On a minimisation problem related to the solenoidal uncertainty

TL;DR

This paper addresses a sharp 1-D minimisation problem arising from the solenoidal (divergence-free) uncertainty principle for vector fields. It situates the problem in a natural weighted Sobolev space $\\bar{\\mathcal{E}}_{\\mu}(\\mathbb{R}_+)$ and shows the interpolation inequality can be upgraded to an exact equality when $\\varepsilon \\\le \\\dfrac{\\mu^{2}}{4}$, with a concrete extremiser structure. The extremisers are given in closed form by confluent hypergeometric functions, $f(x)=C e^{-\\lambda x}\\,_1F_1(b,\\mu,\\lambda x)$ for $\\mu>0$ or $f(x)=C(\\lambda x)^{1-\\mu} e^{-\\lambda x}\\,_1F_1(b+1-\\mu,2-\\mu,\\lambda x)$ for $\\mu<0$, where $b=\\dfrac{\\mu-\\sqrt{\\mu^{2}-4\\varepsilon}}{2}$, and the equality case is tied to a transformed Kummer equation. The results yield sharp constants and explicit extremisers for the solenoidal uncertainty in the 1-D reduction, providing a rigorous foundation that clarifies the validity of the expanding-square approach and removes extraneous parameter constraints.

Abstract

We study Hamamoto's expanding square argument towards a 1-D minimisation problem related to the sharp solenoidal uncertainty principle. Working in the right function space, we recast the involved interpolation type inequality into an exact equality, where the vanishing of the remainder term characterises the extremisers via the confluent hypergeometric functions. In the process we also remove some unnecessary constraints on the prescribed parameters.

Theorems & Definitions (7)

• Lemma 1
• Theorem 2
• Corollary 3
• proof : Proof of Lemma \ref{['lemma1']}
• proof : Proof of Theorem \ref{['theorem2']}
• Remark 4
• proof : Proof of Corollary \ref{['corollary3']}