Localized version of hypergraph Erdős-Gallai Theorem

The weight function of an edge in an n-vertex uniform hypergraph $H$ is defined with respect to the number of edges in the longest Berge path containing the edge. We prove that the summation of the weight function values for all edges in $H$ is at most n, and characterize all extremal hypergraphs that attain this bound. This result strengthens and extends the hypergraph version of the classic Erdős-Gallai Theorem, providing a local version of this theorem.