Be more careful with the exponents of the $L^p$ spaces, in which $f$ and $\hat f$ belong.

If $f\in L^1$, then $\hat f\in L^\infty$, and $\|\hat f\|_\infty\le\|f\|_1$. 
For a dense subset of $L^2$, the $\hat f$ is definable as an element of $L^2$, and $\|f\|_2=\|\hat f\|_2$, and the Fourier transformation is hence definable using a density argument in the whole $L^2$. Hausdorff-Young is an interpolation of the two inequalities.

The inverse Fourier transform has almost identical definition, and thus satisfies the same inequalities, and it can only belong in $L^p$, $p\ge 2$. Note that the inversion, $\mathcal F^{-1}{\mathcal F}f=f$, holds in $L^2$.

Upper bound for the norm of inverse Fourier tansform
Today my ISP blocked my internet due to suspicious activity - outgoing requests to malicious websites. Numerous scans of my machine couldn't reveal the culprit.
However, after doing a quick netstat /f I found the following:

TCP    127.0.0.1:5357         101com.com:49168       TIME_WAIT
TCP    192.168.1.21:49169     THOMSON:netbios-ssn    TIME_WAIT
TCP    192.168.1.21:49170     THOMSON:netbios-ssn    ESTABLISHED

TCP    127.0.0.1:49171        101com.com:49172       ESTABLISHED
TCP    127.0.0.1:49172        101com.com:49171       ESTABLISHED


Coincidentally, only a few days ago I decided to start adding servers to my host file. 101com.com appears to be the first entry on my list. 

So, am I actually sending out requests to 101com even though it is being blacklisted? And if so, how can it be prevented? 

Thanks.
