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.