The CR Nirenberg Problem, Part I: Blow-up Analysis and Compactness

We establish a priori estimates and compactness results for positive solutions of the CR Nirenberg problem under natural weighted flatness conditions. The proofs combine refined blow-up analysis, Pohozaev identities, and the complete classification of positive entire solutions to the critical CR Yamabe equation recently obtained by Li–Liu–Ma \cite{LLM2026} and Li \cite{L2026}. Existence and degree results are established in our subsequent paper \cite{QTWPartII}.