Functions of Nearly Maximal Gowers–Host–Kra Norms on Euclidean Spaces

Abstract

Let k ≥ 2, n ≥ 1 be integers. Let f : R n → C. The kth Gowers-Host-Kra norm of f is defined recursively byThese norms were introduced by Gowers [11] in his work on Szemerédi's theorem, and by Host-Kra [13] in ergodic setting. These norms are also discussed extensively in [17]. It's shown by Eisner and Tao in [10] that for every k ≥ 2 there exist A(k, n) < ∞ and p kThe optimal constant A(k, n) and the extremizers for this inequality are known [10]. In this dissertation, it is shown that if the ratio f U k / f p k is nearly maximal, then f is close in L p k norm to an extremizer.Contents 42 References 48

Fourier duality in the Brascamp–Lieb inequality

“…, and certain off-diagonal Lebesgue bounds were later established by Neuman [22]. We note that if p ω ≥ 2 for each ω then the inequality…”

Anti-uniformity norms, anti-uniformity functions and their algebras on Euclidean spaces

“…As the Brascamp-Lieb-type inequalities have a strong connection to geometric functional analysis on Euclidean spaces, see [2], [3], [4], [10], [9], [11], the author hopes that this initial exploration sets up a longer investigation into the world of the anti-uniform spaces in the future and their geometric implications.…”