TY - JOUR

T1 - Fractional operators of variable order from variable exponent Morrey spaces to variable exponent Campanato spaces on quasi-metric measure spaces with growth condition

AU - Rafeiro, Humberto

AU - Samko, Stefan

N1 - Funding Information:
The research of H. Rafeiro was supported by a Start-up Grant of United Arab Emirates University, Al Ain, United Arab Emirates via Grant G00002994. The research of S. Samko was supported by: (a) Russian Foundation for Basic Research under the grant 19-01-00223, and (b) TUBITAK and Russian Foundation for Basic research under the grant 20-51-46003.
Publisher Copyright:
© 2021, Università degli Studi di Napoli "Federico II".

PY - 2021

Y1 - 2021

N2 - We study fractional potential of variable order on a bounded quasi-metric measure space (X, d, μ) as acting from variable exponent Morrey space Lp(·),λ(·)(X) to variable exponent Campanato space Lp(·),λ(·)(X). We assume that the measure satisfies the growth condition μB(x, r) ⩽ Crγ, 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.

AB - We study fractional potential of variable order on a bounded quasi-metric measure space (X, d, μ) as acting from variable exponent Morrey space Lp(·),λ(·)(X) to variable exponent Campanato space Lp(·),λ(·)(X). We assume that the measure satisfies the growth condition μB(x, r) ⩽ Crγ, 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.

KW - Campanato space

KW - Fractional operator

KW - Morrey space

KW - Quasi-metric measure space

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U2 - 10.1007/s11587-021-00639-4

DO - 10.1007/s11587-021-00639-4

M3 - Article

AN - SCOPUS:85113984861

SN - 0035-5038

JO - Ricerche di Matematica

JF - Ricerche di Matematica

ER -