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Commutators of weighted Hardy operators on Herz-type spaces. (English) Zbl 1230.42023

The authors discuss the Morrey-Herz space boundedness of the commutator U ψ b of the function multiplier M b and the weighted Hardy operator U ψ defined by U ψ f(x)= 0 1 f(tx)ψ(t)dt (x n ), where ψ:[0,1)[0,). When ψ1 and n=1, this reduces to the classical Hardy operator U:Uf(x)=x -1 0 x f(t)dt. They show that when -<α<, λ0, 0<p, 0<q 2 q 1 <, b Lip β ( n ) (0<β<1) and 0 1 t -(α+β+n/q 2 -λ) ψ(t)dt<, then the commutator U ψ b is bounded from the homogeneous Morrey-Herz space MK ˙ p,q 1 α+β+n/q 2 -n/q 1 ,λ ( n ) to MK ˙ p,q 1 α,λ ( n ). Here the homogeneous Morrey-Herz space MK ˙ p,q α,λ ( n ) is defined by

MK ˙ p,q α,λ ( n )={fL loc q ( n {0});f MK ˙ p,q α,λ ( n ) <},

where f MK ˙ p,q α,λ ( n ) =sup k 0 2 -k 0 λ { k=- k 0 2 kαp fχ {2 k-1 <|x|2 k } L q ( n ) p } 1/p . This extends the (L p ,L q ) result for the classical Hardy operator by Z. W. Fu [J. Beijing Norm. Univ., Nat. Sci. 42, No. 4, 342–345 (2006; Zbl 1135.42320)].

42B20Singular and oscillatory integrals, several variables
42B25Maximal functions, Littlewood-Paley theory