  The purpose of this work is to establish a quantitative and constructive
stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev
inequalities which interpolates between the logarithmic Sobolev inequality and
the standard Sobolev inequality (in dimension larger than three), or Onofri's
inequality in dimension two. We develop a new strategy, in which the flow of
the fast diffusion equation is used as a tool: a stability result in the
inequality is equivalent to an improved rate of convergence to equilibrium for
the flow. The regularity properties of the parabolic flow allow us to connect
an improved entropy - entropy production inequality during an initial time
layer to spectral properties of a suitable linearized problem which is relevant
for the asymptotic time layer. Altogether, the stability in the inequalities is
measured by a deficit which controls in strong norms (a Fisher information
which can be interpreted as a generalized Heisenberg uncertainty principle) the
distance to the manifold of optimal functions. The method is constructive and,
for the first time, quantitative estimates of the stability constant are
obtained, including in the critical case of Sobolev's inequality. To build the
estimates, we establish a quantitative global Harnack principle and perform a
detailed analysis of large time asymptotics by entropy methods.
