Naiara Arrizabalaga
We prove some Hardy-Dirac inequalities with two different weights including measure valued and Coulombic ones. Those inequalities are used to construct distinguished self-adjoint extensions of Dirac operators for a class of diagonal potentials related to the weights in the above mentioned inequalities.
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Distinguished selfadjoint extensions of Dirac operators are constructed for a class of potentials including Coulombic ones up to the critical case, $-|x|^{-1}$. The method uses Hardy-Dirac inequalities and quadratic form techniques.
Distinguished selfadjoint extensions of operators which are not semibounded can be deduced from the positivity of the Schur Complement (as a quadratic form). In practical applications this amounts to proving a Hardy-like inequality. Particular cases are Dirac-Coulomb operators where distinguished selfadjoint extensions are obtained for the optimal range of coupling constants.
We consider the problem of essential self-adjointness of the drift-diffusion operator $H=-\frac{1}ρ\nabla\cdot ρ\mathbb D\nabla +V$ on domains $Ω\subset \mathbb{R}^d$ with $\mathcal{C}^2$-boundary $\partial Ω$ and for large classes of coefficients $ρ,\; \mathbb{D}$ and $V$. We give criteria showing how the behavior as $x \rightarrow \partial Ω$ of these coefficients balances to ensure essential self-adjointness of $H$. On the way we prove a weak anisotropic Hardy inequality which is of independent interest.
We show that allowing magnetic fields to be complex-valued leads to an improvement in the magnetic Hardy-type inequality due to Laptev and Weidl. The proof is based on the study of momenta on the circle with complex magnetic fields, which is of independent interest in the context of PT-symmetric and quasi-Hermitian quantum mechanics. We study basis properties of the non-self-adjoint momenta and derive closed formulae for the similarity transforms relating them to self-adjoint operators.
Let $L$ be a nonnegative, self-adjoint operator satisfying Gaussian estimates on $L^2(\RR^n)$. In this article we give an atomic decomposition for the Hardy spaces $ H^p_{L,max}(\R)$ in terms of the nontangential maximal functions associated with the heat semigroup of $L$, and this leads eventually to characterizations of Hardy spaces associated to $L$, via atomic decomposition or the nontangential maximal functions. The proofs are based on a modification of technique due to A. Calderón \cite{C}.
We consider the abstract non-negative self-adjoint operator $L$ acting on $L^2(X)$ which satisfies Davies-Gaffney estimates and the corresponding Hardy spaces $H^p_L(X)$. We assume that doubling condition holds for the metric measure space $X$. We show that a sharp Hörmander-type spectral multiplier theorem on $H^p_L(X)$ follows from restriction type estimates and the Davies-Gaffney estimates. We also describe the sharp result for the boundedness of Bochner-Riesz means on $H^p_L(X)$.
Let $X$ be a metric measure space with a doubling measure and $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$. Assume that $L$ generates an analytic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy Gaussian upper bounds but without any assumptions on the regularity of space variables $x$ and $y$. In this article we continue a study in \cite{SY} to give an atomic decomposition for the Hardy spaces $ H^p_{L,max}(X)$ in terms of the nontangential maximal function associated with the heat semigroup of $L$, and hence we establish characterizations of Hardy spaces associated to an …
In this work we construct self-adjoint extensions of the Dirac operator associated to Hermitian matrix potentials with Coulomb decay and prove that the domain is maximal. The result is obtained by means of a Hardy-Dirac type inequality. In particular, we can work with some electromagnetic potentials such that both, the electric potential and the magnetic one, have Coulomb type singularity.
Distinguished self-adjoint extension and eigenvalues of operators with gaps. Application to Dirac-Coulomb operators
针对兼具块分解与变分间隙性质的无界线性对称算子，提出仅需极弱假设的几何策略，定义带谱间隙的特异自伴延拓并通过极小极大原理刻画间隙内特征值，结果可应用于含变号势或多原子核的类Dirac-Coulomb哈密顿量。
This paper is devoted to the study of the two-dimensional Dirac-Coulomb operator in presence of an Aharonov-Bohm external magnetic potential. We characterize the highest intensity of the magnetic field for which a two-dimensional magnetic Hardy inequality holds. Up to this critical magnetic field, the operator admits a distinguished self-adjoint extension and there is a notion of ground state energy, defined as the lowest eigenvalue in the gap of the continuous spectrum.
Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a non-negative self-adjoint operator on $L^2(\mathcal{X})$ whose heat kernels satisfy the Gaussian upper bound estimates. Assume that the growth function $φ:\ \mathcal{X}\times[0,\infty) \to[0,\infty)$ satisfies that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)\in {\mathbb A}_{\infty}(\mathcal{X})$ (the class of uniformly Muckenhoupt weights). Let $H_{φ,\,L}(\mathcal{X})$ be the Musielak-Orlicz-Hardy space defined via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors characteri…
Dirac operators with infinite mass boundary conditions on unbounded domains with infinite corners
本文研究带无限质量边界条件的二维Dirac算子在含无限角点的无界区域上的自伴性，证明无凹角点时算子自伴，有凹角点时构造并刻画了特选自伴扩张，并分析了其谱。
We establish new Calderón reproducing formulas for self-adjoint operators $D$ that generate strongly continuous groups with finite propagation speed. These formulas allow the analysing function to interact with $D$ through holomorphic functional calculus whilst the synthesising function interacts with $D$ through functional calculus based on the Fourier transform. We apply these to prove the embedding $H^p_D(\wedge T^*M) \subseteq L^p(\wedge T^*M)$, $1\leq p\leq 2$, for the Hardy spaces of differential forms introduced by Auscher, McIntosh and Russ, where $D=d+d^*$ is the Hodge--Dirac operator…
This paper deals with the study of the two-dimensional Dirac operatorwith infinite mass boundary condition in a sector. We investigate the question ofself-adjointness depending on the aperture of the sector: when the sector is convexit is self-adjoint on a usual Sobolev space whereas when the sector is non-convexit has a family of self-adjoint extensions parametrized by a complex number of theunit circle. As a byproduct of this analysis we are able to give self-adjointnessresults on polygones. We also discuss the question of distinguished self-adjointextensions and study basic spectral propert…
We introduce the concept of spherical (as distinguished from planar) reflection positivity and use it to obtain a new proof of the sharp constants in certain cases of the HLS and the logarithmic HLS inequality. Our proofs relies on an extension of a work by Li and Zhu which characterizes the minimizing functions of the type $(1+|x|^2)^{-p}$.
We describe the self-adjoint realizations of the operator $H:=-iα\cdot \nabla + mβ+ \mathbb V(x)$, for $m\in\mathbb R $, and $\mathbb V(x)= |x|^{-1} ( ν\mathbb{I}_4 +μβ-i λα\cdot{x}/{|x|}\,β)$, for $ν,μ,λ\in \mathbb R$. We characterize the self-adjointness in terms of the behaviour of the functions of the domain in the origin, exploiting Hardy-type estimates and trace lemmas. Finally, we describe the distinguished extension.
In this work, we consider the Cauchy problem for $u' - Au = f$ with $A$ the Laplacian operator on some Riemannian manifolds or a sublapacian on some Lie groups or some second order elliptic operators on a domain. We show the boundedness of the operator of maximal regularity $f\mapsto Au$ and its adjoint on appropriate Hardy spaces which we define and study for this purpose. As a consequence we reobtain the maximal $L^q$ regularity on $L^p$ spaces for $p,q$ between 1 and $\infty$.
Hardy Spaces Associated with Non-Negative Self-Adjoint Operators and Ball Quasi-Banach Function Spaces on Doubling Metric Measure Spaces and Their Applications
本文在加倍度量测度空间上结合非负自伴算子与球拟 Banach 函数空间构建 Hardy 空间，证明其原子、分子及 Littlewood–Paley 刻画，并应用于谱乘子和 Schrödinger 群有界性。
Self-adjointness for the MIT bag model on an unbounded cone
该研究针对无界三维圆锥上带MIT袋边界条件的无质量Dirac算子，证明凸锥下其在四分量$H^1$函数域上自伴，给出非凸锥情形的数值证据，还得到凸锥上的Hardy型不等式及扰动稳定性，并讨论了向量子点边界条件的推广。
In this paper we first study the generalized weighted Hardy spaces $H^p_{L,w}(X)$ for $0<p\le 1$ associated to nonnegative self-adjoint operators $L$ satisfying Gaussian upper bounds on the space of homogeneous type $X$ in both cases of finite and infinite measure. We show that the weighted Hardy spaces defined via maximal functions and atomic decompositions coincide. Then we prove weighted regularity estimates for the Green operators of the inhomogeneous Dirichlet and Neumann problems in suitable bounded or unbounded domains including bounded semiconvex domains, convex regions above a Lipschi…
Let $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function satisfying the globally log-Hölder continuous condition and $L$ a one to one operator of type $ω$ in $L^2({\mathbb R}^n)$, with $ω\in[0,\,π/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. In this article, the authors introduce the variable weak Hardy space $W\!H_L^{p(\cdot)}(\mathbb R^n)$ associated with $L$ via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space $W\!T^{p(…
We investigate the properties of self-adjointness of a two-dimensional Dirac operator on an infinite sector with infinite mass boundary conditions and in presence of a Coulomb-type potential with the singularity placed on the vertex. In the general case, we prove the appropriate Dirac-Hardy inequality and exploit the Kato-Rellich theory. In the explicit case of a Coulomb potential, we describe the self-adjoint extensions for all the intensities of the potential relying on a radial decomposition in partial wave subspaces adapted to the infinite-mass boundary conditions. Finally, we integrate ou…
Let $X$ be a space of homogeneous type and let $\mathfrak{L}$ be a nonnegative self-adjoint operator on $L^2(X)$ enjoying Gaussian estimates. The main aim of this paper is twofold. Firstly, we prove the (local) nontangential and radial maximal function charaterization for the local Hardy spaces associated to $\mathfrak{L}$. This deduces the maximal function charaterization for local Hardy spaces in the sense of Coifman and Weiss provided that $\mathfrak{L}$ satisfies certain extra conditions. Secondly, we introduce the local Hardy space associated to the critical function $ρ$ which is motivate…
Littlewood-Paley Characterization for Musielak-Orlicz-Hardy Spaces Associated with Operators
本文基于空间类型X，研究了与非负二阶自伴随算子L相关的穆西尔克-奥里奇-哈代空间H_{φ,L}(X)的利特伍德-帕莱伊函数特征，证明了其与H_{L,G,φ}(X)的等价性。
We consider the two-dimensional Dirac operator with Lorentz-scalar $δ$-shell interactions on each edge of a star-graph. An orthogonal decomposition is performed which shows such an operator is unitarily equivalent to an orthogonal sum of half-line Dirac operators with off-diagonal Coulomb potentials. This decomposition reduces the computation of the deficiency indices to determining the number of eigenvalues of a one-dimensional spin-orbit operator in the interval $(-1/2,1/2)$. If the number of edges of the star graph is two or three, these deficiency indices can then be analytically determine…
Dirac-Coulomb operators with general charge distribution. I. Distinguished extension and min-max formulas
本文研究任意有限带符号电荷分布生成的 Dirac–Coulomb 算子，证明特异自伴延拓的唯一性，并通过上旋量二次型给出能隙特征值的极小极大公式。
We determine explicitly a boundary triple for the Dirac operator $H:=-iα\cdot \nabla + mβ+ \mathbb V(x)$ in $\mathbb R^3$, for $m\in\mathbb R$ and $\mathbb V(x)= |x|^{-1} ( ν\mathbb{I}_4 +μβ-i λα\cdot{x}/|x|\,β)$, with $ν,μ,λ\in \mathbb R$. Consequently we determine all the self-adjoint realizations of $H$ in terms of the behaviour of the functions of their domain in the origin. When $\sup_{x} |x||\mathbb V(x)| \leq 1$, we discuss the problem of selecting the distinguished extension requiring that its domain is included in the domain of the appropriate quadratic form.
Non self-adjoint perturbations of the Heisenberg sublaplacian
本文利用乘子方法和水平 Hardy 不等式证明了 Heisenberg 群上亚拉普拉斯算子在加权 L² 空间中的一致预解式估计，并由此给出复位势的小性与排斥性条件，确保扰动算子无特征值。
The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisly, introducing the two-parameter $n$-dimensional homogeneous scalar differential expressions $T_{α,β} := - Δ+ α|x|^{-2} x \cdot \nabla + β|x|^{-2}$, $α, β\in \mathbb{R}$, $x \in \mathbb{R}^n \backslash \{0\}$, $n \in \mathbb{N}$, $n \geq 2$, and its formal adjoint, denoted by $T_{α,β}^+$, we show that nonnegativity of $T_{α,β}^+ T_{α,β}$ on $C_0^{\infty}(\mathbb{R}^n \backslash \{0\})$…
We consider a Dirac operator in three space dimensions, with an electrostatic (i.e. real-valued) potential $V(x)$, having a strong Coulomb-type singularity at the origin. This operator is not always essentially self-adjoint but admits a distinguished self-adjoint extension $D\_V$. In a first part we obtain new results on the domain of this extension, complementing previous works of Esteban and Loss. Then we prove the validity of min-max formulas for the eigenvalues in the spectral gap of $D\_V$, in a range of simple function spaces independent of $V$. Our results include the critical case $\li…
This paper aims to give a general (possibly compact or noncompact) analog of Strichartz inequalities with loss of derivatives, obtained by Burq, Gérard, and Tzvetkov [19] and Staffilani and Tataru [51]. Moreover we present a new approach, relying only on the heat semigroup in order to understand the analytic connexion between the heat semigroup and the unitary Schrödinger group (both related to a same self-adjoint operator). One of the novelty is to forget the endpoint $L^1-L^\infty$ dispersive estimates and to look for a weaker $H^1-BMO$ estimates (Hardy and BMO spaces both adapted to the hea…
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