I'm trying to understand the hypothesis of Marcinkiewicz-Mihlin-Hörmander multiplier theorem. See for instance Theorem A in following paper of Elias Stain.
Theorem A: Assume that $m: (0, \infty)\to \mathbb R$ satisfies the following $$|m^{(j)}(x)| \leq C x^{-j}\tag{1}$$ for $0 \leq j \leq k$ and $k>d/2.$ Or more generally $$ \sup_{t>0} \left\|\chi m(t \cdot)\right\|_{H^{\alpha}}< \infty\tag{2} $$ where $\chi$ is a non-zero smooth cut-off function of compact support which vanishes near the origin and $\|f\|_{H^s} =\|(1+ |\cdot|^2)^{s/2} \hat{f}\|_{L^2}.$
Then Fourier-multiplier operator $\widehat{Tf}= m(|\xi|^2) \hat{f}$ is bounded on $L^p (\mathbb R^d)$ for $1<p< \infty.$
My questions are:
- Why condition (1) implies (2) in the Theorem?
- How has the theorem been developed historically? (I mean version of Marcinkiewicz theorem 1939, version of Mihim 1957, and finally version of Hormander 1960.)