On the failure of the H\"ormander multiplier theorem in a limiting case

TL;DR

This paper investigates the limitations of the H"ormander multiplier theorem for Fourier multipliers in fractional Sobolev spaces, demonstrating its failure in a specific limiting case where smoothness and integrability parameters align.

Contribution

The paper identifies and proves the failure of the H"ormander multiplier theorem in the critical limiting case where the smoothness parameter matches the difference in reciprocal of p and 1/2.

Findings

The theorem does not hold when |1/p - 1/2| = s/n.

The failure occurs in the limiting case of the fractional Sobolev space.

Provides insight into the boundary conditions of Fourier multiplier boundedness.

Abstract

We discuss the H\"ormander multiplier theorem for $L^p$ boundedness of Fourier multipliers in which the multiplier belongs to a fractional Sobolev space with smoothness $s$. We show that this theorem does not hold in the limiting case $|1/p-1/2|=s/n$.