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 $\left(-△+\partial ∕\partial t\right)$ and $\left(I-△+\partial ∕\partial t\right)$. Here, $△$ is the Laplacian and $I$ is the identity operator. We introduce some generalizations of these potentials, namely, we define the family of operators $A_{\beta ,𝜃}^{\alpha }={\left(𝜃I+{\left(-△\right)}^{\beta ∕2}+\partial ∕\partial t\right)}^{-\alpha }$ for $𝜃\ge 0$ and $\alpha ,\beta >0$, and investigate its behavior in the framework of $L_p\left({ℝ}^{n+1}\right)$-spaces.