An upper bound on the sum of squares of degrees in a hypergraph

We give an upper bound on the sum of squares of ℓ-degrees in a k-uniform hypergraph in terms of ℓ,k and the number of vertices and edges of the hypergraph, where a ℓ-degree is the number of edges of the hypergraph containing a fixed ℓ-element subset of the vertices. For ordinary graphs this bound coincides with one given by de Caen. We show that our bound implies the quadratic LYM-inequality for 2-level antichains of subsets of a finite set.