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Marco Cappiello

Publications and source records attributed to Marco Cappiello.

36 records · Page 2Linked to original sources
arXiv2019

Schrödinger-type equations in Gelfand-Shilov spaces

We study the initial value problem for Schrödinger-type equations with initial data presenting a certain Gevrey regularity and an exponential behavior at infinity. We assume the lower order terms of the Schrödinger operator depending on $(t,x)\in[0,T]\times\R^n$ and complex valued. Under a suitable decay condition as $|x|\to\infty$ on the imaginary part of the first order term and an algebraic growth assumption on the real part, we derive global energy estimates in suitable Sobolev spaces of infinite order and prove a well posedness result in Gelfand-Shilov type spaces. We also discuss by examples the sharpness of the result.

math.AP↗
arXiv2018

Pseudo-differential calculus in anisotropic Gelfand-Shilov setting

We study some classes of pseudo-differential operators with symbols $a$ admitting anisotropic exponential growth at infinity and we prove mapping properties for these operators on Gelfand-Shilov spaces of type S. Moreover, we deduce algebraic and certain invariance properties of these classes.

math.FA↗
arXiv2017

Conormal distributions in the Shubin calculus of pseudodifferential operators

We characterize the Schwartz kernels of pseudodifferential operators of Shubin type by means of an FBI transform. Based on this we introduce as a generalization a new class of tempered distributions called Shubin conormal distributions. We study their transformation behavior, normal forms and microlocal properties.

math-ph↗
arXiv2016

Anisotropic Shubin operators and eigenfunctions expansions in Gelfand-Shilov spaces

We derive new results on the characterization of Gelfand--Shilov spaces $\mathcal{S}^μ_ν(\R^n)$, $μ,ν>0$, $μ+ν\geq 1$ by Gevrey estimates of the $L^2$ norms of iterates of $(m,k)$ anisotropic globally elliptic Shubin (or $Γ$) type operators, $(-Δ)^{m/2} +| x |^k$ with $m,k\in 2\N$ being a model operator, and on the decay of the Fourier coefficients in the related eigenfunction expansions. Similar results are obtained for the spaces $Σ^μ_ν(\R^n)$, $μ,ν>0$, $μ+ν> 1$, cf. \eqref{GSdef}. In contrast to the symmetric case $μ= ν$ and $k=m$ (classical Shubin operators) we encounter resonance type phenomena involving the ratio $κ:=μ/ν$; namely we obtain a characterization of $\mathcal{S}^μ_ν(\R^n)$ and $Σ^μ_ν(\R^n)$ in the case $μ=kt/(k+m), ν= mt/(k+m), t \geq 1$, that is, when $κ=k/m \in \Q$.

math.FA↗
arXiv2016

Pseudo-differential operators in a Gelfand-Shilov setting

We introduce some general classes of pseudodifferential operators with symbols admitting exponential type growth at infinity and we prove mapping properties for these operators on Gelfand-Shilov spaces both in the quasi-analytic and in the non-quasi-analytic case. Moreover, we prove composition theorems and certain invariance properties of these classes.

math.FA↗
arXiv2015

Pointwise decay and smoothness for semilinear elliptic equations and travelling waves

We derive sharp decay estimates and prove holomorphic extensions for the solutions of a class of semilinear nonlocal elliptic equations with linear part given by a sum of Fourier multipliers with finitely smooth symbols at the origin. Applications concern the decay and smoothness of travelling waves for nonlinear evolution equations in fluid dynamics and plasma physics.

math.AP↗
arXiv2015

Sharp decay estimates and smoothness for solutions to nonlocal semilinear equations

We consider semilinear equations of the form p(D)u=F(u), with a locally bounded nonlinearity F(u), and a linear part p(D) given by a Fourier multiplier. The multiplier p(ξ) is the sum of positively homogeneous terms, with at least one of them non smooth. This general class of equations includes most physical models for traveling waves in hydrodynamics, the Benjamin-Ono equation being a basic example. We prove sharp pointwise decay estimates for the solutions to such equations, depending on the degree of the non smooth terms in p(ξ). When the nonlinearity is smooth we prove similar estimates for the derivatives of the solution, as well holomorphic extension to a strip, for analytic nonlinearity.

math.AP↗
arXiv2014

Semilinear pseudodifferential equations in spaces of tempered ultradistributions

We study a class of semilinear elliptic equations on spaces of tempered ultradistributions of Beurling and Roumieu type. Assuming that the linear part of the equation is an elliptic pseudodifferential operator of infinite order with a sub-exponential growth of its symbol and that the non linear part is given by an infinite sum of powers of $u$ with sub-exponential growth with respect to $u,$ we prove a regularity result in the functional setting of the quoted ultradistribution spaces for a weak Sobolev type solution $u$.

math.AP↗
arXiv2014

On the inverse to the harmonic oscillator

Let $b_d$ be the Weyl symbol of the inverse to the harmonic oscillator on $\R^d$. We prove that $b_d$ and its derivatives satisfy convenient bounds of Gevrey and Gelfand-Shilov type, and obtain explicit expressions for $b_d$. In the even-dimensional case we characterize $b_d$ in terms of elementary functions. In the analysis we use properties of radial symmetry and a combination of different techniques involving classical a priori estimates, commutator identities, power series and asymptotic expansions.

math.AP↗
arXiv2014

Some remarks on the radius of spatial analyticity for the Euler equations

We consider the Euler equations on $\mathbb{T}^d$ with analytic data and prove lower bounds for the radius of spatial analyticity $ε(t)$ of the solution using a new method based on inductive estimates in standard Sobolev spaces. Our results are consistent with similar previous results proved by Kukavica and Vicol, but give a more precise dependence of $ε(t)$ on the radius of analyticity of the initial datum.

math.AP↗
arXiv2014

Radial symmetric elements and the Bargmann transform

We prove that a function or distribution on $\rr d$ is radial symmetric, if and only if its Bargmann transform is a composition by an entire function on $\mathbf C$ and the canonical quadratic function from $\cc d$ to $\mathbf C$.

math.FA↗
arXiv2013

Weighted energy estimates for $p$-evolution equations

We prove energy estimates for linear $p$-evolution equations in weighted Sobolev spaces under suitable assumptions on the behavior at infinity of the coefficients with respect to the space variables. As a consequence we obtain well posedness for the related Cauchy problem in the Schwartz spaces $\mathcal{S}$ and $\mathcal{S}'$.

math.AP↗
arXiv2013

Microlocal analysis of quasianalytic Gelfand-Shilov type ultradistributions

We introduce a global wave front set suitable for the analysis of tempered ultradistributions of quasianalytic Gelfand-Shilov type. We study the transformation properties of the wave front set and use them to give microlocal existence results for pull-backs and products. We further study quasianalytic microlocality for classes of localization and ultradifferential operators, and prove microellipticity for differential operators with polynomial coefficients.

math.AP↗
arXiv2013

On the radius of spatial analyticity for semilinear symmetric hyperbolic systems

We study the problem of propagation of analytic regularity for semi-linear symmetric hyperbolic systems. We adopt a global perspective and we prove that if the initial datum extends to a holomorphic function in a strip of radius (=width) ε_0, the same happens for the solution u(t,.) for a certain radius ε(t), as long as the solution exists. Our focus is on precise lower bounds on the spatial radius of analyticity ε(t) as t grows. We also get similar results for the Schroedinger equation with a real-analytic electromagnetic potential.

math.AP↗
arXiv2012

Decay estimates for solutions of nonlocal semilinear equations

We investigate the decay for $|x|\rightarrow \infty$ of weak Sobolev type solutions of semilinear nonlocal equations $Pu=F(u)$. We consider the case when $P=p(D)$ is an elliptic Fourier multiplier with polyhomogeneous symbol $p(ξ)$ and derive sharp algebraic decay estimates in terms of weighted Sobolev norms. In particular, we state a precise relation between the singularity of the symbol at the origin and the rate of decay of the corresponding solutions. Our basic example is the celebrated Benjamin-Ono equation {equation} \label{BO}(|D|+c)u=u^2, \qquad c>0,{equation} for internal solitary waves of deep stratified fluids. Their profile presents algebraic decay, in strong contrast with the exponential decay for KdV shallow water waves.

math.AP↗
arXiv2011

Regularity and decay of solutions of nonlinear harmonic oscillators

We prove sharp analytic regularity and decay at infinity of solutions of variable coefficients nonlinear harmonic oscillators. Namely, we show holomorphic extension to a sector in the complex domain, with a corresponding Gaussian decay, according to the basic properties of the Hermite functions in R^d. Our results apply, in particular, to nonlinear eigenvalue problems for the harmonic oscillator associated to a real-analytic scattering, or asymptotically conic, metric in R^d, as well as to certain perturbations of the classical harmonic oscillator.

math.AP↗
arXiv2010

Holomorphic extension of solutions of semilinear elliptic equations

We prove, for a wide class of semilinear elliptic differential and pseudodifferential equations in $\R^d$, that the solutions which are sufficiently regular and have a certain decay at infinity extend to holomorphic functions in sectors of $\mathbb{C}^d$, improving earlier results where the extension was shown for a strip. Moreover, exponential decay for such extended solutions is also proved. The results apply, in particular, to solitary wave solutions of many classical nonlinear evolution equations.

math.AP↗