Optimal Hypercontractivity and Log-Sobolev inequalities on Cyclic Groups Z_{m·2k}

Title: Optimal Hypercontractivity and Log-Sobolev inequalities on Cyclic Groups Z_{m·2^k}

Speaker: Gan Yao（Harbin Institute of Technology）

Time: 12.17 (Wednesday), 16:00-17:00

Venue: Room 315, Gewu Building

Zoom ID: 976 1746 7912         (Password:  971230)

Link:  https://zoom.us/j/97617467912?pwd=lzTsNI4eE3SYRaAzusNAxfQd3vabHf.1

Abstract:For 1≤p≤q<∞ and n in {3·2^k, 2^k} with k≥1, we prove that the Poisson-like semigroup P_t on Z_n, associated with the word length ψ_n(k)=min(k,n-k), is hypercontractive from L_p to L_q if and only if t≥(1/2)log((q-1)/(p-1)). We establish sharp Log-Sobolev inequalities with the optimal constant 2, by performing a KKT analysis, and lifting from the base cases Z₆ and Z₄ via a Cooley-Tukey n to 2n comparison of Dirichlet forms. The general case for arbitrary n remains open.