# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 39 (2010) pp.85-114

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### Abstract

The $\alpha$-parabolic Bergman space $\berg^{p}(\lambda)$ 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 $\berg^{p}(\lambda)$-functions and study their properties. As an application, we give estimates of tangential derivative norms on $\berg^{p}(\lambda)$.

MSC(Primary) 35K05(MSC2000), 26D10(MSC2000), 42A50(MSC2000) conjugate function; tangential derivative; heat equation; parabolic operator of fractional order; Bergman space;