Weyl transforms, the heat kernel and Green function of a degenerate elliptic operator
Let \(Z\) and \(\overline Z\) be the linear partial differential operators on \(\mathbb R^2\) given by \(\overline Z=(\partial/\partial z)+ (1/2)\overline z\) and \(\overline Z=(\partial/\partial{\overline z})-(1/2)z\), \(z=x+iy\), \((x,y)\in\mathbb R^2\). Let \(L=-(1/2)(Z\overline Z+\overline Z Z)\); this operator is closely related to the sub-Laplacian on the Heisenberg group; moreover, it is elliptic but neither globally elliptic nor Shubin globally hypoelliptic [see \textit{M.~A.\ Shubin}, ``Pseudodifferential operators and spectral theory'' (Nauka, Moscow) (1978; Zbl 0451.47064); English translation: (Springer-Verlag, Berlin--New York) (1987; Zbl 0616.47040)]. The author derives a formula for the heat kernel of \(L\) in terms of Weyl transforms of functions in the Schwartz space \({\mathcal S}(\mathbb R)\) and the Fourier--Wigner transforms of Hermite functions, which leads to a formula for the Green function of~\(L\) (the integral kernel of~\(L^{-1}\)) and helps to show that, if \(u\) is a tempered distribution on \(\mathbb R^2\) such that \(Lu\) is in \({\mathcal S}(\mathbb R^2)\), then \(u\in{\mathcal S}(\mathbb R^2)\); the author refers to this very property as the global hypoellipticity. The author also proves that every element of the strongly continuous one-parameter semigroup generated by \(-L\) defines a bounded operator from \(L^2(\mathbb R^2)\) to \(L^\infty(\mathbb R^2)\) (this property of the semigroup is referred to as ultracontractivity) and a bounded operator from \(L^p(\mathbb R^2)\) to \(L^q(\mathbb R^2)\) for \(1\leq p\leq2\) and \(2\leq q\leq\infty\) (which implies the hypercontractivity of the semigroup). The study of the operator \(L\) seems to be of interest from the point of view of representation theory of the Heisenberg group.
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