Geodesic
Harnack inequality and heat kernel estimates for the Schrödinger operator with Hardy potential
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 16 (2005) no. 3, pp. 171-180.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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 $\lambda = 2 - n + 2 \sqrt{c_{0} - c}$.
In questa Nota preliminare si presentano alcuni risultati del successivo lavoro [11], riguardanti soluzioni positive dell'equazione $\partial_{t} u = \triangle u + \frac{c}{|x^{2}|} u \big( 0 c \frac{(n-2)^{2}}{4}; \, n \ge 3 \big)$. Si dimostra una disuguaglianza di Harnack parabolica, che in particolare implica una stima bilatera sul nucleo del calore associato. Il nostro approccio si basa sull'equivalenza unitaria dell'operatore di Schrödinger $Hu = - \triangle u - \frac{c}{|x|^{2}} u$ con l'opposto dell'operatore di Laplace pesato $\triangle_{\lambda} v = \frac{1}{|x|^{\lambda}} \text{div} (|x|^{\lambda} \nabla v)$ quando $\lambda = 2 - n + 2 \sqrt{c_{0} - c}$. 
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     title = {Harnack inequality and heat kernel estimates for the {Schr\"odinger} operator with {Hardy} potential},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
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Moschini, Luisa; Tesei, Alberto. Harnack inequality and heat kernel estimates for the Schrödinger operator with Hardy potential. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 16 (2005) no. 3, pp. 171-180. http://geodesic.mathdoc.fr/item/RLIN_2005_9_16_3_a2/

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