The Fourier Transform on Rearrangement-Invariant Spaces

Let rho be a rearrangement-invariant (r.i.) norm on the set M(R-n) of Lebesgue-measurable functions on Rn such that the space L-rho(R-n) = {f is an element of M(R-n) : rho(f) < infinity} is an interpolation space between L-2(R-n) and L-infinity(R-n). The principal result of this paper asserts that given such a rho, the inequality rho(f(<^>)) <= C sigma (f) holds for any r.i. norm sigma on M(R-n) if and only if rho<overline>(U f(& lowast;)) <= C sigma(<overline>)(f(& lowast;)). Here, rho<overline> is the unique r.i. norm on M(R+), R+ = (0, infinity), satisfying rho<overline>(f(& lowast;)) = rho(f) and U f(& lowast; )(t) = integral(1/t )(0)f(& lowast;), in which f(& lowast;) is the nonincreasing rearrangement of f on R+. Further, in this case the smallest r.i. norm sigma for which rho(f(<^>)) <= C sigma (f) holds is given by sigma(f) = sigma<overline>(f(& lowast;)) = rho<overline>(U f(& lowast;)), where, necessarily, rho<overline>(integral(1/t )(0)chi((0,a))) = rho(<overline>)(min{1/t, a}) < infinity, for all a > 0. We further specialize and expand these results in the contexts of Orlicz and Lorentz Gamma spaces.