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In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolationinequalities (CKN), with two radial power law weights and exponents in asubcritical range. We address the question of symmetry breaking: are theoptimal functions radially symmetric, or not ? Our intuition comes from aweighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy- entropy production inequality which governs the intermediate asymptotics isindeed equivalent to (CKN), and the self-similar profiles are optimal for(CKN). We establish an explicit symmetry breaking condition by proving thelinear instability of the radial optimal functions for (CKN). Symmetry breakingin (CKN) also has consequences on entropy - entropy production inequalities andon the intermediate asymptotics for (WFD). Even when no symmetry holds in(CKN), asymptotic rates of convergence of the solutions to (WFD) are determinedby a weighted Hardy-Poincar{\'e} inequality which is interpreted as alinearized entropy - entropy production inequality. All our results rely on thestudy of the bottom of the spectrum of the linearized diffusion operator aroundthe self-similar profiles, which is equivalent to the linearization of (CKN)around the radial optimal functions, and on variational methods. Consequencesfor the (WFD) flow will be studied in Part II of this work.

Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities