A note on Trudinger-Moser Functions and Reproducing Kernel Hilbert Spaces

TL;DR

The paper investigates a link between Trudinger–Moser functions and Reproducing Kernel Hilbert Spaces (RKHS) across dimensions. By constructing radial Sobolev-type spaces $H_N$ and showing they are RKHSs with explicit reproducing kernels $k_t$, the authors interpret the evaluation functionals $E_t(u)=u(t)$ as bounded evaluations in these Hilbert spaces, using $\gamma_t$ in 2D and $\gamma_{t,N}$ for $N\ge3$. This yields a concrete RKHS perspective on Trudinger–Moser phenomena, including sharp inequalities and the existence of extremals, and paves the way for further RKHS-based analyses in this context. The results provide explicit kernels and robust connections between TM functions and RKHS theory, suggesting potential extensions such as Carleson–Moser towers and related RKHS constructions.

Abstract

After a brief review of the definition of the Trudinger-Moser functions in dimension $N=2$ and some basic notions in the theory of ``Reproducing Kernel Hilbert Spaces (RKHS)'', we will show that there is a close connection between those two topics. More precisely, among other things, we start by considering a properly chosen multiple of the classical Trudinger-Moser family of functions in dimension $N=2$, which we denote by $ γ_t (r) := \frac{1}{2π}\min\,\{ log \frac{1}{r}, log \frac{1}{t} \}\,, $ where $0 < t , r < 1$, and using the theory of RKHS we will show that $γ_t$ can be seen as a ``bounded'' (linear) evaluation functional $u \longrightarrow u(t)$ for functions $u$ in a suitable Hilbert Space ${\cal H}$. A slightly different definition for a ''Trudinger-Moser'' type function will also be considered for $N\geq 3$.