ABSTRACT.
To each finite subset $B\subset \mathbb{R}^d$, we assign a region $D_B\subset \mathbb{R}^d$ defined by \begin{align*} D_B:=\{x\in\mathbb{R}^d: |x^{\mathfrak{b}}|\le 1\ \text{for $\mathfrak{b}\in B$}\}. \end{align*} For example, $D_{\{{\bf 0}\}}=\mathbb{R}^d$ and $D_{\{e_1\}}=[-1,1]\times \mathbb{R}^{d-1}$. Consider a real valued polynomial \begin{align*} P(x)=\sum_{\mathfrak{m}\in\Lambda(P)}c_{\mathfrak{m}}x^{\mathfrak{m}}\ \text{where $x\in\mathbb{R}^d$ and $\Lambda(P)=\{\mathfrak{m}\in\mathbb{Z}^d_+:c_{\mathfrak{m}}\ne 0\}$.} \end{align*} For, $\lambda>0$, we investigate the following indices $1/\delta$ depending on $(P,D_B)$,
(I.1) the decay rate $1/\delta$ such that $ \left|\{x\in D_B:|\lambda P(x)|\le 1\} \right|\approx\lambda^{-1/\delta} (|\log \lambda |+1)^a, $
(I.2) the range $(a,b)$ of the fractional powers $\rho$ such that $ \int_{D_B} | P(x)|^{-\rho} \ dx<\infty,$
(I.3) the decay rate $1/\delta$ of $ \left|\int e^{i\lambda P(x)} \psi(x)dx \right|\lesssim \lambda^{-1/\delta} (|\log \lambda|+1)^a \ \text{for all $\psi\in C^\infty(D_B)$,} $
with an exact multiplicity index $a$ under a degeneracy type condition of $P$. We describe $\delta$ of (I.1)-(I.3) in terms of the Newton polyhedron ${\bf N}(P,D_B)=\text{Convex hull of}\ (\Lambda(P)+\rm{cone}(B))$. This result is a global extension of Varchenko's theorem (1976). For $D_B=[-1,1]^d$ with $B=\{e_1,\cdots,e_d\}$, he verified that $$\text{ $\delta$ in (I.3) is the Newton distance $ d({\bf N}(P,[-1,1]^d)):=\inf\{t:t\textbf{1}\in{\bf N}(P,[-1,1]^d)\} $}$$ under the non-degeneracy hypothesis of an analytic function $P$ with $P({\bf 0})=\nabla P({\bf 0})=0$. (The same $\delta$ in (I.1) and $(0,1/\delta)$ in (I.2) follow in the similar manner). As applications for $ P(x,y)=x_1y_1+\cdots x_{n}y_{n} + Q(y_1,\cdots,y_{n})$, we obtain the Stein-Tomas Fourier restriction Theorem associated with compact polynomial surfaces and the Strichartz estimates for general dispersive equations.
Analysis, PDE & Probability Seminar 