A pseudodifferential Hormander's inequality

Pseudodifferentialoperatoren mit nichtglatten Koeffizienten auf Mannigfaltigkeiten

A generalization of differential operators are pseudodifferential operators which are used for reasoning about partial differential equations with variable coefficients. A lot of useful properties about classical pseudodifferential operators are already discovered by the works of H. Kumano-Go and L. Hörmander. In this work, we like to transfer their ideas on those operators which symbols are merely Hölder-continous. We focus on the composition of operators, symbol smoothing and operators on manifolds.

Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators

Let $P$ be a classical pseudodifferential operator of complex order $m$ on an $n$-dimensional smooth manifold $Ω_1$. For the truncation $P_Ω$ to a smooth subset $Ω$ there is a well-known theory of boundary value problems when $P_Ω$ has the transmission property (preserves $C^\infty (\barΩ)$) and is of integer order; the calculus of Boutet de Monvel. Many interesting operators, such as for example complex powers of the Laplacian $(-Δ)^μ$ with noninteger mu, are not covered. They have instead the mu-transmission property defined in Hörmander's books, mapping $x_n^μC^\infty (\barΩ)$ into $C^\inft…

Pseudodifferential Operators, Rellich-Kondrachov Theorem and Hardy-Sobolev Spaces

We will present versions of the Rellich-Kondrachov theorem for pseudo-differential operators acting on localizable Hardy spaces. One of the techniques includes boundedness properties for pseudodifferential operators with symbols in the Hörmander class, that might be of interest in his own right, extending results previously obtained by Goldberg, Alvarez and Hounie, Taylor - among others. As application, we obtain compact embedding results for distributions in nonhomogeneous localizable Hardy-Sobolev spaces.

Basic properties of nonsmooth Hormander's vector fields and Poincare's inequality

We consider a family of vector fields defined in some bounded domain of R^p, and we assume that they satisfy Hormander's rank condition of some step r, and that their coefficients have r-1 continuous derivatives. We extend to this nonsmooth context some results which are well-known for smooth Hormander's vector fields, namely: some basic properties of the distance induced by the vector fields, the doubling condition, Chow's connectivity theorem, and, under the stronger assumption that the coefficients belong to C^{r-1,1}, Poincare's inequality. By known results, these facts also imply a Sobole…

A Harnack inequality and Hölder continuity for weak solutions to parabolic operators involving Hörmander vector fields

This paper deals with two separate but related results. First we consider weak solutions to a parabolic operator with Hörmander vector fields. Adapting the iteration scheme of Jürgen Moser for elliptic and parabolic equations in $\mathbb{R}^n$ we show a parabolic Harnack inequality. Then, after proving the Harnack inequality for weak solutions to equations of the form $u_t = \sum X_i (a_{ij} X_j u)$ we use this to show Hölder continuity. We assume the coefficients are bounded and elliptic. The iteration scheme is a tool that may be adapted to many settings and we extend this to nonlinear parab…

A calculus of abstract edge pseudodifferential operators of type $\varrho,δ$

In this paper we expand on B.-W. Schulze's abstract edge pseudodifferential calculus and introduce a larger class of operators that is modeled on Hörmander's $\varrho,δ$ calculus, where $0 \leq δ< \varrho \leq 1$. This expansion is motivated by recent work on boundary value problems for elliptic wedge operators with variable indicial roots by G. Mendoza and the author, where operators of type $1,δ$ for $0 < δ< 1$ appear naturally. Some of the results of this paper also represent improvements over the existing literature on the standard abstract edge calculus of operators of type $1,0$, such as…

Analytic and Gevrey Hypoellipticity for Perturbed Sums of Squares Operators

We prove a couple of results concerning pseudodifferential perturbations of differential operators being sums of squares of vector fields and satisfying Hörmander's condition. The first is on the minimal Gevrey regularity: if a sum of squares with analytic coefficients is perturbed with a pseudodifferential operator of order strictly less than its subelliptic index it still has the Gevrey minimal regularity. We also prove a statement concerning real analytic hypoellipticity for the same type of pseudodifferential perturbations, provided the operator satisfies to some extra conditions (see Theo…

L'inégalité de Melin Hörmander en caractéristiques multiples

We prove here the Melin Hörmander inequality for operators with multiple characteristics.

Representations of almost periodic pseudodifferential operators and applications in spectral theory

The paper concerns algebras of almost periodic pseudodifferential operators on $\mathbb R^d$ with symbols in Hörmander classes. We study three representations of such algebras, one of which was introduced by Coburn, Moyer and Singer and the other two inspired by results in probability theory by Gladyshev. Two of the representations are shown to be unitarily equivalent for nonpositive order. We apply the results to spectral theory for almost periodic pseudodifferential operators acting on $L^2$ and on the Besicovitch Hilbert space of almost periodic functions.

Wigner analysis of operators. Part I: pseudodifferential operators and wave fronts

We perform Wigner analysis of linear operators. Namely, the standard time-frequency representation \emph{Short-time Fourier Transform} (STFT) is replaced by the $\mathcal{A}$-\emph{Wigner distribution} defined by $W_{\mathcal A} (f)=μ({\mathcal A})(f\otimes\bar{f})$, where ${\mathcal A}$ is a $4d\times 4d$ symplectic matrix and $μ({\mathcal A})$ is an associate metaplectic operator. Basic examples are given by the so-called $τ$-Wigner distributions. Such representations provide a new characterization for modulation spaces when $τ\in (0,1)$. Furthermore, they can be efficiently employed in the …

非紧对称空间上的等变拟微分算子的完全符号

Complete Symbols of Equivariant Pseudodifferential Operators on Noncompact Symmetric Spaces

该研究针对非紧对称空间G/K上满足快速非对角衰减条件的G-等变Hörmander拟微分算子，定义Harish-Chandra符号函数并刻画其算子类。

A Nash Type result for Divergence Parabolic Equation related to Hormander's vector fields

In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hormander's vector fields and establish a Nash type result, i.e., the local Holder regularity for weak solutions. After deriving the parabolic Sobolev inequality, (1,1) type Poincaré inequality of Hormander's vector fields and a De Giorgi type Lemma, the Holder regularity of weak solutions to the equation is proved based on the estimates of oscillations of solutions and the isomorphism between parabolic Campanato space and parabolic Holder space. As a consequence, we give the Ha…

Almost periodic pseudodifferential operators and Gevrey classes

We study almost periodic pseudodifferential operators acting on almost periodic functions $G_{\rm ap}^s(\rr d)$ of Gevrey regularity index $s \geq 1$. We prove that almost periodic operators with symbols of Hörmander type $S_{ρ,δ}^m$ satisfying an $s$-Gevrey condition are continuous on $G_{\rm ap}^s(\rr d)$ provided $0 < ρ\leq 1$, $δ=0$ and $s ρ\geq 1$. A calculus is developed for symbols and operators using a notion of regularizing operator adapted to almost periodic Gevrey functions and its duality. We apply the results to show a regularity result in this context for a class of hypoelliptic …

A pseudodifferential calculus for maximally hypoelliptic operators and the Helffer-Nourrigat conjecture

We extend the classical regularity theorem of elliptic operators to maximally hypoelliptic differential operators. More precisely, given vector fields $X_1,\ldots,X_m$ on a smooth manifold which satisfy Hörmander's bracket generating condition, we define a principal symbol for \textit{any} linear differential operator. Our symbol takes into account the vector fields $X_i$ and their commutators. We show that for an arbitrary differential operator, its principal symbol is invertible if and only if the operator is maximally hypoelliptic. This answers affirmatively a conjecture due to Helffer and …

Anisotropic Gohberg Lemmas for Pseudodifferential Operators on Abelian Compact Groups

Classically, Gohberg-type Lemmas provide lower bounds for the distance of suitable pseudodifferential operators acting in a Hilbert space to the ideal of compact operators, in terms of "the behavior of the symbol at infinity". In this article the pseudodifferential operators are associated to a compact Abelian group $X$ and an important role is played by its Pontryagin dual $\widehat X$. Hörmander-type classes of symbols are not always available; they will be replaced by crossed product $C^*$-algebras involving a vanishing oscillation condition, which anyway is more general even in the particu…

Boundedness of Pseudodifferential Operators on Banach Function Spaces

We show that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$, then a pseudodifferential operator $\operatorname{Op}(a)$ is bounded on $X(\mathbb{R}^n)$ whenever the symbol $a$ belongs to the Hörmander class $S_{ρ,δ}^{n(ρ-1)}$ with $0<ρ\le 1$, $0\leδ<1$ or to the the Miyachi class $S_{ρ,δ}^{n(ρ-1)}(\varkappa,n)$ with $0\leδ\leρ\le 1$, $0\leδ<1$, and $\varkappa>0$. This result is applied to the case of variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$.

Positivity results for Weyl's pseudodifferential calculus on the Wiener space

This paper deals with positivity properties for a pseudodifferential calculus, generalizing Weyl's classical quantization, and set on an infinite dimensional phase space, the Wiener space. In this frame, we show that a positive symbol does not, in general, give a positive operator. In order to measure the nonpositivity, we establish a Gårding's inequality, which holds for the symbol classes at hand. Nevertheless, for symbols with radial aspects, additional assumptions ensure the positivity of the associated operator.

Pseudodifferential multi-product representation of the solution operator of a parabolic equation

By using a time slicing procedure, we represent the solution operator of a second-order parabolic pseudodifferential equation on $\R^n$ as an infinite product of zero-order pseudodifferential operators. A similar representation formula is proven for parabolic differential equations on a compact Riemannian manifold. Each operator in the multi-product is given by a simple explicit Ansatz. The proof is based on an effective use of the Weyl calculus and the Fefferman-Phong inequality.

Mutual Absolute Continuity of Harmonic and Surface Measures for Hormander Type Operators

In this paper, we consider the Sub-Laplacian L which consists of sum of squares of smooth vector fields that satisfy Hormander's finite rank condition. We study the Dirichlet problem for this operator on domains that satisfy certain geometric conditions. For such domains, several key results are established. These results consist of 1) A reversed Holder inequality for the Poisson kernel 2) Harmonic measure (corresponding to L) and surface measure (as well as the H-Perimeter measure) are mutually absolutely continuous 3) A representation (hence solvability of the Dirichlet problem) for solution…

Semiclassical second microlocalization at linear coisotropic submanifolds in the torus

We develop a semiclassical second microlocal calculus of pseudodifferential operators associated to linear coisotropic submanifolds $\mathcal{C}\subset T^* \mathbb{T}^n$, where $\mathbb{T}^n = \mathbb{R}^n / \mathbb{Z}^n$. First microlocalization is localization in phase space $T^* \mathbb{T}^n$; second microlocalization is finer localization near a submanifold of $T^* \mathbb{T}^n$. Our second microlocal operators test distributions on $\mathbb{T}^n$ (e.g., Laplace eigenfunctions) for a coisotropic wavefront set, a second microlocal measure of absence of \emph{coisotropic regularity}. This wa…

Weighted multilinear Poincare inequalities for vector fields of Hormander type

As the classical $(p,q)$-Poincaré inequality is known to fail for $0 < p < 1$, we introduce the notion of weighted multilinear Poincaré inequality as a natural alternative when $m$-fold products and $1/m < p$ are considered. We prove such weighted multilinear Poincaré inequalities in the subelliptic context associated to vector fields of Hörmader type. We do so by establishing multilinear representation formulas and weighted estimates for multilinear potential operators in spaces of homogeneous type.

更多关于粗糙和光滑伪微分算子的稀疏界

Some More Sparse Bounds for Rough and Smooth Pseudodifferential Operators

本文扩展了Beltran和Cladek的方法，为仅在空间变量上可测的粗糙伪微分算子获得点wise稀疏界，并提供避免几何衰减稀疏界的替代证明，同时给出了稀疏形式界的充分条件。

An extension of Hörmander's hypoellipticity theorem

Motivated by applications to stochastic differential equations, an extension of Hörmander's hypoellipticity theorem is proved for second-order degenerate elliptic operators with non-smooth coefficients. The main results are established using point-wise Bessel kernel estimates and a weighted Sobolev inequality of Stein and Weiss. Of particular interest is that our results apply to operators with quite general first-order terms.

The Strong Maximum Principle and the Harnack inequality for a class of hypoelliptic divergence-form operators

In this paper we consider a class of hypoelliptic second-order partial differential operators $\mathcal{L}$ in divergence form on $\mathbb{R}^N$, arising from CR geometry and Lie group theory, and we prove the Strong and Weak Maximum Principles and the Harnack Inequality for $\mathcal{L}$. The involved operators are not assumed to belong to the Hörmander hypoellipticity class, nor to satisfy subelliptic estimates, nor Muckenhoupt-type estimates on the degeneracy of the second order part; indeed our results hold true in the infinitely-degenerate case and for operators which are not necessarily …

Harnack inequality and asymptotic lower bounds for the relativistic Fokker-Planck operator

We consider a class of second order degenerate kinetic operators $\mathscr{L}$ in the framework of special relativity. We first describe $\mathscr{L}$ as an Hörmander operator which is invariant with respect to Lorentz transformations. Then we prove a Lorentz-invariant Harnack type inequality, and we derive accurate asymptotic lower bounds for positive solutions to $\mathscr{L} f = 0$. As a consequence we obtain a lower bound for the density of the relativistic stochastic process associated to $\mathscr{L}$.

一类伪微分算子关于伸缩的谱正则性

Spectral regularity with respect to dilations for a class of pseudodifferential operators

本文研究一类伪微分算子在伸缩扰动下的谱性质，去除了慢变假设，证明谱的Hausdorff距离为√|δ|量级，而谱边缘距离按|δ|^ν变化，ν取决于F二阶导数的无穷远衰减速率。

Local and global symbols on compact Lie groups

On the torus, it is possible to assign a global symbol to a pseudodifferential operator using Fourier series. In this paper we investigate the relations between the local and global symbols for the operators in the classical Hörmander calculus and describe the principal symbols, the non-commutative residue and the canonical trace of an operator in terms of its global symbol. We also generalise these results to any compact Lie group.

次椭圆算子的庞加莱不等式与定量De Giorgi方法

Poincaré inequality and quantitative De Giorgi method for hypoelliptic operators

该研究提出基于轨迹的系统方法，证明含任意多个Hörmander交换子的次椭圆方程弱非负下解的庞加莱不等式，进而推导弱哈纳克不等式与赫尔德正则性。

与伪微分算子相关的演化系统的 Littlewood-Paley 型不等式

Littlewood-Paley Type Inequality for Evolution Systems Associated with Pseudo-Differential Operators

本文通过验证卷积核算子的 Hörmander 条件与 Besov 到 $L^q$ 的有界性，并借助向量值 Calderón-Zygmund 定理，建立了伪微分算子相关演化系统的 Littlewood-Paley 型不等式。

One can hear the area and curvature of boundary of a domain by hearing the Steklov eigenvalues

For a given bounded domain $Ω$ with smooth boundary in a smooth Riemannian manifold $(\mathcal{M},g)$, we show that the Poisson type upper-estimate of the heat kernel associated to the Dirichlet-to-Neumann operator, the Sobolev trace inequality, the Log-Sobolev trace inequality, the Nash trace inequality, and the Rozenblum-Lieb-Cwikel type inequality are all equivalent. Upon decomposing the Dirichlet-to-Neumann operator into a sum of the square root of the Laplacian and a pseudodifferntial operator and by applying Grubb's method of symbolic calculus for the corresponding pseudodifferential hea…

Inequalities of Dirichlet eigenvalues for degenerate elliptic partial differential operators

Let ${X_j},{Y_j}(j = 1, \cdot \cdot \cdot,n)$ be vector fields satisfying Hörmander's condition and ${Δ_L} = \sum\limits_{j = 1}^n {(X_j^2 + Y_j^2)}$. In this paper, we establish some inequalities of Dirichlet eigenvalues for degenerate elliptic partial differential operator ${Δ_L}$ and $Δ_L^2$. These inequalities extend Yang's inequalities for Dirichlet eigenvalues of Laplacian to the settings here and the forms of inequalities are more general than Yang's inequalities. To obtain them, we give a generalization of the inequality by Chebyshev.

Fundamental Results for Pseudo-Differential Operators of Type $\mathbf{1},\mathbf{1}$

This paper develops some deeper consequences of an extended definition, proposed previously by the author, of pseudo-differential operators that are of type $1,1$ in Hörmander's sense. Thus, it contributes to the long-standing problem of creating a systematic theory of such operators. It is shown that type $1,1$-operators are defined and continuous on the full space of temperate distributions, if they fulfil Hörmander's twisted diagonal condition, or more generally if they belong to the self-adjoint subclass; and that they are always defined on the temperate smooth functions. As a main tool th…

$L^p$-$L^q$ boundedness of pseudo-differential operators on smooth manifolds and its applications to nonlinear equations

In this paper we study the boundedness of global pseudo-differential operators on smooth manifolds. By using the notion of global symbol we extend a classical condition of Hörmander type to guarantee the $L^p$-$L^q$-boundedness of global operators. First we investigate $L^p$-boundedness of pseudo-differential operators in view of the Hörmander-Mihlin condition. We also prove $L^\infty$-$BMO$ estimates for pseudo-differential operators. Later, we concentrate our investigation to settle $L^p$-$L^q$ boundedness of the Fourier multipliers and pseudo-differential operators for the range $1<p \leq 2…

塞格极限定理与海森堡拉普拉斯算子

Szegö limit theorem and Heisenberg Laplacian

该研究旨在获取经典塞格极限定理的一个版本，在\(L^2(\mathbb{R}^3)\)中用海森堡 - 霍尔曼德 拉普拉斯算子取代圆上二阶导数算子，并推导了凸函数的塞格型不等式。

Spectral inequalities for elliptic pseudo-differential operators on closed manifolds

Let $(M,g)$ be a closed Riemannian manifold. The aim of this work is to prove the Lebeau-Robbiano spectral inequality for a positive elliptic pseudo-differential operator $E(x,D)$ on $M,$ of order $ν>0,$ in the Hörmander class $Ψ^ν_{ρ,δ}(M).$ In control theory this has been an open problem prior to this work. As an application of this fundamental result, we establish the null-controllability of the (fractional) heat equation associated with $E(x,D).$ The sensor $ω\subset M$ in the observability inequality is an open subset of $M.$ The obtained results (that are, the corresponding spectral ineq…

紧致齐次流形上齐次向量丛的伪微分算子

Pseudo-differential operators on Homogeneous vector bundles over compact homogeneous manifolds

本文建立紧致齐次流形上齐次向量丛的亚椭圆伪微分算子全局理论，发展全局符号演算，证明 Calderón-Vaillancourt 定理、Fefferman 定理及 Gårding 不等式，并应用于演化问题适定性，计算外微分与 Dirac 算子的全局符号。