Related papers: On the Gevrey strong hyperbolicity

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…

For a class of weakly hyperbolic systems of the form D_t - A(t,x,D_x), where A(t,x,D_x) is a first-order pseudodifferential operator whose principal symbol degenerates like t^{l_*} at time t=0, for some integer l_* \geq 1, well-posedness of…

We study the Cauchy problem for the equation of the form $$ \ddot{u}(t) + (\aa A + B)\dot{u}(t) + (A+G)u(t) = 0,\tag* $$ where $A$, $B$, and $G$ are \o s in a Hilbert space $\Cal H$ with $A$ selfadjoint, $\sigma(A)=[0,\infty)$, $B\ge0$…

Let $\mathscr{H}^2$ denote the Hardy space of Dirichlet series $f(s) = \sum_{n\geq1} a_n n^{-s}$ with square summable coefficients and suppose that $\varphi$ is a symbol generating a composition operator on $\mathscr{H}^2$ by…

Given a smooth 2-dimensional Riemannian or pseudo-Riemannian manifold $(M, \boldsymbol{g})$ and an ambient 3-dimensional Riemannian or pseudo-Riemannian manifold $(N, \boldsymbol{h})$, one can ask under what circumstances does the exterior…

We give upper bounds for the eigenvalues of the La-place-Beltrami operator of a compact $m$-dimensional submanifold $M$ of $\R^{m+p}$. Besides the dimension and the volume of the submanifold and the order of the eigenvalue, these bounds…

In this article, we study the convolution operator $M_k$ with oscillatory kernel, which is related with solutions to the Cauchy problem for the strictly hyperbolic equations. The operator $M_k$ is associated to the characteristic…

In this paper we consider weakly hyperbolic equations of higher orders in arbitrary dimensions with time-dependent coefficients and lower order terms. We prove the Gevrey well-posedness of the Cauchy problem under $C^k$-regularity of…

We study the Sturm--Liouville operator $$ T(\varepsilon)y=-\frac{1}{\varepsilon}y''+ p(x)y, $$ with concrete $\mathcal{PT}$-- symmetric potential $p(x) = ix$ and Dirichlet boundary conditions on the segment $[-1,1]$. Here $\varepsilon \in…

We assign some kind of invariant manifolds to a given integrable PDE (its discrete or semi-discrete variant). First, we linearize the equation around its arbitrary solution $u$. Then we construct a differential (respectively, difference)…

This article investigates the wave equation for the Schr\"{o}dinger operator on $\mathbb{R}^{n}$, denoted as $\mathcal{H}_0:=-\Delta+V$, where $\Delta$ is the standard Laplacian and $V$ is a complex-valued multiplication operator. We prove…

Let $G=(V,E)$ be a finite graph and $M_G$ be the centered Hardy-Littlewood maximal operator defined there. We find the optimal value $\bf{C}_{G,p}$ such that the inequality $$\text{Var}_{p}(M_{G}f)\leq {\textbf{C}}_{G,p}\text{Var}_{p}(f)$$…

The existence and uniqueness in H\"older spaces of solutions of the Cauchy problem to parabolic integro-differential equation of the order {\alpha}\in(0,2) is investigated. The principal part of the operator has kernel…

We study the local classification problem for differential Pfaffian forms on a supermanifold $M$ that are homogeneous with respect to a given homogeneity structure on $M$. The most familiar examples of homogeneity structures are those…

We study the case of Hermite subdivision operators satisfying a spectral condition of order greater than their size. We show that this can be characterized by operator factorizations involving Taylor operators and difference factorizations…

Results providing conditions on a family of integro-differential operators to determine a formal automorphism are established. Equivalently, the problem can be read in terms of existence and uniqueness of formal solutions of Cauchy problems…

We consider the Cauchy problem for a $3$-evolution operator $P$ with $(t,x)$-depending coefficients and complex valued lower order terms. We assume the initial data to be Gevrey regular and to admit an exponential decay at infinity, that…

We characterize the semigroups of composition operators that are strongly continuous on the mixed norm spaces $H(p,q,\alpha)$. First, we study the separable spaces $H(p,q,\alpha)$ with $q<\infty,$ that behave as the Hardy and Bergman…

Let $\mathbb G$ be a graded Lie group with homogeneous dimension $Q$. In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator $\mathcal{R}$ of homogeneous degree…

In this article we describe the novel method to construct fundamental solutions for operators with variable coefficients. That method was introduced in "A note on the fundamental solution for the Tricomi-type equation in the hyperbolic…