Singular and quasi-bounded functions associated with the heat equation
We develop a theory of singular and quasi-bounded functions associated with the heat equation. We define quasi-bounded and singular functions by means of an operator S defined on the set of nonnegative functions admitting superthermic majorants. We show that a nonnegative temperature can be expressed as a sum of a quasi-bounded and a singular function. We give explicit formulae for the operator S as the integral of a suitable kernel function with respect to a Borel measure obtained from the Widder, respectively, Kemper representation of nonnegative temperatures. In particular, we show that the singular part of a function arises from the singular part of the representing measure with respect to caloric measure. Also we give some other quasiboundedness criteria and discuss potentials.
scientific article; zbMATH DE number 4027864
scientific article