Boundedness of commutators of variable Marcinkiewicz fractional integral operator in grand variable Herz spaces

• Published in: Journal of inequalities and applications Vol. 2024; no. 1; pp. 93 - 16
• Main Authors: Sultan, Babar, Sultan, Mehvish, Khan, Aziz, Abdeljawad, Thabet
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 11.07.2024; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; BMO spaces; Commutators; Fractional calculus; Grand variable Herz spaces; Lebesgue spaces; Marcinkiewicz fractional; Mathematical functions; Mathematics; Mathematics and Statistics; Operators (mathematics); Weighted estimates; Marcinkiewicz fractional; 47B38; BMO spaces; 46E30; Lebesgue spaces; Weighted estimates; Grand variable Herz spaces
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-024-03169-3

Let S n − 1 denote unit sphere in R n equipped with the normalized Lebesgue measure. Let Φ ∈ L s ( S n − 1 ) be a homogeneous function of degree zero such that ∫ S n − 1 Φ ( y ′ ) d σ ( y ′ ) = 0 , where y ′ = y / | y | for any y ≠ 0 . The commutators of variable Marcinkiewicz fractional integral operator is defined as [ b , μ Φ ] β m ( f ) ( x ) = ( ∫ 0 ∞ | ∫ | x − y | ≤ s Φ ( x − y ) [ b ( x ) − b ( y ) ] m | x − y | n − 1 − β ( x ) f ( y ) d y | 2 d s s 3 ) 1 2 . In this paper, we obtain the boundedness of the commutators of the variable Marcinkiewicz fractional integral operator on grand variable Herz spaces K ˙ p ( ⋅ ) α ( ⋅ ) , q ) , θ ( R n ) .