Pawel Kolwicz | Searcharxiv

arXiv2018

Symmetrization, factorization and arithmetic of quasi-Banach function spaces

We investigate relations between symmetrizations of quasi-Banach function spaces and constructions such as Calderon-Lozanovskii spaces, pointwise product spaces and pointwise multipliers. We show that under reasonable assumptions the symmetrization commutes with these operations. We determine also the spaces of pointwise multipliers between Lorentz spaces and Cesaro spaces. Developed methods 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.

math.FA↗

arXiv2012

Pointwise multipliers of Calderón-Lozanovskii spaces

Several results concerning multipliers of symmetric Banach function spaces are presented firstly. Then the results on multipliers of Calderón-Lozanovskii spaces are proved. We investigate assumptions on a Banach ideal space E and three Young functions φ_1, φ_2 and φ, generating the corresponding Calderón-Lozanovskii spaces E_{φ_1}, E_{φ_2}, E_φ so that the space of multipliers M(E_{φ_1}, E_φ) of all measurable x such that x,y \in E_φ for any y \in E_{φ_1} can be identified with E_{φ_2}. Sufficient conditions generalize earlier results by Ando, O'Neil, Zabreiko-Rutickii, Maligranda-Persson and Maligranda-Nakai. There are also necessary conditions on functions for the embedding M(E_{φ_1}, E_φ) \subset E_{φ_2} to be true, which already in the case when E = L^1, that is, for Orlicz spaces M(L^{φ_1}, L^φ) \subset L^{φ_2} give a solution of a problem raised in the book [Ma89]. Some properties of a generalized complementary operation on Young functions, defined by Ando, are investigated in order to show how to construct the function φ_2 such that M(E_{φ_1}, E_φ) = E_{φ_2}. There are also several examples of independent interest.

math.FA↗