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Sharp two-sided heat kernel estimates for critical Schr\"odinger operators on bounded domains

Stathis Filippas·Luisa Moschini·Achilles Tertikas
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Abstract

On a smooth bounded domain \Omega \subset R^N we consider the Schr\"odinger operators -\Delta -V, with V being either the critical borderline potential V(x)=(N-2)^2/4 |x|^{-2} or V(x)=(1/4) dist (x,\partial\Omega)^{-2}, under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Scr\"odinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a serier of new inequalities such as improved Hardy, logarithmic Hardy Sobolev, Hardy-Moser and weighted Poincar\'e. As a byproduct of our technique we are able to answer positively to a conjecture of E.B.Davies.

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Stathis Filippas, Luisa Moschini, Achilles Tertikas. 2006-06-07. Sharp two-sided heat kernel estimates for critical Schr\"odinger operators on bounded domains. https://doi.org/10.1007/s00220-007-0253-z

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