Holder Estimates for Euclidean Models

This chapter presents the Hölder space estimates for Euclidean model problems. It first considers the homogeneous Cauchy problem and the inhomogeneous problem before defining the resolvent operator as the Laplace transform of the heat kernel. It then describes the 1-dimensional kernel estimates that form essential components of the proofs of the Hölder estimates for the general model problems; these include basic kernel estimates, first derivative estimates, and second derivative estimates. The proofs of these estimates are elementary. The chapter concludes by proving estimates on the resolvent and investigating the off-diagonal behavior of the heat kernel in many variables.