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

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Acta Mathematica Sinica,English Series 2002年第2期18卷 375-390页

作者： COHN William S. Department of Mathematics.Wayne State University.Detroit.MI 48202 USA

We derive the explicit fundamental solutions for a class of degenerate(or singular)one- parameter subelliptic differential operators on groups of Heisenberg(H)*** extends the result of Kaplan for the sub-Laplacian on H-type groups,which in turn generalizes Folland's result on the Heisenberg *** an application,we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type *** choosing the parameter equal to the homogeneous dimension Q and using the Mose-Trudinger inequality for the convolutional type operator on stratified groups 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 vector fields and with a q-dimensional *** Q=m+2q.Q'=Q-1/Q and Then. with A_Q as 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-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group [18].

关键词： Heisenberg group Groups of Heisenberg type Sobolev inequalities Moser-Trudinger inequalities best constants One-Parameter representation formulas Fundamental solutions

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Global Poincaré Inequalities on the Heisenberg Group and Applications

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Acta Mathematica Sinica,English Series 2007年第4期23卷 735-744页

作者： Yu Xin DONG Guo Zhen LU Li Jing SUN Department of Mathematics Fudan University Shanghai 200433 P. R. China Department of Mathematics Beijing Normal University Beijing 100875 P. R. China Department of Mathematics Wayne State University Detroit MI 48202 USA

Let f be in the localized nonisotropic Sobolev space Wloc^1,p （H^n） on the n-dimensional Heisenberg group H^n = C^n ×R, where 1≤ p ≤ Q and Q = 2n ＋ 2 is the homogeneous dimension of H^n. Suppose that the subelliptic gradient is gloablly L^p integrable, i.e., fH^n ｜△H^n f｜^p du is finite. We prove a Poincaré inequality for f on the entire space H^n. Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of C0^∞（H^n） under the norm of （∫H^n ｜f｜ Qp/Q-p）^Q-p/Qp ＋（∫ H^n ｜△H^n f｜^p）^1/p. We will also prove that the best constants and extremals for such Poincaré inequalities on H^n are the same as those for Sobolev inequalities on H^n. Using the results of Jerison and Lee on the sharp constant and extremals for L^2 to L（2Q/Q-2） Sobolev inequality on the Heisenberg group, we thus arrive at the explicit best constant for the aforementioned Poincaré inequality on H^n when p=2. We also derive the lower bound of the best constants for local Poincaré inequalities over metric balls on the Heisenberg group H^n.

关键词： Heisenberg group Sobolev inequalities Poincaré inequalities best constants

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