  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 $2$.
