Observable set, observability, interpolation and spectral inequality for heat equation

In this talk, we introduce three kinds of inequalities on the heat equation in the whole space. They are observable inequality, the Holder-type interpolation inequality and the spectral inequality. We present the geometric characteristic of observable sets. In more detail, we show that a measurable set is observable if and only if it is equally distributed. We also show that the above-mentioned three inequalities are equivalent.