Carleson type measures on parabolic Bergman spaces

Abstract

Let $b_{\alpha }^p$,, be the parabolic Bergman space, the Banach space of solutions of parabolic equations $(\partial /\partial t+(-\mathrm{\Delta }{)}^{\alpha })u=0$ on the upper half space $R_{+}^{n+1}$ which have finite $L^p$ norms. We study Carleson type measures on $b_{\alpha }^p$, and give a necessary and sufficient condition for a measure $\mu$ on $R_{+}^{n+1}$ to be of Carleson type on $b_{\alpha }^p$. As an application, we characterize bounded Toeplitz operators in the space $b_{\alpha }^2$.