The fractional Young convolution inequality for regular hypergraphs

Let $M\geq 2$ and write $[M]=\{1,2,\ldots,M\}$. Let $\mathcal{G}$ be a $d$-regular hypergraph on $[M]$, let $r$ and $p_{\mathbf{s}}$ for $\mathbf{s}\in\mathcal{G}$ be real numbers in $(1,\infty)$ satisfying

$$\sum_{\mathbf{s}\in\mathcal{G}}\frac{1}{p_{\mathbf{s}}}=|\mathcal{G}|-\frac{d}{r'},$$

where $r'$ is the Hölder conjugate of $r$. Let $f_j$, $j\in[M]$, be probability density functions on $\mathbb{R}^n$, and let $C_p$ denote the sharp Hausdorff–Young constant. The fractional Young convolution conjecture. One should have

$$\left\|\star_{j\in[M]}f_j\right\|_r\leq \frac{1}{C_r^n}\prod_{\mathbf{s}\in\mathcal{G}}\left[C_{p_{\mathbf{s}}}^n\left\|\star_{j\in\mathbf{s}}f_j\right\|_{p_{\mathbf{s}}}\right]^{1/d}.$$

Furthermore, the inequality should be reversed when $\{p_{\mathbf{s}}:\mathbf{s}\in\mathcal{G}\}\cup\{r\}\subset(0,1)$. This conjecture refines sharp Young's inequality for convolutions of more than two functions and connects convolution inequalities with hypergraph structure, information theory, and probability. The paper proves it when $r\geq2$ and every $p_{\mathbf{s}}\in[1,2]$, but the full range of parameters remains open.