Conjecture de Kato sur les ouverts de $\mathbb{R}$

Abstract

We prove Kato's conjecture for second order elliptic differential operators on an open set in dimension 1 with arbitrary boundary conditions. The general case reduces to studying the operator $T=–\frac{d}{dx}a(x)\frac{d}{dx}$ on an interval, when $a(x)$ is a bounded and accretive function. We show for the latter situation that the domain of $T$ is spanned by an unconditional basis of wavelets with cancellation properties that compensate the action of the non-regular function $a(x)$.