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L₁-uniqueness of degenerate elliptic operators

Derek W. Robinson, Adam Sikora (2011)

Studia Mathematica

Let Ω be an open subset of d with 0 ∈ Ω. Furthermore, let H Ω = - i , j = 1 d i c i j j be a second-order partial differential operator with domain C c ( Ω ) where the coefficients c i j W l o c 1 , ( Ω ̅ ) are real, c i j = c j i and the coefficient matrix C = ( c i j ) satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If 0 d s s d / 2 e - λ μ ( s ) ² < for some λ > 0 where μ ( s ) = 0 s d t c ( t ) - 1 / 2 then we establish that H Ω is L₁-unique, i.e. it has a unique L₁-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique L₂-extension which generates a submarkovian semigroup. Moreover...

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