# A multiplier inclusion theorem on product domains

## TL;DR

This paper establishes a strict inclusion between two classes of multipliers on product domains, showing that multipliers from the Hardy space to L^2 are a proper subset of those from a specific Orlicz space to L^2.

## Contribution

It proves a new multiplier inclusion theorem on product domains, clarifying the relationship between Hardy space multipliers and Orlicz space multipliers.

## Key findings

- Multipliers from Hardy space to L^2 are strictly contained in those from L log^{d/2} L to L^2.
- The inclusion between these multiplier classes is proper, not equal.
- Provides a deeper understanding of multiplier spaces on product domains.

## Abstract

In this note it is shown that the class of all multipliers from the $d$-parameter Hardy space $H^1_{\mathrm{prod}} (\mathbb{T}^d)$ to $L^2 (\mathbb{T}^d)$ is properly contained in the class of all multipliers from $L \log^{d/2} L (\mathbb{T}^d)$ to $L^2(\mathbb{T}^d)$.