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In this brief note, we show that the hypotheses of L\"owner's theorem onmatrix monotonicity in several commuting variables as proved by Agler, McCarthyand Young can be significantly relaxed. Specifically, we extend their theoremfrom continuously differentiable locally matrix monotone functions to arbitrarylocally matrix monotone functions using mollification techniques.

Note on L\"owner's theorem on matrix monotone functions in several commuting variables of Agler, McCarthy and Young