Bounds for the maximal and Riesz potential operators with variable fractionality

We prove $L^{p(\cdot )}$-to-$L^{q(\cdot )}$ bounds for variable versions of the fractional maximal $M^{\alpha (\cdot )}$ and Riesz potential $I^{\alpha (\cdot )}$ operators. The changing fractionality in these operators is given by averaging the function $\alpha (\cdot )$ over balls. The bounds for $M^{\alpha (\cdot )}$ are in terms of a three-exponent Muckenhoupt condition relating $p(\cdot ),q(\cdot ),$ and $\alpha (\cdot )$, while the bounds for $I^{\alpha (\cdot )}$ are in terms of the boundedness of $M^{\alpha (\cdot )}$ and a packing condition on $\alpha (\cdot )\mathrm{.}$ These bounds hold under Hardy–Littlewood maximal function boundedness and Muckenhoupt conditions on the individual exponents $p(\cdot ),q(\cdot ),\alpha (\cdot )\mathrm{.}$ The proofs are based on an adaptation of sparse domination to variable fractionality and an embedding into variable sequential spaces.