A convex function $\Phi$ from $[0,\infty]$ to $[0,\infty]$ with properties
\begin{equation}
\Phi (0)=0,\qquad \lim_{t\to \infty}\Phi (t)=\Phi (\infty )=\infty ,
\end{equation}
is called a *Young function*. For any Young function $\Phi$, the Orlicz space $L^\Phi$ is a Banach space,
and consists of all measurable functions $f$ such that $\Phi (t\cdot |f|)\in L^1$ for some $t>0$. By choosing
$\Phi$ in suitable ways we gain the definition of any (Banach) Lebesgue space $L^p$, as well as sums of such
spaces like $L^p+L^q$, $p,q\in [1,\infty ]$. In particular, the family of Orlicz spaces contain any Lebesgue space.

The Orlicz modulation space $M^{\Phi}$ is obtained by imposing $L^\Phi$ norm conditions of the short-time Fourier transforms of the involved functions and distributions. In the same way we may discuss Orlicz modulation spaces $M^{\Phi ,\Psi}$ of mixed normed types. Again, by choosing the Young functions $\Phi$ and $\Psi$ in suitable ways, $M^{\Phi ,\Psi}$ becomes the classical Feichtinger’s modulation space $M^{p,q}$.

In the talk we explain some basic properties and give some examples on interesting Orlicz spaces and Orlicz modulation spaces. We also explain some classical results on pseudo-differential operators acting on Lebesgue or modulation spaces, and give examples on how such results can be extended to the framework of Orlicz spaces and Orlicz modulation spaces.

As another example we discuss suitable Orlicz modulation spaces and the entropy functional $f\mapsto E_\phi (f)$ with $\phi$ as the coherent state, considered by E. H. Lieb when discussing kinetic energy in quantum systems. Here we find an Orlicz modulation space $M^\Phi$ which satisfies \begin{equation} M^{p_1}\subsetneq M^\Phi \subsetneq M^{p_2},\qquad p_1<\frac 12,\ p_2\ge \frac 12 \end{equation} for which $E_\phi$ is continuous on $M^{p_1}$ and $M^\Phi$, but discontinuous on $M^{p_2}$. We hope that this should shed some light on how to find suitable Banach spaces when dealing with non-linear functionals.

The talk is based on joint works with A. Gumber, E. Nabizadeh Morsalfard, N. Rana, S. Öztop and R. Üster.

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Last updated: 12 Apr 2024