Fractional operators of variable order from variable exponent Morrey spaces to variable exponent Campanato spaces on quasi-metric measure spaces with growth condition

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(center dot),L-lambda(center dot)(X) to variable exponent Campanato space L-p(center dot),L-lambda(center dot)(X). We assume that the measure satisfies the growth condition mu B(x,r) <= Cr-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) <= alpha(x)p(x) <= 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.