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>



Brascamp-Lieb inequalities on the sphere
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](https://arxiv.org/pdf/0907.2858.pdf)


The case of the sphere is discussed in Section 3.1 in that last reference.


