ESTIMATES OF FOURIER-TRANSFORMS IN SOBOLEV SPACES

We investigate the Fourier transforms of functions in the Sobolev spac es W-1(r1,...,rn). It is proved that for any function f is an element of W-1(r1,...,rn) the Fourier transform (f) over cap belongs to the Lo rentz space L-n/r,L-1, where r = n(Sigma(j=1)(n) 1/r(j))(-1) less than or equal to n. Furthermore, we derive from this result that for any m ixed derivative D-s f (f is an element of C-0(infinity), s = (s(1),... ,s(n))) the weighted norm parallel to(D-s f)(boolean AND)parallel to(L 1(omega))(omega) (omega(xi) = \xi\(-n)) can be estimated by the sum of L-1-norms of all pure derivatives of the same order. This gives an an swer to a question posed by A. Pelczynski and M. Wojciechowski.