Some oscillatory integral estimates via real analysis (Q1650184)

The author studies oscillatory integrals \[ I(\lambda):= \int_{\mathbb R^d} e^{i\lambda \phi(x)}\psi(x)dx,\qquad \lambda\in\mathbb R\setminus\{0\}, \] where the phase \(\phi:\mathbb R^d\to \mathbb R\) is real analytic, smooth, or \(C^k\) satisfying a non-degeneracy condition and the amplitude \(\psi\) is smooth or \(C^k\), compactly supported close to the origin. The non-degeneracy condition as well as the estimates in this paper are strongly influenced by the work of \textit{A. N. Varchenko} [Funct. Anal. Appl. 10, 175--196 (1977; Zbl 0351.32011); translation from Funkts. Anal. Prilozh. 10, No. 3, 13--38 (1976)], however the methods are different. In order to give a flavor of these results consider a real analytic \(\phi\) defined in a neighborhood of the origin. The non-degeneracy condition is given through the Newton polyhedron of \(\phi\), which is the convex hull of the union \(\bigcup_\alpha \alpha +\mathbb R^d _+\), the union being taken over all multi-indices \(\alpha \in \mathbb N^n\) corresponding to non-zero Taylor coefficients of \(\phi\). Then \(\phi\) is call analytic non-degenerate if for all compact faces \(F\) of the Newton polyhedron and all \(x\) not contained in any coordinate hyperplane, there exists \(1\leq j\leq d\) such that \[ x_j \partial_{x_j}\Big(\sum_{\alpha\in F}c_\alpha x^\alpha\Big) \neq 0. \] Then if \(\psi\) is a smooth cutoff around the origin the for all multi-indices \(\beta\) one has \[ \Bigg|\int_{\mathbb R^d}e^{-\lambda\phi(x)}x^\beta \psi(x)dx\Bigg|\lesssim \lambda^{-\lfloor\beta+(1,1,\ldots,1)\rfloor}\log^{d_\beta-1}(\lambda); \] In the estimate above \(d_\beta\) is the greatest codimension among all faces of the Newton polyhedron of \(\phi\) containing \((\beta+(1,\ldots,1))/{\lfloor \beta+(1,\ldots,1)\rfloor}\). More general estimates of this type are deduced under suitable assumptions of smooth or \(C^k\)-type for the phase. An important intermediate result used in the proofs of these estimates is the following quantification of boundedness away from zero: if \(\phi\) is analytic non-degenerate, \(\varepsilon\in[0,1]^d\) is sufficiently small, and \(\alpha\) belongs to the Newton polyhedron of \(\phi\) then \[ \sum_{j} |x_j \partial_j \phi(x)|^2 \gtrsim \varepsilon_1 ^{2\alpha_1}\cdots\varepsilon_d ^{2\alpha_d} \qquad \forall x\in [\varepsilon,4\varepsilon]^d. \] In fact the condition above characterizes non-degeneracy for analytic (and more general) phases.