$H^p_+$ is the Hardy space on the upper half-plane of $\mathbb C$ consisting of functions $f$ satisfying $$ \sup_{y_0>0}\left(\int_{-\infty}^{\infty} |f(x+iy_0)|^p dx\right)^{\frac 1p}<\infty $$ For what values of $a$ will the function $f(z)=\frac{e^{iz}}{(z+2i)^{2a}}$ be in $H^p_+$?
This was an exam question for electric engineers, so it shouldn't be too difficult.
If $z=x+i y_0$, then $|e^{iz}|=e^{-y_0}\leq 1$ and: $$ \left|(z+2i)^{2a}\right|=\left|z+2i\right|^{2a}=\left((z+2i)(\bar{z}-2i)\right)^a=(x^2+(y_0+2)^2)^a,\tag{1}$$ hence we just need to find the values of $a$ for which: $$ \int_{\mathbb{R}}\frac{dx}{(x^2+(y_0+2)^2)^{ap}}<+\infty.\tag{2}$$ That leads to $ap>\frac{1}{2}$, or just: $$ a>\frac{1}{2p}.\tag{3}$$