Optimal stability hierarchies of the Heisenberg Uncertainty Principle for solenoidal fields and of the second order Caffarelli--Kohn--Nirenberg inequalities

Abstract

In 2018, V. Maz'ya proposed 75 open problems in analysis and PDEs. One of them is about the best constant for the Heisenberg Uncertainty Principle for solenoidal vector fields motivated by questions in hydrodynamics. This was answered by Cazacu, Flynn and Lam in dimension two and subsequently by Hamamoto in all dimensions $N\ge3$ by establishing sharp inequalities with explicit constants. Nevertheless, the much harder stability problem remains open in all dimensions. The first main result in this paper is to establish a sharp stability hierarchy by proving that its deficit controls the distance to the set of extremals, with the sharp stability constant $\frac12(N-\sqrt{N^2-4N+12})$ for $N\ge4$ and $1$ for $N=3$, together with chains of remainder terms measuring the distance to explicit larger families of poloidal and toroidal fields. The second main result is the second order $L^2$-Caffarelli--Kohn--Nirenberg inequality with the weights $|x|^{-2a}$ and $|x|^{-2b}$ on the line $1+a+b=0$, for which we obtain the stability constant $\frac12\min\{4(1+a),\sqrt{(N+2a)^2+4N-4}-(N+2a)\}$. This CKN inequality is sharp whenever the second number is the smaller one. We develop a novel method: the fourth order one-dimensional problems attached to the spherical modes are transformed, by a Fourier--Hankel transform of real order, into first order inequalities whose sharp stability is that of weighted Gaussian Poincaré inequalities. The inverse transform produces the extremals in terms of Kummer's function. As a byproduct of our new approach, we also obtain substantially simpler proofs, with sharp remainder terms and equality cases, of Hamamoto's one-dimensional inequality and of his sharp uncertainty principle for solenoidal fields.