On cliques in hypergraphs under bounded (j,p)-norm

Let $\mathcal{H}$ be an $r$-uniform hypergraph. For $S\in \binom{V(\mathcal{H})}{j}$, let $\mathrm{deg}(S)$ be the number of edges of $\mathcal{H}$ containing $S$, and define the $(j,p)$-norm of $\mathcal{H}$ by $\|\mathcal{H}\|_{j,p}=\left(\sum_{S\in \binom{V(\mathcal{H})}{j}}\mathrm{deg}(S)^p\right)^{1/p}$. Motivated by a problem of Chao, Dong, Shen and Yang, we determine the maximum number of $t$-cliques in an $n$-vertex $r$-graph with bounded $(j,p)$-norm in the range $p>(t-j)/(r-j)$. The proof uses an entropy argument adapted to hypergraphs, together with a continuous interpolation step. The bound is sharp whenever the corresponding Steiner systems exist.