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

Abstract

We study fractional potential of variable order on a bounded quasi-metric measure space $$(X,d,\\mu )$$\n as acting from variable exponent Morrey space $$ L ^{p(\\cdot ), \\lambda (\\cdot )} (X) $$\n to variable exponent Campanato space $$ \\mathscr {L } ^{p(\\cdot ), \\lambda (\\cdot )} (X) $$\n . We assume that the measure satisfies the growth condition $$ \\mu B(x,r) \\leqslant C r ^{\\gamma } $$\n , the distance is $$ \\theta $$\n -regular in the sense of Macias and Segovia, but do not assume that the space $$ (X,d,\\mu ) $$\n is homogeneous. We study the situation when $$\\gamma -\\lambda (x) \\leqslant \\alpha (x) p(x) \\leqslant \\gamma -\\lambda (x)+\\theta p(x), $$\n 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.