Necessary and Sufficient Conditions for Boundedness of Commutators of the General Fractional Integral Operators on Weighted Morrey Spaces

Zengyan Si and Fayou Zhao

Abstract and Applied Analysis, 2012, vol. 2012, 1-14

Abstract:

We prove that ð ‘ is in L i p ð ›½ ( 𠜔 ) if and only if the commutator [ ð ‘ , ð ¿ âˆ’ ð ›¼ / 2 ] of the multiplication operator by ð ‘ and the general fractional integral operator ð ¿ âˆ’ ð ›¼ / 2 is bounded from the weighted Morrey space ð ¿ ð ‘ , 𠑘 ( 𠜔 ) to ð ¿ ð ‘ž , 𠑘 ð ‘ž / ð ‘ ( 𠜔 1 − ( 1 − ð ›¼ / ð ‘› ) ð ‘ž , 𠜔 ) , where 0 < ð ›½ < 1 , 0 < ð ›¼ + ð ›½ < ð ‘› , 1 < ð ‘ < ð ‘› / ( ð ›¼ + ð ›½ ) , 1 / ð ‘ž = 1 / ð ‘ âˆ’ ( ð ›¼ + ð ›½ ) / ð ‘› , 0 ≤ 𠑘 < ð ‘ / ð ‘ž , 𠜔 ð ‘ž / ð ‘ âˆˆ ð ´ 1 , and ð ‘Ÿ 𠜔 > ( 1 − 𠑘 ) / ( ð ‘ / ( ð ‘ž − 𠑘 ) ) , and here ð ‘Ÿ 𠜔 denotes the critical index of 𠜔 for the reverse Hölder condition.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:929381

DOI: 10.1155/2012/929381

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