I was given a Fourier multiplier $m(\xi)=|\xi|^{-\alpha}$ for $\alpha > 1$, and the operator $L$ associated with $m$ is given by: $$\widehat{Lf}(\xi) = m(\xi) \hat f(\xi).$$ How can we define the order of the operator $L$? Is it different from the order of the multiplier $m$? I received mixed answers: $\alpha$ and $-\alpha$.
I am quite confused. Is there a reference for these definitions?
With order do you mean the least $k$ such that the operator is bounded in the $H^k$ norm?
Your operator is unbounded in the $H^k(\Bbb{R})$ norm for all $k$: that $(1+|\xi|^k) \hat{f}(\xi)\in L^2(\Bbb{R})$ doesn't imply that $(1+|\xi|^k)|\xi|^{-\alpha} \hat{f}(\xi)\in L^2(\Bbb{R})$.