@article {Cohn:2002:1439-8516:375,
author = "Cohn, William and Lu, Guo",
title = "Best Constants for Moser-Trudinger Inequalities, Fundamental Solutions and One-Parameter Representation Formulas on Groups of Heisenberg Type",
journal = "Acta Mathematica Sinica",
volume = "18",
number = "2",
year = "2002",
abstract = "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 results of Kaplan of the sub-Laplacian on H-type groups, which in turn generalizes Folland's result on the Heisenberg group. As an application, we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups. By choosing the parameter equal to the homogeneous dimension Q and using the Moser-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-Trudinger inequality for Sobolev functions in H-type groups.Let <EquationSource Format="TEX"><![CDATA[$${Bbb G}$$]]></EquationSource> be any group of Heisenberg type whose Lie algebra is g enerated by m left invariant vector fields and with a Q-dimensional center. Let <EquationSource Format="TEX"><![CDATA[$$Q = m + 2q,Qprime = {Q over {Q - 1}}$$]]></EquationSource> and <Equation ID="Equ1"> <MediaObject> </MediaObject><EquationSource Format="TEX"><![CDATA[$$ A_{Q} = Q{left[ {{left( {frac{1} {4}} right)}^{{q - frac{1} {2}}} frac{{pi ^{{frac{{q + m}} {2}}} Gamma {left( {frac{{Q + m}} {4}} right)}}} {{QGamma {left( {frac{m} {2}} right)}Gamma {left( {frac{Q} {2}} right)}}}} right]}^{{frac{1} {{Q - 1}}}} $$]]></EquationSource> </Equation>Then,<Equation ID="Equ2"> <MediaObject> </MediaObject><EquationSource Format="TEX"><![CDATA[$$ {mathop {sup }limits_{F in C^{infty }_{0} {left( Omega right)}} }{left{ {frac{1} {{{left| Omega right|}}}{int_Omega {exp } }{left( {A_{Q} {left( {frac{{F{left( u right)}}} {{{left| {abla _{mathbb{G}} F} right|}_{Q} }}} right)}^{{{Q}ifmmode{'}else$'$fi}} } right)}du} right}} < infty , $$]]></EquationSource> </Equation>with A Q as the sharp constant, where <EquationSource Format="TEX"><![CDATA[$$ abla _{{Bbb G}}$$]]></EquationSource> denotes the subellitpic gradient on <EquationSource Format="TEX"><![CDATA[$${Bbb G}$$]]></EquationSource> 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].",
pages = "375-390",
url = "http://www.ingentaconnect.com/content/klu/10114/2002/00000018/00000002/00000159",
doi = "doi:10.1007/s101140200159",
keyword = "Heisenberg group, Groups of Heisenberg type, Sobolev inequalities, Moser-Trudinger inequalities, Best constants, One-Paramenter representation formulas, Fundamental solutions, 35J15, 46E35, 58E35"
}