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Composition operators on summable functions spaces. (English) Zbl 0724.47015

Summary: A composition operator is a linear operator \(C_ T\) on a subspace of \({\mathbb{K}}^ X\) by a point transformation T on a set X (where \({\mathbb{K}}\) denotes the scalar field) by the formula \(C_ Tf(x):=f\circ T(x).\) We give some necessary and/or sufficient conditions under which the map T induces a continuous composition operator on suitable topological subspaces of summable functions of \({\mathbb{K}}^ X\) as \(L^ p\) or \(W^{1,p}\).

MSC:

47B38 Linear operators on function spaces (general)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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