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.
Categories: math.CA