  We introduce sparse versions of function spaces that are relevant to
characterize the solutions of Euler equations without concentration. The
standard Sobolev space $H^{-1}$ is given a sparse structure that allows to
measure the degree of compactness of embeddings into $H^{-1}$ and provides new
quantitative general criteria for $H^{-1}$-stability. Indices of sparseness are
defined, and function spaces whose indices have prescribed decay are
constructed, resulting in an improvement of the classical $H^{-1}$-stability
results: sparse stability. The analysis relies on the introduction of sparse
Riesz-Morrey-Tadmor spaces, that are characterized via maximal operators and
new sparse domination theorems, together with extrapolation techniques. Our
methods also yield improvements on recent results on the conservation of energy
of physically realizable solutions of $2$D-Euler.
