相关论文: Optimal Gevrey Regularity for Certain Sums of Squa…

We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. Treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata…

We study the regularity of Gevrey vectors for H\"ormander operators $$ P = \sum_{j=1}^m X_j^2 + X_0 + c$$ where the $X_j$ are real vector fields and $c(x)$ is a smooth function, all in Gevrey class $G^{s}.$ The principal hypothesis is that…

The sharp Gevrey hypoellipticity is provided for the following generalization of the M\'etivier operator, "Non-hypoellipticit\'e analytique pour $D_{x}^{2}+\left( x^{2} + y^{2}\right)D_{y}^{2}$" by G. M\'etivier, \begin{align*}…

In this paper, we mainly consider the Gevrey regularity and analyticity of the solution to a generalized two-component shallow water wave system with higher-order inertia operators, namely, $m=(1-\partial_x^2)^su$ with $s>1$. Firstly, we…

We prove a couple of results concerning pseudodifferential perturbations of differential operators being sums of squares of vector fields and satisfying H\"ormander's condition. The first is on the minimal Gevrey regularity: if a sum of…

The micro-local Gevrey regularity of a class of "sums of squares" with real analytic coefficients is studied in detail. Some partial regularity result is also given.

We study the computational complexity of approximating the 2->q norm of linear operators (defined as ||A||_{2->q} = sup_v ||Av||_q/||v||_2), as well as connections between this question and issues arising in quantum information theory and…

We prove via FBI-transform a result concerning the microlocal Gevrey regularity of analytic vectors for operators sums of squares of vector fields with real-valued real analytic coefficients of H\"ormander type, thus providing a microlocal…

We study the microlocal regularity of the analytic/Gevrey vectors for the following class of second order partial differential equations \begin{align*} P(x,D) = \sum_{\ell,j=1}^{n} a_{\ell,j}(x) D_{\ell} D_{j} + \sum_{\ell=1}^{n} i…

We establish optimal $C^s$ boundary regularity for the most general class of (linear and translation invariant) nonlocal elliptic operator of order $2s$. Namely, we consider L\'evy operators that are symmetric and its Fourier symbol…

We study a class of sum of squares exhibiting the same Poisson-Treves stratification as the Oleinik-Radkevi\v{c} operator. We find three types of operators having distinct microlocal structures. For one of these we prove a Gevrey…

Hypoellipticity in Gevrey classes $G^s$ is characterized for a simple family of sums of squares of vector fields satisfying the bracket hypothesis, with analytic coefficients. It is shown that hypoellipticity holds if and only if $s$ is…

Let A be a positive self-adjoint linear operator acting on a real Hilbert space H and $\alpha$, c be positive constants. We show that all solutions of the evolution equation u + Au + cA $\alpha$ u = 0 with u(0) $\in$ D(A 1 2), u (0) $\in$ H…

Grothendieck's theorem asserts that every continuous linear operator from $\ell_{1}$ to $\ell_{2}$ is absolutely $\left( 1;1\right) $-summing. In this note we prove that the optimal constant $g_{m}$ so that every continuous $m$-linear…

Let $M$ be an $n(>2)$-dimensional closed orientable submanifold in an $(n+p)$-dimensional space form $\mathbb{R}^{n+p}(c)$. We obtain an optimal upper bound for the second eigenvalue of a class of elliptic operators on $M$ defined by…

Gevrey series are ubiquitous in analysis; any series satisfying some (possibly non-linear) analytic differential equation is Gevrey of some rational order. The present work stems from two observations: 1) the classical Gevrey series, e.g.…

The operator \[ A_{\varepsilon}= D_{1} g_{1}(x_{1}/\varepsilon, x_{2}) D_{1} + D_{2} g_{2}(x_{1}/\varepsilon, x_{2}) D_{2} \] is considered in $L_{2}({\mathbb{R}}^{2})$, where $g_{j}(x_{1},x_{2})$, $j=1,2,$ are periodic in $x_{1}$ with…

Let $x$ be a real number satisfying $x \geq 2$. For any positive integer $n$, we define $s(n)$ as the smallest non-negative integer such that $n + s(n)$ is a perfect square. In this paper, we derive an asymptotic formula for the sum…

The Hardy--Littlewood inequalities for $m$-linear forms have their origin with the seminal paper of Hardy and Littlewood (Q.J. Math, 1934). Nowadays it has been extensively investigated and many authors are looking for the optimal estimates…

This paper is concerned with the study of regularity and stability properties of two Euler-Bernoulli beam equations with localized singular damping. Under suitable regularity assumptions on the damping coefficient, we establish Gevrey…