# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 34 (2005) pp.393-404

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### Abstract

We give a formula for the one-parameter strongly continuous semigroup $e^{-tL},\,t>0$, generated by the Hermite operator $L$ on the Heisenberg group $\H1$ in terms of Weyl transforms, and use it to obtain an $L^2$ estimate for the solution of the initial value problem for the heat equation governed by $L$ in terms of the $L^p$ norm of the initial data for $1\leq p\leq \infty.$

MSC(Primary) 35K05 47G30 Hermite functions, Heisenberg groups, Hermite operators, Wigner transforms,Weyl transforms, Hermite semigroups, heat equations, Weyl-Heisenberg groups, localization operators, $L^p - L^2$ estimates