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Weighted Hardy operators and commutators on Morrey spaces. (English) Zbl 1206.42027
The authors discuss the Morrey space boundedness of the weighted Hardy operators 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 1<q< and -1/q<λ<0, U ψ is bounded on the Morrey space L q,λ ( n ) if and only if 0 1 t nλ ψ(t)dt<, and U ψ op = 0 1 t nλ ψ(t)dt. They also characterize those ψ for which the commutators of U ψ and the function multipliers M b are bounded on L q,λ ( n ) for all BMO( n ) functions b. They give the same results for the Cesàro operators which are adjoint to U ψ , too. Their results generalize the corresponding ones in L q ( n ) spaces.
42B99Fourier analysis in several variables
26D15Inequalities for sums, series and integrals of real functions
42B25Maximal functions, Littlewood-Paley theory
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