A conjugate system and tangential derivative norms on parabolic Bergman spaces

Abstract

The $\alpha$-parabolic Bergman space ${\boldsymbol b}_{\alpha}^{p}(\lambda)$ is the Banach space of solutions of the parabolic equation $L^{(\alpha)} = \partial/\partial t+(-\Delta_{x})^{\alpha}$ on the upper half space $H$ which have finite $L^{p}(H,t^{\lambda}dV)$ norms, where $t^{\lambda}dV$ is the weighted Lebesgue volume measure on $H$. It is known that ${\boldsymbol b}^{p}_{1/2}(\lambda)$ coincide with the harmonic Bergman spaces. In this paper, we introduce the extension of notion of conjugate functions of ${\boldsymbol b}_{\alpha}^{p}(\lambda)$-functions and study their properties. As an application, we give estimates of tangential derivative norms on ${\boldsymbol b}_{\alpha}^{p}(\lambda)$.