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Sharp dispersive estimates for the wave equation on the 5-dimensional lattice graph

Cheng Bi Cheng Bi: School of Mathematical Sciences, Fudan University, Shanghai 200433, China cbi21@m.fudan.edu.cn ,  Jiawei Cheng Jiawei Cheng: School of Mathematical Sciences, Fudan University, Shanghai 200433, China chengjw21@m.fudan.edu.cn  and  Bobo Hua Bobo Hua: School of Mathematical Sciences, LMNS, Fudan University, Shanghai 200433, China; Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China bobohua@fudan.edu.cn
Abstract.

Schultz [24] proved dispersive estimates for the wave equation on lattice graphs ℤd{\mathbb{Z}}^{d} for d=2,3,d=2,3, which was extended to d=4d=4 in [3]. By Newton polyhedra and the algorithm introduced by Karpushkin [15], we further extend the result to d=5:d=5: the sharp decay rate of the fundamental solution of the wave equation on ℤ5\mathbb{Z}^{5} is |t|−116.|t|^{-\frac{11}{6}}. Moreover, we prove Strichartz estimates and give applications to nonlinear equations.

1. introduction

Discrete dispersive equations in the form of difference equations have attracted much attention in the literature of mathematics and physics, since they constitute a natural way to approach numerically real physical laws. Indeed, spatial discretization would be the first step to implement finite difference schemes, transfering an equation on a continuum domain to that on a lattice graph. As for discrete wave equations, they appear in physical applications such as lattice dynamics and can be used to describe the vibrations of atoms inside crystals. A fundamental model is the monotonic chains, see [6, 7, 19]. On general graphs, wave equations have been studied in [8, 12, 20].

In this paper, we consider dispersive and Strichartz estimates for the discrete wave equation

(1.1) {∂t2u​(x,t)−Δ​u​(x,t)=F​(x,t),u​(x,0)=f1​(x),∂tu​(x,0)=f2​(x),(x,t)∈ℤd×ℝ.\left\{\begin{aligned} &\partial_{t}^{2}u(x,t)-\Delta u(x,t)=F(x,t),\\ &u(x,0)=f_{1}(x),\quad\partial_{t}u(x,0)=f_{2}(x),\quad(x,t)\in{\mathbb{Z}}^{d}\times{\mathbb{R}}.\end{aligned}\right.

Here the discrete Laplacian Δ\Delta is defined by

Δ​u​(x,t):=∑j=1d(u​(x+𝒆𝒋,t)+u​(x−𝒆𝒋,t)−2​u​(x,t)),\Delta u(x,t):=\sum_{j=1}^{d}\left(u(x+\boldsymbol{e_{j}},t)+u(x-\boldsymbol{e_{j}},t)-2u(x,t)\right),

where {𝒆𝒋}j=1d\{\boldsymbol{e_{j}}\}_{j=1}^{d} is the standard basis of the lattice ℤd{\mathbb{Z}}^{d}.

By the discrete Fourier transform, the fundamental solution of (1.1) is given by

(1.2) G​(x,t)=1(2​π)d​∫𝕋dei​x⋅ξ​sin⁡(t​ω​(ξ))ω​(ξ)​𝑑ξ,withω​(ξ)=(∑j=1d(2−2​cos⁡ξj))12,G(x,t)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{T}^{d}}e^{ix\cdot\xi}\;\frac{\sin(t\,\omega(\xi))}{\omega(\xi)}\,d\xi,\quad\mbox{with}\ \ \omega(\xi)=\left({\sum_{j=1}^{d}(2-2\cos\xi_{j})}\right)^{\frac{1}{2}},

where 𝕋d=[−π,π]d\mathbb{T}^{d}=[-\pi,\pi]^{d} and x⋅ξ=∑j=1dxj​ξjx\cdot\xi=\sum_{j=1}^{d}x_{j}\xi_{j}, see Section 2.1.

The pioneering work on sharp dispersive estimates for GG was initiated in Schultz [24], where he proves that GG decays like |t|−2/3|t|^{-2/3} and |t|−7/6|t|^{-7/6} when d=2d=2 and 33 respectively. On ℤ4{\mathbb{Z}}^{4}, the authors [3, Theorem 1.1] proved a sharp upper bound of order |t|−3/2​log⁡|t||t|^{-3/2}\log|t| (or 𝒪​(|t|−3/2​log⁡|t|)\mathcal{O}(|t|^{-3/2}\log|t|), in short) as t→∞t\rightarrow\infty. In all cases d=2,3,4d=2,3,4, the following oscillatory integral plays an important role,

(1.3) I​(v,t):=1(2​π)d​∫𝕋dei​t​ϕ​(v,ξ)​1ω​(ξ)​𝑑ξ,ϕ​(v,ξ):=v⋅ξ−ω​(ξ).I(v,t):=\frac{1}{(2\pi)^{d}}\int_{\mathbb{T}^{d}}e^{it\phi(v,\xi)}\,\frac{1}{\omega(\xi)}\,d\xi,\quad\phi(v,\xi):=v\cdot\xi-\omega(\xi).

Note that G​(x,t)G(x,t) is the imaginary part of −I​(x/t,t)-I(x/t,t).

Based on the analysis in [23, 24], we know the main obstacle is to describe long-time asymptotic behaviour for II when |v||v| is small, cf. Section 3. In this case, ϕ​(v,⋅)\phi(v,\cdot) has degenerate critical points and the method of stationary phase breaks down. When d≤3d\leq 3, in the terminology of [2], only stable singularities AkA_{k} (k≤5k\leq 5) and D4D_{4} appear. As dd increases, however, the singularity type becomes complicated. Instead of classifying all kinds of singularities, we seek a suitable way to obtain the stability of II and find its optimal decay rate, uniformly in vv. Here uniformity in vv for the decay is the key issue.

In the sense of V. I. Arnold [1], for an oscillatory integral

(1.4) J​(t,S,ψ):=∫ℝdei​t​S​(ξ)​ψ​(ξ)​𝑑ξ,J(t,S,\psi)\,:=\,\int_{{\mathbb{R}}^{d}}\,e^{itS(\xi)}\,\psi(\xi)\,d\xi,

establishing the uniform estimate is to determine whether the decay estimate of J​(t,S,ψ)J(t,S,\psi) could be extended to J​(t,S+P,ψ)J(t,S+P,\psi) for PP with sufficiently small norm. If d=1d=1, this is Van der Corput lemma, cf. [25, Chapter 8]. If d=2d=2, the answer is also affirmative by [16]. However, Arnold’s conjecture is not always true when d≥3d\geq 3, even in the case that PP is linear. See the counterexamples in [11, 26]. For more results we refer to [10, 13, 22], all these work are closely related to Newton polyhedra.

In general, when d≥3d\geq 3 and SS has degenerate critical points, it is difficult to establish the sharp uniform estimate of J​(t,S+P,ψ)J(t,S+P,\psi). Even for P=0P=0, it is still complicated to determine the oscillation index (cf. (2.3) below) of SS. Nevertheless, Karpushkin proposed an algorithm in [15] to determine the uniform upper bound of J​(t,S+P,ψ)J(t,S+P,\psi) when SS is quasi-homogeneous. This algorithm reduces S+PS+P to a family of polynomial phases and can be iterated several times to get the results, see Section 4.

For odd d≥5d\geq 5, we proved in the previous paper [3, Theorem 1.5] the upper bound 𝒪​(|t|−2​d+16)\mathcal{O}(|t|^{-\frac{2d+1}{6}}) for I​(v,t)I(v,t) with a fixed v=v​(d):=(12​d,⋯,12​d)∈ℝdv=v(d):=(\frac{1}{\sqrt{2d}},\cdots,\frac{1}{\sqrt{2d}})\in{\mathbb{R}}^{d}. This velocity governs the decay rates when d=3,4d=3,4. Due to the lack of uniformity in vv, the dispersive estimate for GG remains open when d≥5d\geq 5. In this paper, we prove the sharp decay estimate using Newton polyhedra and the algorithm of Karpushkin for d=5d=5.

Theorem 1.1.

There exists C>0C>0 independent of x∈ℤ5x\in{\mathbb{Z}}^{5} such that

|G​(x,t)|⩽C​(1+|t|)−116.|G(x,t)|\leqslant C(1+|t|)^{-\frac{11}{6}}.
Remark 1.2.

(a) A related model is the discrete Klein-Gordon equation

∂t2u​(x,t)−Δ​u​(x,t)+m∗2​u​(x,t)=0,\partial_{t}^{2}u(x,t)-\Delta u(x,t)+m_{*}^{2}\,u(x,t)=0,

where m∗>0m_{*}>0 is the mass parameter. This model has been studied in Cuenin and Ikromov [5] for d=2,3,4d=2,3,4 (see also [4]). Theorem 1.1 gives a partial answer to the conjecture in [5]. Indeed, let G∗G_{*} be the corresponding fundamental solution, then GG has additional singularity at the origin compared with G∗G_{*}. When d≥3d\geq 3, by the techniques in [23, 24] and slight modifications of our proof, one can show that G∗G_{*} shares the same decay estimate with GG. This yields sharp dispersive estimates for the discrete Klein-Gordon equation on ℤd{\mathbb{Z}}^{d}.

(b)Theorem 1.1 is sharp in the sense that there exists c0≠0c_{0}\neq 0 such that

(1.5) limt→∞|t116​I​(v​(5),t)|=c0,\lim_{t\rightarrow\infty}\left|t^{\frac{11}{6}}\,I(v(5),t)\right|=c_{0},

where v​(5)=(110,⋯,110)∈ℝ5v(5)=(\frac{1}{\sqrt{10}},\cdots,\frac{1}{\sqrt{10}})\in{\mathbb{R}}^{5}, see Appendix.

For all d≥3d\geq 3, the main ingredient of the proof of the theorem is the uniform estimate of (1.4) with phase of the type

(1.6) 𝐏m​(z)≡𝐏m,dΨ​(z):=(∑j=1mzj)3−∑j=1mzj3+Ψm,d​(zm+1,⋯,zd), 2≤m≤d−1,\mathbf{P}_{m}(z)\equiv\mathbf{P}_{m,d}^{\Psi}(z):=\left(\sum_{j=1}^{m}z_{j}\right)^{3}-\sum_{j=1}^{m}z_{j}^{3}\,+\,\Psi_{m,d}(z_{m+1},\cdots,z_{d}),\ \ 2\leq m\leq d-1,

where Ψm,d\Psi_{m,d} is a nondegenerate quadratic form. This polynomial appears naturally in the study of (1.3). Indeed, if ϕ​(v,⋅)\phi(v,\cdot) has a degenerate critical point ξ0\xi_{0} with corank m∈[2,d−1]m\in[2\,,d-1] (i.e. rankHessϕξ​(v,ξ0)=d−m{}_{\xi}\phi(v,\xi_{0})=d-m), then ϕ​(v,⋅)\phi(v,\cdot) can be expressed as 𝐏m​(z)+𝒪​(|z|4)\mathbf{P}_{m}(z)+\mathcal{O}\left(|z|^{4}\right) near ξ0\xi_{0}. In particular, I​(v​(d),t)I(v(d),t) corresponds to the crucial phase 𝐏d−1\mathbf{P}_{d-1}, see [3, Lemma 3.6]. Note also that 𝐏2\mathbf{P}_{2} has D4D_{4} type singularity, and 𝐏3\mathbf{P}_{3} can be reduced to z1​z2​z3+Ψ3,dz_{1}z_{2}z_{3}+\Psi_{3,d} (in this case ϕ\phi has T4,4,4T_{4,4,4} type singularity, cf. [5]).

In our context d=5d=5, the new case 𝐏4\mathbf{P}_{4} is the most complicated one, which has singularity of class OO, cf. [2, p. 253]. Karpushkin’s algorithm will be applied to this phase, see Section 4. More precisely, we consider the deformation of 𝐏4\mathbf{P}_{4} with rank ≥2\geq 2. Using change of variables, we reduce 𝐏4\mathbf{P}_{4} to homogenous polynomials with three variables, involving many parameters. Then we repeat this process and reduce the problem to oscillatory integrals in ℝ2{\mathbb{R}}^{2}, where the phases are expressed in adapted coordinate systems (cf. Section 2.2). Finally, we use results in [16, 26] to obtain the desired bound.

Note that the key difficulty is to deal with highly degenerate oscillatory integrals, and our approach is different from that in [5] for the discrete Klein-Gordon equation. To the best of the authors’ knowledge, it is the first time to adopt Karpushkin’s algorithm for estimating such oscillatory integrals with phase of corank >3>3. Our results are indeed valid for any analytic perturbation. However, our iteration approach will be tedious as dd increases. Furthermore, as a germ at the origin, 𝐏d−1\mathbf{P}_{d-1} is not finitely determined (cf. [21, Chapter 5]) for even d≥4d\geq 4 since 0 is not its isolated critical point. For these reasons, the dispersive estimates when d≥6d\geq 6 remain open.

By Theorem 1.1 and a well-known result in [18], we obtain the following Strichartz estimate. Also, a standard argument can be used for the global existence of the solutions to nonlinear equation (1.1) with small initial data, see Theorem 5.1.

Theorem 1.3.

Let d=5d=5 and uu be the solution to (1.1). If indices q,r,q~,r~q,r,\widetilde{q},\widetilde{r} satisfy

(1.7) q,r,q~,r~⩾2,1q≤116​(12−1r)and1q~≤116​(12−1r~),q,r,\tilde{q},\tilde{r}\geqslant 2,\quad\frac{1}{q}\leq\frac{11}{6}\left(\frac{1}{2}-\frac{1}{r}\right)\quad\mbox{and}\quad\frac{1}{\tilde{q}}\leq\frac{11}{6}\left(\frac{1}{2}-\frac{1}{\tilde{r}}\right),

then there exists C=C​(q,r,q~,r~)C=C(q,r,\widetilde{q},\widetilde{r}) such that

‖u‖Ltq​ℓr⩽C​(‖f1‖ℓ2+‖f2‖ℓ107+‖F‖Ltq~′​ℓ5​r~′5+r~′),\|u\|_{L^{q}_{t}\ell^{r}}\leqslant C\left(\|f_{1}\|_{\ell^{2}}+\|f_{2}\|_{\ell^{\frac{10}{7}}}+\|F\|_{L_{t}^{\tilde{q}^{\prime}}\ell^{\frac{5\tilde{r}^{\prime}}{5+\tilde{r}^{\prime}}}}\right),

where p′p^{\prime} denotes the conjugate index of pp for any p∈[1,∞]p\in[1,\infty].

The paper is organized as follows. We recall basic facts about the discrete setting and Newton polyhedra in Section 2. We also state some estimates concerning the stability of oscillatory integral in this section. In Section 3, we give the proof of Theorem 1.1. In Section 4, we prove the key result, Proposition 2.8, which is crucial for the proof of Theorem 1.1. In Section 5, we prove Strichartz estimates and give applications to the nonlinear equations. In Appendix we show the sharpness of Theorem 1.1.

Notation. We use |⋅||\cdot| and ⋅\cdot to denote the length and the inner product on Euclidean spaces, respectively, and 𝔸T\mathbb{A}^{T} the transpose of matrix 𝔸\mathbb{A}. Let Bℝd​(ξ,r)B_{{\mathbb{R}}^{d}}(\xi,r) (resp. Bℂd​(ξ,r)B_{{\mathbb{C}}^{d}}(\xi,r)) be the usual open ball in ℝd{\mathbb{R}}^{d} (resp. ℂd{\mathbb{C}}^{d}) with center ξ\xi and radius rr, while B¯ℝd​(ξ,r)\overline{B}_{{\mathbb{R}}^{d}}(\xi,r) (resp. B¯ℂd​(ξ,r)\overline{B}_{{\mathbb{C}}^{d}}(\xi,r)) denotes its closure.

The symbols C,cC,c will be used throughout to denote implicit positive constants independent of (x,t)(x,t), which may vary from one line to the next. For non-negative functions ff and gg, we adopt the notation f≲gf\lesssim g if there exists C>0C>0 such that f≤C​gf\leq Cg. For any ξ∈ℝd\xi\in{\mathbb{R}}^{d}, the translation 𝝉ξ​f​(z):=f​(z+ξ)\boldsymbol{\tau}_{\xi}f(z):=f(z+\xi). Also, we use ∂j\partial_{j} to denote ∂∂ξj\frac{\partial}{\partial\xi_{j}}, ∇\nabla (or ∇ξ\nabla_{\xi}) the usual gradient, and Hess (or Hessξ) the Hessian matrix. Moreover, we use 𝒘​(U)\boldsymbol{w}(U) (resp. 𝒘−1​(U)\boldsymbol{w}^{-1}(U)) to denote the image (resp. preimage) of UU under map 𝒘\boldsymbol{w}.

2. Preliminaries

2.1. The discrete setting

We denote by ℤd{\mathbb{Z}}^{d} the standard dd-dimensional integer latticc graph in ℝd{\mathbb{R}}^{d}, that is, ℤd:={x=(x1,⋯,xd)∈ℝd:xj∈ℤ,j=1,2,⋯,d}{\mathbb{Z}}^{d}:=\{x=(x_{1},\cdots,x_{d})\in{\mathbb{R}}^{d}:x_{j}\in{\mathbb{Z}},\,j=1,2,\cdots,d\}. For p∈[1,∞]p\in[1,\infty], ℓp​(ℤd)\ell^{p}({\mathbb{Z}}^{d}) is the ℓp\ell^{p}-space of functions on ℤd{\mathbb{Z}}^{d} with respect to the counting measure, which is a Banach space endowed with the norm

||f||ℓp:={(∑x∈ℤd|f​(x)|p)1p,p∈[1,∞),supx∈ℤd|f​(x)|,p=∞.||f||_{\ell^{p}}:=\left\{\begin{aligned} &\left(\sum_{x\in{\mathbb{Z}}^{d}}|f(x)|^{p}\right)^{\frac{1}{p}},\ p\in[1,\infty),\\ &\sup_{x\in{\mathbb{Z}}^{d}}|f(x)|,\ p=\infty.\end{aligned}\right.

We shall also use |f|p|f|_{p} to denote the ℓp\ell^{p} norm of ff for notational convenience. Note that ℓp\ell^{p} spaces are nested, that is, ℓp⊂ℓq\ell^{p}\subset\ell^{q} for 1⩽p⩽q⩽∞1\leqslant p\leqslant q\leqslant\infty. Moreover, for functions f,gf,g on ℤd\mathbb{Z}^{d}, the convolution product is given by

f∗g​(x):=∑y∈ℤdf​(x−y)​g​(y),x∈ℤd.f*g(x):=\sum_{y\in\mathbb{Z}^{d}}f(x-y)g(y),\quad x\in{\mathbb{Z}}^{d}.

The discrete Fourier transform of function ff is given by

f^​(ξ)=∑x∈ℤde−i​ξ⋅x​f​(x),ξ∈𝕋d,\hat{f}(\xi)=\sum_{x\in\mathbb{Z}^{d}}e^{-i\xi\cdot x}f(x),\quad\xi\in\mathbb{T}^{d},

while the inverse transform is defined as

fˇ​(x)=1(2​π)d​∫𝕋dei​ξ⋅x​f​(ξ)​𝑑ξ,x∈ℤd.\check{f}(x)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{T}^{d}}e^{i\xi\cdot x}f(\xi)\,d\xi,\quad x\in\mathbb{Z}^{d}.

Applying the Fourier transform to both sides of (1.1), we get

{∂t2u^​(ξ,t)+ω​(ξ)2​u^​(ξ,t)=0,u^​(ξ,0)=f1^​(ξ),∂tu^​(ξ,0)=f2^​(ξ),ξ∈𝕋d,\left\{\begin{aligned} &\partial_{t}^{2}\hat{u}(\xi,t)+\omega(\xi)^{2}\,\hat{u}(\xi,t)=0,\\ &\hat{u}(\xi,0)=\hat{f_{1}}(\xi),~{}~{}\partial_{t}\hat{u}(\xi,0)=\hat{f_{2}}(\xi),\quad\xi\in\mathbb{T}^{d},\end{aligned}\right.

which gives

u​(x,t)=1(2​π)d​∫𝕋dei​ξ⋅x​(cos⁡(t​ω​(ξ))​f1^​(ξ)+sin⁡(t​ω​(ξ))ω​(ξ)​f2^​(ξ))​𝑑ξ,(x,t)∈ℤd×ℝ.u(x,t)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{T}^{d}}e^{i\xi\cdot x}\left(\cos(t\,\omega(\xi))\hat{f_{1}}(\xi)+\frac{\sin(t\,\omega(\xi))}{\omega(\xi)}\hat{f_{2}}(\xi)\right)d\xi,\quad(x,t)\in{\mathbb{Z}}^{d}\times{\mathbb{R}}.

In the notion of operator theory,

(2.1) u​(⋅,t)=cos⁡(t​−Δ)​f1+sin⁡(t​−Δ)−Δ​f2.u(\cdot,t)=\cos(t\sqrt{-\Delta})f_{1}+\frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}}f_{2}.

Without loss of generality, we assume that f1≡0f_{1}\equiv 0 unless otherwise stated. Then we get u=f2∗Gu=f_{2}*G, where the GG is as in (1.2). Moreover, let x=v​tx=vt, the relation ω​(ξ)=ω​(−ξ)\omega(\xi)=\omega(-\xi) yields I​(v,t)=I​(−v,t)I(v,t)=I(-v,t), which gives G​(x,t)=−Im​I​(v,t).G(x,t)=-\mbox{Im}\,I(v,t).

2.2. Newton polyhedra

Let SS be a smooth real-valued function on ℝd{\mathbb{R}}^{d} and real-analytic at 0 such that

(2.2) S​(0)=∇S​(0)=0.S(0)=\nabla S(0)=0.

Let J​(t,S,ψ)J(t,S,\psi) be as in (1.4), where ψ∈C0∞​(ℝd)\psi\in C_{0}^{\infty}({\mathbb{R}}^{d}) with support near the origin. Then the following asymptotic expansion holds (cf. e.g. [2, p. 181]),

(2.3) J​(t,S,ψ)≈∑τ∑ρ=0d−1cτ,ρ,ψ​tτ​logρ⁡t,ast→+∞,J(t,S,\psi)\approx\sum_{\tau}\sum_{\rho=0}^{d-1}c_{\tau,\rho,\psi}\,t^{\tau}\log^{\rho}t,\quad\mbox{as}\ \ t\rightarrow+\infty,

where τ\tau runs through finitely many arithmetic progressions not depending on ψ\psi, which consists of negative rational numbers. Let (τS,ρS)(\tau_{S},\rho_{S}) be the maximum over all pairs (τ,ρ)(\tau,\rho) in (2.3) under the lexicographic ordering such that for any neighborhood UU of the origin, there exists ψ∈C0∞​(U)\psi\in C_{0}^{\infty}(U) for which cτS,ρS,ψ≠0c_{\tau_{S},\rho_{S},\psi}\neq 0. We call τS\tau_{S} the oscillation index of SS at 0 and ρS\rho_{S} its multiplicity.

The pioneer work of Varchenko [26] connects (2.3) with the geometry of Newton polyhedra, which we shall recall in the following part. We will use basic notions from [26], see also [2, 9, 14].

The associated Taylor series of SS at 0 can be written as

(2.4) S​(ξ)=∑γ∈𝒯​(S)sγ​ξγ,where𝒯​(S)={γ∈ℕd:sγ≠0}.S(\xi)=\sum_{\gamma\in\mathcal{T}(S)}s_{\gamma}\,\xi^{\gamma},\quad\ \mbox{where}\ \ \mathcal{T}(S)=\{\gamma\in\mathbb{N}^{d}:s_{\gamma}\neq 0\}.

Without loss of generality, we assume that 𝒯​(S)≠∅\mathcal{T}(S)\neq\emptyset. The Newton polyhedron of SS, denoted by 𝒩​(S)\mathcal{N}(S), is the convex hull of the set

⋃γ∈𝒯​(S)(γ+ℝ+d),whereℝ+d={(ξ1,⋯,ξd)∈ℝd:ξj≥0,j=1,⋯,d}.\bigcup_{\gamma\,\in\,\mathcal{T}(S)}\left(\gamma+\mathbb{R}^{d}_{+}\right),\quad\mbox{where}\ \ {\mathbb{R}}^{d}_{+}=\{(\xi_{1},\cdots,\xi_{d})\in{\mathbb{R}}^{d}:\xi_{j}\geq 0,\,j=1,\cdots,d\}.

The Newton distance dSd_{S} is defined as

dS=inf{ϱ>0:(ϱ,ϱ,⋯,ϱ)∈𝒩​(S)}.d_{S}=\inf\{\varrho>0:(\varrho,\varrho,\cdots,\varrho)\in\mathcal{N}(S)\}.

The principal face 𝒫S\mathcal{P}_{S} is the face on 𝒩​(S)\mathcal{N}(S) of minimal dimension containing (dS,⋯,dS)(d_{S},\cdots,d_{S}). In particular, under certain nondegeneracy condition, it is proved in [26] that the oscillation index of SS at 0 is 1dS\frac{1}{d_{S}} if dS>1d_{S}>1.

Since dSd_{S} depends on the choice of coordinate systems, the height of SS is given by

(2.5) hS:=sup{dS,ξ},h_{S}:=\sup\{d_{S,\xi}\},

where the supremum is taken over all local analytic coordinate systems ξ\xi which preserve the origin, and dS,ξd_{S,\xi} is the Newton distance in coordinates ξ\xi. A given coordinate system ξ~\widetilde{\xi} is said to be adapted to SS if dS,ξ~=hSd_{S,\widetilde{\xi}}=h_{S}.

If d=2d=2, the following results, derived by [26, Proposition 0.7, 0.8] and [9, Lemma 7.0], can recognize whether a given coordinate system is adapted. Note also that adapted coordinates may not exist when d≥3d\geq 3 by the counterexample in [26].

Proposition 2.1.

Let d=2d=2, SS be as in (2.2) and one of the following conditions holds:

  • (a)

    dimℝ2​(𝒫S)=0\mathrm{dim}_{{\mathbb{R}}^{2}}(\mathcal{P}_{S})=0, i.e. 𝒫S\mathcal{P}_{S} is a single point.

  • (b)

    𝒫S\mathcal{P}_{S} is unbounded.

  • (c)

    𝒫S\mathcal{P}_{S} is a compact edge. Moreover,

    𝒫S⊂{ξ:a1​ξ1+ξ2=a2}witha1,a2∈ℕ,\mathcal{P}_{S}\subset\{\xi:a_{1}\xi_{1}+\xi_{2}=a_{2}\}\quad\mbox{with}\ \ a_{1},a_{2}\in{\mathbb{N}},

    and f𝒫S​(⋅,1)f_{\mathcal{P}_{S}}(\cdot\,,1) does not have a real root of multiplicity larger than a21+a1\frac{a_{2}}{1+a_{1}}, where f𝒫S​(ξ)=∑γ∈𝒫Ssγ​ξγf_{\mathcal{P}_{S}}(\xi)=\sum_{\gamma\in\mathcal{P}_{S}}s_{\gamma}\xi^{\gamma}.

Then the coordinate system is adapted.

For instance, one can verify both ξ13​ξ2\xi_{1}^{3}\xi_{2} and ξ12​ξ22\xi_{1}^{2}\xi_{2}^{2} are expressed in adapted coordinates with Newton distance 22, while for 𝐏d−1\mathbf{P}_{d-1} the Newton distance is 62​d+1\frac{6}{2d+1}, cf. [3].

2.3. Results on uniform estimates

As we mentioned in Section 1, it is natural to consider the stability of (1.4). We need some notation initiated from [15].

Definition 2.2.

For any r,s>0r,s>0, the space ℋr​(s)\mathcal{H}_{r}(s) is defined as

ℋr​(s)={P:P​is holomorphic on​Bℂd​(0,r),continuous onB¯ℂd​(0,r),and​|P​(w)|<s,∀w∈B¯ℂd​(0,r)}.\mathcal{H}_{r}(s)=\left\{P:\ \begin{aligned} &P\ \mbox{is holomorphic on}\ B_{{\mathbb{C}}^{d}}(0,r),\mbox{continuous on}\\ &\ \overline{B}_{{\mathbb{C}}^{d}}(0,r),\ \mbox{and}\ |P({w})|<s,\ \forall\,{w}\in\overline{B}_{{\mathbb{C}}^{d}}(0,r)\end{aligned}\ \right\}.
Definition 2.3.

Let (β,p)∈ℝ×ℕ(\beta,p)\in{\mathbb{R}}\times{\mathbb{N}} and f:ℝd→ℝf:\mathbb{R}^{d}\rightarrow\mathbb{R} be real-analytic at 0, we write

M​(f)⋞(β,p)M(f)\curlyeqprec(\beta,p)

if for r>0r>0 sufficiently small, there exist ϵ>0\epsilon>0, C>0C>0 and a neighbourhood U⊂Bℝd​(0,r)U\subset B_{{\mathbb{R}}^{d}}(0,r) of the origin such that

|J​(t,f+P,ψ)|≤C​(1+|t|)β​logp⁡(|t|+2)​‖ψ‖CN​(U)|J(t,f+P,\psi)|\leq C(1+|t|)^{\beta}\log^{p}(|t|+2)\|\psi\|_{C^{N}(U)}

for all ψ∈C0∞​(U)\psi\in C_{0}^{\infty}(U) and P∈ℋr​(ϵ)P\in\mathcal{H}_{r}(\epsilon), where JJ is as in (1.4), N=N​(h)∈ℕN=N(h)\in{\mathbb{N}} and ∥ψ∥CN​(U)=sup{|∂γψ(ξ)|:ξ∈U,γ∈ℕd,∥γ∥ℓ1​(ℕd)≤N}.\|\psi\|_{C^{N}(U)}=\sup\big{\{}|\partial^{\gamma}\psi(\xi)|:\xi\in U,\,\gamma\in{\mathbb{N}}^{d},\|\gamma\|_{\ell^{1}({\mathbb{N}}^{d})}\leq N\big{\}}.

The following theorem is a consequence of [16, Theorem 2.1] and [26, Theorem 0.6].

Theorem 2.4.

Let d=2d=2, SS be as in (2.2) and hSh_{S} be as in (2.5), then there exist coordinate systems that are adapted to SS. Moreover, M​(S)⋞(τS,ρS)M(S)\curlyeqprec(\tau_{S},\rho_{S}) and τS=−hS−1.\tau_{S}=-h_{S}^{-1}.

In the sequel, for ξ∈ℝd\xi\in{\mathbb{R}}^{d}, we write M​(h,ξ)⋞(β,p)M(h,\xi)\curlyeqprec(\beta,p) if M​(𝝉ξ​h)⋞(β,p)M(\boldsymbol{\tau}_{\xi}h)\curlyeqprec(\beta,p). Also, we write M​(h2)⋞M​(h1)+(β2,p2)M(h_{2})\curlyeqprec M(h_{1})+(\beta_{2},p_{2}), if M​(h1)⋞(β1,p1)M(h_{1})\curlyeqprec(\beta_{1},p_{1}) implies that M​(h2)⋞(β1+β2,p1+p2).M(h_{2})\curlyeqprec(\beta_{1}+\beta_{2},p_{1}+p_{2}). Moreover, if M​(h2)⋞M​(h1)+(0,0)M(h_{2})\curlyeqprec M(h_{1})+(0,0), then we write M​(h2)⋞M​(h1)M(h_{2})\curlyeqprec M(h_{1}). And if M​(h)⋞(β2,p2)M(h)\curlyeqprec(\beta_{2},p_{2}), then we write M​(h)+(β1,p1)⋞(β1+β2,p1+p2)M(h)+(\beta_{1},p_{1})\curlyeqprec(\beta_{1}+\beta_{2},p_{1}+p_{2}).

For a given weight

(2.6) α=(α1,⋯,αd),with0<αd≤⋯≤α1<1,\alpha=(\alpha_{1},\cdots,\alpha_{d}),\quad\mbox{with}\quad 0<\alpha_{d}\leq\cdots\leq\alpha_{1}<1,

the one-parameter dilation is defined as

rα​ξ:=(rα1​ξ1,⋯,rαd​ξd),∀r>0,ξ∈ℝd.r^{\alpha}\xi:=(r^{\alpha_{1}}\xi_{1},\cdots,r^{\alpha_{d}}\xi_{d}),\quad\forall\,r>0,\ \,\xi\in{\mathbb{R}}^{d}.
Definition 2.5.

A polynomial ff is called α\alpha-homogeneous of degree ϱ(≥0)\varrho\,(\geq 0), if

f​(rα​ξ)=rϱ​f​(ξ),∀r>0,ξ∈ℝd.f(r^{\alpha}\xi)=r^{\varrho}f(\xi),\quad\forall\,r>0,\ \xi\in{\mathbb{R}}^{d}.
Definition 2.6.

Let ℰα,d\mathcal{E}_{\alpha,d} be the set of α\alpha-homogeneous polynomials of degree 11, ℒα,d\mathcal{L}_{\alpha,d} be the linear space (over ℝ{\mathbb{R}}) of α\alpha-homogeneous polynomials of degree less than 11, and Hα,dH_{\alpha,d} be the set of functions real-analytic at 0 with the associated Taylor’s series having the form ∑γ⋅α>1aγ​ξγ\sum_{\gamma\cdot\alpha>1}a_{\gamma}\xi^{\gamma}, i.e. each monomial is α\alpha-homogeneous of degree greater than 1.

The following lemma is from [3, Section 2], which essentially dates back to [15].

Lemma 2.7.

Let f:ℝd→ℝf:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}} be real analytic at 0.

  • (a)

    If ∇f​(0)≠0\nabla f(0)\neq 0, then for any n∈ℕn\in{\mathbb{N}}, M​(f)⋞(−n,0).M(f)\curlyeqprec(-n,0).

  • (b)

    If f∈ℰα,df\in\mathcal{E}_{\alpha,d} and P∈Hα,dP\in H_{\alpha,d}, then M​(f+P)⋞M​(f).M(f+P)\curlyeqprec M(f).

  • (c)

    If g​(ξ,z)=f​(ξ)+∑j=1mcj​zj2g(\xi,z)=f(\xi)+\sum_{j=1}^{m}c_{j}z_{j}^{2} with all cj≠0c_{j}\neq 0, then M​(g)⋞M​(f)+(−m2,0).M(g)\curlyeqprec M(f)+\left(-\frac{m}{2},0\right).

Proposition 2.8.

Let d=5d=5 and 𝐏m\mathbf{P}_{m} be as in (1.6), it holds that

(i)M​(𝐏3)⋞(−2,1),(i​i)M​(𝐏4)⋞(−116,0).(i)\ \ M(\mathbf{P}_{3})\curlyeqprec(-2,1),\quad(ii)\ \ M(\mathbf{P}_{4})\curlyeqprec\left(-\tfrac{11}{6},0\right).
Remark 2.9.

The index is sharp and is matched with the Newton polyhedra, see [3] for the case 𝐏3\mathbf{P}_{3} and Appendix for the case 𝐏4\mathbf{P}_{4}. This proposition is the key in the proof of Theorem 1.1. We postpone the proof of Proposition 2.8 to Section 4.

3. Proof of Theorem 1.1

The reader is recommended to have [3] at hand, since the following proof is similar to that of [3, Theorem 1.1], and we shall use the results in that paper without repeating all of the proofs here.

The strategy is as follows. For fixed v0v_{0}, we first study the long-time asymptotic behaviour for (1.3) with v=v0v=v_{0}. Then we prove the same decay estimate holds uniformly under (analytic) perturbation PP, that is, when ϕ​(v0,⋅)\phi(v_{0},\cdot) is replaced by ϕ​(v0,⋅)+P\phi(v_{0},\cdot)+P in the integrand of (1.3). In our context, note that

(3.1) |ϕ​(v,ξ)−ϕ​(v0,ξ)|=|(v−v0)⋅ξ|≤πd​|v−v0|,ξ∈𝕋d.\left|\phi(v,\xi)-\phi(v_{0},\xi)\right|=|(v-v_{0})\cdot\xi|\leq\pi^{d}\,|v-v_{0}|,\quad\xi\in\mathbb{T}^{d}.

Therefore, the same estimate holds uniformly for I​(v,t)I(v,t) as long as vv belongs to some small neighborhood of v0v_{0}. Then it suffices to apply a finite covering since (1.1) has finite speed of propagation.

Now we begin the proof, a direct computation gives that |∇ω|<1|\nabla\omega|<1 on 𝕋d\{0}\mathbb{T}^{d}\backslash\{0\}, hence the critical points of ϕ​(v,⋅)\phi(v,\cdot) only appear when |v|<1|v|<1. Thanks to the results [24, Proposition 2.1, Proposition 2.2, Proposition 3.10], there exists 𝐜=𝐜​(d)∈(0,1)\mathbf{c}=\mathbf{c}(d)\in(0,1) such that |G​(t​v,t)|≲(1+|t|)−d2|G(tv,t)|\lesssim(1+|t|)^{-\frac{d}{2}} provided t∈ℝt\in{\mathbb{R}} and |v|>𝐜|v|>\mathbf{c}.

Thus we restrict the attention to small vv. Since ω\omega is periodic, there exists η∈C0∞​((−2​π,2​π)d)\eta\in C_{0}^{\infty}((-2\pi,2\pi)^{d}), η​(0)=1\eta(0)=1 such that the integral in (1.3) can be rewritten as

I​(v,t)=1(2​π)d​∫ℝdei​t​ϕ​(v,ξ)​η​(ξ)​ω​(ξ)−1​𝑑ξ,v​t∈ℤd,\displaystyle I(v,t)=\frac{1}{(2\pi)^{d}}\int_{{\mathbb{R}}^{d}}e^{it\phi(v,\xi)}\eta(\xi)\,\omega(\xi)^{-1}\,d\xi,\quad vt\in{\mathbb{Z}}^{d},

cf. e.g. [3, Section 3]. Choosing χ∈C0∞​(ℝd)\chi\in C_{0}^{\infty}({\mathbb{R}}^{d}) with support near the origin gives

(2π)dI(v,t)=∫ℝdei​t​ϕ​(v,ξ)η(ξ)ω(ξ)−1χ(ξ)dξ+∫ℝdei​t​ϕ​(v,ξ)η(ξ)ω(ξ)−1(1−χ(ξ))dξ=:I1+I2.(2\pi)^{d}I(v,t)=\int_{{\mathbb{R}}^{d}}e^{it\phi(v,\xi)}\eta(\xi)\omega(\xi)^{-1}\chi(\xi)\,d\xi+\int_{{\mathbb{R}}^{d}}e^{it\phi(v,\xi)}\eta(\xi)\omega(\xi)^{-1}(1-\chi(\xi))\,d\xi=:I_{1}+I_{2}.

By [24, Proposition 2.3], we know I1=𝒪​(|t|−d+1)I_{1}=\mathcal{O}(|t|^{-d+1}) as t→∞t\rightarrow\infty. As for I2I_{2}, its asymptotic is determined by the critical points of the phase ϕ​(v,⋅)\phi(v,\cdot).

Let 0≤k≤d0\leq k\leq d be an integer. Note that Hessϕξ​(v,ξ)=−{}_{\xi}\phi(v,\xi)=-Hessω​(ξ)\,\omega(\xi), we set

Σk≡Σk​(d):={ξ∈𝕋d\{0}:∃v∈Bℝd​(0,𝐜)​such that​∇ξϕ​(v,ξ)=0andcorank Hessξ​ϕ​(v,ξ)=k}.\Sigma_{k}\equiv\Sigma_{k}(d):=\left\{\xi\in\mathbb{T}^{d}\backslash\{0\}:\ \begin{aligned} &\exists\,v\in B_{{\mathbb{R}}^{d}}(0,\mathbf{c})\ \mbox{such that}\ \nabla_{\xi}\,\phi(v,\xi)=0\\ &\qquad\ \mbox{and}\ \ \mbox{corank\,Hess}_{\xi}\,\phi(v,\xi)=k\end{aligned}\ \right\}.

Moreover, let Ωk:=∇ω​(Σk)\Omega_{k}:=\nabla\omega(\Sigma_{k}). By [3, Lemma 3.1, Corollary 3.2], we have (only the first quadrant [0,π]5[0,\pi]^{5} is considered by symmetry):

Lemma 3.1.

Let d=5d=5, then

  • (a)

    Σk\Sigma_{k} consists of ξ\xi with exactly (k+1k+1) components equal to π2\frac{\pi}{2} for k=2,3,4k=2,3,4.

  • (b)

    Σ5=∅\Sigma_{5}=\emptyset.

  • (c)

    there exists 𝐜∗∈(𝐜,1)\mathbf{c}_{*}\in(\mathbf{c},1) such that ∪k=14Ωk⊂Bℝ5​(0,𝐜∗)\cup_{k=1}^{4}\Omega_{k}\subset B_{{\mathbb{R}}^{5}}(0,\mathbf{c}_{*}).

  • (d)

    (∇ω)−1​(Ω4)=Σ4(\nabla\omega)^{-1}(\Omega_{4})=\Sigma_{4} and Ωi∩Ωj=∅,∀i,j≥2,i≠j.\Omega_{i}\cap\Omega_{j}=\emptyset,\,\forall\,i,j\geq 2,\ i\neq j.

Due to the compactness and Definition 2.3, in order to obtain the uniform estimate for I2I_{2}, it suffices to establish local bounds in Bℝ5​(0,𝐜∗)×𝒰B_{{\mathbb{R}}^{5}}(0,\mathbf{c}_{*})\times\mathcal{U} and then use partition of unity, where 𝒰\mathcal{U} is the support of η​ω−1​(1−χ)\eta\,{\omega}^{-1}\left(1-\chi\right). Indeed, we have the following lemma, whose proof relies on (3.1) and can be found in [3, p. 13].

Lemma 3.2.

For any q0=(v0,ξ0)∈Bℝ5​(0,𝐜∗)×𝒰q_{0}=(v_{0},\xi_{0})\in B_{{\mathbb{R}}^{5}}(0,\mathbf{c}_{*})\times\mathcal{U}, suppose that

(3.2) M​(ϕ​(v0,⋅),ξ0)⋞(βq0,pq0).M(\phi(v_{0},\cdot),\xi_{0})\curlyeqprec(\beta_{q_{0}},p_{q_{0}}).

Then |I2|≲(1+|t|)β​logp⁡(2+|t|)|I_{2}|\lesssim(1+|t|)^{\beta}\log^{p}(2+|t|) for some (β,p)(\beta,p).

In fact, we only need to handle finite pairs (βq0,pq0)(\beta_{q_{0}},p_{q_{0}}) by a partition of unity, and (β,p)(\beta,p) is their maximum in lexicographic order. The “worst” index appears exactly when q0∈Σ4×Ω4q_{0}\in\Sigma_{4}\times\Omega_{4}. More precisely, to establish (3.2), it suffices to prove:

Theorem 3.3.

Let d=5d=5 and q0=(v0,ξ0)∈Bℝ5​(0,𝐜∗)×𝒰q_{0}=(v_{0},\xi_{0})\in B_{{\mathbb{R}}^{5}}(0,\mathbf{c}_{*})\times\mathcal{U}, then

M​(ϕ​(v0,⋅),ξ0)⋞(β,p),with(β,p)={(−11/6,0),if q0∈Ω4×Σ4;(−2,1),if q0∈Ω3×Σ3;(−13/6,0),if q0∈(Ω2\Ω1)×Σ2;(−2,0),otherwise.M\big{(}\phi(v_{0},\cdot),\,\xi_{0}\big{)}\curlyeqprec(\beta,p),\ \ \mbox{with}\ \ (\beta,p)=\begin{cases}(-11/6,0),&\text{if $~{}q_{0}\in\Omega_{4}\times\Sigma_{4}$};\\ (-2,1),&\text{if $~{}q_{0}\in\Omega_{3}\times\Sigma_{3}$};\\ (-13/6,0),&\text{if $~{}q_{0}\in(\Omega_{2}\backslash\Omega_{1})\times\Sigma_{2}$};\\ (-2,0),&\text{otherwise}.\end{cases}

Once proving Theorem 3.3, we finish the proof of Theorem 1.1. In summary, we have used Lemmas 2.7, 3.1, 3.2 and Theorem 3.3 (and hence Proposition 2.8).

Proof of Theorem 3.3.

We consider each case separately and use Taylor’s formula for ϕ​(v0,ξ)\phi(v_{0},\xi) at ξ=ξ0\xi=\xi_{0}.

Case 1: q𝟎∈𝛀𝟒×𝚺𝟒\boldsymbol{q_{0}\in\Omega_{4}\times\Sigma_{4}}. By [3, Lemma 3.6], there exists invertible linear transformation Φ\Phi which preserves the origin such that

ϕ​(v,Φ​(y)+ξ0)=c+v⋅ξ0+Φ​(y)⋅(v−v0)+(y52+(∑j=14yj)3−∑j=14yj3)+R1​(y)\displaystyle\phi(v,\Phi(y)+\xi_{0})=c+v\cdot\xi_{0}+\Phi(y)\cdot(v-v_{0})+\left(y_{5}^{2}+\left(\sum_{j=1}^{4}y_{j}\right)^{3}-\sum_{j=1}^{4}y_{j}^{3}\right)+R_{1}(y)

holds for yy near 0, where R1∈H𝒘𝟓,5R_{1}\in H_{\boldsymbol{w_{5}},5} (recall Definition 2.6) with 𝒘𝟓=(13,⋯,13,12)\boldsymbol{w_{5}}=(\frac{1}{3},\cdots,\frac{1}{3},\frac{1}{2}). Therefore, taking v=v0v=v_{0} gives

M​(ϕ​(v0,⋅),ξ0)⋞M​(𝐏4)⋞(−116,0),\displaystyle M\left(\phi(v_{0},\cdot),\xi_{0}\right)\curlyeqprec M\left(\mathbf{P}_{4}\right)\curlyeqprec\left(-\frac{11}{6},0\right),

where we used Lemma 2.7 (b)-(c) and Proposition 2.8 (i​i)(ii).

Case 2: q𝟎∈𝛀𝟑×𝚺𝟑\boldsymbol{q_{0}\in\Omega_{3}\times\Sigma_{3}}. By symmetries and Lemma 3.1 (a), we can assume that ξ0=(π2,π2,π2,π2,ξ∗)\xi_{0}=(\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2},\xi_{*}) with ξ∗≠π2\xi_{*}\neq\frac{\pi}{2}. We use a change of variables

{ξj=zj+π2,j=1,2,3,5;ξ4=z4−z1−z2−z3−z5​sin⁡ξ∗+ξ∗.\begin{cases}\xi_{j}=z_{j}+\frac{\pi}{2},\quad j=1,2,3,5;\\ \xi_{4}=z_{4}-z_{1}-z_{2}-z_{3}-z_{5}\sin\xi_{*}+\xi_{*}.\end{cases}

Then a direct computation yields

ϕ​(v0,ξ)\displaystyle\phi(v_{0},\xi) =c+22​ω​(ξ0)3​z42−cos⁡ξ∗ω​(ξ0)​z52−2ω​(ξ0)​(z1+z2)​(z1+z3)​(z2+z3)+R2​(z),\displaystyle=c+\frac{\sqrt{2}}{2\omega(\xi_{0})^{3}}z_{4}^{2}-\frac{\cos\xi_{*}}{\omega(\xi_{0})}z_{5}^{2}-\frac{\sqrt{2}}{\omega(\xi_{0})}(z_{1}+z_{2})(z_{1}+z_{3})(z_{2}+z_{3})+R_{2}(z),

where R2∈H𝒘𝟒,5R_{2}\in H_{\boldsymbol{w_{4}},5} with 𝒘𝟒=(13,13,13,12,12)\boldsymbol{w_{4}}=(\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{2},\frac{1}{2}). Note that

(3.3) (z1+z2)​(z2+z3)​(z1+z3)=13​((z1+z2+z3)3−z13+z23+z33).(z_{1}+z_{2})(z_{2}+z_{3})(z_{1}+z_{3})=\frac{1}{3}\left((z_{1}+z_{2}+z_{3})^{3}-z_{1}^{3}+z_{2}^{3}+z_{3}^{3}\,\right).

By Lemma 2.7 (b)-(c) and Proposition 2.8 (i)(i), we have

M​(ϕ​(v0,⋅),ξ0)⋞M​(𝐏3)⋞(−2,1).M(\phi(v_{0},\cdot),\xi_{0})\curlyeqprec M(\mathbf{P}_{3})\curlyeqprec(-2,1).

Case 3: q𝟎∈(𝛀𝟐\𝛀𝟏)×𝚺𝟐\boldsymbol{q_{0}\in(\Omega_{2}\backslash\Omega_{1})\times\Sigma_{2}}. In this case, a direct computation shows that the zero-eigenvectors of Hessξ​ϕ​(q0)\mbox{Hess}_{\xi}\phi(q_{0}) are 𝜸𝟏=(1,−1,0,0,0)T\boldsymbol{\gamma_{1}}=(1,-1,0,0,0)^{T} and 𝜸𝟐=(1,1,−2,0,0)T.\boldsymbol{\gamma_{2}}=(1,1,-2,0,0)^{T}. Therefore, we let the matrix 𝔸=(𝜸𝟏,𝜸𝟐,𝒆𝟑,𝒆𝟒,𝒆𝟓)\mathbb{A}=(\boldsymbol{\gamma_{1}},\boldsymbol{\gamma_{2}},\boldsymbol{e_{3}},\boldsymbol{e_{4}},\boldsymbol{e_{5}}). By a change of variables ξ=𝔸​z+ξ0\xi=\mathbb{A}z+\xi_{0} and then use a rotation in {z3,z4,z5}\{z_{3},z_{4},z_{5}\}, we get

ϕ​(v0,𝔸​z+ξ0)=c+c1​(z23−z12​z2)+c2​z32+c3​z42+c4​z52+R3​(z),\phi(v_{0},\mathbb{A}z+\xi_{0})=c+c_{1}(z_{2}^{3}-z_{1}^{2}z_{2})+c_{2}z_{3}^{2}+c_{3}z_{4}^{2}+c_{4}z_{5}^{2}+R_{3}(z),

where c1​c2​c3​c4≠0c_{1}c_{2}c_{3}c_{4}\neq 0 and R3∈H𝒘𝟑,5R_{3}\in H_{\boldsymbol{w_{3}},5} with 𝒘𝟑=(13,13,12,12,12)\boldsymbol{w_{3}}=\left(\frac{1}{3},\frac{1}{3},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right). Since z23−z12​z2z_{2}^{3}-z_{1}^{2}z_{2} has D4−D_{4}^{-} type singularity, Lemma 2.7 and Theorem 2.4 give that

M​(ϕ​(v0,⋅),ξ0)⋞M​(𝐏2)⋞(−136,0).\displaystyle M\left(\phi(v_{0},\cdot),\xi_{0}\right)\curlyeqprec M\left(\mathbf{P}_{2}\right)\curlyeqprec\left(-\frac{13}{6},0\right).

Case 4: otherwise. Note that the rank of ϕ\phi at q0q_{0} is at least 4, the splitting lemma (cf. [21]) and Lemma 2.7 (c) imply a rough upper bound 𝒪​(|t|−2)\mathcal{O}(|t|^{-2}), which meets our needs already. ∎

4. Proof of Proposition 2.8

4.1. Preparation

We first give a little notation to state the algorithm (Theorem 4.3) in Karpushkin [15] and some simplification (Lemma 4.4), making it convenient to verify the conditions in his result.

Let d≥2d\geq 2, weight α\alpha be as in (2.6) and h∈ℰα,dh\in\mathcal{E}_{\alpha,d}. We set

h𝕊:=h|𝕊α,ld−1,where𝕊α,ld−1={x∈ℝd:|x|α,l:=(x1lα1+⋯+xdlαd)1l=1},\displaystyle h_{\mathbb{S}}:=h|_{\mathbb{S}_{\alpha,l}^{d-1}},\quad\mbox{where}\quad\mathbb{S}_{\alpha,l}^{d-1}=\left\{x\in{\mathbb{R}}^{d}:|x|_{\alpha,l}:=\left(x_{1}^{\frac{l}{\alpha_{1}}}+\cdots+x_{d}^{\frac{l}{\alpha_{d}}}\right)^{\frac{1}{l}}=1\right\},
while​ 0<l∈ℚsuch that all​l/αj​are even numbers.\displaystyle\mbox{while}\ \ 0<l\in\mathbb{Q}\ \ \mbox{such that all}\ l/\alpha_{j}\ \mbox{are even numbers}.

Notice that h​(0)=∇h​(0)=0h(0)=\nabla h(0)=0. Let

𝒵h={𝔰∈𝕊α,ld−1:h𝕊​(𝔰)=d​h𝕊​(𝔰)=0},\mathcal{Z}_{h}=\left\{\mathfrak{s}\in\mathbb{S}_{\alpha,l}^{d-1}:h_{\mathbb{S}}(\mathfrak{s})=\mathrm{d}h_{\mathbb{S}}(\mathfrak{s})=0\right\},

where d​h𝕊​(𝔰)\mathrm{d}h_{\mathbb{S}}(\mathfrak{s}) is the differential of h𝕊h_{\mathbb{S}} at 𝔰\mathfrak{s}. Moreover, let ℐ∇h\mathcal{I}_{\nabla h} be the Jacobi ideal of hh (cf. e.g. [21, p. 51]). We have the following definition, which is from [15].

Definition 4.1.

A subspace ℬ⊂ℒα,d\mathcal{B}\subset\mathcal{L}_{\alpha,d} is said to be lower (h,α)(h,\alpha)-versal, if

(ℐ∇h∩ℒα,d)⊕ℬ=ℒα,d.\left(\mathcal{I}_{\nabla h}\cap\mathcal{L}_{\alpha,d}\right)\oplus\mathcal{B}=\mathcal{L}_{\alpha,d}.

Recall that 𝒯​(h)\mathcal{T}(h) is defined in (2.4) and 𝝉ξ\boldsymbol{\tau}_{\xi} (ξ∈ℝd)(\xi\in{\mathbb{R}}^{d}) is the translation. Using [15, Proposition 4 on p. 1182, Lemma 21 on p. 1184 ], we have

Lemma 4.2.

Let h∈ℰα,dh\in\mathcal{E}_{\alpha,d} and the first order partial derivatives of hh be linearly independent. Then for any lower (h,α)(h,\alpha)-versal subspace ℬ≠0\mathcal{B}\neq 0, g∈ℬ\{0}g\in\mathcal{B}\backslash\{0\} and any critical point 𝔟\mathfrak{b} of h+gh+g, there exists monomial ι∈ℒα,d\{0}\iota\in\mathcal{L}_{\alpha,d}\backslash\{0\} such that 𝒯​(ι)∈𝒯​(𝛕𝔟​(h+g))\mathcal{T}(\iota)\in\mathcal{T}(\boldsymbol{\tau}_{\mathfrak{b}}(h+g)).

Note that Definition 2.3 carries over to real analytic manifolds. In the sequel, we write ‖α‖1:=‖α‖ℓ1​(ℕd)(=∑j=1dαj)\|\alpha\|_{1}:=\|\alpha\|_{\ell^{1}({\mathbb{N}}^{d})}(=\sum_{j=1}^{d}\alpha_{j}).

Theorem 4.3 (Theorem 1 in [15]).

Let h∈ℰα,dh\in\mathcal{E}_{\alpha,d} and the following two conditions hold.

  • (a)

    There exists a lower (h,α)(h,\alpha)-versal subspace ℬ≠0\mathcal{B}\neq 0, such that for any g∈ℬ\{0}g\in\mathcal{B}\backslash\{0\} and any critical point 𝔟\mathfrak{b} of h+gh+g, it holds that

    M​(h+g,𝔟)⋞(β1,p1).M(h+g,\mathfrak{b})\curlyeqprec(\beta_{1},p_{1}).
  • (b)

    𝒵h≠∅\mathcal{Z}_{h}\neq\emptyset, and

    M​(h𝕊,𝔰)⋞(β2,p2),∀𝔰∈𝒵h.M(h_{\mathbb{S}},\mathfrak{s})\curlyeqprec(\beta_{2},p_{2}),\quad\forall\,\mathfrak{s}\in\mathcal{Z}_{h}.

Then M​(h)⋞(β,p)M(h)\curlyeqprec(\beta,p), where

(β,p)={max​{(β1,p1),(β2,p2),(−‖α‖1,0)},if‖α‖1+β2≠0.max​{(β1,p1),(β2,p2+1)},if‖α‖1+β2=0.(\beta,p)=\left\{\begin{aligned} &\mathrm{max}\{(\beta_{1},p_{1}),(\beta_{2},p_{2}),(-\|\alpha\|_{1},0)\},&\mbox{if}\ \ \|\alpha\|_{1}+\beta_{2}\neq 0.\\ &\mathrm{max}\{(\beta_{1},p_{1}),(\beta_{2},p_{2}+1)\},&\mbox{if}\ \ \|\alpha\|_{1}+\beta_{2}=0.\end{aligned}\right.

If (a)\mathrm{(a)} holds and 𝒵h=∅\mathcal{Z}_{h}=\emptyset, then

M​(h)⋞max​{(β1,p1),(−‖α‖1,0)}.M(h)\curlyeqprec\mathrm{max}\{(\beta_{1},p_{1}),(-\|\alpha\|_{1},0)\}.

Here the maximum is taken in the lexicographic order.

By Lemma 4.2, we know condition (a) can be simplified if the first order partial derivatives of hh are linearly independent, see Section 4.2.

To simplify condition (b), we define the projection

πα,l:ℝd\{0}\displaystyle\pi_{\alpha,l}:\quad{\mathbb{R}}^{d}\backslash\{0\}\ ⟶𝕊α,ld−1\displaystyle\longrightarrow\ \mathbb{S}_{\alpha,l}^{d-1}
ξ\displaystyle\xi\ ⟼𝐫α​ξ,with𝐫=1|ξ|α,l,\displaystyle\longmapsto\ \mathbf{r}^{\alpha}\xi,\ \quad\mbox{with}\quad\mathbf{r}=\frac{1}{|\xi|_{\alpha,l}},

as well as the set

𝒞h:={ξ∈ℝd\{0}:∇h​(ξ)=0}.\mathcal{C}_{h}:=\{\xi\in{\mathbb{R}}^{d}\backslash\{0\}:\nabla h(\xi)=0\}.

Then we have the following relations.

Lemma 4.4.

If h∈ℰα,dh\in\mathcal{E}_{\alpha,d} and 𝒞h≠∅\mathcal{C}_{h}\neq\emptyset, we have

(4.1) πα,l​(𝒞h)=𝒵h.\pi_{\alpha,l}(\mathcal{C}_{h})=\mathcal{Z}_{h}.

Moreover, if all αj\alpha_{j} are equal, then

(4.2) rank​Hess​h𝕊​(πα,l​(ξ0))=rank​Hess​h​(ξ0),∀ξ0∈𝒞h.\mathrm{rank\,Hess\,}h_{\mathbb{S}}(\pi_{\alpha,l}(\xi_{0}))=\mathrm{rank\,Hess\,}h(\xi_{0}),\quad\forall\,\xi_{0}\in\mathcal{C}_{h}.
Proof.

We first prove (4.1). By the α\alpha-homogeneity of hh, we have

(4.3) ∂jh​(rα​ξ)=r1−αj​∂jh​(ξ)forj=1,⋯,d,and𝔈α​(h)​(ξ)=h​(ξ),\displaystyle\partial_{j}h(r^{\alpha}\xi)=r^{1-\alpha_{j}}\partial_{j}h(\xi)\ \ \mbox{for}\ \ j=1,\cdots,d,\quad\mbox{and}\quad\mathfrak{E}_{\alpha}(h)(\xi)=h(\xi),

where 𝔈α=∑j=1dαj​ξj​∂j\mathfrak{E}_{\alpha}=\sum_{j=1}^{d}\alpha_{j}\xi_{j}\partial_{j} is the Euler vector field for α\alpha. Thus, it follows easily that πα,l​(𝒞h)⊂𝒵h\pi_{\alpha,l}(\mathcal{C}_{h})\subset\mathcal{Z}_{h}.

To see the reverse, taking 𝔰=(𝔰1,⋯,𝔰d)=:(𝔰′,𝔰d)∈𝒵h\mathfrak{s}=(\mathfrak{s}_{1},\cdots,\mathfrak{s}_{d})=:(\mathfrak{s}^{\prime},\mathfrak{s}_{d})\in\mathcal{Z}_{h}, we assume that 𝔰d>0\mathfrak{s}_{d}>0, and use the chart ξ′⟼(ξ′,Π​(ξ′))\xi^{\prime}\longmapsto(\xi^{\prime},\Pi(\xi^{\prime})), where

ξ′=(ξ1,…,ξd−1)∈Bℝd−1​(0,1)andΠ​(ξ′)=(1−(ξ1lα1+⋯+ξd−1lαd−1))αdl.\displaystyle\xi^{\prime}=(\xi_{1},\dots,\xi_{d-1})\in B_{{\mathbb{R}}^{d-1}}(0,1)\ \ \mbox{and}\ \ \Pi(\xi^{\prime})=\left(1-\left(\xi_{1}^{\frac{l}{\alpha_{1}}}+\cdots+\xi_{d-1}^{\frac{l}{\alpha_{d-1}}}\right)\right)^{\frac{\alpha_{d}}{l}}.

Then h𝕊​(ξ′)=h​(ξ′,Π​(ξ′))h_{\mathbb{S}}(\xi^{\prime})=h(\xi^{\prime},\Pi(\xi^{\prime})) and d​h𝕊​(𝔰)=0\mathrm{d}h_{\mathbb{S}}(\mathfrak{s})=0 gives

(4.4) ∂jh​(𝔰)=αdαj​𝔰jlαj−1​∂dh​(𝔰)​Π​(𝔰′)1−lαd​for​j=1,⋯,d−1.\partial_{j}h(\mathfrak{s})=\frac{\alpha_{d}}{\alpha_{j}}\,\mathfrak{s}_{j}^{\frac{l}{\alpha_{j}}-1}\partial_{d}h(\mathfrak{s})\,\Pi(\mathfrak{s}^{\prime})^{1-\frac{l}{\alpha_{d}}}\ \mbox{for}\ j=1,\cdots,d-1.

Combining (4.4) and the second equality in (4.3) with ξ=𝔰\xi=\mathfrak{s}, we get (note that 𝔰d=Π​(𝔰′)\mathfrak{s}_{d}=\Pi(\mathfrak{s}^{\prime}))

αd​∂dh​(𝔰)​Π​(𝔰′)1−lαd=h​(𝔰).\alpha_{d}\,\partial_{d}h(\mathfrak{s})\,\Pi(\mathfrak{s}^{\prime})^{1-\frac{l}{\alpha_{d}}}=h(\mathfrak{s}).

Since 𝔰∈𝒵h\mathfrak{s}\in\mathcal{Z}_{h}, we have h​(𝔰)=0h(\mathfrak{s})=0. Then ∂dh​(𝔰)=0\partial_{d}h(\mathfrak{s})=0, which yields ∇h​(𝔰)=0\nabla h(\mathfrak{s})=0 by (4.4) again. Therefore, 𝔰∈𝒞h\mathfrak{s}\in\mathcal{C}_{h} and (4.1) is proved.

Now we prove (4.2). By the proof of (4.1), we know 𝒵h⊂𝒞h\mathcal{Z}_{h}\subset\mathcal{C}_{h}. It suffices to consider ξ0∈𝒵h\xi_{0}\in\mathcal{Z}_{h} since h∈ℰα,dh\in\mathcal{E}_{\alpha,d}. Taking 𝔰∈𝒵h\mathfrak{s}\in\mathcal{Z}_{h}, we can choose l=2​α1l=2\alpha_{1} and then 𝕊α,ld−1=𝕊d−1\mathbb{S}^{d-1}_{\alpha,l}=\mathbb{S}^{d-1}, the standard sphere in ℝd{\mathbb{R}}^{d}. Moreover, we assume 𝔰d>0\mathfrak{s}_{d}>0, then the standard chart is

Π​(ξ′)=(ξ′,1−|ξ′|2),ξ′∈Bℝd−1​(0,1).\Pi(\xi^{\prime})=(\xi^{\prime},\sqrt{1-|\xi^{\prime}|^{2}}),\quad\xi^{\prime}\in B_{{\mathbb{R}}^{d-1}}(0,1).

Since ∇h​(𝔰)=0\nabla h(\mathfrak{s})=0, a direct computation gives

Hess​h𝕊​(𝔰)=𝐇​Hess​h​(𝔰)​𝐇T,where𝐇=(𝕀d−1∇ξ′Π​(𝔰′)),\displaystyle\mbox{Hess}\,h_{\mathbb{S}}(\mathfrak{s})=\mathbf{H}\,\mbox{Hess}\,h(\mathfrak{s})\,\mathbf{H}^{T},\quad\mbox{where}\ \ \mathbf{H}=\left(\begin{array}[]{cc}\mathbb{I}_{d-1}&\nabla_{\xi^{\prime}}\Pi(\mathfrak{s}^{\prime})\end{array}\right),
and𝕀d−1is the(d−1)×(d−1)identity matrix.\displaystyle\mbox{and}\ \ \mathbb{I}_{d-1}\ \ \mbox{is the}\ \ (d-1)\times(d-1)\ \ \mbox{identity matrix.}

Notice that ∂jh​(rα​𝔰)=r​∂jh​(𝔰)=0\partial_{j}h(r^{\alpha}\mathfrak{s})=r\,\partial_{j}h(\mathfrak{s})=0, ∀r>0\forall\,r>0. Taking derivative in rr gives

∂2h​(𝔰)∂ξj​∂ξd=∂1Π​(𝔰′)​∂2h​(𝔰)∂ξj​∂ξ1+⋯+∂d−1Π​(𝔰′)​∂2h​(𝔰)∂ξj​∂ξd−1,j=1,⋯,d.\frac{\partial^{2}h(\mathfrak{s})}{\partial\xi_{j}\partial\xi_{d}}=\partial_{1}\Pi(\mathfrak{s}^{\prime})\,\frac{\partial^{2}h(\mathfrak{s})}{\partial\xi_{j}\partial\xi_{1}}+\cdots+\partial_{d-1}\Pi(\mathfrak{s}^{\prime})\,\frac{\partial^{2}h(\mathfrak{s})}{\partial\xi_{j}\partial\xi_{d-1}},\quad j=1,\cdots,d.

Therefore, we simplify Hessh​(𝔰)\,h(\mathfrak{s}) and get that

Hess​h𝕊​(𝔰)=𝐇𝐇T​(∂2h​(𝔰)∂ξ12⋯∂2h​(𝔰)∂ξ1​∂ξd−1⋮⋱⋮∂2h​(𝔰)∂ξd−1​∂ξ1⋯∂2h​(𝔰)∂ξd−12)​𝐇𝐇T.\mbox{Hess}\,h_{\mathbb{S}}(\mathfrak{s})\,=\,\mathbf{H}\mathbf{H}^{T}\left(\begin{array}[]{ccc}\frac{\partial^{2}h(\mathfrak{s})}{\partial\xi_{1}^{2}}&\cdots&\frac{\partial^{2}h(\mathfrak{s})}{\partial\xi_{1}\partial\xi_{d-1}}\\ \vdots&\ddots&\vdots\\ \frac{\partial^{2}h(\mathfrak{s})}{\partial\xi_{d-1}\partial\xi_{1}}&\cdots&\frac{\partial^{2}h(\mathfrak{s})}{\partial\xi_{d-1}^{2}}\end{array}\right)\mathbf{H}\mathbf{H}^{T}.

Since 𝐇𝐇T=𝕀d−1+∇ξ′Π​(𝔰′)​(∇ξ′Π​(𝔰′))T\mathbf{H}\mathbf{H}^{T}=\mathbb{I}_{d-1}+\nabla_{\xi^{\prime}}\Pi(\mathfrak{s}^{\prime})(\nabla_{\xi^{\prime}}\Pi(\mathfrak{s}^{\prime}))^{T} is non-singular, we finish the proof of (4.2). ∎

With all the tools in hands, we are in a position to prove Proposition 2.8.

4.2. Proof of Proposition 2.8 (i)(i)

Take γ1=(13,13,13)\gamma_{1}=(\frac{1}{3},\frac{1}{3},\frac{1}{3}). By (3.3), a change of coordinates and Lemma 2.7 (c), it suffices to prove

M​(h)⋞(−1,1),withh​(z)=z1​z2​z3.M(h)\curlyeqprec(-1,1),\quad\mbox{with}\ \ h(z)=z_{1}z_{2}z_{3}.

Since (z2​z3,z1​z3,z1​z2)(z_{2}z_{3},z_{1}z_{3},z_{1}z_{2}) are linearly independent, Lemma 4.2 gives that for any lower (h,γ1)(h,\gamma_{1})-versal subspace ℬ≠0\mathcal{B}\neq 0 and g∈ℬ\{0}g\in\mathcal{B}\backslash\{0\}, the rank of h+gh+g at any critical point is nonzero. If the rank of h+gh+g at critical point 𝔟\mathfrak{b} is at least 2, the splitting lemma and Lemma 2.7 (c) yield

M​(h+g,𝔟)⋞(−1,0).M(h+g,\mathfrak{b})\curlyeqprec(-1,0).

If the rank of h+gh+g at critical point 𝔟\mathfrak{b} is 1, it holds that (since ℬ⊂ℒγ1,3\mathcal{B}\subset\mathcal{L}_{\gamma_{1},3})

(4.5) 𝝉𝔟​(h+g)​(z)=c0+c​(z1+a2​z2+a3​z3)2+h,for some​c≠0,c0,a2,a3∈ℝ.\boldsymbol{\tau}_{\mathfrak{b}}(h+g)(z)=c_{0}+c(z_{1}+a_{2}z_{2}+a_{3}z_{3})^{2}+h,\quad\mbox{for some}\ c\neq 0,\ c_{0},a_{2},a_{3}\in{\mathbb{R}}.

A change of variables y1=z1+a2​z2+a3​z3+z2​z32​cy_{1}=z_{1}+a_{2}z_{2}+a_{3}z_{3}+\frac{z_{2}z_{3}}{2c}, y2=z2,y3=z3y_{2}=z_{2},y_{3}=z_{3} gives

𝝉𝔟​(h+g)​(z)=c0+c​y12−y2​y3​(a2​y2+a3​y3)−14​c​y22​y32.\boldsymbol{\tau}_{\mathfrak{b}}(h+g)(z)=c_{0}+c\,y_{1}^{2}-y_{2}y_{3}(a_{2}y_{2}+a_{3}y_{3})-\frac{1}{4c}\,y_{2}^{2}y_{3}^{2}.

By Lemma 2.7 (c) again, it suffices to prove that

M​(y2​y3​(a2​y2+a3​y3)+14​c​y22​y32)⋞(−12,1).M\left(y_{2}y_{3}(a_{2}y_{2}+a_{3}y_{3})+\frac{1}{4c}\,y_{2}^{2}y_{3}^{2}\right)\curlyeqprec\left(-\frac{1}{2},1\right).

We consider two cases a2=a3=0a_{2}=a_{3}=0 and a22+a32>0a_{2}^{2}+a_{3}^{2}>0. It suffices to prove

M​(y12​y2)⋞(−12,0)andM​(y12​y22)⋞(−12,1),respectively.M(y_{1}^{2}y_{2})\curlyeqprec\left(-\frac{1}{2},0\right)\ \ \mbox{and}\ \ M(y_{1}^{2}y_{2}^{2})\curlyeqprec\left(-\frac{1}{2},1\right),\ \ \mbox{respectively}.

These two estimates can be derived by, for instance, a combination of Theorem 2.4 and Proposition 2.1 (alternatively, one can use Theorem 4.3 again). In conclusion, we verify condition (a) in Theorem 4.3 with (β1,p1)=(−1,1)(\beta_{1},p_{1})=(-1,1).

Next, by Lemma 4.4 and the formula of Hess h​(z)h(z), we know

𝒵h={±𝒆𝒋:j=1,2,3}⊂𝕊γ1,232(=𝕊2).\mathcal{Z}_{h}=\{\pm\boldsymbol{e_{j}}:\,j=1,2,3\}\subset\mathbb{S}^{2}_{\gamma_{1},\frac{2}{3}}\,(=\mathbb{S}^{2}).

By Lemma 4.4 again, we know each critical point of h𝕊h_{\mathbb{S}} is non-degenerate. Therefore, (β2,p2)=(−1,0)(\beta_{2},p_{2})=(-1,0).

As a result, Theorem 4.3 gives the desired estimate M​(h)⋞(−1,1)M(h)\curlyeqprec(-1,1).

4.3. Proof of Proposition 2.8 (i​i)(ii)

Now we deal with 𝐏4\mathbf{P}_{4}, which has OO type singularity, see e.g. [2, p. 252]. To begin with, we give two elementary lemmas.

Lemma 4.5.

Suppose that

f​(x1,x2)=∑k=04ak​x14−k​x2k,with(a0,⋯,a4)∈ℝ5\{0}.f(x_{1},x_{2})=\sum_{k=0}^{4}a_{k}\,x_{1}^{4-k}x_{2}^{k},\quad\mbox{with}\ \ (a_{0},\cdots,a_{4})\in{\mathbb{R}}^{5}\backslash\{0\}.

Then M​(f)⋞(−14,0)M(f)\curlyeqprec(-\frac{1}{4},0) if and only if f​(x1,1)f(x_{1},1) or f​(1,x2)f(1,x_{2}) has a real root with multiplicity four. Otherwise, we have M​(f)⋞(−13,0)M(f)\curlyeqprec(-\frac{1}{3},0).

Proof.

Assume that a0≠0a_{0}\neq 0. We discuss the multiplicity for the roots of f​(x1,1)f(x_{1},1) in ℂ{\mathbb{C}} by a similar argument in [21, p. 85]. Under proper linear transforms ff can be reduced to one of the following forms:

(4.6) x14,x13​x2,x12​x22,x12​(x12+x22),x12​x2​(x1+x2),(x12+x22)(x12+a1x1x2+a2x22),a12+a22≠0;x1x2(x12+a1x1x2+a2x22),a2≠0.\displaystyle\begin{split}x_{1}^{4},&\ x_{1}^{3}x_{2},\ x_{1}^{2}x_{2}^{2},\ x_{1}^{2}(x_{1}^{2}+x_{2}^{2}),\ x_{1}^{2}x_{2}(x_{1}+x_{2}),\\ (x_{1}^{2}+x_{2}^{2})(x_{1}^{2}+a_{1}x_{1}&x_{2}+a_{2}x_{2}^{2}),\ a_{1}^{2}+a_{2}^{2}\neq 0\,;\ x_{1}x_{2}(x_{1}^{2}+a_{1}x_{1}x_{2}+a_{2}x_{2}^{2}),\ a_{2}\neq 0.\end{split}

By Proposition 2.1, we check that in each case (except x14x_{1}^{4}), the coordinate system {x1,x2}\{x_{1},x_{2}\} is adapted. Consequently, Van der Corput lemma and Theorem 2.4 yield

M​(x14)⋞(−14,0),M​(x13​x2)⋞(−13,0),\displaystyle M(x_{1}^{4})\curlyeqprec\left(-\frac{1}{4},0\right),\ \ \,\qquad\ \;\quad M(x_{1}^{3}x_{2})\curlyeqprec\left(-\frac{1}{3},0\right),
M​(x12​(x12+x22))⋞(−12,1),M​(x12​x2​(x1+x2))⋞(−12,1),\displaystyle M(x_{1}^{2}(x_{1}^{2}+x_{2}^{2}))\curlyeqprec\left(-\frac{1}{2},1\right),\,\ M(x_{1}^{2}x_{2}(x_{1}+x_{2}))\curlyeqprec\left(-\frac{1}{2},1\right),

while for last two cases in (4.6), the index is (−12,0)(-\frac{1}{2},0). We complete the proof. ∎

Lemma 4.6.

Let (m1,m2,m3)∈ℝ3\{0}(m_{1},m_{2},m_{3})\in{\mathbb{R}}^{3}\backslash\{0\} and (m4,m5,m6)∈ℝ3\{0}(m_{4},m_{5},m_{6})\in{\mathbb{R}}^{3}\backslash\{0\},

f​(r)=m1​r2+m2​r+m3,g​(r)=m4​r2+m5​r+m6.f(r)=m_{1}r^{2}+m_{2}r+m_{3},\ \ g(r)=m_{4}r^{2}+m_{5}r+m_{6}.

If there exists a constant c∈ℝ\{0}c\in{\mathbb{R}}\backslash\{0\} such that f2+c​g2≢0f^{2}+c\,g^{2}\not\equiv 0 and has a real root with multiplicity four, then there exists c0∈ℝ\{0}c_{0}\in{\mathbb{R}}\backslash\{0\} such that f=c0​gf=c_{0}\,g.

Proof.

Without loss of generality, we assume

f​(r)2+c​g​(r)2≡s​(r−r0)4for some​s∈ℝ\{0}.f(r)^{2}+cg(r)^{2}\equiv s(r-r_{0})^{4}\quad\mbox{for some}\ s\in{\mathbb{R}}\backslash\{0\}.

If c>0c>0, then s>0s>0 and r0r_{0} is the root of ff and gg, the conclusion is obvious.

If c<0c<0, since

f​(r)2+c​g​(r)2=(f​(r)+−c​g​(r))​(f​(r)−−c​g​(r)),f(r)^{2}+cg(r)^{2}=\left(f(r)+\sqrt{-c}\,g(r)\right)\left(f(r)-\sqrt{-c}\,g(r)\right),

we know both f+−c​gf+\sqrt{-c}g and f−−c​gf-\sqrt{-c}g have root r0r_{0} with multiplicity 2. Then the proof is finished. ∎

Let 𝐏~4​(z):=(∑j=14zj)3−∑j=14zj3\widetilde{\mathbf{P}}_{4}(z):=(\sum_{j=1}^{4}z_{j})^{3}-\sum_{j=1}^{4}z_{j}^{3} for z∈ℝ4z\in{\mathbb{R}}^{4}. It suffices to show M​(𝐏~4)⋞(−43,0)M(\widetilde{\mathbf{P}}_{4})\curlyeqprec(-\frac{4}{3},0) by Lemma 2.7 (c). We apply Theorem 4.3. Since

∑j=14aj​∂j𝐏~4​(z)≡0⟹(∑j=14aj)​(∑j=14zj)2≡∑j=14aj​zj2,\sum_{j=1}^{4}a_{j}\partial_{j}\widetilde{\mathbf{P}}_{4}(z)\equiv 0\quad\Longrightarrow\quad\left(\sum_{j=1}^{4}a_{j}\right)\left(\sum_{j=1}^{4}z_{j}\right)^{2}\equiv\sum_{j=1}^{4}a_{j}z_{j}^{2},

the coefficient of z1​z2z_{1}z_{2} must be zero. So ∑j=14aj=0\sum_{j=1}^{4}a_{j}=0, which yields aj=0a_{j}=0 for all jj as well. Therefore, the partial derivatives of first order of 𝐏~4\widetilde{\mathbf{P}}_{4} are linearly independent.

Moreover, 𝒵𝐏~4=∅\mathcal{Z}_{\widetilde{\mathbf{P}}_{4}}=\emptyset. Indeed, the stationary equation ∇𝐏~4​(z)=0\nabla\widetilde{\mathbf{P}}_{4}(z)=0 gives

(∑j=14zj)2=z12=z22=z32=z42,\left(\sum_{j=1}^{4}z_{j}\right)^{2}=z_{1}^{2}=z_{2}^{2}=z_{3}^{2}=z_{4}^{2},

which has only zero solution. Then we use Lemma 4.4 to get the conclusion.

We are left with condition (a)\mathrm{(a)} of Theorem 4.3.

Let weight γ2=(13,13,13,13)\gamma_{2}=(\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3}), ‖γ2‖1=43\|\gamma_{2}\|_{1}=\frac{4}{3}. By Lemma 4.2, for any lower (𝐏~4,γ2)(\widetilde{\mathbf{P}}_{4},\gamma_{2})-versal subspace ℬ\mathcal{B} and g∈ℬ\{0}g\in\mathcal{B}\backslash\{0\}, the rank of 𝐏~4+g\widetilde{\mathbf{P}}_{4}+g at its critical point is at least 1. Since 32>43\frac{3}{2}>\frac{4}{3}, it suffices to consider the cases that the rank equals to 1 or 2, by Lemma 2.7 (c) and the splitting lemma.

(1) The rank 1 case. Note that g∈ℒγ2,4g\in\mathcal{L}_{\gamma_{2},4}, by symmetry we can assume that at the critical point 𝐏~4+g\widetilde{\mathbf{P}}_{4}+g is the 5-parameter deformation (which is the counterpart of (4.5)):

ℱ1=ℱ1​(z,𝒂,c,c0):=c0+c​(z1+a2​z2+a3​z3+a4​z4)2+𝐏~4,\mathcal{F}_{1}=\mathcal{F}_{1}(z,\boldsymbol{a},c,c_{0}):=c_{0}+c(z_{1}+a_{2}z_{2}+a_{3}z_{3}+a_{4}z_{4})^{2}+\widetilde{\mathbf{P}}_{4},

where 𝒂=(a2,a3,a4)∈ℝ3\boldsymbol{a}=(a_{2},a_{3},a_{4})\in{\mathbb{R}}^{3}, c0∈ℝc_{0}\in{\mathbb{R}} and c≠0c\neq 0.

It suffices to show M​(ℱ1)⋞(−43,0)M(\mathcal{F}_{1})\curlyeqprec(-\frac{4}{3},0). Let γ2~=(12,13,13,13)\widetilde{\gamma_{2}}=(\frac{1}{2},\frac{1}{3},\frac{1}{3},\frac{1}{3}). We use a change of variables y1=z1+a2​z2+a3​z3+a4​z4y_{1}=z_{1}+a_{2}z_{2}+a_{3}z_{3}+a_{4}z_{4} and yj=zjy_{j}=z_{j} for j=2,3,4j=2,3,4. A direct calculation shows

ℱ1=c0+c​(y1+32​c​(y2+y3+y4)​(y2+y3+y4+2​T))2+ℱ3+ℱ4+ℱ5,\mathcal{F}_{1}=c_{0}+c\left(y_{1}+\frac{3}{2c}(y_{2}+y_{3}+y_{4})(y_{2}+y_{3}+y_{4}+2T)\right)^{2}+\mathcal{F}_{3}+\mathcal{F}_{4}+\mathcal{F}_{5},

where T=−a2​y2−a3​y3−a4​y4T=-a_{2}y_{2}-a_{3}y_{3}-a_{4}y_{4}, ℱ5∈Hγ2~,4\mathcal{F}_{5}\in H_{\widetilde{\gamma_{2}},4} and

(4.7) ℱ3=ℱ3​(y2,y3,y4)=(y2+y3+y4+T)3−(T3+y23+y33+y43),ℱ4=ℱ4​(y2,y3,y4)=−94​c​(y2+y3+y4)2​(y2+y3+y4+2​T)2.\displaystyle\begin{split}\mathcal{F}_{3}&=\mathcal{F}_{3}(y_{2},y_{3},y_{4})=(y_{2}+y_{3}+y_{4}+T)^{3}-(T^{3}+y_{2}^{3}+y_{3}^{3}+y_{4}^{3}),\\ \mathcal{F}_{4}&=\mathcal{F}_{4}(y_{2},y_{3},y_{4})=-\frac{9}{4c}(y_{2}+y_{3}+y_{4})^{2}(y_{2}+y_{3}+y_{4}+2T)^{2}.\end{split}

Using a change of coordinates and Lemma 2.7 (b)-(c), it suffices to prove

(4.8) M​(ℱ3+ℱ4)⋞(−56,0).M(\mathcal{F}_{3}+\mathcal{F}_{4})\curlyeqprec\left(-\frac{5}{6},0\right).

We first focus on the case w.r.t. 𝒂\boldsymbol{a}, in which ∂ℱ3∂yj≡0\frac{\partial\mathcal{F}_{3}}{\partial y_{j}}\equiv 0 for some jj. By symmetry we may assume that j=2j=2. Then a direct computation yields that this happens when

(4.9) 𝒂=(0,1,1)or(1,0,0).\boldsymbol{a}=(0,1,1)\ \ \mbox{or}\ \ (1,0,0).

In these cases ℱ3=3​y3​y4​(y3+y4)\mathcal{F}_{3}=3y_{3}y_{4}(y_{3}+y_{4}), which has D4−D_{4}^{-} type singularity, and

ℱ3+ℱ4=−94​c​(y2+y3+y4)2​(y2−y3−y4)2+3​y3​y4​(y3+y4).\mathcal{F}_{3}+\mathcal{F}_{4}=-\frac{9}{4c}(y_{2}+y_{3}+y_{4})^{2}(y_{2}-y_{3}-y_{4})^{2}+3y_{3}y_{4}(y_{3}+y_{4}).

Consider the weight (14,13,13)(\frac{1}{4},\frac{1}{3},\frac{1}{3}), by Lemma 2.7 (b), in this case to prove (4.8) it suffices to establish the following estimate.

Lemma 4.7.

M​(z14+z2​z3​(z2+z3))⋞(−1112,0).M\big{(}z_{1}^{4}+z_{2}z_{3}(z_{2}+z_{3})\big{)}\curlyeqprec(-\frac{11}{12},0).

Proof.

Using a change of variables z1=w1,z2+z3=w2,z2−z3=w3z_{1}=w_{1},z_{2}+z_{3}=w_{2},z_{2}-z_{3}=w_{3}, the polynomial can be written as a unimodal germ which has U12U_{12} type singularity in the terminology of Arnold, see [2, p. 273]. The uniform estimate associated to this phase has been studied by Karpushkin with index (−1112,0)(-\frac{11}{12},0), which is matched with the oscillation index, see [17] and the reference therein. ∎

Thus, we can assume that ∂ℱ3∂yj≢0\frac{\partial\mathcal{F}_{3}}{\partial y_{j}}\not\equiv 0 for all jj.

Lemma 4.8.

Let ℱ3\mathcal{F}_{3} be as in (4.7). If ∂ℱ3∂yj≢0\frac{\partial\mathcal{F}_{3}}{\partial y_{j}}\not\equiv 0 for j=2,3,4j=2,3,4, then M​(ℱ3)⋞(−56,0)M(\mathcal{F}_{3})\curlyeqprec(-\frac{5}{6},0).

Once proving Lemma 4.8, using Lemma 2.7 (b) again with weight (13,13,13)(\frac{1}{3},\frac{1}{3},\frac{1}{3}), we know M​(ℱ3+ℱ4)⋞M​(ℱ3)M(\mathcal{F}_{3}+\mathcal{F}_{4})\curlyeqprec M(\mathcal{F}_{3}), and we shall get that M​(ℱ1)⋞(−43,0)M(\mathcal{F}_{1})\curlyeqprec(-\frac{4}{3},0) in the rank 1 case.

Proof of Lemma 4.8.

We shall apply Theorem 4.3 on ℱ3\mathcal{F}_{3}. We begin with the linear independence for the partial derivatives. Since

∂ℱ3∂yk=3​(y2+y3+y4+T)2−3​ak​(y2+y3+y4)​(2​T+y2+y3+y4)−3​yk2,k=2,3,4.\frac{\partial\mathcal{F}_{3}}{\partial y_{k}}=3(y_{2}+y_{3}+y_{4}+T)^{2}-3a_{k}(y_{2}+y_{3}+y_{4})(2T+y_{2}+y_{3}+y_{4})-3y_{k}^{2},\ \ k=2,3,4.

Let λ2​∂ℱ3∂y2+λ3​∂ℱ3∂y3+λ4​∂ℱ3∂y4=0\lambda_{2}\frac{\partial\mathcal{F}_{3}}{\partial y_{2}}+\lambda_{3}\frac{\partial\mathcal{F}_{3}}{\partial y_{3}}+\lambda_{4}\frac{\partial\mathcal{F}_{3}}{\partial y_{4}}=0 and display the coefficient of each monomial, we get:

A​(1−aj)2=B​(1−2​aj)2+λj,j=2,3,4,\displaystyle A(1-a_{j})^{2}=B(1-2a_{j})^{2}+\lambda_{j},\ \ j=2,3,4,
A​(1−aj)​(1−ai)=B​(1−2​aj)​(1−2​ai),j≠i, 2≤i,j≤4,\displaystyle A(1-a_{j})(1-a_{i})=B(1-2a_{j})(1-2a_{i}),\ j\neq i,\ \ 2\leq i,j\leq 4,

where A=∑j=24λjA=\sum_{j=2}^{4}\lambda_{j} and B=∑j=24λj​aj.B=\sum_{j=2}^{4}\lambda_{j}a_{j}. A discussion on whether AA (or BB) equals to 0 gives that λj=0\lambda_{j}=0 for j=2,3,4j=2,3,4.

Now we verify condition (a) of Theorem 4.3 for ℱ3\mathcal{F}_{3}. By Lemmas 2.7 (c), it suffices to consider the rank 1 case. Similar to the previous proof, it suffices to prove

M​(ℱ3~)⋞(−56,0),whereℱ3~=c∗​(y2+b3​y3+b4​y4)2+ℱ3,c∗≠0.M(\widetilde{\mathcal{F}_{3}})\curlyeqprec(-\frac{5}{6},0),\quad\mbox{where}\ \ \widetilde{\mathcal{F}_{3}}=c_{*}(y_{2}+b_{3}y_{3}+b_{4}y_{4})^{2}+\mathcal{F}_{3},\ c_{*}\neq 0.

By a change of variables σ2=y2+b3​y3+b4​y4,σ3=y3,σ4=y4\sigma_{2}=y_{2}+b_{3}y_{3}+b_{4}y_{4},\,\sigma_{3}=y_{3},\,\sigma_{4}=y_{4}, we have

ℱ3~=c∗​σ22+3​σ2​Q2+Q3+ℱ6=c∗​(σ2+32​c∗​Q2)2+Q3−94​c∗​Q22+ℱ6,\widetilde{\mathcal{F}_{3}}=c_{*}\sigma_{2}^{2}+3\sigma_{2}Q_{2}+Q_{3}+\mathcal{F}_{6}=c_{*}\left(\sigma_{2}+\frac{3}{2c_{*}}Q_{2}\right)^{2}+Q_{3}-\frac{9}{4c_{*}}Q_{2}^{2}+\mathcal{F}_{6},

where ℱ6∈Hγ3,3\mathcal{F}_{6}\in H_{\gamma_{3},3} with γ3=(12,14,14)\gamma_{3}=(\frac{1}{2},\frac{1}{4},\frac{1}{4}), and

Q2\displaystyle Q_{2} =Q2​(σ3,σ4)=(1−a2)​((1−a3−b3+a2​b3)​σ3+(1−a4−b4+a2​b4)​σ4)2\displaystyle=Q_{2}(\sigma_{3},\sigma_{4})=(1-a_{2})\left((1-a_{3}-b_{3}+a_{2}b_{3})\sigma_{3}+(1-a_{4}-b_{4}+a_{2}b_{4})\sigma_{4}\right)^{2}
+a2​((a3−a2​b3)​σ3+(a4−a2​b4)​σ4)2−(b3​σ3+b4​σ4)2,\displaystyle+a_{2}\left((a_{3}-a_{2}b_{3})\sigma_{3}+(a_{4}-a_{2}b_{4})\sigma_{4}\right)^{2}-(b_{3}\sigma_{3}+b_{4}\sigma_{4})^{2},
Q3\displaystyle Q_{3} =Q3​(σ3,σ4)=((1−a3−b3+a2​b3)​σ3+(1−a4−b4+a2​b4)​σ4)3\displaystyle=Q_{3}(\sigma_{3},\sigma_{4})=\left((1-a_{3}-b_{3}+a_{2}b_{3})\sigma_{3}+(1-a_{4}-b_{4}+a_{2}b_{4})\sigma_{4}\right)^{3}
+((a3−a2​b3)​σ3+(a4−a2​b4)​σ4)3+(b3​σ3+b4​σ4)3−σ33−σ43.\displaystyle+\left((a_{3}-a_{2}b_{3})\sigma_{3}+(a_{4}-a_{2}b_{4})\sigma_{4}\right)^{3}+(b_{3}\sigma_{3}+b_{4}\sigma_{4})^{3}-\sigma_{3}^{3}-\sigma_{4}^{3}.

Note that both Q2Q_{2} and Q3Q_{3} should to be taken into account since, as we shall see, in some cases Q3Q_{3} vanishes and the contribution would come from Q22Q_{2}^{2}.

After a change of variables σ~2=σ2+32​c∗​Q2\widetilde{\sigma}_{2}=\sigma_{2}+\frac{3}{2c_{*}}Q_{2} with σ3\sigma_{3}, σ4\sigma_{4} fixed, and using Lemma 2.7 (c), it remains to show

(4.10) M​(Q3−94​c∗​Q22)⋞(−13,0).M\left(Q_{3}-\frac{9}{4c_{*}}Q_{2}^{2}\right)\curlyeqprec\left(-\frac{1}{3},0\right).

Our strategy is to discuss whether Q3Q_{3} vanishes. If Q3≡0Q_{3}\equiv 0, then we show Q22Q_{2}^{2} has desired estimate. If Q3≢0Q_{3}\not\equiv 0, by the properties of binary cubic forms (see e.g. [21, p. 85]) and the Van der Corput lemma, we also get the desired result.

Through an elementary (but tedious) computation on the coefficients of the monomials, we see that if Q3≡0Q_{3}\equiv 0, it suffices to consider the following two cases (other possible cases are slightly but not essentially different):

(1) a2=a4≠0,a3=1,b3=0,b4=1a_{2}=a_{4}\neq 0,\,a_{3}=1,\,b_{3}=0,\,b_{4}=1, and Q3−94​c∗​Q22=−94​c∗​(a2​σ32−σ42)2.Q_{3}-\frac{9}{4c_{*}}Q_{2}^{2}=-\frac{9}{4c_{*}}(a_{2}\,\sigma_{3}^{2}-\sigma_{4}^{2})^{2}.

(2) a2−a4=b3≠0,b4=a3−a2​b3=1,a_{2}-a_{4}=b_{3}\neq 0,\,b_{4}=a_{3}-a_{2}b_{3}=1, and b3b_{3} is the real root of the equation r2−r−1=0r^{2}-r-1=0. In this case,

Q3−94​c∗​Q22\displaystyle Q_{3}-\frac{9}{4c_{*}}Q_{2}^{2} =−94​c∗​(−a2​b3​σ32+2​(1−a2)​σ3​σ4+b3​σ42)2.\displaystyle=-\frac{9}{4c_{*}}\left(-a_{2}b_{3}\sigma_{3}^{2}+2(1-a_{2})\sigma_{3}\sigma_{4}+b_{3}\sigma_{4}^{2}\right)^{2}.

In both cases, Q3​(1,σ4)−94​c​Q22​(1,σ4)Q_{3}(1,\sigma_{4})-\frac{9}{4c}Q_{2}^{2}(1,\sigma_{4}) cannot process a root with multiplicity 4. Then we get (4.10) by Lemma 4.5. In conclusion, we verify condition (a) of Theorem 4.3 with (β1,p1)=(−56,0)(\beta_{1},p_{1})=(-\frac{5}{6},0).

Now we turn to condition (b) of Theorem 4.3 for ℱ3\mathcal{F}_{3}.

By Lemma 4.4, it suffices to show that for any possible critical point x0(≠0)x_{0}\,(\neq 0) of ℱ3\mathcal{F}_{3}, we have rankHessℱ3​(x0)=2\mathcal{F}_{3}(x_{0})=2. We argue by contradiction that x0=(w1,w2,w3)x_{0}=(w_{1},w_{2},w_{3}) is a critical point such that rankHessF3​(x0)≤1F_{3}(x_{0})\leq 1, then the three principal minors in Hessℱ3​(x0)\mathcal{F}_{3}(x_{0}) are singular. Combining this with the fact ∇ℱ3​(x0)=0\nabla\mathcal{F}_{3}(x_{0})=0, by a direct computation we get the following six equations in (a2,a3,a4,w1,w2,w3)(a_{2},a_{3},a_{4},w_{1},w_{2},w_{3}):

(1−a2)​S2=w12−a2​T2,(1−a3)​S2=w22−a3​T2,(1−a4)​S2=w32−a4​T2,\displaystyle(1-a_{2})S^{2}=w_{1}^{2}-a_{2}T^{2},\quad\ (1-a_{3})S^{2}=w_{2}^{2}-a_{3}T^{2},\quad\ (1-a_{4})S^{2}=w_{3}^{2}-a_{4}T^{2},
((1−a2)2​S−a22​T−w1)​((1−a3)2​S−a32​T−w2)=((1−a2)​(1−a3)​S−a2​a3​T)2,\displaystyle\big{(}(1-a_{2})^{2}S-a_{2}^{2}T-w_{1}\big{)}\big{(}(1-a_{3})^{2}S-a_{3}^{2}T-w_{2}\big{)}=\big{(}(1-a_{2})(1-a_{3})S-a_{2}a_{3}T\big{)}^{2},
((1−a2)2​S−a22​T−w1)​((1−a4)2​S−a42​T−w3)=((1−a2)​(1−a4)​S−a2​a4​T)2,\displaystyle\big{(}(1-a_{2})^{2}S-a_{2}^{2}T-w_{1}\big{)}\big{(}(1-a_{4})^{2}S-a_{4}^{2}T-w_{3}\big{)}=\big{(}(1-a_{2})(1-a_{4})S-a_{2}a_{4}T\big{)}^{2},
((1−a3)2​S−a32​T−w2)​((1−a4)2​S−a42​T−w3)=((1−a3)​(1−a4)​S−a3​a4​T)2,\displaystyle\big{(}(1-a_{3})^{2}S-a_{3}^{2}T-w_{2}\big{)}\big{(}(1-a_{4})^{2}S-a_{4}^{2}T-w_{3}\big{)}=\big{(}(1-a_{3})(1-a_{4})S-a_{3}a_{4}T\big{)}^{2},

where S:=T+w1+w2+w3S:=T+w_{1}+w_{2}+w_{3}. Since x0≠0x_{0}\neq 0, we claim that under the assumption of Lemma 4.8, this system has no solution. In fact, we solve (a2,a3,a4)(a_{2},a_{3},a_{4}) from the first three equations and put them into the left three equations. Then we get a system of (T,w1,w2,w3)(T,w_{1},w_{2},w_{3}),

S3=T3+w13+w23+w33,\displaystyle S^{3}=T^{3}+w_{1}^{3}+w_{2}^{3}+w_{3}^{3},
S​(w2​(T2−w12)2+w1​(T2−w22)2)+(w12−w22)2​S​T=w1​w2​(T2−S2)2\displaystyle S(w_{2}(T^{2}-w_{1}^{2})^{2}+w_{1}(T^{2}-w_{2}^{2})^{2})+(w_{1}^{2}-w_{2}^{2})^{2}ST=w_{1}w_{2}(T^{2}-S^{2})^{2}
+T​((w12−S2)2​w2+(w22−S2)2​w1),\displaystyle+T((w_{1}^{2}-S^{2})^{2}w_{2}+(w_{2}^{2}-S^{2})^{2}w_{1}),
S​(w3​(T2−w12)2+w1​(T2−w32)2)+(w12−w32)2​S​T=w1​w3​(T2−S2)2\displaystyle S(w_{3}(T^{2}-w_{1}^{2})^{2}+w_{1}(T^{2}-w_{3}^{2})^{2})+(w_{1}^{2}-w_{3}^{2})^{2}ST=w_{1}w_{3}(T^{2}-S^{2})^{2}
+T​((w12−S2)2​w3+(w32−S2)2​w1),\displaystyle+T((w_{1}^{2}-S^{2})^{2}w_{3}+(w_{3}^{2}-S^{2})^{2}w_{1}),
S​(w3​(T2−w22)2+w2​(T2−w32)2)+(w22−w32)2​S​T=w2​w3​(T2−S2)2\displaystyle S(w_{3}(T^{2}-w_{2}^{2})^{2}+w_{2}(T^{2}-w_{3}^{2})^{2})+(w_{2}^{2}-w_{3}^{2})^{2}ST=w_{2}w_{3}(T^{2}-S^{2})^{2}
+T​((w22−S2)2​w3+(w32−S2)2​w2).\displaystyle+T((w_{2}^{2}-S^{2})^{2}w_{3}+(w_{3}^{2}-S^{2})^{2}w_{2}).

To solve these equations, we compute the resultant of the first and the jj-th equation for j∈{2,3,4}j\in\{2,3,4\} and enumerate the solutions. Due to the symmetries, we take j=4j=4 as an example and eliminate TT (or w3w_{3}) from the two equations, which gives w12​(w1+w2+Z)​𝒢=0w_{1}^{2}(w_{1}+w_{2}+Z)\,\mathcal{G}=0, where

𝒢=Z2​(Z+w1+w2)​(w12+w1​w2+w22)+w1​w2​(w1+w2)​(Z​(w1+w2)+w1​w2),Z=T​or​w3.\mathcal{G}=Z^{2}(Z+w_{1}+w_{2})(w_{1}^{2}+w_{1}w_{2}+w_{2}^{2})+w_{1}w_{2}(w_{1}+w_{2})(Z(w_{1}+w_{2})+w_{1}w_{2}),\quad Z=T\ \mbox{or}\ w_{3}.

It is easy to check that w1=0w_{1}=0 or w1+w2+Z=0w_{1}+w_{2}+Z=0 gives either x0=0x_{0}=0 or 𝒂\boldsymbol{a} is of type (4.9). The latter case leads to ∂ℱ3∂yj≡0\frac{\partial\mathcal{F}_{3}}{\partial y_{j}}\equiv 0 for some j∈{2,3,4}j\in\{2,3,4\}, contradicting to the assumption in Lemma 4.8. For left possible solutions, (w3,T)(w_{3},T) should be two distinct real roots of 𝒢\mathcal{G} in ZZ (the case w3=Tw_{3}=T implies above trivial solutions). However, a calculation on the discriminant of this equation gives that 𝒢\mathcal{G} never has all roots real.

As a consequence, we get (β2,p2)=(−1,0)(\beta_{2},p_{2})=(-1,0) in condition (b) of Theorem 4.3. Finally, taking α=γ1=(13,13,13)\alpha=\gamma_{1}=(\frac{1}{3},\frac{1}{3},\frac{1}{3}) in Theorem 4.3, we have

M​(ℱ3)⋞max​{(−56,0),(−1,1)}=(−56,0).M(\mathcal{F}_{3})\curlyeqprec\mathrm{max}\left\{\left(-\frac{5}{6},0\right),\ (-1,1)\right\}=\left(-\frac{5}{6},0\right).

The proof of Lemma 4.8 is completed. ∎

Finally, in the rank 1 case we get (β1,p1)=(−43,0)(\beta_{1},p_{1})=(-\frac{4}{3},0) in Theorem 4.3.

(2) The rank 2 case. Like the rank 1 case, by symmetry we may assume that at the critical point 𝐏~4+g\widetilde{\mathbf{P}}_{4}+g is the 9-parameter deformation,

W1:=W1​(z,𝒂,𝒃,𝒄)=c0+c1​(z1+a2​z2+a3​z3+a4​z4)2+c2​(z2+b1​z1+b3​z3+b4​z4)2+𝐏~4,W_{1}:=W_{1}(z,\boldsymbol{a},\boldsymbol{b},\boldsymbol{c})=c_{0}+c_{1}(z_{1}+a_{2}z_{2}+a_{3}z_{3}+a_{4}z_{4})^{2}+c_{2}(z_{2}+b_{1}z_{1}+b_{3}z_{3}+b_{4}z_{4})^{2}+\widetilde{\mathbf{P}}_{4},

where 𝒂=(a2,a3,a4)\boldsymbol{a}=(a_{2},a_{3},a_{4}), 𝒃=(b1,b3,b4)\boldsymbol{b}=(b_{1},b_{3},b_{4}), c1​c2≠0c_{1}c_{2}\neq 0 and a2​b1≠1a_{2}b_{1}\neq 1 (since the rank is 2).

In the sequel, the strategy is the same as the proof of M​(ℱ3~)⋞(−56,0)M(\widetilde{\mathcal{F}_{3}})\curlyeqprec(-\frac{5}{6},0) in the proof of Lemma 4.8 (under (4.10)). We first introduce a little notation, let

μ1:=a2​b1−1,(μ2,μ3,μ4,μ5):=1μ1​(a3​b1−b3,a4​b1−b4,a2​b3−a3,a2​b4−a4).\mu_{1}:=a_{2}b_{1}-1,\ \ (\mu_{2},\mu_{3},\mu_{4},\mu_{5}):=\frac{1}{\mu_{1}}(a_{3}b_{1}-b_{3},\,a_{4}b_{1}-b_{4},\,a_{2}b_{3}-a_{3},\,a_{2}b_{4}-a_{4}).

As before, in W1W_{1} we set y1=z1+a2​z2+a3​z3+a4​z4y_{1}=z_{1}+a_{2}z_{2}+a_{3}z_{3}+a_{4}z_{4}, y2=z2+b1​z1+b3​z3+b4​z4y_{2}=z_{2}+b_{1}z_{1}+b_{3}z_{3}+b_{4}z_{4} and yk=zky_{k}=z_{k} for k=3,4k=3,4. By a direct computation parallel to that of ℱ3~\widetilde{\mathcal{F}_{3}}, it suffices to prove the following counterpart of (4.10),

(4.11) M​(W3+c1~​W42+c2~​W52)⋞(−13,0),M\left(W_{3}+\widetilde{c_{1}}W_{4}^{2}+\widetilde{c_{2}}W_{5}^{2}\right)\curlyeqprec\left(-\frac{1}{3},0\right),

where c1~​c2~≠0\widetilde{c_{1}}\widetilde{c_{2}}\neq 0,

W3=y3​y4​(y3+y4)−\displaystyle W_{3}=y_{3}y_{4}(y_{3}+y_{4})- ((μ2​y3+μ3​y4)+(μ4​y3+μ5​y4))\displaystyle\Big{(}(\mu_{2}y_{3}+\mu_{3}y_{4})+(\mu_{4}y_{3}+\mu_{5}y_{4})\Big{)}
×((1−μ2)​y3+(1−μ3)​y4)​((1−μ4)​y3+(1−μ5)​y4),\displaystyle\times\Big{(}(1-\mu_{2})y_{3}+(1-\mu_{3})y_{4}\Big{)}\Big{(}(1-\mu_{4})y_{3}+(1-\mu_{5})y_{4}\Big{)},

and

W4=\displaystyle W_{4}=\ (b1−1)​((μ2​y3+μ3​y4)+(μ4​y3+μ5​y4)−(y3+y4))2\displaystyle(b_{1}-1)\Big{(}(\mu_{2}y_{3}+\mu_{3}y_{4})+(\mu_{4}y_{3}+\mu_{5}y_{4})-(y_{3}+y_{4})\Big{)}^{2}
+(μ4​y3+μ5​y4)2−b1​(μ2​y3+μ3​y4)2,\displaystyle+(\mu_{4}y_{3}+\mu_{5}y_{4})^{2}-b_{1}(\mu_{2}y_{3}+\mu_{3}y_{4})^{2},
W5=\displaystyle W_{5}=\ (a2−1)​((μ2​y3+μ3​y4)+(μ4​y3+μ5​y4)−(y3+y4))2\displaystyle(a_{2}-1)\Big{(}(\mu_{2}y_{3}+\mu_{3}y_{4})+(\mu_{4}y_{3}+\mu_{5}y_{4})-(y_{3}+y_{4})\Big{)}^{2}
+(μ2​y3+μ3​y4)2−a2​(μ4​y3+μ5​y4)2.\displaystyle+(\mu_{2}y_{3}+\mu_{3}y_{4})^{2}-a_{2}(\mu_{4}y_{3}+\mu_{5}y_{4})^{2}.

If W3≢0W_{3}\not\equiv 0, then we finish by Van der Corput lemma. If W3≡0W_{3}\equiv 0, using Lemma 4.6 we shall show that c1~​W42+c2~​W52\widetilde{c_{1}}W_{4}^{2}+\widetilde{c_{2}}W_{5}^{2} do not possess a root with multiplicity four. Thus (4.11) is valid by Lemma 4.5.

We classify all cases that W3≡0W_{3}\equiv 0. A direct calculation shows that, except for the trivial solutions, it suffices to consider the following two cases:

(1)μ3=1,μ2=−μ4=−μ5,μ2​(μ22−(μ2+1))=0.\displaystyle(1)\quad\mu_{3}=1,\quad\mu_{2}=-\mu_{4}=-\mu_{5},\quad\mu_{2}(\mu_{2}^{2}-(\mu_{2}+1))=0.
(2)μ2=μ5=1,μ3=μ4,μ3​(μ32−(μ3+1))=0.\displaystyle(2)\quad\mu_{2}=\mu_{5}=1,\quad\mu_{3}=\mu_{4},\quad\mu_{3}(\mu_{3}^{2}-(\mu_{3}+1))=0.

We only consider (1), since (2) is similar. If μ2≠0\mu_{2}\neq 0, then μ52+μ5=1\mu_{5}^{2}+\mu_{5}=1 and

W4=(b1−1)​μ5​y32+2​y3​y4−b1​μ5​y42,W5=(a2−1)​μ5​y32−2​a2​y3​y4+μ5​y42,W_{4}=(b_{1}-1)\mu_{5}\,y_{3}^{2}+2y_{3}y_{4}-b_{1}\mu_{5}y_{4}^{2},\quad W_{5}=(a_{2}-1)\mu_{5}\,y_{3}^{2}-2a_{2}y_{3}y_{4}+\mu_{5}y_{4}^{2},

with the discriminants

ΔW4=4​(μ52​b1​(b1−1)+1),ΔW5=4​(a22−μ52​a2+μ52).\Delta_{W_{4}}=4\big{(}\mu_{5}^{2}\,b_{1}(b_{1}-1)+1\big{)},\quad\Delta_{W_{5}}=4\big{(}a_{2}^{2}-\mu_{5}^{2}a_{2}+\mu_{5}^{2}\big{)}.

Regard them as the quadratic form in b1b_{1} and a2a_{2}, respectively. Since μ52<4\mu_{5}^{2}<4, we know both ΔW4\Delta_{W_{4}} and ΔW5\Delta_{W_{5}} are positive. Therefore, noting that a2​b1≠1a_{2}b_{1}\neq 1, a combination of Lemma 4.6 and Lemma 4.5 gives (4.11).

If μ5=0\mu_{5}=0, then

W4=(b1−1)​y32−b1​y42,W5=(a2−1)​y32+y42.W_{4}=(b_{1}-1)y_{3}^{2}-b_{1}y_{4}^{2},\quad W_{5}=(a_{2}-1)y_{3}^{2}+y_{4}^{2}.

We use Lemmas 4.5, 4.6 again to get the conclusion.

As a result, we obtain (4.11), thus M​(𝐏~4)⋞(−43,0)M(\widetilde{\mathbf{P}}_{4})\curlyeqprec(-\frac{4}{3},0) by Theorem 4.3.

5. space-time estimates and nonlinear equations

To begin with, for 1⩽q,r<∞1\leqslant q,r<\infty, the mixed space-time Lebesgue spaces Ltq​ℓrL^{q}_{t}\ell^{r} are Banach spaces endowed with the norms

‖F~‖Ltq​ℓr:=(∫ℝ(∑x∈ℤd|F~​(x,t)|r)qr​𝑑t)1q,||\widetilde{F}||_{L^{q}_{t}\ell^{r}}:=\left(\int_{{\mathbb{R}}}\left(\sum_{x\in{\mathbb{Z}}^{d}}|\widetilde{F}(x,t)|^{r}\right)^{\frac{q}{r}}dt\right)^{\frac{1}{q}},

with natural modifications for the case q=∞q=\infty or r=∞r=\infty. By an abuse of notations, for any function F~=F~​(x,t)\widetilde{F}=\widetilde{F}(x,t) defined on ℤd×ℝ{\mathbb{Z}}^{d}\times{\mathbb{R}}, we write F~=F~​(t)\widetilde{F}=\widetilde{F}(t) sometimes for simplicity. Recall that −Δ\sqrt{-\Delta} has been defined in (2.1).

In this section, the proofs follow the same line as those of [3, Theorems 1.4, 5.9].

Proof of Theorem 1.3.

By Duhamel’s formula,

(5.1) u​(x,t)=cos⁡(t​−Δ)​f1​(x)+sin⁡(t​−Δ)−Δ​f2​(x)+∫0tsin⁡(t−s)​−Δ−Δ​F​(x,s)​𝑑s.u(x,t)=\cos(t\sqrt{-\Delta})f_{1}(x)+\frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}}f_{2}(x)+\int_{0}^{t}\frac{\sin(t-s)\sqrt{-\Delta}}{\sqrt{-\Delta}}F(x,s)ds.

Considering the space ℓ2​(ℤ5)\ell^{2}({\mathbb{Z}}^{5}), we set the operators U±​(t):=χ[0,∞)​(t)​e±i​t​−ΔU_{\pm}(t):=\chi_{[0,\infty)}(t)\,e^{\pm it\sqrt{-\Delta}}, where χ\chi is the characteristic function. From Theorem 1.1, we know that |(G∗f)​(t)|ℓ∞≲(1+|t|)−11/6​|f|1|(G*f)(t)|_{\ell^{\infty}}\lesssim(1+|t|)^{-11/6}|f|_{1} provided f∈ℓ1f\in\ell^{1}. Then as a direct consequence of Keel and Tao [18, Theorem 1.2], we have

‖ei​t​−Δ​F1‖Ltq​ℓr≲|F1|2and‖∫0tei​(t−s)​−Δ​F2​(s)​𝑑s‖Ltq​ℓr≲‖F2‖Ltq~′​ℓr~′\|e^{it\sqrt{-\Delta}}F_{1}\|_{L^{q}_{t}\ell^{r}}\lesssim|F_{1}|_{2}\quad\mbox{and}\quad\left\|\int_{0}^{t}e^{i(t-s)\sqrt{-\Delta}}F_{2}(s)\,ds\right\|_{L^{q}_{t}\ell^{r}}\lesssim\|F_{2}\|_{L^{\tilde{q}^{\prime}}_{t}\ell^{\tilde{r}^{\prime}}}

for indices satisfying (1.7) and (F1,F2)∈ℓ2​(ℤd)×Ltq~′​ℓr~′(F_{1},F_{2})\in\ell^{2}({\mathbb{Z}}^{d})\times L_{t}^{\tilde{q}^{\prime}}\ell^{\tilde{r}^{\prime}}. This together with (5.1) gives

‖u‖Ltq​ℓr≲|f1|2+|1−Δ​f2|2+‖1−Δ​F‖Ltq~′​ℓr~′.\|u\|_{L^{q}_{t}\ell^{r}}\lesssim|f_{1}|_{2}+\left|\frac{1}{\sqrt{-\Delta}}f_{2}\right|_{2}+\left\|\frac{1}{\sqrt{-\Delta}}F\right\|_{L_{t}^{\tilde{q}^{\prime}}\ell^{\tilde{r}^{\prime}}}.

Finally, we utilize the boundedness of the operator (−Δ)−1(\sqrt{-\Delta})^{-1}, see e.g. [3, Lemma 5.5], to finish the proof. ∎

Theorem 5.1.

In (1.1), let F=|u|k−1​uF=|u|^{k-1}u with k≥3k\geq 3 and f1≡0f_{1}\equiv 0. If f2∈ℓ1​(ℤ5)f_{2}\in\ell^{1}({\mathbb{Z}}^{5}) with |f2|1|f_{2}|_{1} sufficiently small, then for any p∈[2,+∞]p\in[2,+\infty], the global solution u(k)u_{(k)} exists in ℓp\ell^{p} with

|u(k)​(t)|p≲(1+|t|)−116​(1−2p).|u_{(k)}(t)|_{p}\lesssim(1+|t|)^{-\frac{11}{6}\left(1-\frac{2}{p}\right)}.
Proof.

For the convenience of notations we write ζp=116​(1−2p)\zeta_{p}=\frac{11}{6}\left(1-\frac{2}{p}\right). We only prove the theorem for p=kp=k, while the case for general p≥2p\geq 2 is similar.

For the contraction mapping principle to work, we need to estimate the ℓq\ell^{q} norm of the solution, i.e. G​(t)∗f2G(t)*f_{2}, in terms of time and the ℓp\ell^{p} norm of initial data f2f_{2}. By interpolation between Theorem 1.1 and a trivial inequality |G​(t)|≤C|G(t)|\leq C, we deduce

|G​(t)|k≤Ck​(1+|t|)−ζk.|G(t)|_{k}\leq C_{k}(1+|t|)^{-\zeta_{k}}.

Therefore, by Young’s inequality it holds that

(5.2) |G​(t)∗f2|q≤|G​(t)|r​|f2|p≤Cr​(1+|t|)−ζr​|f2|p|G(t)*f_{2}|_{q}\leq|G(t)|_{r}|f_{2}|_{p}\leq C_{r}(1+|t|)^{-\zeta_{r}}|f_{2}|_{p}

where 1r=1+1q−1p\frac{1}{r}=1+\frac{1}{q}-\frac{1}{p}.

Now we consider the metric space

ℳ:={F~:ℤ5×ℝ→ℂ,‖F~‖ℳ=supt∈ℝ(1+|t|)ζk​|F~​(⋅,t)|k≤2​C0​|f2|1},\mathcal{M}:=\left\{\widetilde{F}:{\mathbb{Z}}^{5}\times{\mathbb{R}}\rightarrow{\mathbb{C}},\|\widetilde{F}\|_{\mathcal{M}}=\sup_{t\in{\mathbb{R}}}(1+|t|)^{\zeta_{k}}|\widetilde{F}(\cdot,t)|_{k}\leq 2C_{0}|f_{2}|_{1}\right\},

with C0=C0​(k)C_{0}=C_{0}(k) to be determined later and the map Λ\Lambda on ℳ\mathcal{M},

Λ​F~:=Λ​F~​(t)=G​(t)∗f2+∫0tG​(t−s)∗F​(F~​(s))​𝑑s.\Lambda\widetilde{F}:=\Lambda\widetilde{F}(t)=G(t)*f_{2}+\int_{0}^{t}G(t-s)*F(\widetilde{F}(s))\,ds.

Given that F~∈ℳ\widetilde{F}\in\mathcal{M}, we see that

(1+|t|)ζk​|Λ​F~​(t)|k≤|f2|1+∫0t(1+|t|1+|t−s|)ζk​|F~​(s)|kk​𝑑s≤|f2|1+‖F~‖ℳk​𝐔k,(1+|t|)^{\zeta_{k}}|\Lambda\widetilde{F}(t)|_{k}\leq|f_{2}|_{1}+\int_{0}^{t}\left(\frac{1+|t|}{1+|t-s|}\right)^{\zeta_{k}}|\widetilde{F}(s)|_{k}^{k}\,ds\leq|f_{2}|_{1}+\|\widetilde{F}\|_{\mathcal{M}}^{k}\mathbf{U}_{k},

where we have used (5.2) and 𝐔k=∫ℝ(1+|s|)(1−k)​ζk​𝑑s<∞\mathbf{U}_{k}=\int_{{\mathbb{R}}}\,(1+|s|)^{(1-k)\zeta_{k}}\,ds<\infty, verified by k≥3k\geq 3. Taking the supremum, choosing proper C0C_{0} and supposing |f2|1|f_{2}|_{1} sufficiently small in order, we know Λ​F~∈ℳ\Lambda\widetilde{F}\in\mathcal{M}. One can also check that

‖Λ​u1−Λ​u2‖ℳ≤Ck​𝐔k​(2​C0​|f2|1)k−1​‖u1−u2‖ℳ.\|\Lambda u_{1}-\Lambda u_{2}\|_{\mathcal{M}}\leq C_{k}\mathbf{U}_{k}(2C_{0}|f_{2}|_{1})^{k-1}\|u_{1}-u_{2}\|_{\mathcal{M}}.

Therefore, Λ\Lambda is a contraction as long as |f2|1|f_{2}|_{1} is sufficiently small. Finally, Λ\Lambda admits a fixed point in ℳ\mathcal{M}, which is the global solution to (1.1). ∎

6. Appendix

Let d=5d=5 and 𝐏4\mathbf{P}_{4} be as in (1.6), note that 𝐏4\mathbf{P}_{4} is finitely determined (cf. [21, Chapter 5]). We prove that the oscillation index (cf. Section 2.2) of 𝐏4\mathbf{P}_{4} at 0 is −116-\frac{11}{6} with multiplicity 0. Then one can verify (1.5) by the proof of [3, Lemma 3.6] and the following proof with slight modifications.

Recall that 𝐏~4​(ξ)=(∑j=14ξj)3−∑j=14ξj3\widetilde{\mathbf{P}}_{4}(\xi)=(\sum_{j=1}^{4}\xi_{j})^{3}-\sum_{j=1}^{4}\xi_{j}^{3} for ξ∈ℝ4\xi\in{\mathbb{R}}^{4}, the Newton distance d𝐏~4=34d_{\widetilde{\mathbf{P}}_{4}}=\frac{3}{4}. By the additivity of the oscillation index, it suffices to prove the following assertion.

Proposition 6.1.

There exist ψ∈C0∞​(ℝ4)\psi\in C_{0}^{\infty}({\mathbb{R}}^{4}) with ψ​(0)≠0\psi(0)\neq 0, and cψ≠0c_{\psi}\neq 0 such that

∫ℝ4ei​λ​𝐏~4​(ξ)​ψ​(ξ)​𝑑ξ=cψ​λ−43+𝒪​(λ−32​log⁡λ),asλ→+∞.\int_{{\mathbb{R}}^{4}}e^{i\lambda\widetilde{\mathbf{P}}_{4}(\xi)}\psi(\xi)\,d\xi=c_{\psi}\,\lambda^{-\frac{4}{3}}+\mathcal{O}\left(\lambda^{-\frac{3}{2}}\log\lambda\right),\quad\mbox{as}\ \ \lambda\rightarrow+\infty.
Proof.

Using a change of variables

ξ1+ξ2=2​w1,ξ1−ξ2=2​w2,ξ3+ξ4=2​w3,ξ3−ξ4=2​w4,\xi_{1}+\xi_{2}=2w_{1},\ \xi_{1}-\xi_{2}=2w_{2},\ \xi_{3}+\xi_{4}=2w_{3},\ \xi_{3}-\xi_{4}=2w_{4},

it suffices to consider J​(λ,f,ψ)J(\lambda,f,\psi) for proper ψ\psi (see (6.2) below), where

f​(w)=4​(w1+w3)3−(w13+w33)−3​w1​w22−3​w3​w42,w∈ℝ4.f(w)=4(w_{1}+w_{3})^{3}-(w_{1}^{3}+w_{3}^{3})-3w_{1}w_{2}^{2}-3w_{3}w_{4}^{2},\quad w\in{\mathbb{R}}^{4}.

Let C≫1C\gg 1 be a constant, and

D1={w:|w1|≤C​λ−1,|w3|≤C​λ−1},D2={w:|w1|≥C​λ−1,|w3|≥C​λ−1},\displaystyle D_{1}=\left\{w:|w_{1}|\leq C\lambda^{-1},|w_{3}|\leq C\lambda^{-1}\right\},\ \ D_{2}=\left\{w:|w_{1}|\geq C\lambda^{-1},|w_{3}|\geq C\lambda^{-1}\right\},
D3={w:|w1|>C​λ−1,|w3|<C​λ−1},D4={w:|w1|<C​λ−1,|w3|>C​λ−1}.\displaystyle D_{3}=\left\{w:|w_{1}|>C\lambda^{-1},|w_{3}|<C\lambda^{-1}\right\},\ \ D_{4}=\left\{w:|w_{1}|<C\lambda^{-1},|w_{3}|>C\lambda^{-1}\right\}.

We write

J(λ,f,ψ)=∫D1+∫D2+∫D3+∫D4=:J1+J2+J3+J4.J(\lambda,f,\psi)=\int_{D_{1}}+\int_{D_{2}}+\int_{D_{3}}+\int_{D_{4}}=:J_{1}+J_{2}+J_{3}+J_{4}.

It is easy to see that J1=𝒪​(λ−2)J_{1}=\mathcal{O}(\lambda^{-2}), while on D3D_{3}, the method of stationary phase gives

(6.1) ∫−∞+∞ei​λ​w1​w22​ψ​(w)​𝑑w2=cλ​|w1|​ψ​(w1,0,w3,w4)+𝒪​(λ−1​|w1|−1),asλ→+∞,\int_{-\infty}^{+\infty}e^{i\lambda w_{1}w_{2}^{2}}\,\psi(w)\,dw_{2}=\frac{c}{\sqrt{\lambda|w_{1}|}}\psi(w_{1},0,w_{3},w_{4})+\mathcal{O}(\lambda^{-1}|w_{1}|^{-1}),\quad\mbox{as}\ \ \lambda\rightarrow+\infty,

Inserting this to J3J_{3} yields an upper bound |J3|≲λ−32|J_{3}|\lesssim\lambda^{-\frac{3}{2}}. The same estimate is valid for J4J_{4}. Now it suffices to consider J2J_{2}. Similar to (6.1), we obtain that

(6.2) J2=c​λ−1​∫𝒱ei​λ​(4​(w1+w3)3−(w13+w33))​|w1​w3|−12​ψ1​(w1,w2)​𝑑w1​𝑑w2+𝒪​(λ−2​log2⁡λ),J_{2}=c\lambda^{-1}\int_{\mathcal{V}}\,e^{i\lambda(4(w_{1}+w_{3})^{3}-(w_{1}^{3}+w_{3}^{3}))}|w_{1}w_{3}|^{-\frac{1}{2}}\psi_{1}(w_{1},w_{2})\,dw_{1}dw_{2}+\mathcal{O}(\lambda^{-2}\log^{2}\lambda),

provided λ\lambda large enough, where 𝒱={(w1,w3)∈ℝ2:|w1|>C​λ−1,|w3|>C​λ−1}\mathcal{V}=\{(w_{1},w_{3})\in{\mathbb{R}}^{2}:|w_{1}|>C\lambda^{-1},|w_{3}|>C\lambda^{-1}\}, and ψ1=ψ0​((w1+w3)6,w16​w36)\psi_{1}=\psi_{0}((w_{1}+w_{3})^{6},w_{1}^{6}w_{3}^{6}) for some ψ0∈C∞​(ℝ2)\psi_{0}\in C^{\infty}({\mathbb{R}}^{2}) supported in Bℝ2​(0,1)B_{{\mathbb{R}}^{2}}(0,1).

Denote by 𝐊\mathbf{K} the integrand in (6.2), by symmetries it suffices to estimate

𝒦1=∫𝒱∩𝒟𝐊​𝑑w1​𝑑w2and𝒦2=∫𝒱∩{(w1,w3)∈ℝ2:w1>0,w3<0}𝐊​𝑑w1​𝑑w2,\mathcal{K}_{1}=\int_{\mathcal{V}\,\cap\,\mathcal{D}}\,\mathbf{K}\,dw_{1}dw_{2}\quad\mbox{and}\ \quad\mathcal{K}_{2}=\int_{\mathcal{V}\,\cap\,\{(w_{1},w_{3})\in{\mathbb{R}}^{2}:w_{1}>0,\,w_{3}<0\}}\,\mathbf{K}\,dw_{1}dw_{2},

where 𝒟\mathcal{D} is the triangle with vertices (C​λ−1,C​λ−1),(C\lambda^{-1},C\lambda^{-1}), (1/2,1/2)(1/2,1/2) and (1−C​λ−1,C​λ−1)(1-C\lambda^{-1},C\lambda^{-1}). We begin with 𝒦1\mathcal{K}_{1}, a change of variables s1=w1+w3s_{1}=w_{1}+w_{3}, s2=w1​w3s_{2}=w_{1}w_{3} gives

(6.3) 𝒦1=∫2λ1∫1λ2s124e3​i​λ​s1​(s12+s2)s2​(s12−4​s2)​ψ1​(s1,s2)​𝑑s2​𝑑s1+𝒪​(λ−12​log⁡λ),λ→+∞,\mathcal{K}_{1}=\int_{\frac{2}{\lambda}}^{1}\int_{\frac{1}{\lambda^{2}}}^{\frac{s_{1}^{2}}{4}}\frac{e^{3i\lambda s_{1}(s_{1}^{2}+s_{2})}}{\sqrt{s_{2}(s_{1}^{2}-4s_{2})}}\,\psi_{1}(s_{1},s_{2})\,ds_{2}ds_{1}+\mathcal{O}(\lambda^{-\frac{1}{2}}\log\lambda),\quad\lambda\rightarrow+\infty,

where the remainder comes from the integral on the triangle with vertices (2​C​λ−1,C2​λ−2),(2C\lambda^{-1},C^{2}\lambda^{-2}), (1,λ−1​(1−λ−1))(1,\lambda^{-1}(1-\lambda^{-1})) and (1,1/4)(1,1/4) in (s1,s2)(s_{1},s_{2})-plane.

A change of variable r=4​s2/s12r=4s_{2}/s_{1}^{2} gives

𝒦1\displaystyle\mathcal{K}_{1} =12​∫4λ21(∫2λ​r1e3​i​λ​s13​(1+r/4)​ψ1​(s1,s12​r/4)​𝑑s1)​d​rr​(1−r)+𝒪​(λ−12​log⁡λ)\displaystyle=\frac{1}{2}\int_{\frac{4}{\lambda^{2}}}^{1}\left(\int_{\frac{2}{\lambda\sqrt{r}}}^{1}e^{3i\lambda s_{1}^{3}(1+r/4)}\,\psi_{1}(s_{1},s_{1}^{2}r/4)\,ds_{1}\right)\frac{dr}{\sqrt{r(1-r)}}+\mathcal{O}(\lambda^{-\frac{1}{2}}\log\lambda)
=12​∫4λ21(∫01e3​i​λ​s13​(1+r/4)​ψ1​(s1,s12​r/4)​𝑑s1)​d​rr​(1−r)+𝒪​(λ−12​log⁡λ).\displaystyle=\frac{1}{2}\int_{\frac{4}{\lambda^{2}}}^{1}\left(\int_{0}^{1}\,e^{3i\lambda s_{1}^{3}(1+r/4)}\,\psi_{1}(s_{1},s_{1}^{2}r/4)\,ds_{1}\right)\frac{dr}{\sqrt{r(1-r)}}+\mathcal{O}(\lambda^{-\frac{1}{2}}\log\lambda).

Using the method of stationary phase (cf. [25, Chapter 8  §5]) to the inner integral yields the asymptotic 𝒦1=c1​λ−13+𝒪​(λ−12​log⁡λ)\mathcal{K}_{1}=c_{1}\lambda^{-\frac{1}{3}}+\mathcal{O}(\lambda^{-\frac{1}{2}}\log\lambda) as λ→+∞\lambda\rightarrow+\infty.

Similar argument can be applied to 𝒦2\mathcal{K}_{2}. Indeed, a simple computation gives that

𝒦2=∫{(w1,w3)∈ℝ2: 0<w1<1,−1<w3<0}𝐊​𝑑w1​𝑑w2+𝒪​(λ−12).\mathcal{K}_{2}=\int_{\{(w_{1},w_{3})\in{\mathbb{R}}^{2}:\,0<w_{1}<1,\,-1<w_{3}<0\}}\,\mathbf{K}\,dw_{1}dw_{2}+\,\mathcal{O}(\lambda^{-\frac{1}{2}}).

In coordinate systems (s1,s2)(s_{1},s_{2}) where s1=w1+w3s_{1}=w_{1}+w_{3} and s2=−w1​w3s_{2}=-w_{1}w_{3}, we have

𝒦2=∫0+∞(∫−∞+∞ei​λ​s1​(s12−s2)s12+4​s2ψ1(s1,s2)ds1)d​s2s2+𝒪(λ−12)=:𝒦3+𝒪(λ−12).\displaystyle\mathcal{K}_{2}=\int_{0}^{+\infty}\left(\int_{-\infty}^{+\infty}\frac{e^{i\lambda s_{1}(s_{1}^{2}-s_{2})}}{\sqrt{s_{1}^{2}+4s_{2}}}\,\psi_{1}(s_{1},s_{2})\,ds_{1}\right)\frac{ds_{2}}{\sqrt{s_{2}}}+\,\mathcal{O}(\lambda^{-\frac{1}{2}})=:\mathcal{K}_{3}+\,\mathcal{O}(\lambda^{-\frac{1}{2}}).

Writing r=s1/s2r=s_{1}/\sqrt{s_{2}} gives

𝒦3\displaystyle\mathcal{K}_{3} =∫0+∞(∫−∞+∞ei​λ​s232​r​(r2−1)r2+4​ψ1​(s2​r,s2)​𝑑r)​d​s2s2(let​τ=s232)\displaystyle=\int_{0}^{+\infty}\left(\int_{-\infty}^{+\infty}\frac{e^{i\lambda s_{2}^{\frac{3}{2}}r(r^{2}-1)}}{\sqrt{r^{2}+4}}\psi_{1}(\sqrt{s_{2}}\,r,s_{2})\,dr\right)\frac{ds_{2}}{\sqrt{s_{2}}}\quad(\mbox{let}\ \tau=s_{2}^{\frac{3}{2}})
=∫−∞+∞(∫0+∞ei​λ​τ​r​(r2−1)​τ−23​ψ1​(τ13​r,τ23)​𝑑τ)​d​rr2+4.\displaystyle=\int_{-\infty}^{+\infty}\left(\int_{0}^{+\infty}e^{i\lambda\tau\,r(r^{2}-1)}\,\tau^{-\frac{2}{3}}\,\psi_{1}(\tau^{\frac{1}{3}}r,\tau^{\frac{2}{3}})\,d\tau\right)\frac{dr}{\sqrt{r^{2}+4}}.

Noting that r​(r2−1)r(r^{2}-1) has zeros 0,±10,\pm 1, we split

𝒦3=∫|r|<C​λ−1+∫C​λ−1<|r|<1−C​λ−1+∫|r−1|<C​λ−1+∫|r|>1+C​λ−1=:+𝒦4+𝒦5+𝒦6+𝒦7.\mathcal{K}_{3}=\int_{|r|<C\lambda^{-1}}+\int_{C\lambda^{-1}<|r|<1-C\lambda^{-1}}+\int_{|r-1|<C\lambda^{-1}}+\int_{|r|>1+C\lambda^{-1}}=:+\mathcal{K}_{4}+\mathcal{K}_{5}+\mathcal{K}_{6}+\mathcal{K}_{7}.

Obviously, 𝒦4=𝒪​(λ−1)\mathcal{K}_{4}=\mathcal{O}(\lambda^{-1}) and 𝒦6=𝒪​(λ−1)\mathcal{K}_{6}=\mathcal{O}(\lambda^{-1}). For the left two terms we use stationary phase method as in the estimate of 𝒦1\mathcal{K}_{1} (for 𝒦7\mathcal{K}_{7} we use a change of variable y=r3​τy=r^{3}\tau in addition), and get that both of them have asymptotics λ−13\lambda^{-\frac{1}{3}} as λ→∞\lambda\rightarrow\infty.

As a consequence, we check that there exists c0≠0c_{0}\neq 0 such that J2=c0​λ−43+𝒪​(λ−12​log⁡λ)J_{2}=c_{0}\lambda^{-\frac{4}{3}}+\mathcal{O}(\lambda^{-\frac{1}{2}}\log\lambda), which completes the proof of Proposition 6.1. ∎

Acknowledgement

C.Bi is grateful to Prof. James Montaldi and Titus Piezas III for helpful discussions and suggestions. B.Hua is supported by NSFC, No. 12371056, and by Shanghai Science and Technology Program [Project No. 22JC1400100].

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