Sobolev Inequality On The Heisenberg Group And Isoperimetric Inequality,

In this paper, we study the equivalence of the Gagliardo-Nirenberg-Sobolev inequality and the isoperimetric inequality on the Heisenberg group H", also give the proof of the equivalence. We prove that the C-C balls are not isoperimetric sets on the one-dimensional Heisenberg group H1 .Under the hypothesis that there were isoperimetric sets in the class A, we obtain the best constant in the isoperimetric inequality.