Best Constants for Moser-Trudinger Inequalities,Fundamental Solutions and One-Parameter Representation Formulas on Groups of Heisenberg Type

We derive the explicit fundamental solutions for a class of degenerate(or singular)one-parameter subelliptic differential operators on groups of Heisenberg(H)type.This extends the resultof Kaplan for the sub-Laplacian on H-type groups,which in turn generalizes Folland’s result on theHeisenberg group.As an application,we obtain a one-parameter representation formula for Sobolevfunctions of compact support on H-type groups.By choosing the parameter equal to the homogeneousdimension Q and using the Mose-Trudinger inequality for the convolutional type operator on stratifiedgroups obtained in[18].we get the following theorem which gives the best constant for the Moser-Trudiuger inequality for Sobolev functions on H-type groups.Let G be any group of Heisenberg type whose Lie algebra is generated by m left invariant vectorfields and with a q-dimensional center.Let Q=m+2q.Q’=Q-1/Q andThen.with Aas the sharp constant,where ▽G denotes the subelliptic gradient on G.This continues the research originated in our earlier study of the best constants in Moser-Trudingerinequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group[18].