Symmetrization, factorization and arithmetic of quasi-Banach function spaces

We investigate relations between symmetrizations of quasi-Banach function spaces and constructions such as Calderón–Lozanovskiĭ spaces, pointwise product spaces and pointwise multipliers. We show that under reasonable assumptions the symmetrization commutates with these operations. We determine also the spaces of pointwise multipliers between Lorentz spaces and Cesàro spaces. The methods that we develop may be regarded as an arithmetic of quasi-Banach function spaces and proofs of Theorems 3, 4 and 6 give a kind of tutorial for these methods. Finally, the above results will be used in proofs of some factorization results.