Estimates of Fourier transforms in Sobolev spaces

We investigate the Fourier transforms of functions in the Sobolev spaces $W_1^{r_1,..., r_n}$. It is proved that for any function $f ∈ W_1^{r_1,...,r_n}$ the Fourier transform f̂ belongs to the Lorentz space $L^{n/r,1}$, where $r = n(∑_{j=1}^n 1/r_{j})^{-1} ≤ n$. Furthermore, we derive from this result that for any mixed derivative $D^{s}f (f ∈ C_0^∞, s=(s_1,... ,s_n))$ the weighted norm $∥(D^{s}f)^∧∥_{L^1(w)} (w(ξ) = |ξ|^{-n})$ can be estimated by the sum of $L^1$-norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.