Gorenstein duality for Real spectra

Following Hu and Kriz, we study the $C_2$-spectra $BP$\mathbb{R}$\langle n \rangle$and$E$\mathbb{R}$(n)$ that refine the usual truncated Brown-Peterson and the Johnson-Wilson spectra. In particular, we show that they satisfy Gorenstein duality with a representation grading shift and identify their Anderson duals. We also compute the associated local cohomology spectral sequence in the cases $n=1$ and $2$.