Distribution Kernel of Analytic Semigroups in the Hypoelliptic Case

In this chapter, by applying the Schwartz kernel theorem to the analytic semigroup \(e^{t\,\mathfrak {A}_{p}}\) of time-depending families of operators we can prove a functional analytic version of Chisato Iwasaki (Osaka J. Math. 31:663–728, 1994, Theorem I) (see Theorems 13.1 and 13.2). Moreover, we can obtain the explicit representation formula for the analytic semigroup \(e^{t\,\mathfrak {A}_{p}}\) , which is a generalization of Greiner (Arch. Ration. Mech. Anal. 41:163–218, 1971, Theorem 2.5.2) to the hypoelliptic case (Theorem 13.3). By virtue of Theorem 13.3, we can apply Theorem 4.8 to obtain the heat trace theorem, which is a generalization of Greiner (Arch. Ration. Mech. Anal. 41:163–218, 1971, Theorem 2.6.1) to the hypoelliptic case (Theorem 13.5). Table 13.1 gives a bird’s-eye view of analytic semigroups, resolvents, Radon measures and heat Green kernels.