Spectrum of Small Sets and Dubovitskii ̌–Federer Theorems with Unconditionally Saturated Banach Spaces

Abstract

The Mordell exponential sum estimates and sets of large trigonometric sums are presented. The exposition of Bourgain 2-source extractor and subspace of the Bourgain-Dellaen space are given. We study the generalized N-property and Morse-Sard theorem for the sharp case of Sobolev mappings and the trace theorem with Luzin N and Morse-Sard properties for the sharp case of Sobolev-Lorentz mappings. We also study Dubovitskiı ̌-Sard and Dubovitskiı ̌–Federer theorems in Sobolev spaces and the coarea formula. The operators in tight by support Banach spaces and an additive combinatorics approach relating rank to communication complexity with the structure of the spectrum of small sets and the uniform structure of the separable essential Lebesgue spaces are introduced. The Hereditarily indecomposable essential Lebesgue spaces and unconditionally saturated Banach space and that solves the scalar–plus–compact problem and property are discissed.