Sobolev-type theorem for commutators of Hardy operators in grand Herz spaces

UDC 517.5 The higher-order commutators of fractional Hardy-type operators of variable order ζ ( z ) are shown to be bounded from the grand variable Herz spaces K ˙ p ( ⋅ ) a ( ⋅ ) , u ) , θ ( ℝ n ) into the weighted space K ˙ ρ , q ( ⋅ ) a ( ⋅ ) , u ) , θ ( ℝ n ) , where ρ = ( 1 + | z 1 | ) - λ and 1 q ( z ) = 1 p ( z ) - ζ ( z ) n if p ( z ) is not necessarily constant at infinity.