Is Brascamp-Lieb inequality on the sphere applicable for these functions for some $1\leq p<2$

Question

My question is on Brascamp-Lieb-inequality on the Euclidean sphere (which can be viewed as an analogue of Young's inequality on the sphere) obtained in [1].

(See also this question: Brascamp-Lieb inequalities on the sphere )

The inequality reads:

Let $e_{i},i=1,\dots,N$, be orthogonal vectors in $\mathbb{R}^{N}$. If $f_j$ are positive functions on $[-1,1]$, $N \geq 3$, and $p\geq 2$ then $$ \int_{S^{N-1}} \prod_{j=1}^N f_j(x\cdot e_j) d\mu(x) \leq \prod_{j=1}^N \left(\int_{S^{N-1}} f_j(x\cdot e_j)^p d\mu(x)\right)^{1/p}, $$ where $\mu$ is the uniform Borel probability measure on the sphere $S^{N-1}$.

Fix $(x_{1},\dots,x_{N})\in \mathbb{R}^{N}\setminus S^{N-1}$. Let $0<\alpha_{j}<1$ and $\sum_{j=1}^{N}\alpha_{j}<N-1$. Suppose that $$f_{j}(t)=|t-x_{j}|^{-\alpha_{j}},\quad x_{i}\neq x_{j},\;\forall i\neq j.$$ Does the inequality $(1)$ hold true for these particular functions $f_{j}$ for some $1\leq p<2$ ?

That is, is it true that

$$ \int_{S^{N-1}} \prod_{j=1}^N |\theta\cdot e_j-x_{j}|^{-\alpha_{j}} d\mu(\theta) \leq C\prod_{j=1}^N \left(\int_{S^{N-1}} |\theta\cdot e_j-x_{j}|^{-p\alpha_{j}} d\mu(\theta)\right)^{\frac{1}{p}}\qquad (*)$$ for some $1\leq p<2$ ?

[1] 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

Answer 1

Unfortunately, the answer to the question as stated is "No" even if $N=2$. Let $k=1,\alpha_1=\alpha_2=\frac 12$. Choose $x=(x_1,x_2)$ on the unit circle (say $(3/5,4/5)$). Then the integrals on the right are finite but the integral on the left diverges (you basically integrate inverse distance to $x$). Perhaps, you meant something else?