A study of $CP$ violation in $B^\mp \to Dh^\mp$ ($h=K,\pi$) with the modes $D \to K^\mp \pi^\pm \pi^0$, $D \to \pi^+\pi^-\pi^0$ and $D \to K^+K^-\pi^0$

One of the objectives of the LHCb experiment at CERN is a precision determination of the angle $\gamma$ from the CKM Unitarity Triangle, a $CP$-violating parameter of the Standard Model. Sensitivity to $\gamma$ can be obtained by exploiting the interference effects in $B^\mp \to Dh^\mp$ ($h=K,\pi$) decays, which can lead to measurable charge asymmetries. This thesis presents an analysis of the decays $B^\mp \to DK^\mp$ and $B^\mp \to D\pi^\mp$ where the $D$ meson is reconstructed in the three-body final states of $K^\mp \pi^\pm \pi^0$, $\pi^+ \pi^- \pi^0$ and $K^+ K^- \pi^0$. The analysis is performed using 3.0 fb$^{-1}$ of $pp$ collision data collected by LHCb in 2011 and 2012. It represents the first time that these particular decay modes are studied at the LHCb experiment. Several $CP$ observables are measured using a simultaneous fit to the invariant $B^\mp$ mass distributions of the decay channels of interest. A first observation is obtained of the decays $B^\mp \to [\pi^\mp K^\pm \pi^0]_D \pi^\mp$ and $B^\mp \to [K^+ K^- \pi^0]_D \pi^\mp$ at $5.3\sigma$ and greater than $10\sigma$ significance levels, respectively. In addition, first evidence of the decay $B^\mp \to [K^+ K^- \pi^0]_D K^\mp$ is obtained at a significance of $4.5\sigma$. The results are interpreted in the context of $\gamma$ and related parameters