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A density theorem. (Russian) Zbl 1021.46025

The author studies the question whether the space of infinitely-differentiable functions is dense in a Sobolev anisotropic space.
Let \(K\) be a compact set in \(\mathbb R^n\) and let \(\left(L_p^l\right)_K\) denote the set of functions in \(L_p^l(\mathbb R^n)\) with compact supports in \(K\). Assuming that \(K\) satisfies some additional geometrical properties, the author proves that \(C_0^{\infty}(K)\) is dense in \(\left(L_p^l\right)_K\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E15 Banach spaces of continuous, differentiable or analytic functions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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