Function spaces of parabolic Bloch type

Abstract

The $L^{(\alpha)}$-harmonic function is the solution of the parabolic operator $L^{(\alpha)}= \partial_{t}+(-\Delta_{x})^{\alpha}$. We study a function space $\widetilde{{\cal B}}_{\alpha}(\sigma)$ consisting of $L^{(\alpha)}$-harmonic functions of parabolic Bloch type. In particular, we give a reproducing formula for functions in $\widetilde{{\cal B}}_{\alpha}(\sigma)$. Furthermore, we study the fractional calculus on $\widetilde{{\cal B}}_{\alpha}(\sigma)$. As an application, we also give a reproducing formula with fractional orders for functions in $\widetilde{{\cal B}}_{\alpha}(\sigma)$. Moreover, we investigate the dual and pre-dual spaces of function spaces of parabolic Bloch type.