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_ or a non-degenerate fractional singular integral operator T_, 0 1. Finally, we prove a powerful pointwise estimate for T_ that relates T_ to M_ 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.