A sharp variant of the Marcinkiewicz theorem with multipliers in Sobolev spaces of Lorentz type

TL;DR

This paper refines the Marcinkiewicz multiplier theorem by identifying optimal Lorentz space conditions for boundedness of Fourier multipliers in Sobolev spaces, especially addressing cases with equal smoothness parameters.

Contribution

It extends previous results by handling cases where multiple smoothness parameters are equal, providing near-optimal Lorentz space conditions for multiplier boundedness.

Findings

Established a new Lorentz space framework for multipliers with equal smoothness parameters.

Proved the boundedness of operators under these refined Lorentz space conditions.

Achieved near-optimal results up to a small logarithmic factor.

Abstract

Given a bounded measurable function $\sigma$ on $\mathbb{R}^n$, we let $T_\sigma $ be the operator obtained by multiplication on the Fourier transform by $\sigma $. Let $0<s_1\le s_2\le \cdots \le s_n<1$ and $\psi$ be a Schwartz function on the real line whose Fourier transform $\widehat{\psi}$ is supported in $[-2,-1/2]\cup[1/2,2]$ and which satisfies $\sum_{j \in \mathbb{Z}} \widehat{\psi}\left(2^{-j} \xi\right)=1$ for all $\xi \neq 0$. In this work we sharpen the known forms of the Marcinkiewicz multiplier theorem by finding an almost optimal function space with the property that, if the function \begin{equation*} (\xi_1,\dots, \xi_n)\mapsto \prod_{i=1}^n (I-\partial_i^2)^{\frac {s_i}2} \Big[ \prod_{i=1}^n \widehat{\psi}(\xi_i) \sigma(2^{j_1}\xi_1,\dots , 2^{j_n}\xi_n)\Big] \end{equation*} belongs to it uniformly in $j_1,\dots , j_n \in \mathbb Z$, then $T_{\sigma}$ is bounded on $ {L}^p(\mathbb R^n)$ when $ |\frac{1}{p}-\frac{1}{2} | < s_1$ and $1<p<\infty$. In the case where $s_i\neq s_{i+1}$ for all $i$, it was proved in [Grafakos, Israel J. Math., to appear] that the Lorentz space $L ^{\frac{1}{s_1},1} (\mathbb{R}^n) $ is the function space sought. In this work we address the significantly more difficult general case when for certain indices $i$ we might have $s_i=s_{i+1}$. We obtain a version of the Marcinkiewicz multiplier theorem in which the space $L ^{\frac{1}{s_1},1}$ is replaced by an appropriate Lorentz space associated with a certain concave function related to the number of terms among $s_2,\dots , s_n$ that equal $s_1$. Our result is optimal up to an arbitrarily small power of the logarithm in the defining concave function of the Lorentz space.