The Semigroup and the Inverse of the Laplacian on the Heisenberg Group

• Main Authors: APARAJITA DASGUPTA, M.W WONG
• Format: Article
• Language: English
• Published: Universidad de La Frontera 2010-01-01
• Series: Cubo
• Subjects: Heisenberg group; Laplacian; parametrized partial differential operators; Hermite functions; Fourier-Wigner transforms; heat equation; one parameter semigroup; inverse of Laplacian; Sobolev spaces
• Online Access: http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462010000300006
• ISSN: 0716-7776; 0719-0646

By decomposing the Laplacian on the Heisenberg group into a family of parametrized partial differential operators Lt ,t &#8712; R {0}, and using parametrized Fourier-Wigner transforms, we give formulas and estimates for the strongly continuous one-parameter semigroup generated by Lt, and the inverse of Lt . Using these formulas and estimates, we obtain Sobolev estimates for the one-parameter semigroup and the inverse of the Laplacian.<br>Mediante descomposición del Laplaceano sobre el grupo de Heisenberg en una familia de operadores diferenciales parciales parametrizados Lt, t &#8712; R {0}, y usando transformada de Fourier-Wigner parametrizada, damos fórmulas y estimativas para la continuidad fuerte del semigrupo generado por Lt, y la inversa de Lt. Usando esas fórmulas y estimativas obtenemos estimativas de Sobolev para el semigrupo a un parámetro y la inversa del Laplaceano.