Abstract: The Fourier transform on R^n enjoys well-known continuity properties: It maps integrable functions (L^1) to bounded functions, and it extends to a unitary isomorphism on L^2. The problem of continuity gets more complicated in general spaces, namely in weighted L^p spaces and weighted Lorentz spaces.

We will introduce the Lorentz norm based on the decreasing rearrangement (f*) and the maximal function (f**) of a measurable function f. Then we will present Sinnamon's sufficient and necessary conditions for boundedness of Fourier transform between weighted Lorentz spaces. The conditions are stated in terms of level function and averaging operators.

Finally, we will discuss the problem in the case of Fourier series in Lorentz spaces.