Interpolation of compact Lipschitz operators
Abstract
Let (A0,A1) and (B0,B1) be Banach couples such that A0 is contained in A1 and (B0,B1) satisfies Arne Persson's approximation condition (H). Let T:A1 --> B1 be a possibly nonlinear Lipschitz mapping which also maps A0 into B0 and satisfies the following quantitative compactnesss condition: Ta ∈ ||a||A0 K for each a ∈ A0, where K is a fixed compact subset of B0. We show that T maps the real interpolation space (A0,A1)θ,p compactly into its counterpart (B0,B1)θ,p for each θ ∈ (0,1) and p ∈ [1,∞].
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