Continuum limit of 2D fractional nonlinear Schrödinger equation

We prove that the solutions to the discrete nonlinear Schrödinger equation with non-local algebraically decaying coupling converge strongly in L2(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2({\mathbb {R}}^2)$$\end{document} to those of the continuum fractional nonlinear Schrödinger equation, as the discretization parameter tends to zero. The proof relies on sharp dispersive estimates that yield the Strichartz estimates that are uniform in the discretization parameter. An explicit computation of the leading term of the oscillatory integral asymptotics is used to show that the best constants of a family of dispersive estimates blow up as the non-locality parameter α∈(1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2)$$\end{document} approaches the boundaries.

The mathematical description of physical phenomena, in many instances, results in the formulation of partial differential equations (PDEs) describing state variables in continuum media.Despite the fact that it is highly unlikely to find exact solutions of many linear or nonlinear PDEs, advances in numerical analysis and scientific computing open the door to find approximate solutions to complex problems.In particular, numerical approximations based on finite difference schemes are constructed by discretizing spatial variables, leading to a system of coupled ordinary differential equations.In this line of research, the objective is then to determine how well the approximate solution evaluated in the grid approximates the solutions of the corresponding PDE.
On the other hand, there are well known universal models that are inherently discrete.Generically referred to as coupled oscillator systems, they describe phenomena such as localization or synchronization, characteristic of its discrete nature.Best known examples are the Fermi-Pasta-Ulam-Tsingou model, the discrete nonlinear Schrödinger Equation and the Kuramoto model.The first two describe dynamics in a lattice with nearest neighbor interactions, whereas the Kuramoto model addresses synchronization for globally coupled oscillators.These and similar models continue to be studied given their applicability in photonics, lasers, and networks such as the power grid to name some.For such models a suitable approximation named the long-wave approximation assumes a "smooth" variation of the state variable amongst neighbor lattices.Specifically in a 1-dimensional lattice, this means u n±1 ≈ u n .In this regime, it is reasonable to consider continuum approximation.For a 1-d lattice model, the continuum approximation u n±1 → U (x ± h), where h > 0 is small, with nearest neighbor coupling C(u n+1 + u n−1 ) leads to a term proportional to ∂ 2 U ∂x 2 and in return, the system of ODEs is then approximated by a PDE.
Recently, there has been an increased interest in the models based on FNLSE.While most of the research deals with continuum models, including numerical computations of solutions in the nonlinear regime, less is known about discrete systems showing global coupling with algebraic decay on the coupling strength with respect to the distance between nodes in the lattice.This work considers such a case in a two-dimensional lattice and centers on the question of the validity of a suitable continuum approximation.This is not always a trivial task as, for instance, invariances and symmetries may arise or be lost.In contrast to the (continuum) nonlinear Schrödinger equation that admits the Galilean boost from which traveling wave solutions emerge, many lattice systems lack translational invariance.It is known that highly localized solutions in a lattice system do not propagate due to the presence of the Peirels-Nabarro potential [26,10]; for a recent work on FNLSE in this context, see [22].All this is to point out the challenges and open problems that need to be studied by a combination of analytical and numerical tools.In this contribution, we report what we think are first analytic results on the underlying fundamental question of determining the continuum approximation on the FNLSE in more than one dimension.
2 Statement of the problem.This work concerns the continuum limit of the discrete fractional nonlinear Schrödinger equation (FNLSE) to the continuum FNLS i∂ t u = (−∆) as h → 0+ where α ∈ (0, 2] \ {1}, p > 1, µ = ±1, and u : R 2+1 → C, u h : hZ 2 × R → C. Let (2.1), (2.2) be well-posed in some Banach spaces X, X h , respectively, where 0 < h ≤ 1 denotes a discretization parameter.Suppose u 0,h ∈ X h is the discretized u 0 ∈ X.Given an interpolation operator p h : X h → X and T > 0 such that u(t), u h (t) denote the well-posed solutions on [0, T ], the main problem then reduces to identifying values of α, p that allows lim The study of evolution equations on R with a general class of interaction kernel was done in [25] where the continuum limit was proved in the weak sense.By applying the analytic tools in [15] that yield dispersive estimates for the discrete Schrödinger evolution that are uniform in h, [16] extended the aforementioned weak convergence to strong convergence in the L 2 -setting (with convergence rates) for α = 2 in R d , d = 1, 2, 3 and α ∈ (0, 2) \ {1} on R. The central perspective in [16], upon which we develop, that sharp dispersive estimates that are uniform in h control the difference p h u h − u, at least in the scaling-subcritical regime, proved to be fruitful as can be illustrated in various works such as [11] that studied the case α = 2 on T 2 as the spatial domain, [12] that studied the large box limit for α = 2 in R d , d = 2, 3, and [13] that showed the rigorous derivation of the KdV equation from the FPU system.Using a similar idea, the continuum limit of the space-time FNLS was investigated in [7].Furthermore see the works of Ignat and Zuazua [17,18,19,31,20] where novel approaches such as the Fourier filtering and the two-grid algorithm were used.
In practice, obtaining appropriate dispersive estimates reduces to oscillatory integral estimations, which is of central concern in our approach.Unlike the continuum case, the dispersion relation for the discrete evolution has degenerate critical points, which results in weaker dispersion than the continuum Schrödinger evolution.This in return admits weaker Strichartz estimates, which limits the class of nonlinearities that leads to the well-posedness of the corresponding nonlinear equation via the contraction mapping argument.To be more quantitative, let where the implicit constant blows up as h → 0+, which contrasts with Our objective is to obtain Strichartz estimates for the discrete evolution that are uniform in h.
For α < 2, [16, Proposition 3.1] obtained the set of degenerate critical points on hZ consists of isolated points whose corresponding oscillatory integrals cannot be estimated directly by the Van der Corput Lemma.In higher dimensions, the set of degeneracy is geometrically more complicated.In fact, our analysis shows that the degenerate critical points define a one-dimensional embedded smooth submanifold in the torus [−π, π] 2 where each singular point admits a unique direction along which the third derivative does not vanish (fold) except at four points (cusp) at which the fourth derivative does not vanish.This observation that a singular point is at worst a cusp is consistent with [1].It is expected that more severe singularities exist in higher dimensions as the structure of the Hessian of the dispersion relation becomes more complicated.This dimension-dependent geometric complication is purely a remnant of non-locality since the linear evolution of classical Schrödinger operator on hZ d splits as the d-fold tensor product on each dimension.Consider the dispersion relation and the quantity of interest where 2) is w h,0 .[27] showed that when m = 0, α = 1, which corresponds to the dispersion relation of the discrete wave equation, then the quantity of interest decays as 6 ) in d = 3.When m > 0, α = 1, which corresponds to the discrete Klein-Gordon equation, [1] showed that the quantity of interest decays as O(t − 3  4 ) in d = 2, and the result was extended to higher dimensions (d = 3, 4) in [5].When m = 0, α = 2, the time decay of the fundamental solution of the classical discrete Schrödinger equation was shown to be O(t − d 3 ) in [28].Our objective is to obtain the sharp time decay of the quantity of interest for m = 0, α ∈ (1, 2) in d = 2.In particular, it is shown that the oscillatory integral decays as O(t − 3  4 ).The main tool that we adopt is the analysis of Newton polyhedron generated by the Taylor expansion of the phase function x • ξ − tw h,0 (ξ) in an adapted coordinate system, a method pioneered in [30].Furthermore the asymptotics in both regimes α → 1+ (wave limit) and α → 2− (Schrödinger limit) are studied.To our knowledge, the dependence on the non-local parameter has not been clearly investigated in previous works.To obtain the asymptotics of the leading term of O(t − 3 4 ) as a function of α, we represent the phase function in a superadapted coordinate system to apply results of [8].
The relation of our work to the theory of stability of degenerate oscillatory integrals is subtle.A cursory observation might suggest that a degenerate integral (our quantity of interest) would be stable under a small perturbation in the non-local parameter.However the phase fails to be smooth for α < 2, and therefore becomes large in appropriate norm(s) as the support of η becomes arbitrarily close to the origin.In our approach, it suffices to invoke the stability result [21] under linear perturbations in phase.For more general stability results under analytic or smooth perturbations, see [23,9].For the support of η close to the origin, sin z ∼ z by the small angle approximation, after which one might wish to invoke [3] that obtained sharp dispersive estimates for radial dispersion relations.However such approximation is not a linear perturbation and hence we handle that case by direct computation.
The paper is organized as follows.Notations and main results are presented in Section 3. Assuming the results hold, the desired continuum limit is shown in Section 4. The proof of our main proposition is in Section 5, followed by a concluding remark in Section 6.
To discuss continuum limit, the parameters that yield the well-posedness of (2.1), (2.2) must be identified.For the discrete equation, the linear operator defines a bounded, non-negative, self-adjoint operator on L 2 h , and so are its fractional powers given by functional calculus.Equivalently (−∆ h ) α 2 is given by the Fourier multiplier where the discrete Fourier transform is defined as h → L ∞ h , which yields an immediate well-posedness of (2.1) in L 2 h via the contraction mapping argument.For the continuum case, consider the family of self-similar solutions and observing that {u λ (•, t)} λ>0 leaves Ḣsc (R d ) invariant for all t, one obtains the Sobolev-critical regularity Our analysis is in the scaling-subcritical regime where the time of existence depends on the Sobolev norm of data.Moreover suppose the power of nonlinearity is at least cubic.
More specifically, our set-up is in the mass supercritical and energy subcritical regime, or equivalently, in which every u 0 ∈ H α 2 (R 2 ) has a local solution but not necessarily global; for blow-up criteria in the focusing mass supercritical case via localized virial estimates, see [2,6].
We specify the discretization described in the introduction.For h > 0, define where {e i } d i=1 generates Z d .The discretization converges to the continuum solution.
Theorem 3.1.Let p ≥ 3 and max( 8 7 , 2(p−1) p+1 ) < α < 2. For any arbitrary h ) be the well-posed solutions from Lemma 3.1 where (3.4) Remark 3.1.To estimate the nonlinear part of p h u h − u uniformly in h, we show that an appropriate space-time Lebesgue norm of u h is uniformly bounded in [0, T (∥u 0 ∥ H α 2 )] (see Lemma 4.2).However our proof is insufficient to conclude that a similar uniform bound holds in the energy-critical case, and therefore our method does not extend, at least directly, when α = 2(p−1) p+1 .Remark 3.2.The result is local in time, and thus it is of interest to extend (3.4) such that the estimate holds for t ∈ [0, T e ) where 0 < T e ≤ ∞ is the maximal time of existence of (2.2).This extension is not straightforward due to the existence of finite-time blow up solutions in the mass supercritical regime.For 2) were discretized by another means.Let A h be a self-adjoint linear operator on

4) and the triangle inequality.
It is expected that our approach would apply to a general class of discrete models governed by {A h }.A priori, A h is assumed to act on L 2 h and thus its extension to L 2 (R d ) needs to be defined, after which, the limit of Since the Fourier coefficients are absolutely integrable, ν h can be interpreted as a complex Borel measure on R d given by where δ y is the Dirac mass at y , and therefore where ∥ν h ∥ T V measures the total variation.
Proof.Since A h is a convolution against a finite measure with bounded symbol m h , A h is a translationinvariant bounded linear operator on ) by duality.By the Riesz-Thorin Theorem, we have The last equality is given by the fact that any translation-invariant bounded linear operator on given by a bounded multiplier on the Fourier space.
As an example, consider two classes of multipliers where . It can be verified that m h defines and A h , where A h is as (3.9), are divergent as h → 0 in the space of bounded linear operators on L 2 (R d ) in the uniform operator topology.On the other hand, , and hence the divergence by Proposition 3.1.The statements on strong convergence in L 2 follows as [25, Lemma 3.9], noting that followed by the Dominated Convergence Theorem to interchange the limit as h tends to zero and the integral in ξ, justified by A potential issue with m h in (3.8) is that the Euclidean metric | • | does not yield the physically-relevant distance between two points on hZ c|ξ| α for some non-zero constant c if and only if q = 2.For all q ∈ [1, ∞] \ {2}, an operator A q h defined by m q h is non-negative and self-adjoint on L 2 (R d ), defined on the dense domain H α (R d ).
Proof.Let ξ = |ξ|ξ ′ where ξ ′ ∈ S d−1 , the unit sphere.Let ρ(ξ ′ ) ∈ R d×d be a rotation operator that takes the d th standard basis vector to ξ ′ , i.e., ξ ′ = ρ(ξ ′ )e d .Then α < 2 justifies the limit of Riemann sum, and by changing variables, A priori, the integral in the last expression, call it I(ξ), reduces to a continuous function on S d−1 that is constant if and only if q = 2, observing the norm-invariance under rotation if and only if q = 2.The domain That A q h is non-negative and self-adjoint follows from I ≥ 0.
The continuum limit in higher dimensions, therefore, depends on the geometry of the underlying discrete model.This would potentially lead to complications on a spatial domain with an irregular lattice structure, which we leave as an open-ended thought.To this end, our analysis is restricted to (−∆ h ) where h and altogether resolves the identity The sum has an upper bound in N since hξ = O d (1).
Adopting the notations in [1], define subsets of M := T 2 \ {0} given by where w(ξ) The main proposition concerns a family of frequency-localized dispersive estimates with sharp time decay.Furthermore the lower bounds of implicit constants blow up both in the wave and Schrödinger limit. and For more details on the domain of N ∈ 2 Z that satisfy (3.10), see (5.18).By interpolating the estimates in (3.10), one obtains Corollary 3.1.Assume the hypotheses of Proposition 3.4.Then, Remark 3.4.Assuming Proposition 3.4, it is straightforward to obtain the Strichartz estimates for the linear evolution by averaging in t, N , which we briefly describe.Suppose ∥U h (t) As in [3, p.1127 Squaring both sides and summing in N , ) and hence by (3.13), The derivative loss occurs for α < 8  3 or α < 2 on hZ 2 (for all h > 0) or R 2 , respectively.
where τ > 0 without loss of generality, for τ < 0 amounts to taking the complex conjugate of J Φv,ζ .To show (3.10), observe that by the Young's inequality applied to the convolution in hZ d where Change variables ξ → ξ h and define τ = 2 α t h α , v = x hτ to obtain A priori since x ∈ hZ d , it follows that v ∈ τ −1 Z d , which we consider as a subset of R d .If σ 0 > 0 is the sharp decay rate for J Φv,ζ in the sense that sup holds for all τ ∈ R \ {0} for some C(ζ) > 0 and no bigger σ ′ > 0 satisfies (3.14), then Hence our goal reduces to obtaining (3.14) for a dyadic family of Littlewood-Paley functions.Outside of a neighborhood of the origin, Φ v is analytic, and therefore the major contributions to J are due to critical points ξ ∈ R d that satisfy ∇Φ v (ξ) = 0, or equivalently, v = ∇w(ξ) =: v ξ .In any arbitrary dimension,  d,h 1 (spacelike event), then J decays faster than τ −n for any n ∈ N, by integration by parts, with the implicit constant dependent on n and the distance between v and the light cone (see Lemma 5.1).Inside the light cone (including the boundary), J undergoes an algebraic decay due to critical points.For such v, it is generically true that the corresponding critical point(s) ξ ∈ T d are non-degenerate, and therefore J decays as τ − d 2 .However there exists a low-dimensional subset of T d that retards even further the decay rate of d 2 .We consider this problem of resolution of singularities for d = 2.
To systematically study the decay and asymptotics of J as a function of v and ζ, consider the Taylor series expansion of Φ around ξ such that ξ is the unique critical point in the support.Then J Φv ξ ,ζ has an asymptotic expansion as τ → ∞ where σ 0 , or the oscillatory index, is chosen to be the minimal number such that for any neighborhood of ξ, say U , there exists in particular, σ 0 depends only on the phase, not the smooth bump function.Under some hypotheses, σ 0 , d 0 are deduced from the higher order Taylor expansion of Φ v ξ (see Lemma 5.6), a process that we briefly describe.
Let Φ be a real-valued analytic function on a small neighborhood of the origin.Assume Φ(0) = 0, ∇Φ(0) = 0 and therefore the Taylor expansion of Φ at the origin in the multi-index notation is Define the Taylor support T = {α ∈ N d : c α ̸ = 0} and assume that Φ is of finite type, i.e., T ̸ = ∅.Define the Newton polyhedron of Φ, call it N , to be the convex hull of Let d = d(Φ) = inf{t : (t, t, . . ., t) ∈ N } be the distance from the origin to N .Since Φ is of finite type, 0 < d < ∞.Note that T is not invariant under analytic coordinate transformations.Let d x be the distance computed in the x coordinate system and define the height of Φ as h(Φ) = sup x d x where the supremum is over all analytic coordinate systems.The coordinate system (x) is adapted if d x = h.In R 2 , see [30, Proposition 0.7,0.8]for sufficient conditions for (x) to be adapted.An adapted system need not be unique.To obtain the asymptotics of oscillatory integrals, we work in a superadapted coordinate system defined specifically in dimension two in [8] as a coordinates system in which Φ pr (x, ±1) have no real roots of order greater than or equal to d x (Φ), possibly except x = 0.In particular, if d(Φ) > 1 and Φ pr (x, ±1) is a quadratic polynomial with no repeated roots, then (x, y) is superadapted.See the introductions of [1,8,21,30], from which this paper adopts all relevant terminologies, for a brief survey of the relationship between oscillatory integrals and Newton polyhedra.To illustrate these ideas, consider an example.Let Φ(x, y) = x 2 + y 2 + x 3 .Then Since Φ pr (x, ±1) = x 2 + 1 has no real root, the given coordinates system is superadapted.

Continuum limit.
The proof of Theorem 3.1 is given.
The implicit constants in the following estimates are independent of h > 0 and dependent only on β, d.
Remark 4.1.From (3.3), we have from which the definition of q in Lemma 4.2 is motivated.
Lemma 4.3.Assume the hypothesis of Theorem 3.1 and let q be given by Lemma 4.2.Given . By Lemma 3.1, let u, u h be the well-posed solutions corresponding to initial data u 0 , u 0,h .Then u, u h satisfy ∥u∥ Proof.The estimate for u is derived from the proof of local well-posedness by the contraction mapping argument.Similarly the time of existence for the discrete evolution that ensures the estimate (4.1) is . By Lemma 4.1, T h ≳ T uniformly in h, and therefore u, u h are well-defined on , unique in a (smaller) Strichartz space (see [14,Theorem 1.1]), satisfying h and the global solution given by Lemma 3.1.Similarly as above, Following the proof of [16, Theorem 1.1], we have and altogether, By the Gronwall's inequality, Non-degenerate critical points are treated by the method of stationary phase.A well-known asymptotics [29, Chapter 8, Proposition 6] is given.Lemma 5.2.Let ξ ∈ M, v ∈ R 2 satisfy v = ∇w(ξ) and det D 2 w(ξ) ̸ = 0. Then there exists a small neighborhood around ξ such that for all ζ ∈ C ∞ c supported in the neighborhood, as τ → ∞ where If D 2 w(ξ) is singular, i.e., if ξ ∈ M satisfies H(ξ, α) := det D 2 w(ξ) = 0, then the asymptotic formula of Lemma 5.2 is not applicable; ξ = 0 is not considered since the oscillatory integrals from Proposition 3.4 have test functions supported outside of the origin.Note that Since h, h are symmetric under ξ 1 → ±ξ 1 , ξ 2 → ±ξ 2 , (ξ 1 , ξ 2 ) → (ξ 2 , ξ 1 ), the domain of analysis could be restricted to the first quadrant either above or below the identity ξ 1 = ξ 2 .By definition, h is nonzero on M and therefore the roots of H correspond to those of h.Following the approach in [1], the representation of h is in polynomials under the change of variables a = cos ξ 1 , b = cos ξ 2 , and is given by For brevity, let 2 , E α is a smooth one-dimensional embedded submanifold by the Implicit Function Theorem.This is false for α = 2; a cusp (a, b) = (0, 0), which corresponds to ξ = (± π 2 , ± π 2 ), appears when α = 2 since ∇ (a,b) h(0, 0) = 0.By direct computation, Hence there exists a smooth map E\{0} ∋ (a, b) → α ∈ (1, 2) satisfying h(a, b, α) = 0 by the Implicit Function Theorem, observing that ∂ α h(a, b, α) = (a + b)(ab − 1) is non-vanishing on E \ {0}.Observe that E α consists of two connected components; one component, say Γ 1 (a,b) (α), passes through the origin whereas the other, say Γ 2 (a,b) (α), does not.As α → 1+, Γ 2 (a,b) (α) becomes arbitrarily close to (a, b) = (1, 1) whose corresponding point in T 2 , the origin, is not in M .In fact, See Figure 1 for the contour plots of E α .Let {e 1 , e 2 } be the standard basis of R 2 , and with a slight abuse of notation, consider {e 1 , e 2 } as a global orthonormal frame of T 2 .For each ξ ∈ M , let {k 1 (ξ), k 2 (ξ)} be an orthonormal basis of T ξ M , the tangent space at ξ, with the coordinate system (x, y), i.e., for all v ∈ T ξ M , there exists unique (x, where The contour plots of E α and its correspondence in (ξ 1 , ξ 2 ) ∈ M are given for different values of α.
To further investigate the higher order directional derivatives, [1, Lemma 5.4] is extended by induction and the product rule for derivatives whose proof is immediate and hence omitted.
The inflection points of sin 2 ( ξi 2 ) persist to exist as singular points of D 2 w(ξ) even when α < 2. By symmetry, the qualitative behavior of J Φv,ζ is the same near each point in K 3 .
Take the former; the proof for the latter is similar and thus is omitted.First assume sin ξ 2 ̸ = 0.
In the (a, b) coordinates, ∇ ξ h = −(sin ξ 1 ∂ a h, sin ξ 2 ∂ b h), and our task reduces to solving by applying Lemma 5.3 with m = 2. Modulo sin ξ 2 , (5.3) is a polynomial equation of degree 3 in b (or a) and therefore can be solved explicitly.The intersection of h(a, b, α) = 0 and (5.3) occurs at (a, b) = (0, 0).If ξ lies at the intersection of sin ξ 2 = 0 and h(ξ, α) = 0, then it can be directly verified that the left vector of (5.2) is zero whereas the right vector is a scalar multiple of 1 0 .Then the claim can be proved similarly as before.
The higher-order derivatives at critical points determine the height of the phase function.
Lemma 5.5.For all ξ ∈ M , Proof.Let ξ / ∈ E α .In any given coordinate system, say (x, ỹ), if xw(ξ), ∂ 2 ỹ w(ξ) ̸ = 0 due to non-degeneracy.In either case, d (x,ỹ) = 1.Taking supremum over all such coordinate systems, the first claim has been shown.The rest follows similarly as [1, Lemma 3.1] using Lemma 5.4.In particular, the Newton diagrams for ξ ∈ E α \ K 3 and ξ ∈ K 3 are given by The computation of heights depends only on the non-zero Taylor coefficients of Φ v , and therefore does not reflect the variations on α.However the leading terms of asymptotics (3.16) depend on α.Define As in (3.16),I has an asymptotic expansion as ϵ → 0+, where {r j } is an increasing arithmetic sequence of positive rational numbers such that the (minimal) r 0 is determined only by the phase function that renders at least one of c 0 , c ′ 0 , C 0 , C ′ 0 non-zero.For 0 ≤ m ≤ ∞, let − 1 m be the slope of the subset of N d that the bisectrix intersects.Define Φ + v ξ ,pr (x, ±1) = Φ v ξ ,pr (x, ±1) if Φ v ξ ,pr (x, ±1) > 0 and zero otherwise.A summary of [8, Theorem 1.1,1.2]that applies to our case is given.Lemma 5.6.Let ξ ∈ E α .Then Φ v ξ = Φ v ξ (x, y) in the coordinate system defined by {k 1 (ξ), k 2 (ξ)} is superadapted.The slowest decay of the asymptotics is given by σ 0 = r 0 = 1 h(Φv ξ ) .The leading terms have vanishing logarithmic terms, i.e., c ′ 0 = C ′ 0 = 0, and moreover where C 0 is computed similarly by replacing Φ v ξ by its negative.
Since ∂ 3 y w(P ) = 0 for all P ∈ K 3 (see Lemma 5.4), the decay of J is the slowest on K 3 .Recalling that E α , it is of interest to determine d 0 (α) on K 3 .Lemma 5.7.Let ξ ∈ K 3 and let ζ ∈ C ∞ c be supported in a small neighborhood around ξ in which ξ is the unique critical point of Φ v ξ .Then where c ∈ C \ {0} is independent of ζ and α.
Proof.Without loss of generality, let ξ = ( π 2 , π 2 ).Define where the linear span of {k 1 , k 2 } is coordinatized in (x, y).Using (ξ 1 , ξ 2 ) as a coordinate for {e 1 (0), e 2 (0)}, we have We first claim (x, y) defines a superadapted system for Φ v ξ at (x, y) = (0, 0).Note that Φ v ξ ,pr = −w pr and The second order derivatives are and the third order derivatives are By direct computation, it can be verified that ∂ j y w(ξ) = 0 for all j ≥ 2. These derivatives determine the Newton polyhedron, Newton diagram, and w pr given by (5.6) The bisectrix intersects by using (5.6); the computation of C 0 , which amounts to replacing Φ v ξ by −Φ v ξ , is similar and thus is omitted.The conclusion of lemma follows immediately from the explicit form of d 0 in (5.4) and (5.7).By analogy with the proof of Lemma 5.7, it is expected that the reciprocal of the distance of this diagram, 2  3 , yields the sharp decay rate of the corresponding oscillatory integral.Indeed, this expectation coincides with the result obtained in [28].
It is insightful to apply (5.10) Several comments regarding B P , B are summarized below.The following lemma can be verified by direct computation using (5.10).
Let η N,j (•) = η( • N )ϕ j (•).Then ∥η N,j ∥ C 3 ≲ N −3 where the implicit constant depends on the given partition of unity.Since N > N α , we have N and the uniform bound sup

α
+ {x ∈ R d : x i ≥ 0}, and the Newton diagram N d to be the union of all compact faces of N .Let N pr , the principal part of Newton diagram, be the subset of N d that intersects the bisectrix {x 1 = x 2 = • • • = x d }.Define the principal part of Φ (or the normal form) as Φ pr (x) = |α|≥2, α∈Npr c α x α .

. 2 )
Applying (4.1) to (4.2), we obtain(3.4)where C 1 , C 2 > 0 depend on various parameters including ∥u 0 ∥ H α 2 , but not h.This completes the proof.5 Proof of Proposition 3.4.Let α ∈ (1, 2) unless otherwise specified.For ζ ∈ C ∞ c (M ), the quantity J Φv,ζ is at worst a non-degenerate integral almost everywhere with respect to the Lebesgue measure on R 2 1) is well-defined for ξ ∈ M and blows up (in the sense of determinant) as ξ → 0. It can be shown directly that D 2 w(ξ) is not the zero matrix for any ξ ∈ M .If D 2 w(ξ) is of full rank, then the decay of J Φv ξ ,ζ can be analyzed via Lemmas 5.1 and 5.2, and henceforth suppose rank(D 2 w( are the directional derivatives.Suppose ∂ yy w(ξ) = 0 for all ξ ∈ E, or equivalently, k 2 (ξ) is the direction along which D 2 w(ξ) is degenerate.Then it follows that ∂ xx w(ξ) ̸ = 0 since D 2 w(ξ) is not a zero matrix.Due to diagonalization, ∂ xy w(ξ) = 0.

( 5 . 8 )
to obtain the series expansion of d 0 .For {(a, b) : h(a, b, α) = 0}, it suffices to consider a ≥ b or a ≤ b by the symmetry of h under (a, b) → (b, a).Define the two roots of h(a, b, α) = 0 in terms of a, α as