We report about joint work in progress with Angela A. Albanese and Werner J. Ricker. Let $X$ be a Banach space of analytic functions on the open unit disc $\mathbb{D}\subset \mathbb{C}$ which contains the polynomials and such that the inclusion $X\subset H(\mathbb{D})$ is continuous. Given a continuous linear operator $T\colon X\to X$, we study the largest domain space $[T,X]$ of $T$. That is, $[T,X]\subset H(\mathbb{D})$ is the largest Banach space of analytic functions containing $X$ to which $T$ has a continuous, linear, $X$-valued extension $T\colon [T,X]\to X$. The class of operators considered consists of generalized Volterra operators $T=V_g$ acting in the Korenblum growth Banach spaces $X:=A^{-\gamma}$, for $\gamma>0$. The Volterra operator $V_g\colon H(\mathbb{D})\to H(\mathbb{D})$ is the linear operator, defined for $g \in H(\mathbb{D})$, by $$ (V_gf)(z):=\int_0^z f(\xi)g'(\xi)\,d\xi,\quad f\in H(\mathbb{D}), \ z\in\mathbb{D}. $$ The operator $V_g$ acts continuously in $H(\mathbb{D})$. For each $\gamma>0$ the Korenblum growth Banach spaces are defined by \[ A^{-\gamma}:=\{f\in H(\mathbb{D}):\ \|f\|_{-\gamma}:=\sup_{z\in\mathbb{D}}(1-|z|)^\gamma |f(z)|<\infty\} \] and its (proper) closed subspace by \[ A^{-\gamma}_0:=\{f\in H(\mathbb{D}):\ \lim_{|z|\to 1^-}(1-|z|)^\gamma |f(z)|=0\}. \] Previous studies dealt with the classical Cesàro operator $T:=C$ acting in the Hardy spaces $H^p$, $1\leq p<\infty$ by Curbera and Ricker, in $A^{-\gamma}$, by Albanese, Bonet and Ricker, and more recently, generalized Volterra operators $T$ acting in $X:=H^p$ by Bellavita, Daskalogiannis, Nikolaidis and Stylogiannis.

In the study of the surjectivity of the Borel mapping in Carleman-Roumieu ultraholomorphic classes, defined on sectors of the Riemann surface of the logarithm and in terms of weight sequences $\mathbf{M}=(M_p)_{p\ge 0}$, the new condition $$ \exists C_0>0,\ H>1\forall p \in \mathbb{N}_0,\ \log(m_{p+1}/m_p) \leq C_0H^{p+1}, \quad (m_p:=M_{p+1}/M_p)$$ has appeared recently, allowing for the construction of linear continuous extension operators from optimal flat functions. It turns out that this condition is the key for presenting a new Stieltjes moment problem in Gelfand-Shilov classes defined by weight sequences. Unlike the classical situation dealing with regular sequences (in the sense of Dyn'kin), the Stieltjes moment sequence of a function in such classes will have its growth controlled by the shifted sequence $\mathbf{M}_{+1}:=(M_{p+1})_{p\ge 0}$ instead of by $\mathbf{M}$ itself. We will present results concerning the injectivity and surjectivity of the Stieltjes moment mapping in this new framework. Time permitting, we will comment on results in Beurling-like classes and on some possible extensions of our results. This is a joint work with J. Jiménez-Garrido, I. Miguel-Cantero and G. Schindl.

We characterize the compactness of the Weyl operator when acting from $\mathcal{S}_\omega$ into itself in terms of the short-time Fourier transform of its symbol. Some examples are given. This is a joint work with Vicente Asensio (Universitat Politècnica de València), Chiara Boiti (Università degli Studi di Ferrara) and Alessandro Oliaro (Università di Torino)

For a sequence $\mathbf{M}=\{M_p\}_{p\in\mathbb{N}_0}$ of real positive numbers, its associated function is defined by $$\omega_{\mathbf{M}}(t):=M_0\sup_{p\in\mathbb{N}_0}\log\frac{t^p}{M_p},\qquad t>0.$$ S. Mandelbrojt proved that if $\lim_{p\rightarrow+\infty}M_p^{1/p}=+\infty$, then \begin{equation*} (\star)\;\;\;\;M_p=M_0\sup_{t>0}\frac{t^p}{\exp\omega_M(t)},\qquad p\in\mathbb{N}_0, \end{equation*} if and only if $\{M_p\}_{p\in\mathbb{N}_0}$ is logarithmically convex, i.e. $$M_p^2\leq M_{p-1}M_{p+1},\qquad\forall p\in\mathbb{N}.$$ However, $(\star)$ had never been generalized to the $d$-dimensional anisotropic case, since the classical coordinate-wise logarithmic convexity condition $$M_\alpha^2\leq M_{\alpha-e_j}M_{\alpha+e_j}, \qquad\alpha\in\mathbb{N}_0^d,\ 1\leq j \leq d,\ \alpha_j\geq1,$$ is not sufficient. Assuming the stronger condition that $\{M_{\alpha}\}_{\alpha\in\mathbb{N}_0^d}$ is log-convex on the globality of its variables, in the sense that $\log M_{\alpha}=F(\alpha)$ for a convex function $F [0,+\infty)^d\rightarrow\mathbb{R}$, we extend Mandelbrojt's identity $(\star)$ to \begin{equation*} M_{\alpha}=M_0\sup_{s\in(0,+\infty)^d}\frac{s^{\alpha}}{\exp\omega_{\mathbf{M}}(s)}, \qquad\forall\alpha\in\mathbb{N}_0^d. \end{equation*} Our construction of the (optimal) convex minorant of $\{a_{\alpha}\}_{\alpha\in\mathbb{N}^d_0}$ (then $a_{\alpha}=\log M_{\alpha}$) is made by taking the supremum of hyperplanes approaching from below the given sequence and leads to the notion of convexity in the sense that $a_{\alpha}=F(\alpha)$ for a convex function $F$. This result is a very useful tool for working in the anisotropic setting, and we expect several applications, that could be object of future works. This is a joint work with D. Jornet, A. Oliaro, and G. Schindl.

In this talk we consider different quadratic representations based on the Fourier transform, such as the Wigner transform, the Spectrogram, and Cohen class representations. In signal analysis they are usually interpreted as joint distributions of the energy of a signal in time and frequency, and each one of them has good features and inconveniences, due to (various forms of) the uncertainty principle that prevents to find a unique representation satisfying all ''ideal'' desired properties. This is the reason why many different quadratic forms are considered. On the other hand, since different representations have a common meaning, it is worth analyzing the relations between them, and we could expect that different representations have, for example, comparable ''sizes'' in some sense. In this work we show that, when the size of a representation is measured by its $L^p$ norm, it is not always true that different representations have ''comparable size'', but this becomes true when considering weighted norms, and this leads to use, as a natural functional framework, spaces of ultradifferentiable type. As an application, some uncertainty principles for representations are derived. The talk is based on a joint work with Prof. A. Albanese and C. Mele (University of Salento, Italy).

This presentation focuses on the study of the regularity of random wavelet series. We first study their belonging to certain functional spaces and we compare these results with long-established results related to random Fourier series. Next, we show how the study of random wavelet series leads to precise pointwise regularity properties of processes like fractional Brownian motion. Additionally, we explore how these series helps create Gaussian processes with random Hölder exponents. This talk is based on joint works with S. Jaffard, L. Loosveldt and B. Vedel.

We discuss recent results on ultradistributional boundary values of zero solutions of an elliptic and parabolic, respectively, constant coefficient partial differential operator. These results unify and considerably extend various classical results of Komatsu and Matsuzawa about boundary values of holomorphic functions, harmonic functions and zero solutions of the heat equation in ultradistribution spaces. This is joint work with Andreas Debrouwere (Vrije Universiteit Brussel).

In this talk, I will discuss global analytic hypoellipticity for a class of differential operators that can be expressed as $P = \sum_{j=1}^\nu X_j^2 + X_0 + a$ with real-analytic coefficients on compact Lie groups. To obtain global analytic hypoellipticity, we assume that the vector fields satisfy Hörmander's finite type condition and that there exists a closed subgroup whose action leaves the vector fields invariant. We further assume the operator is elliptic in directions transversal to the action of the subgroup. This paves the way for further studies on the regularity of sums of squares on principal fiber bundles.