Quasi - Bounded Functions Associated with the Heat Equation

A theory of singular and quasi-bounded temperatures associated with the heat equation is developed. Guided by the harmonic case [1] an operator S is defined on the set M of non-negative functions on an open set of Rn+l which admit superthermic majorants. Quasi-bounded and singular functions in M are defined from the operator S . The basic properties of S are discussed. Non-negative temperatures can be uniquely decomposed as a sum of a quasi-bounded temperature and a singular 'temperature. Explicit formulae for the operator S are developed in the case of a half space, an unbounded strip, and a bounded Lipschitz domain. These constitute the main results of the work. No corresponding results in the harmonic case, to our knowledge, have been published else",here, although it is now obvious that similar explicit formulae for the operator S in ~che harmonic case can be proved without difficul'ty. The decomposi,tion of a non-negative temperature u in ,the half space as a sum of a singular and a quasi·-bounded temperature corresponds to taking the convolution with the fundamental solution of the singular and 'the absolutely continuous parts, respectively, of the Borel measure \! which, from Widder's Theorem, represen ts u