  We consider the problem of testing a particular type of composite null
hypothesis under a nonparametric multivariate regression model. For a given
quadratic functional $Q$, the null hypothesis states that the regression
function $f$ satisfies the constraint $Q[f]=0$, while the alternative
corresponds to the functions for which $Q[f]$ is bounded away from zero. On the
one hand, we provide minimax rates of testing and the exact separation
constants, along with a sharp-optimal testing procedure, for diagonal and
nonnegative quadratic functionals. We consider smoothness classes of
ellipsoidal form and check that our conditions are fulfilled in the particular
case of ellipsoids corresponding to anisotropic Sobolev classes. In this case,
we present a closed form of the minimax rate and the separation constant. On
the other hand, minimax rates for quadratic functionals which are neither
positive nor negative makes appear two different regimes: "regular" and
"irregular". In the "regular" case, the minimax rate is equal to $n^{-1/4}$
while in the "irregular" case, the rate depends on the smoothness class and is
slower than in the "regular" case. We apply this to the issue of testing the
equality of norms of two functions observed in noisy environments.
