Generalized vector potential and Trace Theorem for Lipschitz domains

The vector potential is a fundamental concept widely applied across various fields. This paper presents an existence theorem of a vector potential for divergence-free functions in Wm,p(RN,T) with general $m,p,N$. Based on this theorem, one can establish the space decomposition theorem for functions in Wm,p₀(curl;Ω,RN) and the trace theorem for functions in Wm,p(Ω) within the Lipschitz domain Ω ⊂ RN. The methods of proof employed in this paper are straightforward, natural, and consistent.