Heat Kernel Asymptotics for the Hypoelliptic Robin Problem

This chapter is devoted to the proof of Theorem 1.8 (see Theorem 25.11). More precisely, inspired by formula ( 24.1 ) for the compensating operator \(W_{G}(t;a.b)\) we introduce the compensating operator \(W_{S}(t,a,b)\) for the hypoelliptic Robin problem by formula (25.1). Theorem 25.11 applies to the regular Robin–Dirichlet case and we obtain the short time trace formula (25.48) for the analytic semigroup operator \(e^{t\,\mathfrak {A}_{p}}\) (Corollary 25.12). Similarly, Theorem 25.11 applies to the Dirichlet–regular Robin case and we obtain the short time trace formula (25.49) for the analytic semigroup \(e^{t\,\mathfrak {A}_{p}}\) (Corollary 25.14). This chapter ends with a list of hypoelliptic Robin heat kernels \(W_{S}(t;a,b)\) and \(V_{S}(t;a,b)\) in Sect. 25.6.