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In this note, we consider negative $k$ and show that the span of the positive cone in $W^{k,p}(\\mathbb R^d)$ is a vector lattice in this case.\n  We also prove a related abstract result: if $(T(t))_{t \\in [0,\\infty)}$ is a positive $C_0$-semigroup on a Banach lattice $X$ with order continuous norm, then the span of the cone $X_{-1,+}$ in the extrapolation space $X_{-1}$ is a vector lattice. 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Schwenninger, Jochen Gl\\\"uck, Sahiba Arora","submitted_at":"2024-04-02T17:20:12Z","abstract_excerpt":"It is well-known that the Sobolev spaces $W^{k,p}(\\mathbb R^d)$ are vector lattices with respect to the pointwise almost everywhere order if $k \\in \\{0,1\\}$, but not if $k \\ge 2$. In this note, we consider negative $k$ and show that the span of the positive cone in $W^{k,p}(\\mathbb R^d)$ is a vector lattice in this case.\n  We also prove a related abstract result: if $(T(t))_{t \\in [0,\\infty)}$ is a positive $C_0$-semigroup on a Banach lattice $X$ with order continuous norm, then the span of the cone $X_{-1,+}$ in the extrapolation space $X_{-1}$ is a vector lattice. 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