On The Heisenberg Group (p, ¦Õ) - Capacity

In chapter 2, we generalize the definition of capacity on Euclid space to the Heisenberggroup:LetΩbe a domain in the Heisenberg group, let F be a compact subset inΩand letφ(x,ξ) be a continuous function onΩ×Hn which is nonnegative and positive homoge-neous of the first degree with respect toξ.The number inf {integral from n=Ω[φ(x,▽Hu)]pdx, u∈(?)(F,Ω)},is called the (p,φ)-capacity of F relative toΩand denoted by (p,φ)-cap(F,Ω).Here (?)(F,Ω)={u∈C0∞(Ω):u≥1 on F}.And then we get the formula for the p-capacity as an integral over level surfacesaccording the Coarea formula on the Stratified group: p-cap(F,Ω)=(?) {integral from n=0 to 1 dτ/((integral from n=Eτ|▽Hu|p-1 dSHnQ-1)1/p-1)}1-p,whereΩis a domain in the Heisenberg group and F is any compact subset toΩ,p≥1,Q=2n+2.Then we get the lower estates for the p-capacity by using the isoperimetric in-equality on Heisenberg group:for some constant C≥0.In chapter 3, we make some estimates for the p-capacity, especially of the estimatefor the integral containing the p-capacity of the set Nt. We also obtain the Sobolev-typeinequality in Orlicz norm on the Heisenberg group and the conditions for the validityof the multiplicative Sobolev-type inequality.