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

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Consider a monomial curve γ : R Ñ R d and a family of truncated Hilbert transforms along γ, H γ . This paper addresses the possibility of the pointwise sparse domination of the r-variation of H γ -namely, whether the following is true:where f is a nonnegative measurable function, r ą 2 and Sf pxq " ř QPQ xf yQ,pχQpxq for some p and some sparse collection Q depending on f, p.It's these properties that enables one to have a Calderon-Zygmund decomposition using these dyadic γ-cubes.Dilation with cubes. If Q Ă R d is a cube with side-lengths lpQq " pl 1 , ¨¨¨, l d q, then λQ is another cube with the same center as Q and side-lengths lpλQq " pλl 1 , ¨¨¨, λl d q. One should be careful that if Q is a γ-cube or a dyadic γ-cube then λQ might not be a γ-cube nor a dyadic γ-cube. Monotonic functionsOrder vectors in R d in the following manner, x ě y if x i ě y i . This is not a complete ordering. Consider functions f : R d Ñ R such that exactly one of the following holds, f p xq ě f p yq whenever x ě y, f p xq ď f p yq whenever x ě y.