Elliptic theta function and the best constants of Sobolev-type inequalities

摘要

We obtained the best constants of Sobolev-type inequalities corresponding to higher-order partial differential operators $L=(partial_t-Delta+a_0)cdots(partial_t-Delta+a_{M-1})$ and $L_0=(-Delta+a_0)cdots(-Delta+a_{M-1})$ with positive distinct characteristic roots $a_0,dots,a_{M-1}$, under the suitable assumption on $M$ and $n$. The best constants are given by $L^2$-norm of Green's functions of the boundary value problem $Lu=f(x,t)$ and $L_0 u=f(x)$. The Green's functions are expressed by the elliptic theta function.