Suppose I have a domain $\Omega$ such that the trace operator $T^p:W^{1,p}(\Omega)\rightarrow L^p(\partial \Omega)$. Now suppose I have a function $u\in W^{1,q}(\Omega)\cap W^{1,p}(\Omega)$.
Will we have that $T^p(u)=T^q(u)$?
That is, are the different trace operators equal in the intersection of the Sobolev spaces?