Hardy-Littlewood优控制问题中的三项幂等反例
Three-term idempotent counterexamples in the Hardy-Littlewood majorant problem
摘要：
Hardy-Littlewood优问题是在30年代提出的，它可以表述为：无论何时$\hat{f}\ge|\hat g|$，是否$\int |f|^p\ge \int|g|^p$。它只对偶数的指数$p$有一个肯定的答案。Montgomery指出，即使在幂等多项式中也必须存在一些反例，即存在一些有限的指数集和一些$\pm$符号，其中有符号的指数和具有比指数和中所有符号都选择+获得的幂等更大的$p^{\rm th}$范数。最近Mockenhaupt和Schlag证明了这个猜想。\comment{Their construction was used by Bonami and Révész to find analogous examples among bivariate idempotents, which were in turn used to show integral concentration properties of univariate idempotents.}然而，一个自然的问题是，即使是经典的$1+e^{2πi x} \pm e^{2πi (k+2)x}$三项指数和，用于$p=3$和$k=1$已经由哈代和利特尔伍德，应该在这方面的工作。这一点还没有得到证明，因为Mockenhaupt和Schlag的构造是用四项幂等元的。我们调查尖锐的问题，并表明，至少在某些情况下确实存在三项幂等反例的Hardy-Littlewood优问题，这是我们为$0<p<6, p \notin 2\NN$ $\int_0^{\frac12}|1+e^{2πix}-e^{2πi([\frac p2]+2)x}|^p > \int_0^{\frac12}|1+e^{2πix}+e^{2πi([\frac p2]+2)x}|^p$。证明结合了精细的微积分与数值积分和精确的误差估计。
英文摘要：
The Hardy-Littlewood majorant problem was raised in the 30's and it can be formulated as the question whether $\int |f|^p\ge \int|g|^p$ whenever $\hat{f}\ge|\hat g|$. It has a positive answer only for exponents $p$ which are even integers. Montgomery conjectured that even among the idempotent polynomials there must exist some counterexamples, i.e. there exists some finite set of exponentials and some $\pm$ signs with which the signed exponential sum has larger $p^{\rm th}$ norm than the idempotent obtained with all the signs chosen + in the exponential sum. That conjecture was proved recently by Mockenhaupt and Schlag. \comment{Their construction was used by Bonami and Révész to find analogous examples among bivariate idempotents, which were in turn used to show integral concentration properties of univariate idempotents.}However, a natural question is if even the classical $1+e^{2πi x} \pm e^{2πi (k+2)x}$ three-term exponential sums, used for $p=3$ and $k=1$ already by Hardy and Littlewood, should work in this respect. That remained unproved, as the construction of Mockenhaupt and Schlag works with four-term idempotents. We investigate the sharpened question and show that at least in certain cases there indeed exist three-term idempotent counterexamples in the Hardy-Littlewood majorant problem; that is we have for $0<p<6, p \notin 2\NN$ $\int_0^{\frac12}|1+e^{2πix}-e^{2πi([\frac p2]+2)x}|^p > \int_0^{\frac12}|1+e^{2πix}+e^{2πi([\frac p2]+2)x}|^p$. The proof combines delicate calculus with numerical integration and precise error estimates.
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