operator norm of this multiplier operator I am having some trouble with some basic properties of a given operator.
Firstly, the operator T is defined as taking the fourier inverse transform of the function $(1-|\zeta|)1_{[-1,1]}(\zeta)\hat{f}(\zeta)$. 
(a) Show T is bounded on $L^2(R)$, and compute the operator norm. 
(b) Further, show that $T$ is a bounded operator on $L^p(R)$. The hint of b is that T is in fact convolution with a function g s.t $|g(x)|＜ C/1+x^2$. Lp convolution inequality is needed. 
Just guess a) requires plancherel theorem to show $||T||_2 = ||u||_∞$ ,where u is $(1-|\zeta|)1_{[-1,1]}(\zeta)$. But cant figure out how to do it . Also, I dont know how to do with (b).
Could someone help with it? Thanks.
 A: Hint for a:
${\large|}1-|\zeta|{\large|}\le1$ on $[-1,1]$
Hint for b:
$$
\begin{align}
\int_{-1}^1(1-|x|)e^{-2\pi ix\xi}\,\mathrm{d}x
&=2\int_0^1(1-x)\cos(2\pi x\xi)\,\mathrm{d}x\\
&=\frac{\sin(2\pi\xi)}{\pi\xi}-2\int_0^1x\cos(2\pi x\xi)\,\mathrm{d}x\\
&=\frac{\sin(2\pi\xi)}{\pi\xi}-\frac1{\pi\xi}\int_0^1x\,\mathrm{d}\sin(2\pi x\xi)\\
&=\frac{\sin(2\pi\xi)}{\pi\xi}-\frac{\sin(2\pi\xi)}{\pi\xi}+\frac1{\pi\xi}\int_0^1\sin(2\pi x\xi)\,\mathrm{d}x\\
&=\frac{1-\cos(2\pi\xi)}{2\pi^2\xi^2}
\end{align}
$$

Complete Answer for a:
The $L^2$ norm for a Fourier Multiplier Operator is the $L^\infty$ norm of the multiplier. This follows easily by Plancherel's Theorem. Since $\left\|1-|\zeta|\right\|_{L^\infty[-1,1]}=1$, the $L^2$ norm of $T$ is $1$.

Complete Answer for b: By the Convolution Theorem, $\widehat{m\hat{f}}=\widehat{m}\ast\tilde{f}$, where $\tilde{f}(x)=f(-x)$. Since the Fourier Transform of $(1-|\zeta|)1_{[-1,1]}(\zeta)$ is $\frac{1-\cos(2\pi\xi)}{2\pi^2\xi^2}$ and
$$
\left\|\frac{1-\cos(2\pi\xi)}{2\pi^2\xi^2}\right\|_{L^1}=1
$$
Young's Inequality for Convolutions shows that the $L^p$ norm for this multiplier operator is $1$ for all $p$.
A: Since this is homework, only hints:
(a) Start with $\|Tf\|_2^2$, use Plancherel and estimate by "a constant times $\|f\|^2$ (in fact, use Plancherel twice). This constant is an upper bound for $\|T\|$. Then, find some $f$ such that the estimate is sharp.
(b) I presume that $M$ is $T$? However, this hint you have is already saying quite a lot. You also need to use the convolution theorem (twice), i.e. that the Fourier transform takes convolutions to pointwise products.
