Cubature formulas and Sobolev inequalities

We study a problem in the theory of cubature formulas on the sphere: given $\theta \in (0,1)$, determine the infimum of $|\nu|\theta = \sum{i = 1}^n \nu_i^\theta$overcubatureformulas$\nu$ofstrength$t$,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 $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\theta }$ for any $\theta \in (0,1)$. In particular, we completely characterize the cubature measures of strength $3$ minimizing $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\theta }$, showing that these are precisely the tight spherical $3$-designs.