Truncated smooth function spaces

We introduce truncated Besov and Triebel–Lizorkin function spaces and study their main properties: embeddings, interpolation, duality, lifting, traces. These new scales allow us to improve several known results in functional analysis and PDE’s. In particular, we obtain a full solution to the trace/extension problem in the critical case as well as sharp Sobolev-type embeddings with critical smoothness parameters.