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by T Eisner, T Tao

Venue: | J. Anal. Math |

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by
Michael Christ

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...ted in part by NSF grant DMS-0901569 and by the Mathematical Sciences Research Institute. 1 2 MICHAEL CHRIST This conclusion is false if one or more of the three exponents equal 1 or∞. Eisner and Tao =-=[9]-=- have shown that for arbitrary locally compact Abelian groups, if the optimal constant equals 1, then all near-extremizers are close to scalar multiples of indicator functions of cosets of compact ope...

by
Xuancheng Shao

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...oung inequality see [5, 6]. Finally, note that the q = 4 case of the Hausdorff-Young inequality gives a sharp upper bound for the Gowers U2-norm of functions on Rd. In a recent work of Eisner and Tao =-=[8]-=-, this is generalized to sharp upper bounds for Gowers Uk-norm of functions on Rd for k > 2. It is an interesting problem to investigate sharp upper bounds for Gowers Uk-norm of compact sets in Rd wit...

by
Pablo Candela, Olof Sisask

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...) both at most 1 and ‖f − g‖Uk+1(G) ≤ δ, we have |SL,G(f)− SL,G(g)| ≤ ǫ. Here the notation ‖f‖Uk(G) refers to the kth Gowers uniformity norm, which is defined on L∞(G) for any compact abelian group G =-=[3]-=-. Using Theorem 1.2, the main convergence result from [1] can be extended as follows. Theorem 5.2. Let F be a finite family of full-rank integer-matrices of complexity 1, and let dF(Zp) denote the max...

by
Michael Christ

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...en ‖Ff‖LpxLqθ ≥ (1− δ)‖f‖p. Near extremizers of the Hausdorff-Young inequality, for arbitrary discrete Abelian groups, were characterized implicitly by Fournier [12], and explicitly by Eisner and Tao =-=[11]-=-. The following more precise statement was shown in [4], where it was also observed that an equivalent formulation in terms of Young’s convolution inequality extends to all discrete groups, not necess...

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unknown authors

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...imit as follows from the characterization of these seminorms via cube spaces [HK05, §3.5] and the mean ergodic theorem. It follows by induction on l ∈ N that (2.5) ‖ · ‖U l+1(X ) ≤ ‖ · ‖L2l (X ), see =-=[ET12]-=- for subtler analysis. Moreover, if µ = ∫ µxdµ(x) is the ergodic decomposition then ‖ f ‖2 l U l (X ,µ) = ∫ ‖ f ‖2 l U l (X ,µx ) dµ(x) for all f ∈ L∞(µ). If (X ,µ, T ) is ergodic then for each l ther...

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