  We study nonuniform Sobolev spaces, i.e., spaces of functions whose partial
derivatives lie in possibly different Lebesgue spaces. Although standard proofs
do not apply, we show that nonuniform Sobolev spaces share similar properties
as the classical ones. These spaces arise naturally in the study of certain
PDEs. For instance, we illustrate that nonuniform fractional Sobolev spaces are
useful in the study of local estimates for solutions of heat equations and the
convergence of Schr\"odinger operators. In this work we extend recent advances
on local energy estimates for solutions of heat equations and the convergence
of Schr\"odinger operators to nonuniform fractional Sobolev spaces.
