Sobolev's logarithmic inequality and proof of the Gaussian maximizer conjecture for the capacity of Gaussian measuring channels

The proof is given that the capacity of the Gaussian approximate measurement of the position (as well as the position-momentum) with an energy constraint is always achieved using Gaussian encoding. The proof is based on the general principles of convex analysis. It is noteworthy that for these basic models, the method reduces the solution of the optimization problem to a generalization of the famous logarithmic Sobolev inequality. We hope that this method should work for other models that go beyond the "threshold condition", in which the upper limit of capacity in the form of the difference between the maximum and minimum output entropies is achievable.