Extremal of Log-Sobolev Functionals and Li-Yau Estimate on RCD∗(K,N) Spaces

In this work, we study the extremal functions of the log-Sobolev functional on compact metric measure spaces satisfying the RCD ∗ ( K , N ) condition for K in R and N in ( 2 , ∞ ) . We show the existence, regularity and positivity of non-negative extremal functions. Based on these results, we prove a Li-Yau type estimate for the logarithmic transform of any non-negative extremal functions of the log-Sobolev functional. As applications, we show a Harnack type inequality as well as lower and upper bounds for the non-negative extremal functions.