I'm trying to understand the hypothesis of   Marcinkiewicz-Mihlin-H\"ormander multiplier theorem.  See for instance **Theorem A** in following [paper](http://www.numdam.org/article/JEDP_1995____A16_0.pdf) of Elias Stain.


**Theorem A**: Assumes that $m: (0, \infty)\to \mathbb R$ satisfies the following
(1) 
$$|m^{(j)}(x)| \leq C x^{-j}$$ for $0 \leq j \leq k$ and $k>d/2.$
Or more generally

(2)
 
$$ \sup_{t>0} \left\|\chi m(t \cdot)\right\|_{L^2_{\alpha}}< \infty $$
where $\chi$ is a non-zero smooth cut-off function of compact support which vanishes near the origin.

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: (A)  How to define $\|\cdot\|_{L^2_{\alpha}}$? Is it standard notation? (It seems that Stein has not defined this notation in his paper. Thus I guess it must be standard)

>(B) Why condition (1) implies (2) in the Theorem?

> (C) I'm interested to check symbol $m(\xi)=e^{i|\xi|^2}$ Satisfies the condition (2) of the Theorem A? (with this symbol we have solution to Schrodinger equations)

>(D) How theorem has been developed historically? (I mean version of Marcinkiewicz theorem  1939, version of  Mihim  1957, and finally version of Hormander  1960.)


My heuristic  Guess is:  we  can define $\|f\|_{L^2_{\alpha}}^2= \int_{\mathbb R^d} |f(x)|^2 (1+|x|^2)^{\alpha}  dx$ or $\|f\|_{L^2_{\alpha}}^2= \int_{\mathbb R^d} |\widehat{f}(\xi)|^2 (1+|\xi|^2)^{\alpha}  dx$  Is this correct or am I missing something?