A three term sublevel set inequality

Abstract

Let $B$ be a ball in ${\mathbb R}^2$. For $j=1,2,3$ let $\varphi_j:B\to{\mathbb R}^1$ 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 $S = S(f_1,f_2,f_3,\varepsilon) = \{x\in B: |\sum_{j=1}^3 a_j(x)(f_j\circ\varphi_j)(x)|<\varepsilon\}$. Let $S' = \{x\in S: \max_j|f_j\circ\varphi_j(x)|\ge 1\}$. Our main result is an upper bound, under certain hypotheses on the data $\varphi_j,a_j$ for the Lebesgue measure of $S'$ of the form $|S'| \le c\varepsilon^\gamma$ for some constants $c,\gamma>0$ that depend on the data $a_j,\varphi_j$ 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 $\sum_j a_j(x)(f_j\circ\varphi_j)(x)\equiv 0$ is the trivial solution $f_k=0\ \forall\,k$. Certain auxiliary hypotheses, which hold for generic $\varphi_j,a_j$, 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 $\varphi_j$ are linear.