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

TL;DR

This paper explores the connection between Trudinger-Moser functions in dimension 2 and Reproducing Kernel Hilbert Spaces, showing how these functions can be viewed as evaluation functionals within a suitable RKHS framework.

Contribution

It establishes a novel link between Trudinger-Moser functions and RKHS theory, providing a new perspective on their properties and interpretations.

Findings

Trudinger-Moser functions can be represented as evaluation functionals in RKHS.

A specific Hilbert space framework is constructed for these functions.

Extension of the concept to higher dimensions is also discussed.

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 $ \gamma_t (r) := \frac{1}{2\pi}\min\,\{ log \frac{1}{r}, log \frac{1}{t} \}\,, $ where $0 < t , r < 1$, and using the theory of RKHS we will show that $\gamma_t$ 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 $N\geq 3$.