Perturbative renormalization of $\phi_4^4$ theory on the half space $\mathbb{R}^+ \times\mathbb{R}^3$ with flow equations

In this paper, we give a rigorous proof of the renormalizability of the massive $\phi_4^4$ theory on a half-space, using the renormalization group flow equations. We find that five counter-terms are needed to make the theory finite, namely $\phi^2$, $\phi\partial_z\phi$, $\phi\partial_z^2\phi$, $\phi\Delta_x\phi$ and $\phi^4$ for $(z,x)\in\mathbb{R}^+\times\mathbb{R}^3$. The amputated correlation functions are distributions in position space. We consider a suitable class of test functions and prove inductive bounds for the correlation functions folded with these test functions. The bounds are uniform in the cutoff and thus directly lead to renormalizability.