Respiratory motion is one of the main sources of motion artifacts in positron emission tomography (PET) imaging. The emission image and patient motion can be estimated simultaneously from respiratory gated data through a joint estimation framework. However, conventional motion estimation methods based on registration of a pair of images are sensitive to noise. The goal of this study is to develop a robust joint estimation method that incorporates a deep learning (DL)-based image registration approach for motion estimation. We propose a joint estimation framework by incorporating a learned image registration network into a regularized PET image reconstruction. The joint estimation was formulated as a constrained optimization problem with moving gated images related to a fixed image via the deep neural network. The constrained optimization problem is solved by the alternating direction method of multipliers (ADMM) algorithm. The effectiveness of the algorithm was demonstrated using simulated and real data. We compared the proposed DL-ADMM joint estimation algorithm with a monotonic iterative joint estimation. Motion compensated reconstructions using pre-calculated deformation fields by DL-based (DL-MC recon) and iterative (iterative-MC recon) image registration were also included for comparison. Our simulation study shows that the proposed DL-ADMM joint estimation method reduces bias compared to the ungated image without increasing noise and outperforms the competing methods. In the real data study, our proposed method also generated higher lesion contrast and sharper liver boundaries compared to the ungated image and had lower noise than the reference gated image.