic subelliptic operators allowing for step higher than two and perturbation of the coefficients. The proof hinges on precise estimates for eigenvalues and eigenfunctions of one-dimensional Schrödinger operators, which are stable under perturbations of the potential.

math.AP↗
arXiv2019

Exponential decay of Bergman kernels on complete Hermitian manifolds with Ricci curvature bounded from below

Given a smooth positive measure $μ$ on a complete Hermitian manifold with Ricci curvature bounded from below, we prove a pointwise Agmon-type bound for the corresponding Bergman kernel, under rather general conditions involving the coercivity of an associated complex Laplacian on $(0,1)$-forms. Thanks to an appropriate version of the Bochner--Kodaira--Nakano basic identity, we can give explicit geometric sufficient conditions for such coercivity to hold. Our results extend several known bounds in the literature to the case in which the manifold is neither assumed to be Kähler nor of "bounded geometry". The key ingredients of our proof are a localization formula for the complex Laplacian (of the kind used in the theory of Schrödinger operators) and a mean value inequality for subsolutions of the heat equation on Riemannian manifolds due to Li, Schoen, and Tam. We also show in an appendix that the so-called "twisted basic identities" are standard basic identities with respect to conformally Kähler metrics.

math.CV↗
arXiv2017

On noncompactness of the $\overline\partial$-Neumann problem on pseudoconvex domains in $\mathbb{C}^3$

In this paper we deal with the following question: is it true that any bounded smooth pseudoconvex domain in $\mathbb{C}^n$ whose boundary contains a $q$-dimensional complex manifold $M$ necessarily has a noncompact $\overline\partial$-Neumann operator $N_q$ ($1\leq q\leq n-1$)? We prove that a smooth bounded pseudoconvex domain $Ω\subseteq\mathbb{C}^3$ with a one-dimensional complex manifold $M$ in its boundary has a noncompact Neumann operator on $(0,1)$-forms, under the additional assumption that $bΩ$ has finite regular D'Angelo $2$-type at a point of $M$, improving previous results of Fu, Şahutoğlu, and Straube.

math.CV↗
arXiv2015

Discreteness of the spectrum of Schrödinger operators with non-negative matrix-valued potentials

We prove three results giving sufficient and/or necessary conditions for discreteness of the spectrum of Schrödinger operators with non-negative matrix-valued potentials, i.e., operators acting on $ψ\in L^2(\mathbb{R}^n,\mathbb{C}^d)$ by the formula $H_Vψ:=-Δψ+Vψ$, where the potential $V$ takes values in the set of non-negative Hermitian $d\times d$ matrices. The first theorem provides a characterization of discreteness of the spectrum when the potential $V$ is in a matrix-valued $A_\infty$ class, thus extending a known result in the scalar case ($d=1$). We also discuss a subtlety in the definition of the appropriate matrix-valued $A_\infty$ class. The second result is a sufficient condition for discreteness of the spectrum, which allows certain degenerate potentials, i.e., such that $\det(V)\equiv0$. To formulate the condition, we introduce a notion of oscillation for subspace-valued mappings. Our third and last result shows that if $V$ is a $2\times2$ real polynomial potential, then $-Δ+V$ has discrete spectrum if and only if the scalar operator $-Δ+λ$ has discrete spectrum, where $λ(x)$ is the minimal eigenvalue of $V(x)$.

math.SP↗
arXiv2015

Coercivity of weighted Kohn Laplacians: the case of model monomial weights in $\mathbb{C}^2$

The weighted Kohn Laplacian $\Box_φ$ is a natural second order elliptic operator associated to a weight $φ:\mathbb{C}^n\rightarrow\mathbb{R}$ and acting on $(0,1)$-forms, which plays a key role in several questions of complex analysis. We consider here the case of model monomial weights in $\mathbb{C}^2$, i.e., $ φ(z,w):=\sum_{(α,β)\inΓ}|z^αw^β|^2, $ where $Γ\subseteq \mathbb{N}^2$ is finite. Our goal is to prove coercivity estimates of the form $\Box_φ\geq μ^2$, where $μ:\mathbb{C}^n\rightarrow\mathbb{R}$ acts by pointwise multiplication on $(0,1)$-forms, and the inequality is in the sense of self-adjoint operators. We recently proved (arxiv.org:1502.00865) how to derive from $μ$-coercivity estimates for $\Box_φ$ pointwise bounds for the weighted Bergman kernel associated to $φ$. Here we introduce a technique to establish $μ$-coercivity with $ μ(z,w)=c(1+|z|^a+|w|^b) \qquad(a,b\geq0),$ where $a,b\geq0$ depend (and are easily computable from) $Γ$. As a corollary we also prove that, for a wide class of model monomial weights, the spectrum of $\Box_φ$ is discrete if and only if the weight is not decoupled, i.e. $Γ$ contains at least a point $(α,β)$ with $α\neq0\neqβ$. Our methods comprise a new holomorphic uncertainty principle and linear optimization arguments.

math.CV↗
arXiv2015

Pointwise estimates of weighted Bergman kernels in several complex variables

We prove new pointwise bounds for weighted Bergman kernels in $\mathbb{C}^n$, whenever a coercivity condition is satisfied by the associated weighted Kohn Laplacian on $(0,1)$-forms. Our results extend the ones obtained in $\mathbb{C}$ by Christ. Our main idea is to develop a version of Agmon theory (originally introduced to deal with Schrödinger operators) for weighted Kohn Laplacians on $(0,1)$-forms, inspired by the fact that these are unitarily equivalent to certain generalized Schrödinger operators.

math.CV↗
arXiv2015

Matrix Schrödinger Operators and Weighted Bergman Kernels

The aim of the present thesis is twofold: to study the problem of discreteness of the spectrum of Schrödinger operators with matrix-valued potentials in ${\mathbb R}^d$ (Chapter 1), and to prove new pointwise bounds for weighted Bergman kernels in ${\mathbb C}^n$ (chapters 2 and 3). These two themes are connected by the observation that the weighted Bergman kernel may be effectively studied by means of the analysis of the weighted Kohn Laplacian, which in turn is unitarily equivalent to a Schödinger operator with matrix-valued potential.

math.CV↗