Concentration inequalities for holomorphic functions

Functional inequalities play a crucial role in several mathematical fields, including partial differential equations, calculus of variations and mathematical physics. Let us illustrate the topic with the following rough question:

Given some notion of concentration on a function space, under what conditions is such concentration optimal?

This question may immediately bring to mind uncertainty principles, considered in harmonic (Fourier) analysis, quantum mechanics or signal processing.

If the concentration is given in terms of the Wehrl entropy, the question on its optimality goes back to the late 1970s. Lieb proved that the Wehrl entropy of quantum Glauber states is minimum when it is a coherent state. However, the uniqueness of these minimizers, which in turn answered positively the original Wehrl’s conjecture in 1979, has been proved two years ago. Even more recently, the stability or quantitative counterpart of the optimality has been addressed.

Analogous questions can be formulated in other settings, like for Bloch states and the corresponding space of one-dimensional complex polynomials of bounded degree. Moreover, a local version of the entropy has also been considered, that is, the concentration is measured in terms of the localization in a subset. This in particular coincides with the energy concentration for the short-time.

Fourier transform (STFT) or the wavelet transform, in the suitable function spaces. The course would introduce all the aforementioned results on concentration inequalities, covering both the cases in different spaces of holomorphic functions (Fock, Bergman and polynomials) and the two notions of concentration in terms of the localization in a subset and of the (generalized) Wehrl entropy.

● Introduction to uncertainty principles.
● Spaces of holomorphic functions; reproducing kernels and coherent states.
● Superlevel sets of subharmonic functions.
● Inequalities for the norm of the localization operator.
● Lieb conjecture on the generalized Wehrl entropy.
● Uniqueness of the optimizers.
● Stability of the inequalities.

• Rupert L. Frank, “Sharp inequalities for coherent states and their optimizers,” Adv. Nonlinear Stud. 23 (2023), 28.
• María Ángeles García-Ferrero and Joaquim Ortega-Cerdà, “Stability of the concentration inequality on polynomials,” Commun. Math. Phys. (accepted), 2025.
• Jaime Gómez, André Guerra, João P. G. Ramos, and Paolo Tilli, “Stability of the Faber–Krahn inequality for the short-time Fourier transform,” Invent. Math. 236 (2024), 779–836.
• Jaime Gómez, David Kalaj, Petar Melentijević, and João P. G. Ramos, “Uniform stability of concentration inequalities and applications,” arXiv:2411.16010v1, 2024.
• Aleksei Kulikov, Fabio Nicola, Joaquim Ortega-Cerdà, and Paolo Tilli, “A monotonicity theorem for subharmonic functions on manifolds,” arXiv:2212.14008v2, 2023.
• Fabio Nicola, Federico Riccardi, Paolo Tilli, “The Wehrl-type entropy conjecture for symmetric SU(N) coherent states: cases of equality and stability” (2024), arXiv:2412.10940
  
• Fabio Nicola and Paolo Tilli, “The Faber–Krahn inequality for the short-time Fourier transform,” Invent. Math. 230 (2022), no. 1, 1–30.
• João P. G. Ramos and Paolo Tilli, “A Faber–Krahn inequality for wavelet transforms,” Bull. Lond. Math. Soc. 55 (2023), no. 4, 2018–2034.

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