Towards topological Hochschild homology of Johnson–Wilson spectra

We offer a complete description of $THH(E(2))$ under the assumption that the Johnson-Wilson spectrum $E(2)$ at a chosen odd prime carries an $E_\infty$-structure. We also place $THH(E(2))$ in a cofiber sequence $E(2) \rightarrow THH(E(2))\rightarrow \overline{THH}(E(2))$ and describe $\overline{THH}(E(2))$ under the assumption that $E(2)$ is an $E_3$-ring spectrum. We state general results about the $K(i)$-local behaviour of $THH(E(n))$ for all $n$ and $0 \leq i \leq n$. In particular, we compute $K(i)_*THH(E(n))$.