Topological Hochschild homology of truncated Brown-Peterson spectra I

We compute topological Hochschild homology of sufficiently structured forms of truncated Brown--Peterson spectra with coefficients. In particular, we compute $operatorname{THH}_*(operatorname{taf}^D;M)$ for $Min { Hmathbb{Z}_{(3)},k(1),k(2)}$ where $operatorname{taf}^D$ is the $E_{infty}$ form of $BPlangle 2rangle$ constructed by Hill--Lawson. We compute $operatorname{THH}_*(operatorname{tmf}_1(3);M)$ when $Min { Hmathbb{Z}_{(2)},k(2)}$ where $operatorname{tmf}_1(3)$ is the $E_{infty}$ form of $BPlangle 2rangle$ constructed by Lawson--Naumann. We also compute $operatorname{THH}_*(Blangle nrangle;M)$ for $M=Hmathbb{Z}_{(p)}$ and certain $E_3$ forms $Blangle nrangle$ of $BPlangle nrangle$. For example at $p=2$, this result applies to the $E_3$ forms of $BPlangle nrangle$ constructed by Hahn--Wilson.