In the study of Boolean functions, the hypercontractive inequality enables one to bound from above the norm of $Tf$ by some norm of $f$, where $T$ is the noise operator depending on the noise parameter. This can be written in terms of the Fourier transform of $f$ as (in a special case of $q=2$):

$\left( \sum\limits_{S \subseteq [n]} (p - 1)^{|S|} \widehat{f}(S)^2 \right)^{1/2} \leq \left( \frac{1}{2^n} \sum\limits_{x \in (0,1)^n} |f(x)|^p \right)^{1/p}$

However, this involves all Fourier levels of $f$. In the application I have in mind, I'm interested only in bounding the norm of the first level of $f$, i.e. restricting the sum on the left to $|S| = 1$. Is it possible to give any inequality of this kind, probably with a different right hand side (but still involving some information about the norm of $f$) and some additional assumptions on $f$? If it's impossible for some trivial reasons, let me know anyway.

Here I'd be mostly interested in matrix-valued Boolean functions (see for example http://arxiv.org/abs/0705.3806), although any answer would be appreciated.