We prove an existence theorem for the pointwise limit of a wide class of one-parameter $BMO$-type seminorms as the parameter goes to zero. We generalize results appearing in the literature by introducing novel $BMO$-type seminorms and extending our analysis to vector-valued functions.
We then prove a characterization result of some Sobolev-type spaces of vector-valued maps based on the asymptotic behavior of suitable $BMO$-type seminorms.
As an application of these results, given an open set $\Omega\subseteq\mathbb{R}^{n}$ and $p\in(1,\infty)$, we establish a non-distributional characterization of both the Sobolev space $W^{1,p}(\Omega;\mathbb{R}^{m})$ and
the space $E^{1,p}(\Omega;\mathbb{R}^{n})$ of $L^{p}$ functions whose distributional symmetric gradient is $p$-integrable.
Moreover, when $p\in[1,\infty)$, for both spaces we show how the corresponding $BMO$-type seminorms can be used to approximate certain positive, convex and $ p $-homogenous functionals closely related to the $ L^{p} $ norms of the distributional (symmetric) gradients.