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 $\widehat{\psi}$ 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.