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April 16, 2026Open Access

Sobolev-type theorem for commutators of variable Marcinkiewicz fractional in grand variable Herz-Morrey spaces

Key Points

This research aims to explore the boundedness of commutators of variable Marcinkiewicz fractional operators in grand variable Herz-Morrey spaces.
Defined the variable Marcinkiewicz fractional integral operator using a specific integral formulation.
Utilized the normalized Lebesgue measure on the unit sphere Rn.
Analyzed the boundedness of the commutators on grand variable Herz-Morrey spaces.
Demonstrated that commutators of the Marcinkiewicz fractional operator are bounded on the specified function spaces.
Provided a new perspective on the behavior of these operators in variable spaces.

Abstract

Let Sn? 1 denote unit sphere in Rn equipped with the normalized Lebesgue measure. Let? ? Lr (Sn? 1) be a homogeneous function of degree zero. The variable Marcinkiewicz fractional integral operator is defined as _ b,? ? ₘ? (f) (z1) = (Z? 0 | Z |z1? z2|? s? (z1? z2) b (z1)? b (z2) m |z1? z2|n? 1? ? (z1) f (z2) dz2 | 2 ds s3) 1/2. The commutators of Marcinkiewicz fractional operator of variable order is shown to be bounded on the grand variable Herz-Morrey spaces M? K? (? ), u),? ? , p (? ) (Rn).

Sobolev-type theorem for commutators of variable Marcinkiewicz fractional in grand variable Herz-Morrey spaces

Key Points

Abstract

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