  Motivated by the recent characterization of Sobolev spaces due to Brezis-Van
Schaftingen-Yung we prove new weak-type inequalities for one parameter families
of operators connected with mixed norm inequalities. The novelty here comes
from the fact that the underlying measure space incorporates the parameter as a
variable. The connection to classical and fractional order Sobolev spaces is
shown through the use of generalized Riesz potential spaces and the
Caffarelli-Silvestre extension principle. Higher order inequalities are also
considered. We indicate many examples and applications to PDE's and different
areas of Analysis, suggesting a vast potential for future research. In a
different direction, and inspired by methods originally due to Gagliardo and
Garsia, we obtain new maximal inequalities which combined with mixed norm
inequalities are applied to obtain Brezis--Van Schaftingen--Yung type
inequalities in the context of Calder\'{o}n-Campanato spaces. In particular,
Log versions of the Gagliardo-Brezis-Van Schaftingen-Yung spaces are introduced
and \ compared with corresponding limiting versions of Calder\'{o}n-Campanato
spaces, resulting in a sharpening of recent inequalities due to Crippa-De
Lellis and Bru\'{e}-Nguyen.
