Harnack inequality and heat kernel estimates for the Schrödinger operator with Hardy potential

Moschini, Luisa ; Tesei, Alberto

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 16 (2005) no. 3, pp. 171-180

Résumé

In this preliminary Note we outline some results of the forthcoming paper [11], concerning positive solutions of the equation $\partial_{t} u = \triangle u + \frac{c}{|x^{2}|} u \big( 0 c \frac{(n-2)^{2}}{4}; \, n \ge 3 \big)$. A parabolic Harnack inequality is proved, which in particular implies a sharp two-sided estimate for the associated heat kernel. Our approach relies on the unitary equivalence of the Schrödinger operator $Hu = - \triangle u - \frac{c}{|x|^{2}} u$ with the opposite of the weighted Laplacian $\triangle_{\lambda} v = \frac{1}{|x|^{\lambda}} \text{div} (|x|^{\lambda} \nabla v)$ when $\lambda = 2 - n + 2 \sqrt{c_{0} - c}$.