The heat equation for the Hermite operator on the Heisenberg group

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 H 1 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 ≤ p ≤ ∞.

Weyl Transforms, the Heat Kernel and Green Function of a Degenerate Elliptic Operator

“…Ultracontractivity and hypercontractivity of the strongly continuous semigroup generated by L are proved. These results supplement the L 2 estimates of solutions of the initial value problem for the heat equation governed by L in terms of the L p norms of the initial data, 1 p 2, studied in the paper [15] by Wong. The results in this paper are valid for the operator L on R 2n given by…”

“…In the paper [15] by Wong, a formula for the strongly continuous one-parameter semigroup generated by L in terms of Weyl transforms is given. The kernel of the integral operator e −t L , t > 0, is known as the heat kernel, which is derived in this section.…”

“…In order to bring out the significance of the ultracontractivity as given by Corollary 7.2, we need the following theorem, which is a special case of Theorem 6.2 in the paper [15] by Wong. Remark 7.5.…”

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

“…While the semigroup and the inverse can be studied in the framework of functional analysis as explained in [3,4,5,8,9,16], the results and methods in this paper are based on explicit formulas in hard analysis and are related to the works in [1,2,6,7,10,12,14,15].…”

The Heat Equation for the Generalized Hermite and the Generalized Landau Operators

“…Following Wong's point of view (see [6], by Wong), we give a formula for the one-parameter strongly continuous semigroup e −tL λ , t > 0, generated by the generalized Hermite operator L λ , for a fixed λ ∈ R \ {0}, in terms of the Weyl transforms. Then we use it to obtain an L 2 estimate for the solution of the initial value problem for the heat equation governed by L λ , in terms of the L p norm, 1 ≤ p ≤ ∞, of the initial data.…”