# Multiplier conditions for Boundedness into Hardy spaces

## TL;DR

This paper establishes explicit necessary and sufficient conditions for certain multilinear multiplier operators to be bounded into Hardy spaces, focusing on the vanishing of symbols and their derivatives on specific hyperplanes.

## Contribution

It provides new explicit conditions for the boundedness of multilinear multipliers of Coifman-Meyer type into Hardy spaces, extending previous results.

## Key findings

- Conditions involve vanishing of symbols and derivatives on hyperplanes
- Applicable to linear and multilinear operators of Coifman-Meyer type
- Results include intermediate types of operators

## Abstract

In the present work, we find useful and explicit necessary and sufficient conditions for linear and multilinear multiplier operators of Coifman-Meyer type, finite sum of products of Calder\'on-Zygmund operators, and also of intermediate types to be bounded from a product of Lebesgue or Hardy spaces into a Hardy space. These conditions state that the symbols of the multipliers $\sigma(\xi_1,\dots , \xi_m)$ and their derivatives vanish on the hyperplane $\xi_1+\cdots+\xi_m=0$.