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Composition operators of summable functions spaces

A composition operator is a linear operator C T on a subspace of K X  by a point transformation T on a set X (where K denotes the scalar field) by the formula C T f(x) :=f ∘ 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 K X as L p or W 1,p .