论文标题
三个任期的超级套件不等式
A three term sublevel set inequality
论文作者
论文摘要
令$ b $为$ {\ mathbb r}^2 $的球。对于$ j = 1,2,3 $，让$φ_j：b \ to {\ mathbb r}^1 $为真实的分析订阅，让$ a_j $为真实的分析系数函数。到任何$ \ varepsilon> 0 $和任何lebesgue可测量的功能$ f_j：{\ mathbb r}^1 \ 1 \ to {\ mathbb c} $关联sublevel set $ s = s = s（f_1，f_2，f_2，f_2，f_3，f_3，f_3，\ varepsilon） a_j（x）（f_j \ crocroum_j）（x）| <\ varepsilon \} $。令$ s'= \ {x \ in s：\ max_j | f_j \circφ_j（x）| \ ge 1 \} $。我们的主要结果是在数据$φ_j上的某些假设下，a_j $的上限，用于$ | s'|对于某些常数$ c，γ> 0 $的\ le c \ varepsilon^γ$，取决于数据$ a_j，φ_j$，而不是函数$ f_j $或parameter $ \ varepsilon $。主要假设是，在任何连接的$ b $的开放子集中，$ \ sum_j a_j（x）（f_j \ circc或crocroum_j）（x）（x）\ equiv 0 $的唯一真实的分析解决方案$（f_1，f_2，f_3）$是微不足道的解决方案。还强加了某些适用于通用$φ_J的辅助假设。以前知道所有系数$ a_j $的情况。 该结果是在同伴论文中分析的主要成分，该分析具有与四个因素$ f_j $的隐式振荡积分。 还讨论了某些相关的结果。特别是，对于所有映射$φ_J$是线性的特殊情况，都获得了任意许多汇总的概括。
Let $B$ be a ball in ${\mathbb R}^2$. For $j=1,2,3$ let $φ_j:B\to{\mathbb R}^1$ be real analytic submersions, and let $a_j$ be real analytic coefficient functions. To any $\varepsilon>0$ and any Lebesgue measurable functions $f_j:{\mathbb R}^1\to {\mathbb C}$ associate the sublevel set $S = S(f_1,f_2,f_3,\varepsilon) = \{x\in B: |\sum_{j=1}^3 a_j(x)(f_j\circφ_j)(x)|<\varepsilon\}$. Let $S' = \{x\in S: \max_j|f_j\circφ_j(x)|\ge 1\}$. Our main result is an upper bound, under certain hypotheses on the data $φ_j,a_j$ for the Lebesgue measure of $S'$ of the form $|S'| \le c\varepsilon^γ$ for some constants $c,γ>0$ that depend on the data $a_j,φ_j$ but not on the functions $f_j$ or parameter $\varepsilon$. The main hypothesis is that in any connected open subset of $B$, the only real analytic solution $(f_1,f_2,f_3)$ of $\sum_j a_j(x)(f_j\circφ_j)(x)\equiv 0$ is the trivial solution $f_k=0\ \forall\,k$. Certain auxiliary hypotheses, which hold for generic $φ_j,a_j$, are also imposed. The case in which all coefficients $a_j$ are constant was previously known. This result is a principal ingredient in an analysis, in a companion paper, of related implicitly oscillatory integrals with four factors $f_j$. Certain related results are also discussed. In particular, a generalization to arbitrarily many summands f_j is obtained for the special case in which all mappings $φ_j$ are linear.
