# A sharp version of the H\"ormander multiplier theorem

## TL;DR

This paper improves the H"ormander multiplier theorem by replacing the classical Sobolev space with a Lorentz space-based Sobolev space, allowing for broader applicability in harmonic analysis.

## Contribution

It introduces a sharper version of the H"ormander multiplier theorem using Lorentz space-based Sobolev spaces, enhancing the theorem's scope and precision.

## Key findings

- The new theorem extends the class of multipliers for which boundedness holds.
- It replaces the integrability condition with a Lorentz space-based smoothness condition.
- The result broadens the applicability of multiplier theorems in harmonic analysis.

## Abstract

We provide an improvement of the H\"ormander multiplier theorem in which the Sobolev space $L^r_s(\mathbb R^n)$ with integrability index $r$ and smoothness index $s>n/r$ is replaced by the Sobolev space with smoothness $s$ built upon the Lorentz space $L^{n/s,1}(\mathbb R^n)$.