Functional inequalities for time-changed symmetric α-stable processes

We establish sharp functional inequalities for time-changed symmetric α-stable processes on ℝd with d ⩾ 1 and α ∈ (0, 2), which yield explicit criteria for the compactness of the associated semigroups. Furthermore, when the time change is defined via the special function W(x) = (1 + ∣x∣)β with β > α, we obtain optimal Nash-type inequalities, which in turn give us optimal upper bounds for the density function of the associated semigroups.