Basic Estimates of Stability Rate for One-dimensional Diffusions

In the context of one-dimensional diffusions, we present basic estimates (having the same lower and upper bounds with a factor of 4 only) for four Poincar'e-type (or Hardy-type) inequalities. The derivation of two estimates have been open problems for quite some time. The bounds provide exponentially ergodic or decay rates. We refine the bounds and illustrate them with typical examples.