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

Abstract: 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 $\pp$ for a fractional maximal operator $M_\alpha$ or a non-degenerate fractional singular integral operator $T_\alpha$, $0 \leq \alpha < n$, to satisfy weak $(\pp,\qq)$ inequalities or strong $(\pp,\qq)$ inequalities, with $\qq$ being defined pointwise almost everywhere by % [ \frac{1}{p(x)} - \frac{1}{q(x)} = \frac{\alpha}{n}. ] % We first prove preliminary results linking fractional averaging operators and the $K_0^\alpha$ condition, a qualitative condition on $\pp$ related to the norms of characteristic functions of cubes, and show some useful implications of the $K_0^\alpha$ condition. We then show that if $M_\alpha$ satisfies weak $(\pp,\qq)$ inequalities, then $\pp \in K_0^\alpha(\R^n)$. We use this to prove that if $M_\alpha$ satisfies strong $(\pp,\qq)$ inequalities, then $p_->1$. Finally, we prove a powerful pointwise estimate for $T_\alpha$ that relates $T_\alpha$ to $M_\alpha$ 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.