Multilinear Hardy–Cesàro operator and commutator on the product of Morrey–Herz spaces

Abstract

We obtain necessary and sufficient conditions on functions s ₁(t),..., s _m (t) and ψ(t) such that the weighted multilinear Hardy–Cesàro operator

$$\left( {f_1 , \ldots f_m } \right) \mapsto \int_{[0,1]^n } {\left( {\prod\limits_{k = 1}^m {f_k } \left( {s_k \left( t \right)x} \right)} \right)\psi (t)dt}$$

is bounded from $\dot K_{q_1 }^{\alpha _1 ,p_1 } \left( {\omega _1 } \right) \times \cdots \times \dot K_{q_m }^{\alpha _m ,p_m } \left( {\omega _m } \right)$ to $\dot K_q^{\alpha ,p} \left( \omega \right)$ and from $M\dot K_{p_1 ,q_1 }^{\alpha _1 ,\lambda _1 } \left( {\omega _1 } \right) \times \cdots \times M\dot K_{p_m ,q_m }^{\alpha _m ,\lambda _m } \left( {\omega _m } \right)$ to $M\dot K_{p,q}^{\alpha ,\lambda } \left( \omega \right)$. Sharp bounds are also obtained for both cases 0 < p < 1 and 1 ≤ p < ∞. Provided b ₁, ..., b _m are Lipschitz functions we give a sufficient condition on functions s ₁(t), ..., s _m (t), ψ(t) such that the commutator of the weighted Hardy–Cesàro operator

$$\left( {f_1 , \ldots f_m } \right) \mapsto \int_{[0,1]^n } {\left( {\prod\limits_{k = 1}^m {f_k \left( {s_k \left( t \right)x} \right)} } \right)} \left( {\prod\limits_{k = 1}^m {\left( {b_k \left( x \right) - b_k \left( {s_k \left( t \right)x} \right)} \right)} } \right)\gamma (t)dt$$

is bounded from $M\dot K_{p_1 ,q_1 }^{\alpha _1 ,\lambda _1 } \left( {\omega _1 } \right) \times \cdots \times M\dot K_{p_m ,q_m }^{\alpha _m ,\lambda _m } \left( {\omega _m } \right)$ to $M\dot K_{p,q}^{\alpha ',\lambda } \left( \omega \right)$ for both cases 0 < p < 1 and 1 ≤ p < ∞. As a consequence, when m = n = 1 and s ₁(t) = t, we obtain an improvement of a recent result by Tang, Xue and Zhou.