A generalization of parabolic Riesz and parabolic Bessel potentials

Abstract

Classical parabolic Riesz and parabolic Bessel type potentials are interpreted as negative fractional powers of the differential operators (−△+∂∕∂t) and (I−△+∂∕∂t). Here, △ is the Laplacian and I is the identity operator. We introduce some generalizations of these potentials, namely, we define the family of operators Aβ,𝜃α=(𝜃I+(−△)β∕2+∂∕∂t)−α for 𝜃≥0 and α,β>0, and investigate its behavior in the framework of Lp(ℝn+1)-spaces.