In my recent studies (fourier multipliers on weighted Lp spaces) I have to deal with this kind of transformation: if w is a measurable function on $R^n$, define
$w^\*(x)=\sup\_y \frac{w(x+y)}{w(y)}$.
Has anyone seen this transformation before? Has this interesting properties?
In particular I'm interested in functions such that w=w\* or $w^\*\leq cw$, like the exponential.
Thanks in advance.


UPDATE: I'm interested in $w$ continuous, positive and even



Is anything known about $w^*(x)=\sup_y w(x+y)/w(y)$ for measurable functions w on $R^n$
Sort of a random thought: if $w = w^\*$, then $w$ must be positive. So take logs on both side. Since log is monotonic, it commutes with sup. Then we have $a = \log w$, and $$a(x) = \sup\_y a(x+y) - a(y)$$
which implies that 
$$ a(x) + a(y) \geq a(x+y)$$
so $a$ must be sub-additive. Helge's comment tells us that $a(0) = 0$. 


This looks necessary and sufficient. (As for sub-additive functions $a$ such that $a(0) = 0$, $a(x) + a(0) = a(x+0)$ is satisfied, so the sup is attained.)


