On Compactness of Operators in Variable Exponent Lebesgue Spaces
We give a short discussion of known statements on compactness of operators in variable exponent Lebesgue spaces L p (·)(Ω, ϱ) and show that the existence of a radial integrable decreasing dominant of the kernel of a convolution operator guarantees its compactness in the space L p (·)(Ω, ϱ) whenever the maximal operator is bounded in this space, where |Ω| < ∞ and ϱ is an arbitrary weight such that L p (·)(Ω, ϱ) is a Banach function space. In the non-weighted case ϱ = 1 we also give a modification of this statement for Ω = ℝn.
KeywordsVariable exponent spaces compact operator integral operators convolution operators radial decreasing dominants
Mathematics Subject Classification (2000)Primary 46E30 Secondary 47B38 47G10
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