Endpoint multiplier theorems of Marcinkiewicz type

TL;DR

This paper proves sharp endpoint multiplier theorems of Marcinkiewicz type for rough one-dimensional multipliers, establishing optimal mapping properties between specific function spaces.

Contribution

It introduces new sharp endpoint estimates for Marcinkiewicz multipliers, extending classical results to more delicate function space mappings.

Findings

Marcinkiewicz multipliers map H^1 to L^{1, Infty}

L log^{1/2} L maps to L^{1, Infty}

Estimates are proven to be sharp

Abstract

We establish sharp (H^1, L^{1,q}) and local (L \log^r L, L^{1,q}) mapping properties for rough one-dimensional multipliers. In particular, we show that the multipliers in the Marcinkiewicz multiplier theorem map H^1 to L^{1,\infty} and L \log^{1/2} L to L^{1,\infty}, and that these estimates are sharp.