Let $f: [0,1] \rightarrow \mathbb{R}$ be a bounded measurable function. For some real non-negative numbers $a\_1, a\_2, b\_1, b\_2$ with $a\_1+b\_1=a\_2+b\_2=1$ consider the quantity
$$N(f)=\int\_{[0,1]} \int\_{[0,1]} f(a\_1x+b\_1y)f(a\_2x+b\_2y) \, dx \, dy.$$


If $a\_1=a\_2=1$ and $b\_1=b\_2=0$ then $N(f)^{1/2}$ is just the $L\_2$ norm of $f$. I am interested to understand when the quantity $N(f)^{1/2}$ defines a norm in other situations. Is anything like that known? Maybe at least some clear necessary conditions for the thing to be a norm?


Naturally, if this is known and easy, the general question would for a $k$-wise product of $f$ with some affine combination of the integrating variables plugged in each $f$ and then the question would be whether or not $N(f)^{1/k}$ is a norm.



Which averages of products of a function give a norm?
It is a norm when $a\_1=a\_2$, $b\_1=b\_2$. Otherwise making the change of variables $u=a\_1x+b\_1y, v=a\_2x+b\_2y$ we get the integral of $f(u)f(v)$ over certain parallelogram $P$ with a diagonal joining $(0, 0) $ and $(1, 1) $. Assume that $f$ has a small support $\Delta$ so that $\Delta^2\subset P$. Then the integral is just $(\int f)^2$ and this may be equal to 0.


