Mgr. Martin Křepela, Ph.D.

We characterize the rearrangement -invariant hull, with respect to a given measure mu , of weighted Lebesgue spaces. The solution leads us to first consider when this space is contained in the sum of ( L 1 + L infinity )( R, mu ) and the final condition is given in terms of embeddings for weighted Lorentz spaces. (c) 2024 Elsevier Inc. All rights reserved.

An equivalent expression of Orlicz modulars in terms of measure of level sets of difference quotients is established. The result in a sense complements the famous Maz'ya-Shaposhnikova formula for the fractional Gagliardo-Slobodeckij seminorm and its recent extension to the setting of Orlicz functions.

We improve the discretization technique for weighted Lorentz norms by eliminating all "non-degeneracy" restrictions on the involved weights. We use the new method to provide equivalent estimates on the optimal constant C such that the inequality

Based on the Gale-Ryser theorem [2, 6], for the existence of suitable (0, 1)-matrices for different partitions of a natural number, we revisit the classical result of Lorentz [4] regarding the characterization of a plane measurable set, in terms of its cross-sections, and extend it to general measure spaces.