  In a previous work (arXiv:2306.05948), we constructed by convex integration
examples of energy dissipating solutions to the 2D Euler equations on
$\mathbb{R}^2$ with vorticity in the real Hardy space $H^p(\mathbb{R}^2)$. In
the present paper, we develop tools that significantly improve that result in
two ways: Firstly, we achieve vorticities in $H^p(\mathbb{R}^2)$ in the optimal
range $p\in (0,1)$ compared to $(2/3,1)$ in our previous work. Secondly, the
solutions constructed here possess compact support and in particular preserve
linear and angular momenta.
