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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.

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    I rolled back the last edit since it removed the statement of the problem.2012-11-15

2 Answers 2