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.

Algebras with polynomial identities generalize commutative and finite dimensional algebras. This generalization is not only formal. PI-algebras share many structural properties with commutative and finite dimensional algebras. We first recall the notion of a PI-algebra and give some key examples. We then expand the class of PI-algebras to include algebras with certain permutational and rewritable properties. After discussing the Kurosh's problems in ring theory, we present a quantitative version of Shirshov's notion of height which leads to purely combinatorial proofs of the Kurosh's problems for PI-algebras. This method also yields a quantitative proof of Berele's theorem that the Gelfand-Kirillov dimension of the finitely generated PI-algebras is finite. On the other hand, this method works with algebras only having the permutational property, so Kurosh's problems and Berele's theorem have positive solutions for these algebras.