Hardy's Theorem for the $(k,rac{2}{n})-$Fourier Transform

By comparing a function and its $(k, \frac{2}{n})-$Fourier transform to a Gaussian analogue, $e^{-na|x|^\frac{2}{n}}$, we establish a Hardy-type uncertainty principle using Phragmén-Lindlöf lemma. Furthermore, we investigate the heat equation in this context, deriving a dynamical version of Hardy’s theorem that illustrates the temporal evolution of the uncertainty principle. We also extend our results to $L^p-L^q$ versions, proving Miyachi-type and Cowling-Price-type theorems for the $(k,\frac{2}{n})$-Fourier transform.

💡 Research Summary

This paper establishes a Hardy-type uncertainty principle for the one-dimensional (k, 2/n)-generalized Fourier transform, introduced by Ben Saïd, Kobayashi, and Ørsted. The classical Hardy theorem states that a function and its Fourier transform cannot both decay faster than a Gaussian; specifically, if |f(x)| ≤ C exp(-a|x|^2) and |̂f(ξ)| ≤ C exp(-b|ξ|^2), then f must be zero if ab > 1/4, and proportional to the Gaussian if ab = 1/4.

The authors tackle the significant challenge posed by the kernel of the F_k,n transform, which involves fractional powers like |λx|^(1/n), preventing F_k,n f from being an entire function—a crucial property for traditional complex-analytic proofs. To overcome this, they ingeniously decompose a function into its even and odd parts and introduce two new integral transforms, T1 and T2. These transforms are designed to be closely related to F_k,n (satisfying F_k,n f(x^n) = T1 f_e(x) + T2 f_o(x)) but crucially yield entire functions when applied to functions with Gaussian-like decay of the form exp(-na|x|^(2/n)). Using Poisson integral representations and careful estimates, the authors prove that T1 f and T2 f are entire and satisfy growth bounds of the form |T_l f(z)| ≤ C exp((n/(4a)) (ℑ(z))^2).

Leveraging these properties and a Phragmén-Lindelöf type lemma, the main theorem (Theorem 1.1) is proven: If a measurable function f satisfies |f(x)| ≤ C exp(-na|x|^(2/n)) and |F_k,n f(x)| ≤ C exp(-nb|x|^(2/n)), then f ≡ 0 if ab > 1/4. If ab = 1/4, f is a constant multiple of the optimal function exp(-na|x|^(2/n)). The case a=1, n=1 recovers the classical and Dunkl transform results.

The study extends beyond this static uncertainty principle. First, the authors investigate a dynamical version by analyzing the heat equation associated with the (k, 2/n)-Laplacian, H_k,n u(t,x)=0. This provides a characterization of how the uncertainty principle evolves over time, linking the decay of the initial condition u(0,x) to the decay of the solution u(t,x) at later times. Second, the results are generalized to L^p-L^q settings, proving Miyachi-type and Cowling-Price-type theorems. These versions replace pointwise Gaussian decay conditions with integrability conditions (e.g., requiring f to belong to a weighted L^p space), significantly broadening the applicability of the uncertainty principle.

In summary, this work successfully generalizes Hardy’s fundamental uncertainty principle to a challenging non-classical Fourier transform setting by introducing novel technical tools (T1, T2 transforms). It further enriches the theory by deriving a dynamical interpretation via a heat equation and establishing more versatile L^p-L^q formulations, contributing substantially to harmonic analysis and PDE theory.