  We study a problem in the theory of cubature formulas on the sphere: given
$\theta \in (0, 1)$, determine the infimum of $\|\nu\|_\theta = \sum_{i = 1}^n
\nu_i^\theta$ over cubature formulas $\nu$ of strength $t$, where $\nu_i$ are
the weights of the formula $\nu$. This problem, which generalizes the classical
problem of bounding the minimal cardinality of a cubature formula -- the case
$\theta = 0$ -- was introduced in recent work of Hang and Wang
(arXiv:2010.10654), who showed the problem to be related to optimal constants
in Sobolev inequalities. Using the elementary theory of reproducing kernel
Hilbert spaces on $S^{n - 1}$, we extend the best known upper and lower bounds
for the minimal cardinality of strength-$t$ cubature formulas to bounds for the
infimum of $\|\cdot\|_\theta$ for any $\theta \in (0, 1)$. In particular, we
completely characterize the cubature measures of strength $3$ minimizing
$\|\cdot\|_\theta$, showing that these are precisely the tight spherical
$3$-designs.
