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,μ) as acting from variable exponent Morrey space L^p(·),λ(·)(X) to variable exponent Campanato space L^p(·),λ(·)(X). We assume that the measure satisfies the growth condition μB(x,r)⩽Cr^γ, the distance is θ-regular in the sense of Macías and Segovia, but do not assume that the space (X,d,μ) is homogeneous. We study the situation when γ-λ(x)⩽α(x)p(x)⩽γ-λ(x)+θp(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.