Interpolation and Uniform Interpolation in Quantifier-Free Fragments of Combined First-Order Theories

Abstract

In this survey, we report our recent work concerning combination results for interpolation and uniform interpolation in the context of quantifier-free fragments of first-order theories. We stress model-theoretic and algebraic aspects connecting this topic with amalgamation, strong amalgamation, and model-completeness. We give sufficient (and, in relevant situations, also necessary) conditions for the transfer of the quantifier-free interpolation property to combined first-order theories; we also investigate the non-disjoint signature case under the assumption that the shared theory is universal Horn. For convex, strong-amalgamating, stably infinite theories over disjoint signatures, we also provide a modular transfer result for the existence of uniform interpolants. Model completions play a key role in the whole paper: They enter into transfer results in the non-disjoint signature case and also represent a semantic counterpart of uniform interpolants.

Kakeya Inequalities by Maximal Functions in Hardy Spaces

“…In 1991, Bourgain in [4] improved this result for each n ≥ 3 to some pðdÞ ∈ ððd + 1Þ/2, ðd + 2Þ/2Þ. By the interpolation theory, (see [6][7][8] and reference therein),…”