I'm trying to understand the hypothesis of the Marcinkiewicz-Mihlin-Hörmander multiplier theorem. See for instance Theorem A in this paper of Elias Stein.
Theorem A: Assume that $m: (0, \infty)\to \mathbb R$ satisfies the following equation
$$
|m^{(j)}(x)| \leq C x^{-j}\quad0 \leq j \leq k,\; k>\frac{d}{2},\label{1}\tag{1}
$$
rr more generally
$$
\sup_{t>0} \|\chi m(t \cdot)\|_{L^2_{\alpha}}< \infty \label{2}\tag{2}
$$
where $\chi$ is a non-zero smooth cut-off function of compact support that 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 \eqref{1} implies \eqref{2} in Theorem A?
(C) Does symbol $m(\xi)=e^{i|\xi|^2}$ satisfy condition \eqref{2} of Theorem A? (I am interested in this problem since, by using this symbol, we can solve Schrodinger's equation)
(D) How this theorem has been developed historically? I mean is it correct to say that the first version of this result goes back to Marcinkiewicz in 1939, and later versions were Mihlin's in 1957 and finally Hormander's in 1960?
About question A above, my heuristic guess is that we can define $\|f\|_{L^2_{\alpha}}^2$ as
$$
\|f\|_{L^2_{\alpha}}^2= \int_{\mathbb R^d} |f(x)|^2 (1+|x|^2)^{\alpha} \mathrm{d}x
$$
or
$$
\|f\|_{L^2_{\alpha}}^2= \int_{\mathbb R^d} |\widehat{f}(\xi)|^2 (1+|\xi|^2)^{\alpha} \mathrm{d}x
$$
Is my guess correct or am I missing something?
