Characterizations of Orlicz-Sobolev spaces by means of generalized Orlicz-Poincaré inequalities (Q416322)

Summary: Let \(\Phi\) be an \(N\)-function. We show that a function \(u \in L^\Phi(\mathbb R^n)\) belongs to the Orlicz-Sobolev space \(W^{1,\Phi}(\mathbb R^n)\) if and only if it satisfies the (generalized) \(\Phi\)-Poincaré inequality. Under more restrictive assumptions on \(\Phi\), an analog of the result holds in a general metric measure space setting.