For self-adjoint operators comparable to a positive-homogeneous semi-elliptic operator, the heat kernel satisfies off-diagonal bounds with the Legendre-Fenchel transform of the symbol, under three hypotheses.
Ultracontractivity and Nash Type Inequalities
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Davies' method for heat-kernel estimates: An extension to the semi-elliptic setting
For self-adjoint operators comparable to a positive-homogeneous semi-elliptic operator, the heat kernel satisfies off-diagonal bounds with the Legendre-Fenchel transform of the symbol, under three hypotheses.