A Novel Schwartz Space for the $\left(k,\frac{2}{n}\right)-$Generalized Fourier Transform
Nelson Faustino, Selma Negzaoui
TL;DR
This work constructs a Schwartz-type space $\mathcal{S}_{k,n}(\mathbb{R})$ tailored to the $(k,\frac{2}{n})$-generalized Fourier transform $\mathcal{F}_{k,n}$, restoring invariance that the classical Schwartz space loses for many transforms. By embedding $|x|^{2/n}$ and $|x|^{2-2/n}\Delta_k$ into the defining seminorms and exploiting the $\mathfrak{sl}(2,\mathbb{R})$-structure, the authors prove $\mathcal{F}_{k,n}(\mathcal{S}_{k,n})=\mathcal{S}_{k,n}$, mirroring the classical Fourier invariance. They also identify a dense, bounded-support subspace $\mathcal{D}_{k,n}(\mathbb{R})$ with a continuous embedding into $\mathcal{S}_{k,n}(\mathbb{R})$, and establish density of $\mathcal{S}_{k,n}$ in $L^p(d\mu_{k,n})$ for $1\le p<\infty$, using an approximation-identity framework. The paper further outlines a higher-dimensional conjecture for $\mathcal{S}_{k,a}(\mathbb{R}^N)$ and discusses current obstacles in kernel estimates, suggesting a path toward extending the theory beyond $N=1$ and $a=\tfrac{2}{n}$.
Abstract
The Schwartz space is the natural ambient space for dealing with the classical Fourier transform beyond $L^2-$spaces. However, when studying more general transforms, such as the $(k,a)-$generalized Fourier transform, these spaces lose several important properties, such as invariance. To address this issue, we introduce a Schwartz-type space, denoted by $\mathcal{S}_{k,n}(\mathbb{R})$, which is tailored for the $\left(k,\frac{2}{n}\right)-$generalized Fourier transform. This space retains the invariance property of the classical Fourier transform and is useful in cases where the standard Schwartz space is not. To prove afterwards that $\mathcal{S}_{k,n}(\mathbb{R})$ is dense in $L^p(dμ_{k,n})$, for $1 \leq p < \infty$, we identify the underlying subspace of functions with bounded support, denoted by $\mathcal{D}_{k,n}(\mathbb{R})$. Finally, we discuss a conjecture regarding extensions to higher dimensions.