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

Moschini, Luisa, and Tesei, Alberto. "Harnack inequality and heat kernel estimates for the Schrödinger operator with Hardy potential." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 16.3 (2005): 171-180. <http://eudml.org/doc/252396>.

@article{Moschini2005,
abstract = {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 $\lambda = 2 - n + 2 \sqrt\{c_\{0\} - c\}$.},
author = {Moschini, Luisa, Tesei, Alberto},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Harnack inequality; Moser iteration technique; Weighted Laplace-Beltrami operator; Sharp heat kernel estimate; Hardy potential; weighted Laplace-Beltrami operator; sharp heat kernel estimate},
language = {eng},
month = {9},
number = {3},
pages = {171-180},
publisher = {Accademia Nazionale dei Lincei},
title = {Harnack inequality and heat kernel estimates for the Schrödinger operator with Hardy potential},
url = {http://eudml.org/doc/252396},
volume = {16},
year = {2005},
}

TY - JOUR
AU - Moschini, Luisa
AU - Tesei, Alberto
TI - Harnack inequality and heat kernel estimates for the Schrödinger operator with Hardy potential
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2005/9//
PB - Accademia Nazionale dei Lincei
VL - 16
IS - 3
SP - 171
EP - 180
AB - 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 $\lambda = 2 - n + 2 \sqrt{c_{0} - c}$.
LA - eng
KW - Harnack inequality; Moser iteration technique; Weighted Laplace-Beltrami operator; Sharp heat kernel estimate; Hardy potential; weighted Laplace-Beltrami operator; sharp heat kernel estimate
UR - http://eudml.org/doc/252396
ER -