A priori bounds for the $\Phi^4$ equation in the full sub-critical regime

We derive a priori bounds for the $\Phi^4$ equation in the full sub-critical regime using Hairer's theory of regularity structures. The equation is formally given by \begin{equation} \label{e}(\partial_t-\Delta)\phi = -\phi^3 + \infty \phi +\xi, \tag{$\star$} \end{equation} where the term $+\infty \phi$ represents infinite terms that have to be removed in a renormalisation procedure. We emulate fractional dimensions $d<4$ by adjusting the regularity of the noise term $\xi$, choosing $\xi \in C^{-3+\delta}$. Our main result states that if $\phi$ satisfies this equation on a space-time cylinder $P= (0,1) \times \{ |x| \leq 1 \}$, then away from the boundary $\partial P$ the solution $\phi$ can be bounded in terms of a finite number of explicit polynomial expressions in $\xi$, and this bound holds uniformly over all possible choices of boundary data for $\phi$. The derivation of this bound makes full use of the super-linear damping effect of the non-linear term $-\phi^3$. A key part of our analysis consists of an appropriate re-formulation of the theory of regularity structures in the specific context of \eqref{e}, which allows to couple the small scale control one obtains from this theory with a suitable large scale argument. Along the way we make several new observations and simplifications. Instead of a model $(\Pi_x)_x$ and the family of translation operators $(\Gamma_{x,y})_{x,y}$ we work with just a single object $(\mathbb{X}_{x, y})$ which acts on itself for translations, very much in the spirit of Gubinelli's theory of branched rough paths. Furthermore, we show that in the specific context of \eqref{e} the hierarchy of continuity conditions which constitute Hairer's definition of a \emph{modelled distribution} can be reduced to the single continuity condition on the "coefficient on the constant level".