Abstract

In collaboration with Marco Cappiello, Sérgio Luís Zani and Adalberto Panobianco Bergamasco.

This work aims to investigate global solvability in time-periodic Gelfand-Shilov spaces for the first-order operator $L=\partial_t+a(x)\partial_x$ on $\mathbb{T}\times\mathbb{R}$, where $a(x)$ is a smooth real-valued function satisfying global Gevrey estimates and suitable growth conditions at infinity. We establish a complete characterization of global solvability in time-periodic Gelfand-Shilov spaces in terms of the structure of the zero set of $a(x)$. The time-periodic Gelfand-Shilov space $\mathcal{S}_{\sigma,\mu}(\mathbb{T}\times\mathbb{R})$, as introduced in [AC22, AC25], consists of smooth functions that are uniformly Gevrey regular of order $\sigma$ in the periodic variable and exhibit Gelfand–Shilov behavior of order $\mu$ in the real variable. Our findings build on previous work by Bergamasco and Petronilho [BP]. 

This is a work in progress.

Acknowledgments: We would like to thank CAPES for Financial Support.

References 

[AC22] F. Ávila Silva, M. Cappiello, Time-periodic Gelfand-Shilov spaces and global hypoellipticity on $\mathbb{T}\times\mathbb{R}^n$, J. Funct. Anal. 282 (2022), no. 9, 29 pp.

[AC25] F. Ávila Silva, M. Cappiello, Globally solvable time-periodic evolution equations in Gelfand-Shilov classes, Math. Ann. 391 (2025), no. 1, 399–430.

[BP] A. P. Bergamasco, G. Petronilho, Closedness of the range for vector fields on the torus, J. Differential Equations 154 (1999), 132–139.

[GS] I. M. Gelfand, G. E. Shilov, Generalized Functions. Vol. 2: Spaces of Fundamental and Generalized Functions, Academic Press, New York-London, 1968.

[NR] F. Nicola, L. Rodino, Global Pseudo-Differential Calculus on Euclidean Spaces, Pseudo-Differential Operators. Theory and Applications, 4, Birkhäuser, Basel, 2010.

Abstract

The talk consists of two parts. First, I will discuss global solvability for first-order complex vector fields and systems in mixed Schwartz spaces. Specifically, for operators of the form $L = \partial_t + \sum_{j=1}^{n}c_{j}(t)\partial_{x_j}$ on $\mathbb{T}^1\times\mathbb{R}^n$ and $\mathbb{L} = d_{t} + c(t)\wedge \partial_x$ on $\mathbb{T}^n\times\mathbb{R}$, where the coefficients are smooth periodic functions, we show that solvability in the Schwartz space depends on the sign of the imaginary parts and on connectivity properties of certain sublevel and superlevel sets. These results extend classical periodic solvability theorems (cf. [3,4]) to the mixed Schwartz setting. This work is in progress as part of my PhD thesis, under the supervision of Adalberto Bergamasco and Sérgio Zani (University of São Paulo).

Second, global solvability and global hypoellipticity for systems of differential operators of the form $\mathbb{L}=d_t+\omega(t)\wedge P$ on $\mathbb{T}^n\times X$, where $X$ is a non-compact manifold endowed with a scattering structure and $P\in\Psi_{\mathrm{sc}}^{m,\mu}(X)$ is a positive self-adjoint elliptic scattering operator. We establish a complete characterization of global solvability and global hypoellipticity for $\mathbb{L}$ in terms of the spectral properties of $P$ and the behavior of families of periodic ordinary differential equations. The Schwartz space $\mathcal{S}(\mathbb{T}^n\times X)$, characterized via eigenfunction expansions of $P$, cf. [1,2], consists of smooth functions whose expansion coefficients decay rapidly with respect to the eigenvalues of $P$. Our results rely on the spectral framework for $P$, recalled, e.g., [1,2]. This is work in progress jointly with Sandro Coriasco (University of Turin).

Acknowledgments: We thank CAPES for financial support during the author’s stay in Turin.

References 

[1] F. de Avila Silva, M. Bonino, S. Coriasco,  Global hypoellipticity and solvability for a class of evolution operators in time-periodic weighted Sobolev spaces. Math. Ann. 394, 87 (2026).

[2] F. de Avila Silva, M. Bonino, S. Coriasco,  Global hypoellipticity for a class of time-periodic operators on asymptotically Euclidean manifolds, Preprint, 2025.

[3] A. P. Bergamasco, C. de Medeira, S. L. Zani, Globally solvable systems of complex vector fields, J. Differential Equations, 2012.

[4] A. P. Bergamasco, P. L. Dattori da Silva, R. B. Gonzalez, Existence and Regularity of Periodic Solutions to Certain First-Order Partial Differential Equations, J. Fourier Anal. Appl., 2017.

[5] R. Melrose, Geometric Scattering Theory, Stanford Lectures, Cambridge University Press, Cambridge, 1995.

[6] M. A. Shubin, Pseudodifferential operators and spectral theory, Springer-Verlag, 1987.

[7] F. Treves, Topological Vector Spaces, Academic Press, New York, 1967.