# Equivalence of (quasi-)norms on a vector-valued function space and its applications to multilinear operators

## TL;DR

This paper establishes (quasi-)norm equivalences on vector-valued function spaces, extending to Triebel-Lizorkin spaces, and applies these results to improve multilinear multiplier theorems and boundedness of bilinear pseudo-differential operators.

## Contribution

It introduces new (quasi-)norm equivalences on vector-valued function spaces and extends these to Triebel-Lizorkin spaces, enhancing existing multilinear operator theorems.

## Key findings

- Extended (quasi-)norm equivalence to Triebel-Lizorkin spaces.
- Improved multilinear Hormander's multiplier theorem.
- Enhanced boundedness results for bilinear pseudo-differential operators.

## Abstract

In this paper we present (quasi-)norm equivalence on a vector-valued function space $L^p_A(l^q)$ and extend the equivalence to $p=\infty$ and $0<q<\infty$ in the scale of Triebel-Lizorkin space, motivated by Fraizer-Jawerth. By applying the results, we improve the multilinear Hormander's multiplier theorem of Tomita, that of Grafakos-Si, and the boundedness results for bilinear pseudo-differential operators, given by Koezuka-Tomita.