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

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 $\hat{\psi }$ is supported in $[-2,-1/2]\cup [1/2,2]$ and which satisfies ${\sum }_{j\in \mathbb{Z}}\hat{\psi }\left(2^{-j}\xi \right)=1$ for all $\xi \ne 0$. 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*} (ξ_1,\dots, ξ_n)\mapsto \prod_{i=1}^n (I-\partial_i^2)^{\frac {s_i}2} \Big[ \prod_{i=1}^n \widehatψ(ξ_i) σ(2^{j_1}ξ_1,\dots , 2^{j_n}ξ_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 $s_i\ne s_{i+1}$ 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.