The general methods which are powerful for the necessity of bounded commutators are given. As applications, some necessary conditions for bounded commutators are first obtained in certain endpoint cases, and several new characterizations of BM O spaces, Lipschitz spaces and their weighted versions via boundedness of commutators in various function spaces are deduced.2010 Mathematics Subject Classification. 42B20; 42B25.

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In this note we give a simple proof of the endpoint regularity for the uncentred Hardy–Littlewood maximal function on $\mathbb{R}$. Our proof is based on identities for the local maximum points of the corresponding maximal functions, which are of interest in their own right.