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 $\text{pp}$ for a fractional maximal operator $M_{\alpha }$ or a non-degenerate fractional singular integral operator $T_{\alpha }$, $0\le \alpha <n$, to satisfy weak $(\text{pp},\text{qq})$ inequalities or strong $(\text{pp},\text{qq})$ inequalities, with $\text{qq}$ being defined pointwise almost everywhere by % $$\frac{1}{p(x)}-\frac{1}{q(x)}=\frac{\alpha }{n}\mathrm{.}$$ % We first prove preliminary results linking fractional averaging operators and the $K_0^{\alpha }$ condition, a qualitative condition on $\text{pp}$ 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 $(\text{pp},\text{qq})$ inequalities, then $\text{pp}\in K_0^{\alpha }({\mathbb{R}}^n)$. We use this to prove that if $M_{\alpha }$ satisfies strong $(\text{pp},\text{qq})$ 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.