arXiv subjects

Luigi Rodino

Publications and source records attributed to Luigi Rodino.

41 records · Page 3Linked to original sources

arXiv2011

Global regularity for ordinary differential operators with polynomial coefficients

For a class of ordinary differential operators $P$ with polynomial coefficients, we give a necessary and sufficient condition for $P$ to be globally regular in $\R$, i.e. $u\in\cS^\prime(\R)$ and $Pu\in\cS(\R)$ imply $u\in \cS(\R)$ (this can be regarded as a global version of the Schwartz' hypoellipticity notion). The condition involves the asymptotic behaviour, at infinity, of the roots $ξ=ξ_j(x)$ of the equation $p(x,ξ)=0$, where $p(x,ξ)$ is the (Weyl) symbol of $P$.

math.CA↗

arXiv2011

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 operators.

math.FA↗

arXiv2008

Boundedness of Fourier Integral Operators on $\mathcal{F} L^p$ spaces

We study the action of Fourier Integral Operators (FIOs) of H{ö}rmander's type on ${\mathcal{F}} L^p({\mathbb {R}}^d_{comp}$, $1\leq p\leq\infty$. We see, from the Beurling-Helson theorem, that generally FIOs of order zero fail to be bounded on these spaces when $p\not=2$, the counterexample being given by any smooth non-linear change of variable. Here we show that FIOs of order $m=-d|1/2-1/p|$ are instead bounded. Moreover, this loss of derivatives is proved to be sharp in every dimension $d\geq1$, even for phases which are linear in the dual variables. The proofs make use of tools from time-frequency analysis such as the theory of modulation spaces.

math.AP↗

arXiv2008

On the global boundedness of Fourier integral operators

We consider a class of Fourier integral operators, globally defined on $\mathbb{R}^{d}$, with symbols and phases satisfying product type estimates (the so-called $SG$ or scattering classes). We prove a sharp continuity result for such operators when acting on the modulation spaces $M^p$. The minimal loss of derivatives is shown to be $d|1/2-1/p|$. This global perspective produces a loss of decay as well, given by the same order. Strictly related, striking examples of unboundedness on $L^p$ spaces are presented.

math.FA↗

arXiv2007

Time-Frequency Analysis of Fourier Integral Operators

We use time-frequency methods for the study of Fourier Integral operators (FIOs). In this paper we shall show that Gabor frames provide very efficient representations for a large class of FIOs. Indeed, similarly to the case of shearlets and curvelets frames, the matrix representation of a Fourier Integral Operator with respect to a Gabor frame is well-organized. This is used as a powerful tool to study the boundedness of FIOs on modulation spaces. As special cases, we recapture boundedness results on modulation spaces for pseudo-differential operators with symbols in $M^{\infty,1}$, for some unimodular Fourier multipliers and metaplectic operators.

math.AP↗