Abstract
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 Macías 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.
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The research of H. Rafeiro was supported by a Start-up Grant of United Arab Emirates University, Al Ain, United Arab Emirates via Grant \(\mathcal {N}\) G00002994. The research of S. Samko was supported by: (a) Russian Foundation for Basic Research under the grant \(\mathcal {N}\) 19-01-00223, and (b) TUBITAK and Russian Foundation for Basic research under the grant \(\mathcal {N}\) 20-51-46003.
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Rafeiro, H., Samko, S. Fractional operators of variable order from variable exponent Morrey spaces to variable exponent Campanato spaces on quasi-metric measure spaces with growth condition. Ricerche mat 73, 803–818 (2024). https://doi.org/10.1007/s11587-021-00639-4
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DOI: https://doi.org/10.1007/s11587-021-00639-4