LAUSR

We study fractional potential of variable order on a bounded quasi-metric measure space $$(X,d,\mu )$$ as acting from variable exponent Morrey space $$ L ^{p(\cdot ), \lambda (\cdot )} (X) $$ to variable exponent Campanato space $$ \mathscr {L } ^{p(\cdot ), \lambda (\cdot )} (X) $$ . We assume that the measure satisfies the growth condition $$ \mu B(x,r) \leqslant C r ^{\gamma } $$ , the distance is $$ \theta $$ -regular in the sense of Macias and Segovia, but do not assume that the space $$ (X,d,\mu ) $$ is homogeneous. We study the situation when $$\gamma -\lambda (x) \leqslant \alpha (x) p(x) \leqslant \gamma -\lambda (x)+\theta p(x), $$ and pay special attention to the cases of bounds of this interval. The left bound formally corresponds to the BMO target space. In the case of right bound a certain “correcting factor” of logarithmic type should be introduced in the target Campanato space.