Assumption in Hörmander-Mikhlin multiplier theorem

The well-known Hörmander-Mikhlin multiplier theorem is the following statement:Let $m \in C^N(\mathbb{R}^n \setminus \{0\})$ with $N = [n/2] + 1$ andsuppose for each multi-index $\alpha$ with $|\alpha| \leq N$, there exists a constant $c_{\alpha} > 0$ such that$$|\partial^{\alpha} m(\xi)| \leq c_{\alpha} |\xi|^{-|\alpha|}, \quad \forall \xi \in \mathbb{R}^n \setminus \{0\}.$$Then, the Fourier multiplier operator defined by $T_mf = \mathcal{F}^{-1}[m \hat{f}]$ is a bounded operator on $L^p(\mathbb{R}^n)$ for all $1 < p < \infty$.

I am wondering if the multiplier theorem holds under the following slightly weaker assumptions: For each multi-index $\alpha$ with $|\alpha| \leq N$,$$\sup_{\xi \in \mathbb{R}^n \setminus \{0\}} |\xi^{\alpha} \partial^{\alpha}m(\xi)| < +\infty$$or, equivalently, there exists a constant $c_{\alpha} > 0$ such that$$|\partial^{\alpha} m(\xi)| \leq c_{\alpha} |\xi^{\alpha}|^{-1}$$whenever $\xi^{\alpha} = \xi_1^{\alpha_1} \cdots \xi_n^{\alpha_n} \not=0$.

Any ideas are welcome.