$\alpha$-parabolic bergman spaces

Abstract

The $\alpha$-parabolic Bergman space $\bm{b}^p_\alpha$ is the set of all $p$-th integrable solutions $u$ of the equation $(\partial/\partial t + (-\Delta)^{\alpha})u = 0$ on the upper half space, where $0 < \alpha \leq 1$ and $1 \leq p \leq \infty$. The Huygens property for the above $u$ will be obtained. After verifying that the space $\bm{b}^p_\alpha$ forms a Banach space, we discuss the fundamental properties. For example, as for the duality, $(\bm{b}^p_\alpha)^* \cong \bm{b}^q_\alpha$ for $p > 1$ and $(\bm{b}^1_\alpha)^* \cong \mathcal{B}_\alpha/ \mathbf{R}$ are shown, where $q$ is the exponent conjugate to $p$ and $\mathcal{B}_\alpha$ is the $\alpha$-parabolic Bloch space.