Functional inequalities for time-changed symmetric α-stable processes

We establish sharp functional inequalities for time-changed symmetric alpha-stable processes on Double-struck capital R-d with d > 1 and alpha is an element of (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 + divide x divide )(beta) with beta > alpha, we obtain optimal Nash-type inequalities, which in turn give us optimal upper bounds for the density function of the associated semigroups.