Abstract
In the limiting case of Sobolev-Adams theorem for Morrey spaces of variable order we prove that the fractional operator of variable order maps the corresponding vanishing Morrey space into VMO.
Acknowledgements
The research of H. Rafeiro was supported by a Research Start-up Grant of United Arab Emirates University, UAE, via Grant No. G00002994. The research of S. Samko was supported by Russian Foundation for Basic Research under Grants 19-01-00223 and 18-01-00094-a.
References
[1] P.W. Jones, Extension theorems for BMO. Indiana Univ. Math. J. 29, No 1 (1980), 41–66.10.1512/iumj.1980.29.29005Search in Google Scholar
[2] V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral Operators in Non-Standard Function Spaces. Volume 1: Variable Exponent Lebesgue and Amalgam Spaces. Birkhäuser/Springer, Basel (2016).10.1007/978-3-319-21015-5_1Search in Google Scholar
[3] V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral Operators in Non-Standard Function Spaces. Volume 2: Variable Exponent Hölder, Morrey-Campanato and Grand Spaces. Birkhäuser/Springer, Basel (2016).Search in Google Scholar
[4] M. Lifshits and W. Linde, Fractional integration operators of variable order: continuity and compactness properties. Math. Nachr. 287, No 8-9 (2014), 980–1000.10.1002/mana.201200337Search in Google Scholar
[5] H. Rafeiro, S. Samko, Fractional integrals and derivatives: mapping properties. Fract. Calc. Appl. Anal. 19, No 3 (2016), 580–607; 10.1515/fca-2016-0032; https://www.degruyter.com/view/j/fca.2016.19.issue-3/issue-files/fca.2016.19.issue-3.xml.Search in Google Scholar
[6] H. Rafeiro, S. Samko, BMO–VMO results for fractional integrals in variable exponent Morrey spaces. Nonlinear Analysis184 (2019), 35–43; 10.1016/j.na.2019.01.020.Search in Google Scholar
[7] S. Samko, A note on Riesz fractional integrals in the limiting case α(x)p(x) ≡ n. Fract. Calc. Appl. Anal. 16, No 2 (2013), 370–377; 10.2478/s13540-013-0023-x; https://www.degruyter.com/view/j/fca.2013.16.issue-2/issue-files/fca.2013.16.issue-2.xml.Search in Google Scholar
[8] S. Samko, Remark to the paper of S. Samko, “A note on Riesz fractional integrals in the limiting case α(x)p(x) ≡ n”, from FCAA, Vol. 16, No 2, 2013. Fract. Calc. Appl. Anal. 17, No 1 (2013), 277–278; 10.2478/s13540-014-0167-3; https://www.degruyter.com/view/j/fca.2014.17.issue-1/issue-files/fca.2014.17.issue-1.xml.Search in Google Scholar
[9] S. Samko, Fractional integration and differentiation of variable order: an overview. Nonlinear Dyn. 71 (2013), 653–662; 10.1007/s11071-012-0485-0.Search in Google Scholar
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