We prove a hypercontractive inequality for matrix-valued functions defined over large alphabets. In order to do so, we prove a generalization of the powerful $2$-uniform convexity inequality for trace norms of Ball, Carlen, Lieb (Inventiones Mathematicae'94). Using our hypercontractive~inequality, we present upper and lower bounds for the communication complexity of the Hidden Hypermatching problem defined over large alphabets. We then consider streaming algorithms for approximating the value of Unique Games on a hypergraph with $t$-size hyperedges. By using our communication lower bound, we show that every streaming algorithm in the adversarial model achieving an $(r-\varepsilon)$-approximation of this value requires $\Omega(n^{1-2/t})$ quantum space, where $r$ is the alphabet size. We next present a lower bound for locally decodable codes (LDC) $\mathbb{Z}_r^n\to \mathbb{Z}_r^N$ over large alphabets with recoverability probability at least $1/r + \varepsilon$. Using hypercontractivity, we give an exponential lower bound $N = 2^{\Omega(\varepsilon^4 n/r^4)}$ for $2$-query (possibly non-linear) LDCs over $\mathbb{Z}_r$ and using the non-commutative Khintchine inequality we prove an improved lower bound of $N = 2^{\Omega(\varepsilon^2 n/r^2)}$.

翻译：我们证明了定义在大字母表上的矩阵值函数的超压缩不等式。为此，我们推广了Ball、Carlen、Lieb（Inventiones Mathematicae'94）关于迹范数的强有力$2$-一致凸性不等式。利用我们的超压缩不等式，我们给出了定义在大字母表上的隐超匹配问题通信复杂度的上界和下界。接着，我们研究了在具有$t$大小超边的超图上近似唯一博弈值的流算法。通过使用我们的通信下界，我们证明了在对抗模型中，每个实现$(r-\varepsilon)$近似该值的流算法需要$\Omega(n^{1-2/t})$的量子空间，其中$r$是字母表大小。随后，我们针对恢复概率至少为$1/r + \varepsilon$的大字母表局部可解码码（LDC）$\mathbb{Z}_r^n\to \mathbb{Z}_r^N$给出了下界。利用超压缩性，我们得到了$\mathbb{Z}_r$上$2$查询（可能非线性）LDC的指数下界$N = 2^{\Omega(\varepsilon^4 n/r^4)}$，并通过非交换Khintchine不等式证明了一个改进的下界$N = 2^{\Omega(\varepsilon^2 n/r^2)}$。