I'm working through Stein Singular Integrals, and can't figure out how to prove this lemma (and find Stein's proof either wrong or un-readable). It is the following:
Lemma 3.3
Suppose $K:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfies the following three properties: $$(a)\ \ \ |K(x)| \leq B|x|^{-n} $$ $$(b)\ \ \ \int_{|x|>2|y|}{|K(x-y) - K(x)|dx} \leq B\ \ \ \forall y\in\mathbb{R}^n $$ $$(c)\ \ \ \int_{R_1 < |x| < R_2}{K(x)dx} = 0\ \ \ \forall 0 < R_1 < R_2 < \infty $$
For $\varepsilon > 0$ define $K_\varepsilon(x) = K(x)\chi_{B_\varepsilon}(x)$ where $\chi_{B_\varepsilon}$ is the characteristic function of the ball $B_\varepsilon$ of radius $\varepsilon$ centered at the origin. Then we have
$$ \sup_{y\in\mathbb{R}^n}{|\mathcal{F}(K_\varepsilon)(y)|} \leq CB $$
where the $\mathcal{F}$ is the Fourier Transform, and $C$ is a dimensional constant not depending on $K$ or $\varepsilon$.
(end of statement of Lemma 3.3)
First of all, I don't even think this is true unless we interpret the supremum as an essential supremum, but I'm not even concerned about that. We are to first prove it for the case $\varepsilon = 1$ and then use a scaling argument. This is fine with me, but I do not understand the case $\varepsilon = 1$. The main maneuver suggested by Stein is the following:
$$ \int_{\mathbb{R}^n}{e^{2\pi i y \cdot x} K_1(x) dx} = \cfrac{1}{2} \int_{\mathbb{R}^n}{e^{2\pi i y \cdot x} [K_1(x) - K_1(x-z)]dx} $$
where $z = \cfrac{y}{2|y|^2}$. Then, how I understand it is, we should split this into an integral over $\{ |x| > 2|z| \}$ and one over $\{ |x| \geq 2|z| \}$. The first can be estimated with the help of assumption (b), but I do not seem to agree with Stein that the second can be estimated, or I mis-understood what he wrote. If anyone can give a reference or a hint in the right direction, I'll appreciate it. I've been tinkering with this for far too long, so if I can't get help, I'm going to have to leave this theorem alone...