Brascamp-Lieb inequalities on the sphere In the paper [CLL], Carlen, Lieb, and Loss demonstrate a version of the Young inequality on the sphere $S^{N-1}$ in $\mathbb{R}^N$.  For positive functions $f_j$ on $[-1,1]$, the following bound holds:
$$
\int_{S^{N-1}} \prod_{j=1}^N f_j(x\cdot e_j) \,d\mu(x)
\leq \prod_{j=1}^N \left(\int_{S^{N-1}} f_j(x\cdot e_j)^2\,d\mu(x)\right)^{1/2},
$$
where $\mu$ is the uniform measure on the sphere $S^{N-1}$ and $e_j$ are orthogonal vectors.  The proof exploits monotonicity under a nonlinear heat flow, and this technique also works to prove the generalized Young inequality on $\mathbb{R}^N$.  
In the paper [BCCT] of Bennett, Carbery, Christ, and Tao, this heat flow method is generalized to prove the "geometric" Brascamp Lieb inequalities on $\mathbb{R}^N$, which is the inequality 
$$
\int_{\mathbb{R}^N} \prod_{j=1}^m (f_j\circ B_j)^{p_j}(x)\,dx
\leq \prod_{j=1}^m \left(\int_{\mathbb{R}^{n_j}} f_j(y)\,dy\right)^{p_j}
$$
in the case that the linear maps $B_j:\mathbb{R}^N\to\mathbb{R}^{n_j}$ are orthogonal projections and satisfy
$$
\sum_{j=1}^m p_j B_j^* B_j = id_{N}.
$$
My question is whether this sort of generalization can also be demonstrated to hold on the sphere.  And if so, what is the corresponding "loss" in the $L^p$ exponent? (Observe that in the first inequality, an $L^2$ norm appears on the right, whereas one would expect an $L^1$ norm if one were stating the inequality on $\mathbb{R}^N$)  Moreover, do the more general Brascamp-Lieb inequalities hold on the sphere, and what is the relationship between the Euclidean BL constants and the spherical ones?
[BCCT] J Bennett, A Carbery, M Christ, and T Tao. "The Brascamp-Lieb Inequalities: Finiteness, Structure, and Extremals."  https://arxiv.org/abs/math/0505065
[CLL] EA Carlen, EH Lieb, and M Loss.  "A Sharp Analogue of Young's Inequality on $S^N$ and Related Entropy Inequalities."  https://arxiv.org/abs/math/0408030
 A: Indeed, there are extensions to spheres:
F. Barthe, D. Cordero-Erausquin, B. Maurey. Entropy of spherical marginals and related inequalities. J. Math. Pures Appl., 86: 89–99 (2006).
It reads as follows: If $x ∈ S^{n−1} \subset R^n$ (the standard (n − 1)-sphere), consider the projection  $P_{E_i}(x)$, $i = 1,...,m,$ where $E_i \subset R^n$ are subspaces for which we have

$
\sum_{i=1}^m c_i P_{E_i} \le Id_{R^n}. 
$
Then, whenever $f_i$ are non-negative measurable functions on the sphere, such that $f_i$ depends only on $E_i$ (that is $f_i(x) = g_i(P_{E_i} (x))$, for the uniform probability measure $\sigma$ on $S^{n−1}$ we have
$$
\int_{S^{n-1}} \Pi_{i=1}^m f_i^{c_i/2}d\sigma \le \Pi_{i=1}^m \left( \int_{S^{n-1}} f_i d\sigma\right)^{c_i/2}
$$
Those geometric Brascamp-Lieb inequalities are discussed in great generality in:
F. Barthe, D. Cordero-Erausquin, M. Ledoux and B. Maurey: Correlation and Brascamp-Lieb inequalities for Markov semigroups, arxiv version
The case of the sphere is discussed in Section 3.1 in that last reference.
