# fa.functional analysis – Decay estimate of Fourier transform of a compactly supported function

Assume $$f(x), x in mathbb{R}$$ is a function with a compact support such that its Fourier transform $$hat{f}(xi)$$ has a decay rate
$$hat{f}(xi) lesssim frac{1}{|xi|^gamma + 1}$$
for some $$gamma ge 1$$.
Now set $$h(x) = xf(x).$$ Since $$f$$ has a compact support, $$h$$ should have similar or better regularity than $$f$$. Can we now get the following decay estimate of the Fourier transform of $$h$$ ?
$$hat{h}(xi) lesssim frac{1}{|xi|^gamma + 1}$$
I know now we have $$hat{h}(xi) = -ipartial_xi hat{f}(xi)$$, but it seems hard to only use this relation to get the decay estimate.