A new Carleson measure adapted to multi-level ellipsoid covers

We develop highly anisotropic Carleson measure over multi-level ellipsoid covers \begin{document}$ \Theta $\end{document} of \begin{document}$ \mathbb{R}^n $\end{document} that are highly anisotropic in the sense that the ellipsoids can change rapidly from level to level and from point to point. Then we show that the Carleson measure \begin{document}$ \mu $\end{document} is sufficient for which the integral defines a bounded operator from \begin{document}$ H^p(\Theta) $\end{document} to \begin{document}$ L^p(\mathbb{R}^{n+1}, \, \mu),\ 0<p\leq 1 $\end{document} . Finally, we give several equivalent Carleson measures adapted to multi-level ellipsoid covers and obtain a specific Carleson measure induced by the highly anisotropic BMO functions.