Necessary conditions for the boundedness of fractional operators on variable Lebesgue spaces

In this paper we prove necessary conditions for the boundedness of fractional operators on the variable Lebesgue spaces. More precisely, we find necessary conditions on an exponent function for a fractional maximal operator M_α or a non-degenerate fractional singular integral operator T_α, 0 ≤ α < n, to satisfy weak (,) inequalities or strong (,) inequalities, with being defined pointwise almost everywhere by % \[ 1/p(x) - 1/q(x) = α/n. \] % We first prove preliminary results linking fractional averaging operators and the K₀^α condition, a qualitative condition on related to the norms of characteristic functions of cubes, and show some useful implications of the K₀^α condition. We then show that if M_α satisfies weak (,) inequalities, then ∈ K₀^α(ⁿ). We use this to prove that if M_α satisfies strong (,) inequalities, then p₋>1. Finally, we prove a powerful pointwise estimate for T_α that relates T_α to M_α along a carefully chosen family of cubes. This allows us to prove necessary conditions for fractional singular integral operators similar to those for fractional maximal operators.