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Chiara Boiti

Publications and source records attributed to Chiara Boiti.

At least 19 recordsLinked to original sources
arXiv2026

Umbrella theorems for time-frequency representations

An uncertainty principle due to H.S. Shapiro, the so-called Umbrella Theorem, asserts that there is no square integrable function uniformly dominating all the elements of an orthonormal family in $L^2(\mathbb{R})$ and their Fourier transforms, unless the sequence is finite. In this paper we present some results on Umbrella Theorems in $L^2(\mathbb{R}^d)$ related to time-frequency representations. We further extend the analysis to the case of $L^2(\mathbb{R}^+)$, by means of the Mellin transform.

math.FA↗
arXiv2026

Stability of global wave front sets by perturbations of frames

In this paper we consider the Gabor wave front set of ultradistributions in the frame of ultradifferentiable functions. We prove that such a wave front set, defined through a Gabor frame on a regular lattice, is not affected by perturbations of the frame, in two different cases: when we consider $\varepsilon$-perturbations of Christensen type, and when we consider nonstationary Gabor frames.

math.FA↗
arXiv2025

A third-order conservation law for the Kirchhoff-Pokhozhaev equation

We prove that the special Kirchhoff equation studied by Pokhozhaev admits a third-order conservation law. We further show that if the energy of the solution is sufficiently small, then the $L^2$-norms of the derivatives up to third order of the solution remain uniformly bounded with respect to time.

math.AP↗
arXiv2024

On the compactness of the Weyl operator in $\mathcal{S}_ω$

We characterize, using time-frequency analysis, the continuity and compactness of the Weyl operator in global classes of ultradifferentiable functions $\mathcal{S}_ω$, for weight functions $ω$ in the sense of Braun, Meise and Taylor. As a consequence, we give results about the compactness of the localization operator in $\mathcal{S}_ω$, in relation with the spaces of $ω$-multipliers and $ω$-convolutors of $\mathcal{S}_ω$. Moreover, we provide several examples that complement our investigation.

math.FA↗
arXiv2024

Construction of the log-convex minorant of a sequence $\{M_α\}_{α\in\mathbb{N}_0^d}$

We give a simple construction of the log-convex minorant of a sequence $\{M_α\}_{α\in\mathbb{N}_0^d}$ and consequently extend to the $d$-dimensional case the well-known formula that relates a log-convex sequence $\{M_p\}_{p\in\mathbb{N}_0}$ to its associated function $ω_M$, that is $M_p=\sup_{t>0}t^p\exp(-ω_M(t))$. We show that in the more dimensional anisotropic case the classical log-convex condition $M_α^2\leq M_{α-e_j}M_{α+e_j}$ is not sufficient: convexity as a function of more variables is needed (not only coordinate-wise). We finally obtain some applications to the inclusion of spaces of rapidly decreasing ultradifferentiable functions in the matrix weighted setting.

math.FA↗
arXiv2024

On the inclusion relations of global ultradifferentiable classes defined by weight matrices

We study and characterize the inclusion relations of global classes in the general weight matrix framework in terms of growth relations for the defining weight matrices. We consider the Roumieu and Beurling cases, and as a particular case we also treat the classical weight function and weight sequence cases. Moreover, we construct a weight sequence which is oscillating around any weight sequence which satisfies some minimal conditions and, in particular, around the critical weight sequence $(p!)^{1/2}$, related with the non-triviality of the classes. Finally, we also obtain comparison results both on classes defined by weight functions that can be defined by weight sequences and conversely.

math.FA↗
arXiv2023

Notes on a paper of Pokhozhaev

We prove a second order identity for the Kirchhoff equation which yields, in particular, a simple and direct proof of Pokhozhaev's second order conservation law when the nonlinearity has the special form $(C_1 s +C_2)^{-2}$. As applications, we give: an estimate of order $\varepsilon^{-4}$ for the lifespan $T_\varepsilon$ of the solution of the Cauchy problem with initial data of size $\varepsilon$ in Sobolev spaces when the nonlinearity is given by any $C^2$ function $m(s)>0$; a necessary and sufficient condition for boundedness of a second order energy of the solutions.

math.AP↗
arXiv2023

Mean-dispersion principles and the Wigner transform

Given a function $f\in L^2(\mathbb R)$, we consider means and variances associated to $f$ and its Fourier transform $\hat{f}$, and explore their relations with the Wigner transform $W(f)$, obtaining a simple new proof of Shapiro's mean-dispersion principle. Uncertainty principles for orthonormal sequences in $L^2(\mathbb R)$ involving linear partial differential operators with polynomial coefficients and the Wigner distribution, or different Cohen class representations, are obtained, and an extension to the case of Riesz bases is studied.

math.AP↗
arXiv2022

A simple proof of Kotake-Narasimhan theorem in some classes of ultradifferentiable functions

We give a simple proof of a general theorem of Kotake-Narasimhan for elliptic operators in the setting of ultradifferentiable functions in the sense of Braun, Meise and Taylor. We follow the ideas of Komatsu. Based on an example of Métivier, we also show that the ellipticity is a necessary condition for the theorem to be true. The present new version of the paper modifies the proof of Theorem 1.4 for an observation by Hoepfner and Rampazo who pointed out that an induction hypothesis depends on a constant $C_q$ that changes in the induction process, and hence the argument might not work as it was written. However, the statement of the result was originally correct and modifying $C_q$ with a more concrete expression in the induction hypothesis, the induction procedure is easily clarified with almost the same proof. Moreover, we eliminate the condition that the weight is identically zero in the interval [0,1], showing that the statements hold true with very similar arguments.

math.AP↗
arXiv2020

Nuclear global spaces of ultradifferentiable functions in the matrix weighted setting

We prove that the Hermite functions are an absolute Schauder basis for many global weighted spaces of ultradifferentiable functions in the matrix weighted setting and we determine also the corresponding coefficient spaces, thus extending previous work by Langenbruch. As a consequence we give very general conditions for these spaces to be nuclear. In particular, we obtain the corresponding results for spaces defined by weight functions.

math.FA↗
arXiv2019

Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis

We use techniques from time-frequency analysis to show that the space $\mathcal S_ω$ of rapidly decreasing $ω$-ultradifferentiable functions is nuclear for every weight function $ω(t)=o(t)$ as $t$ tends to infinity. Moreover, we prove that, for a sequence $(M_p)_p$ satisfying the classical condition $(M1)$ of Komatsu, the space of Beurling type $\mathcal S_{(M_p)}$ when defined with $L^{2}\,$norms is nuclear exactly when condition $(M2)'$ of Komatsu holds.

math.FA↗
arXiv2019

About the nuclearity of ${\mathcal S}_{(M_{p})}$ and ${\mathcal S}_ω$

We use an isomorphism established by Langenbruch between some sequence spaces and weighted spaces of generalized functions to give sufficient conditions for the (Beurling type) space ${\mathcal S}_{(M_p)}$ to be nuclear. As a consequence, we obtain that for a weight function $ω$ satisfying the mild condition: $2ω(t)\leq ω(Ht)+H$ for some $H>1$ and for all $t\geq0$, the space ${\mathcal S}_ω$ in the sense of Björck is also nuclear.

math.FA↗
arXiv2019

Real Paley-Wiener theorems in spaces of ultradifferentiable functions

We develop real Paley-Wiener theorems for classes ${\mathcal S}_ω$ of ultradifferentiable functions and related $L^{p}$-spaces in the spirit of Bang and Andersen for the Schwartz class. We introduce results of this type for the so-called Gabor transform and give a full characterization in terms of Fourier and Wigner transforms for several variables of a Paley-Wiener theorem in this general setting, which is new in the literature. We also analyze this type of results when the support of the function is not compact using polynomials. Some examples are given.

math.FA↗
arXiv2017

The Gabor wave front set in spaces of ultradifferentiable functions

Given a non-quasianalytic subadditive weight function $ω$ we consider the weighted Schwartz space $\mathcal{S}_ω$ and the short-time Fourier transform on $\mathcal{S}_ω$, $\mathcal{S}'_ω$ and on the related modulation spaces with exponential weights. In this setting we define the $ω$-wave front set $WF'_ω(u)$ and the Gabor $ω$-wave front set $WF^G_ω(u)$ of $u\in\mathcal{S}'_ω$, and we prove that they coincide. Finally we look at applications of this wave front set for operators of differential and pseudo-differential type.

math.FA↗
arXiv2016

Semilinear p-evolution equations in Sobolev spaces

We prove local in time well-posedness in Sobolev spaces of the Cauchy problem for semi-linear p-evolution equations of the first order with real principal part, but complex valued coefficients for the lower order terms, assuming decay conditions on the imaginary parts as |x| goes to infinity.

math.AP↗