Publications
Journal Articles
8.
Mazzucchi, Sonia; Nicola, Fabio; Trapasso, S. Ivan
Phase space analysis of higher-order dispersive equations with point interactions Journal Article
In: Comm. Partial Differential Equations, vol. 51, iss. 5, pp. 564–584, 2026.
Abstract | Links | BibTeX | Tags: dispersive equations, modulation spaces, partial differential equations, point interactions
@article{MNT_hodisp_24,
title = {Phase space analysis of higher-order dispersive equations with point interactions},
author = {Sonia Mazzucchi and Fabio Nicola and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2407.15521, arXiv
https://www.tandfonline.com/doi/epdf/10.1080/03605302.2026.2702344?needAccess=true, PDF},
doi = {10.1080/03605302.2026.2702344},
year = {2026},
date = {2026-06-23},
urldate = {2026-06-23},
journal = {Comm. Partial Differential Equations},
volume = {51},
issue = {5},
pages = {564–584},
abstract = {We investigate nonlinear, higher-order dispersive equations with measure (or even less regular) potentials and initial data with low regularity. Our approach is of distributional nature and relies on the phase space analysis (via Gabor wave packets) of the corresponding fundamental solution - in fact, locating the modulation/amalgam space regularity of such generalized Fresnel-type oscillatory functions is a problem of independent interest in harmonic analysis.},
keywords = {dispersive equations, modulation spaces, partial differential equations, point interactions},
pubstate = {published},
tppubtype = {article}
}
We investigate nonlinear, higher-order dispersive equations with measure (or even less regular) potentials and initial data with low regularity. Our approach is of distributional nature and relies on the phase space analysis (via Gabor wave packets) of the corresponding fundamental solution – in fact, locating the modulation/amalgam space regularity of such generalized Fresnel-type oscillatory functions is a problem of independent interest in harmonic analysis.
7.
Mazzucchi, Sonia; Nicola, Fabio; Trapasso, S. Ivan
Phase space analysis of finite and infinite dimensional Fresnel integrals Journal Article
In: J. Funct. Anal., vol. 289, iss. 8, no. 111009, pp. 1–51, 2025.
Abstract | Links | BibTeX | Tags: Feynman path integrals, harmonic analysis, infinite dimensional analysis, mathematical physics, modulation spaces
@article{MNT_JFA_25,
title = {Phase space analysis of finite and infinite dimensional Fresnel integrals},
author = {Sonia Mazzucchi and Fabio Nicola and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2403.20082, arXiv
https://www.sciencedirect.com/science/article/pii/S0022123625001910/pdfft?md5=2323ac7b3fed88926bde9dafa29aba1c&pid=1-s2.0-S0022123625001910-main.pdf, PDF},
doi = {10.1016/j.jfa.2025.111009},
year = {2025},
date = {2025-04-15},
urldate = {2025-04-15},
journal = {J. Funct. Anal.},
volume = {289},
number = {111009},
issue = {8},
pages = {1–51},
abstract = {The full characterization of the class of Fresnel integrable functions is an open problem in functional analysis, with significant applications to mathematical physics (Feynman path integrals) and the analysis of the Schrödinger equation. In finite dimension, we prove the Fresnel integrability of functions in the Sjöstrand class (M^{infty,1}) - a family of continuous and bounded functions, locally enjoying the mild regularity of the Fourier transform of an integrable function. This result broadly extends the current knowledge on the Fresnel integrability of Fourier transforms of finite complex measures, and relies upon ideas and techniques of Gabor wave packet analysis. We also discuss the problem of designing infinite-dimensional extensions of this result, obtaining the first, non-trivial concrete realization of a general framework of projective functional extensions introduced by Albeverio and Mazzucchi. As an interesting byproduct, we obtain the exact (M^{infty,1} to L^infty) operator norm of the free Schrödinger evolution operator.},
keywords = {Feynman path integrals, harmonic analysis, infinite dimensional analysis, mathematical physics, modulation spaces},
pubstate = {published},
tppubtype = {article}
}
The full characterization of the class of Fresnel integrable functions is an open problem in functional analysis, with significant applications to mathematical physics (Feynman path integrals) and the analysis of the Schrödinger equation. In finite dimension, we prove the Fresnel integrability of functions in the Sjöstrand class \(M^{\infty,1}\) – a family of continuous and bounded functions, locally enjoying the mild regularity of the Fourier transform of an integrable function. This result broadly extends the current knowledge on the Fresnel integrability of Fourier transforms of finite complex measures, and relies upon ideas and techniques of Gabor wave packet analysis. We also discuss the problem of designing infinite-dimensional extensions of this result, obtaining the first, non-trivial concrete realization of a general framework of projective functional extensions introduced by Albeverio and Mazzucchi. As an interesting byproduct, we obtain the exact \(M^{\infty,1} \to L^\infty\) operator norm of the free Schrödinger evolution operator.
6.
Trapasso, S. Ivan
Phase space analysis of spectral multipliers for the twisted Laplacian Journal Article
In: Trans. Amer. Math. Soc., vol. 378, no. 2, pp. 967-999, 2025.
Abstract | Links | BibTeX | Tags: harmonic analysis, harmonic oscillator, modulation spaces, partial differential equations, spectral multipliers
@article{T_TAMS_24,
title = {Phase space analysis of spectral multipliers for the twisted Laplacian},
author = {S. Ivan Trapasso},
url = {https://arxiv.org/abs/2306.00592, arXiv},
doi = {10.1090/tran/9224},
year = {2025},
date = {2025-02-01},
urldate = {2025-02-01},
journal = {Trans. Amer. Math. Soc.},
volume = {378},
number = {2},
pages = {967-999},
abstract = {We prove boundedness results on modulation and Wiener amalgam spaces for some families of spectral multipliers for the twisted Laplacian. We exploit the metaplectic equivalence relating the twisted Laplacian with a partial harmonic oscillator, leading to a general transference principle for the corresponding spectral multipliers. Our analysis encompasses powers of the twisted Laplacian and oscillating multipliers, with applications to the corresponding Schrödinger and wave flows. On the other hand, elaborating on the twisted convolution structure of the eigenprojections and its connection with the Weyl product of symbols, we obtain a complete picture of the boundedness of the heat flow for the twisted Laplacian. Results of the same kind are established for fractional heat flows via subordination.},
keywords = {harmonic analysis, harmonic oscillator, modulation spaces, partial differential equations, spectral multipliers},
pubstate = {published},
tppubtype = {article}
}
We prove boundedness results on modulation and Wiener amalgam spaces for some families of spectral multipliers for the twisted Laplacian. We exploit the metaplectic equivalence relating the twisted Laplacian with a partial harmonic oscillator, leading to a general transference principle for the corresponding spectral multipliers. Our analysis encompasses powers of the twisted Laplacian and oscillating multipliers, with applications to the corresponding Schrödinger and wave flows. On the other hand, elaborating on the twisted convolution structure of the eigenprojections and its connection with the Weyl product of symbols, we obtain a complete picture of the boundedness of the heat flow for the twisted Laplacian. Results of the same kind are established for fractional heat flows via subordination.
5.
Bhimani, Divyang G.; Manna, Ramesh; Nicola, Fabio; Thangavelu, Sundaram; Trapasso, S. Ivan
Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness Journal Article
In: Adv. Math., vol. 392, pp. 107995, 2021.
Abstract | Links | BibTeX | Tags: harmonic oscillator, modulation spaces, partial differential equations, semigroups, time-frequency analysis
@article{BMNTT_AIM_21,
title = {Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness},
author = {Divyang G. Bhimani and Ramesh Manna and Fabio Nicola and Sundaram Thangavelu and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2008.01226, arXiv},
doi = {10.1016/j.aim.2021.107995},
year = {2021},
date = {2021-12-03},
urldate = {2021-12-03},
journal = {Adv. Math.},
volume = {392},
pages = {107995},
publisher = {Academic Press},
abstract = {We study the Hermite operator (H=-Delta+|x|^2) in (mathbb{R}^d) and its fractional powers (H^beta), (beta>0) in phase space. Namely, we represent functions (f) via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform (V_g f) ((g) being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of (V_g f), that is in terms of membership to modulation spaces (M^{p,q}), (0< p,qleq infty). We prove the complete range of fixed-time estimates for the semigroup (e^{-tH^beta}) when acting on (M^{p,q}), for every (0< p,qleq infty), exhibiting the optimal global-in-time decay as well as phase-space smoothing. As an application, we establish global well-posedness for the nonlinear heat equation for (H^{beta}) with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay (e^{-c t}) as the solution of the corresponding linear equation, where (c=d^beta) is the bottom of the spectrum of (H^beta). This is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data - hence in (M^{infty,1}).},
keywords = {harmonic oscillator, modulation spaces, partial differential equations, semigroups, time-frequency analysis},
pubstate = {published},
tppubtype = {article}
}
We study the Hermite operator \(H=-\Delta+|x|^2\) in \(\mathbb{R}^d\) and its fractional powers \(H^\beta\), \(\beta>0\) in phase space. Namely, we represent functions \(f\) via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform \(V_g f\) (\(g\) being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of \(V_g f\), that is in terms of membership to modulation spaces \(M^{p,q}\), \(0< p,q\leq \infty\). We prove the complete range of fixed-time estimates for the semigroup \(e^{-tH^\beta}\) when acting on \(M^{p,q}\), for every \(0< p,q\leq \infty\), exhibiting the optimal global-in-time decay as well as phase-space smoothing. As an application, we establish global well-posedness for the nonlinear heat equation for \(H^{\beta}\) with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay \(e^{-c t}\) as the solution of the corresponding linear equation, where \(c=d^\beta\) is the bottom of the spectrum of \(H^\beta\). This is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data – hence in \(M^{\infty,1}\).
4.
Trapasso, S. Ivan
Time-frequency analysis of the Dirac equation Journal Article
In: J. Differential Equations, vol. 269, no. 3, pp. 2477–2502, 2020.
Abstract | Links | BibTeX | Tags: Dirac equation, modulation spaces, partial differential equations, pseudodifferential operators, time-frequency analysis
@article{T_JDE_20,
title = {Time-frequency analysis of the Dirac equation},
author = {S. Ivan Trapasso},
url = {https://arxiv.org/abs/1909.09842, arXiv
https://www.sciencedirect.com/science/article/pii/S0022039620300577/pdfft?md5=9529d30135284078192dbd0cbc114288&pid=1-s2.0-S0022039620300577-main.pdf, PDF},
doi = {10.1016/j.jde.2020.02.002},
year = {2020},
date = {2020-07-15},
urldate = {2020-07-15},
journal = {J. Differential Equations},
volume = {269},
number = {3},
pages = {2477–2502},
publisher = {Academic Press},
abstract = {The purpose of this paper is to investigate several issues concerning the Dirac equation from a time-frequency analysis perspective. More precisely, we provide estimates in weighted modulation and Wiener amalgam spaces for the solutions of the Dirac equation with rough potentials. We focus in particular on bounded perturbations, arising as the Weyl quantization of suitable time-dependent symbols, as well as on quadratic and sub-quadratic non-smooth functions, hence generalizing the results in a recent paper by Kato and Naumkin. We then prove local well-posedness on the same function spaces for the nonlinear Dirac equation with a general nonlinearity, including power-type terms and the Thirring model. For this study we adopt the unifying framework of vector-valued time-frequency analysis as developed by Wahlberg; most of the preliminary results are stated under general assumptions and hence they may be of independent interest.},
keywords = {Dirac equation, modulation spaces, partial differential equations, pseudodifferential operators, time-frequency analysis},
pubstate = {published},
tppubtype = {article}
}
The purpose of this paper is to investigate several issues concerning the Dirac equation from a time-frequency analysis perspective. More precisely, we provide estimates in weighted modulation and Wiener amalgam spaces for the solutions of the Dirac equation with rough potentials. We focus in particular on bounded perturbations, arising as the Weyl quantization of suitable time-dependent symbols, as well as on quadratic and sub-quadratic non-smooth functions, hence generalizing the results in a recent paper by Kato and Naumkin. We then prove local well-posedness on the same function spaces for the nonlinear Dirac equation with a general nonlinearity, including power-type terms and the Thirring model. For this study we adopt the unifying framework of vector-valued time-frequency analysis as developed by Wahlberg; most of the preliminary results are stated under general assumptions and hence they may be of independent interest.
3.
Cordero, Elena; Nicola, Fabio; Trapasso, S. Ivan
Almost diagonalization of τ-pseudodifferential operators with symbols in Wiener amalgam and modulation spaces Journal Article
In: J. Fourier Anal. Appl., vol. 25, no. 4, pp. 1927–1957, 2019.
Abstract | Links | BibTeX | Tags: almost diagonalization, harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis
@article{CNT_JFAA_19,
title = {Almost diagonalization of τ-pseudodifferential operators with symbols in Wiener amalgam and modulation spaces},
author = {Elena Cordero and Fabio Nicola and S. Ivan Trapasso},
url = {https://arxiv.org/abs/1802.10314, arXiv},
doi = {10.1007/s00041-018-09651-z},
year = {2019},
date = {2019-08-15},
urldate = {2019-08-15},
journal = {J. Fourier Anal. Appl.},
volume = {25},
number = {4},
pages = {1927–1957},
publisher = {Springer US},
abstract = {In this paper we focus on the almost-diagonalization properties of (tau)-pseudodifferential operators using techniques from time-frequency analysis. Our function spaces are modulation spaces and the special class of Wiener amalgam spaces arising by considering the action of the Fourier transform of modulation spaces. A particular example is provided by the Sjöstrand class, for which Gröchenig exhibited the almost diagonalization of Weyl operators. We shall show that such result can be extended to any (tau)-pseudodifferential operator, for (tau in [0,1]), also with symbol in weighted Wiener amalgam spaces. As a consequence, we infer boundedness, algebra and Wiener properties for (tau)-pseudodifferential operators on Wiener amalgam and modulation spaces.},
keywords = {almost diagonalization, harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis},
pubstate = {published},
tppubtype = {article}
}
In this paper we focus on the almost-diagonalization properties of \(\tau\)-pseudodifferential operators using techniques from time-frequency analysis. Our function spaces are modulation spaces and the special class of Wiener amalgam spaces arising by considering the action of the Fourier transform of modulation spaces. A particular example is provided by the Sjöstrand class, for which Gröchenig exhibited the almost diagonalization of Weyl operators. We shall show that such result can be extended to any \(\tau\)-pseudodifferential operator, for \(\tau \in [0,1]\), also with symbol in weighted Wiener amalgam spaces. As a consequence, we infer boundedness, algebra and Wiener properties for \(\tau\)-pseudodifferential operators on Wiener amalgam and modulation spaces.
2.
Cordero, Elena; D'Elia, Lorenza; Trapasso, S. Ivan
Norm estimates for τ-pseudodifferential operators in Wiener amalgam and modulation spaces Journal Article
In: J. Math. Anal. Appl., vol. 471, no. 1-2, pp. 541–563, 2019.
Abstract | Links | BibTeX | Tags: harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis, Wiener amalgam spaces
@article{CDT_JMAA_19,
title = {Norm estimates for τ-pseudodifferential operators in Wiener amalgam and modulation spaces},
author = {Elena Cordero and Lorenza D'Elia and S. Ivan Trapasso},
url = {https://arxiv.org/abs/1803.07865, arXiv
https://www.sciencedirect.com/science/article/pii/S0022247X18309193/pdfft?md5=56e69bc9745c316f260d37420041913d&pid=1-s2.0-S0022247X18309193-main.pdf, PDF},
doi = {10.1016/j.jmaa.2018.10.090},
year = {2019},
date = {2019-03-01},
urldate = {2019-03-01},
journal = {J. Math. Anal. Appl.},
volume = {471},
number = {1-2},
pages = {541–563},
publisher = {Academic Press},
abstract = {We study continuity properties on modulation spaces for (tau)-pseudodifferential operators with symbols (a) in Wiener amalgam spaces. We obtain boundedness results for (tau in (0,1)) whereas, in the end-points (tau=0) and (tau=1), the corresponding operators are in general unbounded. Furthermore, for (tau in (0,1)), we exhibit a function of (tau) which is an upper bound for the operator norm.
The continuity properties of (tau)-pseudodifferential operators, for any (tauin [0,1]), with symbols (a) in modulation spaces are well known. Here we find an upper bound for the operator norm which does not depend on the parameter (tau in [0,1]), as expected. Key ingredients are uniform continuity estimates for (tau)-Wigner distributions.},
keywords = {harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis, Wiener amalgam spaces},
pubstate = {published},
tppubtype = {article}
}
We study continuity properties on modulation spaces for \(\tau\)-pseudodifferential operators with symbols \(a\) in Wiener amalgam spaces. We obtain boundedness results for \(\tau \in (0,1)\) whereas, in the end-points \(\tau=0\) and \(\tau=1\), the corresponding operators are in general unbounded. Furthermore, for \(\tau \in (0,1)\), we exhibit a function of \(\tau\) which is an upper bound for the operator norm.
The continuity properties of \(\tau\)-pseudodifferential operators, for any \(\tau\in [0,1]\), with symbols \(a\) in modulation spaces are well known. Here we find an upper bound for the operator norm which does not depend on the parameter \(\tau \in [0,1]\), as expected. Key ingredients are uniform continuity estimates for \(\tau\)-Wigner distributions.
The continuity properties of \(\tau\)-pseudodifferential operators, for any \(\tau\in [0,1]\), with symbols \(a\) in modulation spaces are well known. Here we find an upper bound for the operator norm which does not depend on the parameter \(\tau \in [0,1]\), as expected. Key ingredients are uniform continuity estimates for \(\tau\)-Wigner distributions.
1.
D'Elia, Lorenza; Trapasso, S. Ivan
Boundedness of pseudodifferential operators with symbols in Wiener amalgam spaces on modulation spaces Journal Article
In: J. Pseudo-Differ. Oper. Appl., vol. 9, pp. 881–890, 2018.
Abstract | Links | BibTeX | Tags: harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis, Wiener amalgam spaces
@article{DT_JPDO_18,
title = {Boundedness of pseudodifferential operators with symbols in Wiener amalgam spaces on modulation spaces},
author = {Lorenza D'Elia and S. Ivan Trapasso},
url = {https://arxiv.org/abs/1703.08989, arXiv},
doi = {10.1007/s11868-017-0220-1},
year = {2018},
date = {2018-12-01},
urldate = {2018-12-01},
journal = {J. Pseudo-Differ. Oper. Appl.},
volume = {9},
pages = {881–890},
publisher = {Springer International Publishing},
abstract = {This paper provides sufficient conditions for the boundedness of Weyl operators on modulation spaces. The Weyl symbols belong to Wiener amalgam spaces, or generalized modulation spaces, as recently renamed by their inventor Hans Feichtinger. This is the first result which relates symbols in Wiener amalgam spaces to operators acting on classical modulation spaces.},
keywords = {harmonic analysis, modulation spaces, pseudodifferential operators, time-frequency analysis, Wiener amalgam spaces},
pubstate = {published},
tppubtype = {article}
}
This paper provides sufficient conditions for the boundedness of Weyl operators on modulation spaces. The Weyl symbols belong to Wiener amalgam spaces, or generalized modulation spaces, as recently renamed by their inventor Hans Feichtinger. This is the first result which relates symbols in Wiener amalgam spaces to operators acting on classical modulation spaces.