  We characterize one-sided weighted Sobolev spaces
$W^{1,p}(\mathbb{R},\omega)$, where $\omega$ is a one-sided Sawyer weight, in
terms of a.e.~and weighted $L^p$ limits as $\alpha\to1^-$ of Marchaud
fractional derivatives of order $\alpha$. Similar results for weighted Sobolev
spaces $W^{2,p}(\mathbb{R}^n,\nu)$, where $\nu$ is an $A_p$-Muckenhoupt weight,
are proved in terms of limits as $s\to1^-$ of fractional Laplacians
$(-\Delta)^s$. These are Bourgain--Brezis--Mironescu-type characterizations for
weighted Sobolev spaces. We also complement their work by studying a.e.~and
weighted $L^p$ limits as $\alpha,s\to0^+$.
