On the continuity of Fourier multipliers on the homogeneous Sobolev spaces $\dot{W}^1_1(R^d)$

TL;DR

This paper proves that all Fourier multipliers on the homogeneous Sobolev space ot{W}_1^1(\u211d^d) are continuous functions, extending previous results for homogeneous degree-zero multipliers.

Contribution

It generalizes prior work by Bonami and Poornima, establishing the continuity of all Fourier multipliers on ot{W}_1^1(^d), regardless of homogeneity degree.

Findings

All Fourier multipliers on ot{W}_1^1(^d) are continuous functions.

Extends previous results limited to degree-zero homogeneous multipliers.

Provides a broader understanding of Fourier multiplier behavior on Sobolev spaces.

Abstract

In this paper we proof that every Fourier multiplier on homogeneous sobolev space $\dot{W}_1^1(\mathbb{R}^d)$ is a continuous function. Our theorem is generalization of A. Bonami and S. Poornima result for Fourier multipliers, which are homogeneous functions of degree zero.