Traces of functions with majorizable derivatives

We study the limiting values ($y\to+0$) of functions $f(x,y)$: $x\in R_n$, $y>0$, for which $\left|{\partial f}/{\partial y}\right|\leqslant M\varphi(y)$; $\left|{\partial f}/{\partial x_k}\right|\leqslant M\psi_k(y)$, $M=M[f]$, in the case of arbitrary weight functions. It is shown that the space of traces can be described as the set of all functions $f(x,0)$ which satisfy a Lipschitz condition in some metric $\omega(x,\tilde{x})$ associated with the weights.