For functions on a metric measure space, we introduce a notion of
``gradient at a given scale''. This allows us to define Sobolev
inequalities at a given scale. We prove that satisfying a Sobolev
inequality at a large enough scale is invariant under
large-scale equivalence, a metric-measure version of coarse
equivalence. We prove that for a Riemmanian manifold satisfying a
local Poincaré inequality, our notion of Sobolev inequalities at
large scale is equivalent to its classical version. These notions
provide a natural and efficient point of view to study the
relations between the large time on-diagonal behavior of random
walks and the isoperimetry of the space. Specializing our main
result to locally compact groups, we obtain that the
$L^p$-isoperimetric profile, for every $1\leq p\leq \infty$ is
invariant under quasi-isometry between amenable unimodular
compactly generated locally compact groups. A qualitative
application of this new approach is a very general
characterization of the existence of a spectral gap on a
quasi-transitive measure space $X$, providing a natural point of
view to understand this phenomenon.