Weighted multilinear Hardy operators and commutators

TL;DR

This paper introduces weighted multilinear Hardy operators, establishes their bounds on Lebesgue and Morrey spaces, and characterizes the boundedness of their commutators with symbols in BMO, with applications to classical inequalities.

Contribution

It provides the first sharp bounds for these operators and necessary and sufficient conditions for the boundedness of their commutators on Morrey spaces.

Findings

Sharp bounds for weighted multilinear Hardy operators on Lebesgue and Morrey spaces.

Necessary and sufficient conditions for the boundedness of commutators with BMO symbols.

Applications to sharp estimates of Riemann-Liouville and Weyl inequalities.

Abstract

In this paper, we introduce a type of weighted multilinear Hardy operators and obtain their sharp bounds on the product of Lebesgue spaces and central Morrey spaces. In addition, we obtain sufficient and necessary conditions of the weight functions so that the commutators of the weighted multilinear Hardy operators (with symbols in central BMO space) are bounded on the product of central Morrey spaces. These results are further used to prove sharp estimates of some inequalities due to Riemann-Liouville and Weyl.