Generalized BMO-type seminorms and vector-valued Sobolev functions

We establish a pointwise limit theorem for a broad class of pa-ra-me-ter--de-pen-dent BMO-type seminorms as the parameter tends to zero. By introducing novel BMO-type seminorms, we provide a unified framework that extends several existing results and yields non-distributional characterizations of Sobolev-type spaces, both in the scalar and in the vector-valued setting. More precisely, for any open set $\mathrm{\Omega }\subset {\mathbb{R}}^n$ and any $p\in (1,\mathrm{\infty })$, we provide a characterization of the Sobolev space $W^{1,p}(\mathrm{\Omega };{\mathbb{R}}^m)$. In addition, we characterize the space $E^{1,p}(\mathrm{\Omega };{\mathbb{R}}^n)$ of $L^p$ maps with $p$-integrable distributional symmetric gradient.\ Finally, for all $p\in [1,\mathrm{\infty })$, we show that these seminorms converge to integral functionals with convex, $p$-homogeneous integrands associated with the distributional gradient and the symmetric gradient.