$L^{p}$ estimates for joint quasimodes of semiclassical pseudodifferential operators whose characteristic sets have $k$th order contact
Melissa Tacy


  
In this paper we develop L^p estimates for functions u which are joint quasimodes of semiclassical pseudodifferential operators p_1(x,hD) and p_2(x,hD) whose characteristic sets meet with kth order contact, k≥1. As part of the technical development we use  Fourier integral operators to adapt a flat wavelet analysis to the curved level sets of p_1(x,ξ).

Data-Driven Robust Stabilization with Robust Domain of Attraction Estimate for Nonlinear Discrete-Time Systems footnoteinfo
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    Received May 29, 2019; revised July 5, 2019; accepted
July 10, 2019. Date of publication July 31, 2019 
 date of current version
August 21, 2019
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Let (M,g) be a two dimensional compact, boundaryless Riemannian manifold and p_1(x,hD), p_2(x,hD) be two semiclassical pseudodifferential operators L^2(M)→ L^2(M). In this paper we consider the question of the concentration properties of a function u that approximately solves

    p_1(x,hD)u=0  and   p_2(x,hD)u=0.

In particular we ask about the growth rate of u_L^p compared to u_L^2. A key example to keep in mind is when one of the equations requires that u be an approximate solution to the eigenfunction equation,

    -Δ_g u=λ^2u.

Equation (<ref>) can be converted to a semiclassical equation by dividing through by λ^2 and setting h=λ^-1. Then, we require that u satisfies the semiclassical equation

    (-h^2Δ_g-1)u=0.

In <cit.> Sogge shows that eigenfunctions (in fact more generally spectral clusters) of the Laplacian obey

    u_L^p≲λ^δ(p)u_L^2

where

    δ(p)=1/2-2/p   6≤p≤∞
    1/4-1/2p   2≤ p≤ 6.

Koch, Tataru and Zworski <cit.> extend this result to approximate solutions of any semiclassical equation p(x,hD)u=0 where the symbol p(x,ξ) displays sufficiently Laplace-like behaviour. These results are sharp in the sense that there exist examples that saturate the estimates given by (<ref>). 

In his letter to Morawetz <cit.> Sarnak poses the question of potential improvements when u is (in addition to being a Laplacian eigenfunction) an eigenfunction of r other differential operators. He obtains L^∞ results under the assumption that M is a rank r symmetric space. Marshall then <cit.> extends Sarnak's result to a full set of L^p estimates. 

In <cit.> Tacy examines this problem from the perspective of contact between the characteristic hypersurfaces {ξ| p_j(x,ξ)=0} and obtains sharp results under the condition that if ν_j(x,ξ) is the normal to {ξ| p_j(x,ξ)=0} then ν_1,…,ν_r are linearly independent. In the two dimensional case this means that, for all x, the sets {ξ| p_1(x,ξ)=0} and {ξ| p_2(x,ξ)=0} meet with order 0 contact. The results of <cit.> give that in that case

    u_L^p≲u_L^2  ∀ p≥2.

In this paper we address the case where {ξ| p_1(x,ξ)=0} and {ξ| p_2(x,ξ)=0} meet with higher order contact. 

 Before we state our main theorem, let us consider first what estimates we might reasonably expect. We study the flat model example where p_1(x,ξ)=|ξ|^2-1 and p_2(x,ξ)=p_2(ξ) is a smooth curve such that {p_2(ξ)=0} intersects the circle at (1,0) with order k contact. We will assume that u is an order h joint quasimode, that is
 
    (-h^2Δ_^n-1)u_L^2≲ hu_L^2  and  p_2(hD)u_L^2≲ hu_L^2.

 This level of error is a very natural choice (as discussed in Section <ref>). We will use the Fourier transform method to analyse such functions u. In particular we work with the semiclassical Fourier transform
 
    [u](ξ)=1/(2π h)^n/2∫ e^-i/h⟨ x,ξ⟩u(x)dx.

 With this normalisation  is still an isometry on L^2 and has the property that
 
    [hD_x_iu]=ξ_i[u]  and  [p(hD)u]=p(ξ)[u].

 So from the first semiclassical equation we have that
 
    (|ξ|^2-1)[u]=O_L^2(h).

 That is [u] should live, predominately, in the annulus of width h around |ξ|^2=1.  The second equation gives us that
 
    p_2(ξ)[u]=O_L^2(h).


So in addition [u] cannot be too large away from the set p(ξ)=0. We have assumed that {p_1(ξ)=0} and {p_2(ξ)=0} meet at a single point where they have kth order contact. The order of the contact controls how close the curves are to each other. If we write them locally as graphs around ξ_2=0,

    ξ_1=a_1(ξ_2)

and

    ξ_1=a_2(ξ_2)

then to have kth order contact the first k derivatives must agree.  Once |a_1(ξ_2)-a_2(ξ_2)|>ch it is impossible for the Fourier transform of u to be simultaneously located in a order h region of both curves. Therefore, since the contact is of order k, we would expect [u] to be small outside the region |ξ_2|≤ h^1/k+1, see Figure <ref>.
 
 Now let's consider what kinds of quasimodes we can construct within such restrictions. We will use the family of examples, T^h_α(x) from <cit.> given by
 
    T^h_α(x)=h^-3+α/2e^i/hx_1/2π∫_^2e^i/h(x_1(ξ_1-1)+x_2,ξ_2)χ_α(ξ)dξ

 where  ω_0 corresponds to the unit vector in the ξ_1 direction and
 
    χ_α(ξ)=χ_α(r,ω)=1    |r-1|<h and |ω-ω_0|<h^α
    
     0    otherwise.
 
 
 



 Clearly T^h_α(x) is an order h quasimode of (-h^2Δ_^2-1).  In <cit.> it is demonstrated that T^h_α_L^2=1 and for some ϵ>0 sufficiently small,
 
    |T^h_α(x)|>ch^-1/2+α/2

 on a ϵ h^1-2α×ϵ h^1-α tube about (0,0) with long direction e_1. The contact condition means that we should only consider the behaviour of the examples where h^α≤ h^1/k+1. Since the case where k=0 is treated in <cit.> we focus on k≥1. Notice that h^1/2≤ h^1/k+1 for  k≥ 1. Therefore the T^h_1/2(x) are always joint quasimodes. These examples have the property that
 
    T_1/2^h_L^p>c h^-1/4+1/2p.

 So for 2≤ p≤ 6 we can expect no improvement over Sogge's estimate. To saturate the high p estimates one usually picks α so that the support of [u] is spread through the largest possible region. In this case that is α=α_k so that h^α_k=h^1/k+1. That is α_k=1/k+1 (if α is any smaller than α_k, T_α^h(x) will not be a good quasimode of p_2(hD)). Now

    T_α_k^h_L^p   >ch^-1/2+1/2(k+1)(h^1-2/k+1h^1-1/k+1)^1/p
       =ch^-1/2+2/p+1/k+1(1/2-3/p).

This tells us that we could never expect a better upper bound than (<ref>). In Theorem <ref> we see indeed that this is the highest growth rate possible. 


Suppose u is a semiclassically localised,  strong joint  quasimode of order h (see Definitions <ref> and <ref> for the definitions of semiclassically localised and strong joint quasimode) for a pair of semiclasscial pseudodifferential operators p_1(x,hD), p_2(x,hD) where the symbols p_j(x,ξ) obey the following admissibility conditions.


  * For each x_0 and j=1,2 the set {ξ| p_j(x_0,ξ)} is a smooth hypersurface.

  * For each x_0 the sets {ξ| p_j(x_0,ξ)} meet at a single point ξ_0 and at that point have kth order contact.

  * There is some j such that for all x_0, the sets {ξ| p_j(x_0,ξ)=0} have non-degenerate second fundamental form.



Then

    u_L^p≲ h^-δ(p,k)u_L^2,

where

    δ(p,k)=
    -(1/2-2/p)+1/k+1(1/2-3/p)   6≤ p≤∞
    
    -1/4+1/2p   2≤ p≤ 6.





It is necessary only to consider the case where u is a strong joint quasimode of order h of two operators. Suppose u is a strong joint quasimode of order h  of r operators p_i(x,hD) with orders of contact k_ij. Then the best estimate for the L^p norms of u will come from applying Theorem <ref> to the pair p_i(x,hD),p_j(x,hD) with smallest contact order k_ij. 


In this paper we will many times rely on a number of standard results from semiclassical analysis. In particular those regarding the composition and invertibility properties of semiclassical pseudodifferential operators and the development of parametrix constructions for propagators. The relevant results are listed in Appendix A for the convenience of the reader. 



§ COMBINING WAVELETS WITH FOURIER INTEGRAL OPERATORS


To obtain the results of Theorem <ref> we develop a way to combine the theory of Fourier integral operators with that of wavelet analysis. The basic idea dates back to some of the earliest successes in microlocal analysis. In <cit.> Fefferman describes “the algorithm of the 70s” for understanding the L^2 theory of PDEs with variable coefficients. Consider a partial differential equation

    Pu=0  where   P=∑_|γ|≤Nc_γ(x)D^γ.

This can be expressed as a pseudodifferential equation p(x,D)u=0 where

    p(x,D)u=1/(2π)^n∫ e^i⟨ x-y,ξ⟩p(x,ξ)u(y)dξ dy

and

    p(x,ξ)=∑_|γ|≤Nc_γ(x)ξ^γ.

Standard considerations about the invertibility of pseudodifferential operators ensure that if Pu=0, the microsupport of u must lie inside the set {p(x,ξ)=0}. The simplest manifestation of “the algorithm of the 70s” is the case where {p(x,ξ)=0} is a smooth hypersurface. The idea is to (after localisation) straighten {p(x,ξ)=0} out to become the hypersurface {ξ_1=0}. The solutions to D_x_1u=0 are simple to understand, this algorithm allows  information to be carried back to the more complicated Pu=0. In particular one develops a unitary Fourier integral operator W that has the property that

    D_x_1W=Wp(x,D).

Then if v=Wu and Pu=0  we have that v is a solution to D_x_1v=0 (or an approximate solution if u is only an approximate solution to Pu=0). Exactly the same technique can be applied to semiclassical pseudodifferential operators, except in this case we have

    hD_x_1W=Wp(x,hD).

Since W is unitary the L^2 theory for u follows from the L^2 theory for v. However it is immediate that the L^p theory cannot so directly follow. Take for example p(x,ξ)=ξ_1-ξ_2^2 localised near (0,0). The hypersurface {ξ_1-ξ_2^2=0} can be flattened out to {ξ_1=0} by a suitable Fourier integral operator. Note that {ξ_1=0} is flat and {ξ_1-ξ_2^2=0} is curved. Classical theory on the Fourier restriction/extension problem (see for example Stein <cit.> Chapter 9 for an authoritative source) tell us that the L^p theory of solutions to Pu=0 depend crucially on the curvature of the characteristic set. This information about the curvature is encoded in the L^2→ L^p mapping properties of W^-1. 

In this paper we will use “the algorithm of the 70s” but incorporate some wavelet theory. Suppose we have an operator W so that

    hD_x_1W=Wp_1(x,hD)

and we assume that p_1(x,ξ) is the symbol that satisfies the curvature condition of Theorem <ref>. Then if v=Wu, we have that v is an order h quasimode of hD_x_1u=0. That is 

    hD_x_1v_L^2≲ hu_L^2.

Therefore the semiclassical Fourier transform of v, [v] must be supported near {ξ_1=0}. This property makes v very suitable for a wavelet decomposition in the x_1 variable. To that end let f be smooth compactly supported function with

    ∫ f(τ)dτ=0  and   C_f=∫|f̂(ξ)|^2/|ξ|dξ<∞.

Then using the continuous wavelet transform we can write

    v=1/C_f∫1/|a|^5/2X_v(a,b,x_2)f(a^-1(x_1-b))dadb

where

    X_v(a,b,x_2)=1/|a|^1/2∫ f(a^-1(y_1-b))v(y_1,x_2)dy_1.

Since v is a good approximate solution to hD_x_1v=0 we expect that the main contribution to the integral comes from where |a|∼ 1. We write

    u=W^-1v=1/C_f∫1/|a|^5/2W^-1[X_v(a,b,x_2)f(a^-1(x_1-b))]dadb.

We also know that u is a quasimode of p_2(x,hD). Heuristically W straightens out {p_1(x,ξ)=0} to become {ξ_1=0} so we would expect that {p_2(x,ξ)=0} transforms to a curve that meets {ξ_1=0} with order k contact (as depicted in Figure <ref>). We may as well assume that this intersection point is (0,0). Then near (0,0), p_2(x,ξ) should have local form p_2(x,ξ)=ξ^k+1g(x,ξ) where g(0,0)≠0. So we would expect that v is an order h quasimode of h^k+1D^k+1_x_2 in addition to hD_x_1=0. 

This is in fact what happens (see Proposition <ref>). Therefore to incorporate the fact that h^k+1D^k+1_x_2v=O_L^2(h) we take a Fourier transform of v in x_2 and dyadically decompose into regions |ξ_2|∼ 2^jh^1/k+1. 

Another way to think of this is to define

    [X](a,b,ξ_2)   =[X(a,b,·)](ξ_2)
       =1/(2π h)^1/2∫ e^-i/h⟨ x_2,ξ_2⟩1/|a|^1/2f(a^-1(x_1-b))v(x_1,x_2)dx_2dx_2.

Then

    u   =W^-1v
       =W^-1[1/(2π h)^1/2∫ e^i/h⟨ x_2,ξ_2⟩1/|a|^5/2[X](a,b,ξ_2)f(a^-1(x_1-b))dadbdξ_2]
       =1/(2π h)^1/2∫1/|a|^5/2[X](a,b,ξ_2)W^-1[e^i/h⟨ x_2,ξ_2f(a^-1(x_1-b))]dadbdξ_2
       =1/(2π h)^1/2∫1/|a|^5/2[X](a,b,ξ_2)ψ_a,b,ξ_2(x_1,x_2)dadbdξ_2

where

    ψ_a,b,ξ_2=W^-1[e^i/h⟨· ,ξ_2⟩f(a^-1(· -b))].

Here we are seeing (<ref>) as synthesis for u in terms of ψ_a,b,ξ_2. This can be understood as a specific example of more general phenomena. Suppose Λ is a parameter space and T_v(λ) is an analysis operator

    T_v(λ)=⟨ v,ϕ_λ⟩

with synthesis

    v=∫ T_v(λ)ϕ_λ(x)dμ(λ)

for some functions ϕ_λ and measure μ on Λ. Then if

    (T∘ W)_u(λ)=T_Wu(λ)=⟨ Wu,ϕ_λ⟩=⟨ u,W^⋆ϕ_λ⟩=⟨ u,W^-1ϕ_λ⟩.
 
We can see (T∘ W)_u(λ) as an analysis operator for u with synthesis

    u=∫ (T∘ W)_u(λ)ψ_λ(x)dμ(λ)

where ψ_λ=W^-1(ϕ_λ). Therefore we may take  an appropriate analysis/synthesis of L^2(^n) and use W^-1 to produce a new analysis/synthesis that is adapted to the operator p(x,hD). 





§ QUASIMODES AND JOINT QUASIMODES

The focus of this paper is on u that satisfy

    (-h^2Δ_g-1)u=0

or some similar semiclassical equation. By working in coordinate charts and associating each patch with a patch on ^n, we can write the operator -h^2Δ_g-1 as a semiclassical quantisation of a symbol p(x,ξ) which has principal symbol |ξ|_g^2-1. Here we use the left quantisation

    (-h^2Δ_g-1)u=p(x,hD)u=1/(2π h)^n∫ e^i/h⟨ x-y,ξ⟩p(x,ξ)u(y)dydξ.

Since we must localise to make sense of (<ref>) it is reasonable to only consider those u which are semiclassically localised in phase space.


We say that u is semiclassically localised if there is a smooth, compactly supported function χ:T^⋆M→ so that

    u=χ(x,hD)u+O(h^∞).



Localisation means that we will need to work with approximate solutions rather than exact ones. The commutation property for semiclassical pseudodifferential operators, Proposition <ref>, tells us that even if p(x,hD)u=0,

    p(x,hD)χ(x,hD)u=χ(x,hD)p(x,hD)u+hr(x,hD)u=hr(x,hD)u

that is χ(x,hD)u is only an approximate solution with error O_L^2(hu_L^2). Therefore it make sense to work with approximate solutions (quasimodes) with error O_L^2(hu_L^2) from the start. 


We say that u is a quasimode of  order h^β (sometimes written as O_L^2(h^β) or O(h^β))  of p(x,hD) if

    p(x,hD)u_L^2≲h^βu_L^2.

If u is a joint quasimode of order h^β  of p_1(x,hD) and p_2(x,hD) then

    p_i(x,hD)u_L^2≲h^βu_L^2   i=1,2.



Definition <ref> is enough to produce the L^p estimates for quasimodes considered in <cit.>, <cit.> and <cit.>. However for this work as in <cit.> we will need a slightly stronger kind of quasimode. As discussed in <cit.> if we start with a exact solution u we could produce a quasimode v by taking

    v=u+hf

for some function f_L^2=1. However such  examples are rather artificial. To avoid this, as in <cit.>, we will work with strong quasimode. Strong quasimodes have the property  that repeated application of p(x,hD) continues to improve the quasimode error. 


We say that u is a strong quasimode of order h^β (O^str_L^2(h^β) or O^str(h^β))  of p(x,hD) if

    p^M(x,hD)u_L^2≲h^β Mu_L^2,   M=1,2,…

If u is a  strong joint quasimode of order h^β of p_1(x,hD) and p_2(x,hD) then

    p^M_1_1(x,hD)∘ p^M_2_2(x,hD)u_L^2≲h^β (M_1+M_2)u_L^2,   i=1,2  M_i=1,2,…

In some cases (where the comparison to u_L^2 is important) we will write O^str_L^2(h^βu_L^2) to indicate (<ref>) holds. 


Clearly an exact solution u,

    p(x,hD)u=0

is a strong quasimode. As discussed in <cit.> spectral clusters (a major example of quasimodes) are also strong quasimodes. A spectral cluster of window width W=W(λ) is given by,

    u=∑_λ_j∈[λ,λ-W(λ)]c_jϕ_j  with  -Δϕ_j=λ_j^2ϕ_j.

Such functions are strong order W(h^-1)h quasimodes of (-h^2Δ_g-1) when h=λ^-1.

We have seen that the commutation identity implies that the property of being an order h quasimode is preserved under localisation.  That is, if u is an order h quasimode of p(x,hD), χ(x,hD)u is also an O_L^2(h) quasimode of p(x,hD). This property also holds for strong joint quasimodes (see Proposition  1.4 in <cit.>). If u is a  strong joint quasimode of order h of p_1(x,hD) and p_2(x,hD) then χ(x,hD)u is also a  strong joint quasimode of order h of p_1(x,hD) and p_2(x,hD). Similarly, in Lemma <ref>, we will see that if u is a  strong joint quasimode of order h of p_1(x,hD) and p_2(x,hD) it is also a  strong joint quasimode of order h of p_1(x,hD) with linear combinations of p_2(x,hD) and p_1(x,hD).




Suppose u is a strong joint  quasimode of order h for p_1(x,hD) and p_2(x,hD) and e_1(x,hD), e_2(x,hD) are any semiclassical pseudodifferential operator with smooth symbols. Then u is also a  strong joint quasimode  of order h for any pairs consisting of elements of 
    {p_1(x,hD), p_2(x,hD), (p_1(x,hD)e_1(x,hD) + p_2(x,hD)e_2(x,hD))}.




Given any composition q(x,hD)p(x,hD) the semiclassical calculus allows us to commute with loss of an O_L^2(h) term (Proposition <ref>) so

    q(x,hD)p(x,hD)=p(x,hD)q(x,hD)+hr(x,hD)

where r(x,hD) maps L^2→ L^2 with norm bounded independent of h.  Expanding 

    (p_1(x,hD)e_1(x,hD) + p_2(x,hD)e_2(x,hD))^M

 and commuting terms as necessary we have that

    (p_1(x,hD)e_1(x,hD)+p_2(x,hD)e_2(x,hD))^M=∑_m=0^M∑_j=0^M-mh^mr_j,m(x,hD)p_1(x,hD)^jp_2(x,hD)^M-m-j

where all the r_j,m(x,hD) map L^2→ L^2 with norm bounded independent of h. So u is indeed a  strong joint  quasimode of order h of p_1(x,hD)e_1(x,hD)+p_2(x,hD)e_2(x,hD) and either p_1(x,hD) or p_2(x,hD). 






Now suppose we have the hypotheses of Theorem <ref>. That is, u is a  strong joint quasimode of order h of p_1(x,hD) and p_2(x,hD). We will see that, as in <cit.>, it is enough to estimate χ(x,hD)u_L^p where χ(x,ξ) is a smooth function supported in a small (but h independent) region about a point (x_0,ξ_0),

    (x_0,ξ_0)∈⋂_i=1^2{(x,ξ)| p_i(x,ξ)=0}.

To see that such a localisation is valid  consider χ(x,hD)u where χ(x,ξ) is supported near a point (x_0,ξ_0) such that p_i(x_0,ξ_0)≠0.   From Proposition <ref> we know that if |p_i(x,ξ)|>c>0, the operator p_i(x,hD) is invertible and its inverse (p_i(x,hD))^-1 has bounded mapping norm L^2→L^2. By choosing the support of χ small enough we may assume that p_i(x,ξ) is bounded away from zero on the support of χ and therefore so is p^M_i(x,ξ). Proposition <ref> tells us that p^M_i(x,ξ) is the principal symbol of p^M_i(x,hD) so by Proposition <ref> we can produce an inverse (p^M_i(x,hD))^-1. Therefore if 

    p^M_i(x,hD)χ(x,hD)u=h^Mf  f_L^2≲u_L^2,

we can invert p_i(x,hD) to obtain

    χ(x,hD)u=h^M(p^M(x,hD))^-1f

and
 
    χ(x,hD)u_L^2≲ h^Mu_L^2.
 
 By applying semiclassical Sobolev estimates <cit.> we obtain

    χ(x,hD)u_L^p≲h^-n/2+n/p+Mu_L^2.

Choosing M large enough we obtain better estimates than those of Theorem <ref>. So we need only consider χ(x,hD)u where χ(x,ξ) is supported in a neighbourhood of some point (x_0,ξ_0) where both of  the p_i(x_0,ξ_0)=0. We may as well assume this point is (0,0).

Assume that p_1(x,ξ) satisfies the curvature condition.   Since {ξ| p_1(x_0,ξ)} is a smooth hypersurface in ^2 we may assume that (after a suitable change of coordinate system) that ∂_ξ_1p_1(0,0)≠ 0. By taking a suitable localisation we can extend this to the property that ∂_ξ_1p_1(x,ξ) is bounded away from zero on the support of χ(x,ξ). Therefore we may write

    p_1(x,ξ)=e_1(x,ξ)(ξ_1-a(x,ξ_2))

where |e_1(x,ξ)|>c>0. So the invertibility of e_1(x,hD) ensures that u is also a quasimode of (hD_x_1-a(x,hD_x_2)). Since the characteristic sets of meet with contact of at least order one at (0,0) we can also factorise p_2(x,ξ) as

    p_2(x,ξ)=e_2(x,ξ)(ξ_1-q(x,ξ_2))

where |e_2(x,ξ)|>c>0. Therefore (since e_2(x,hD) is invertible) u is a quasimode of (hD_x_1-q(x,hD_x_2)).
Now let W(x_1):L^2() → L^2() be the operator so that

    hD_x_1W(x_1)=-W(x_1)a(x,hD_x_2)

or equivalently 

    (hD_x_1-a(x,hD_x_2))W^⋆=0.

We can write (see Proposition <ref>)

    Wg=1/2π h∫ e^i/h( x_2ξ_2-ϕ(x_1,y_2,ξ_2))b(x_1,y_2,ξ_2)g(y_2)dξ_2dy_2

where 

    ∂_x_1ϕ+a(x_1,y_2,∂_y_2ϕ)=0  ϕ(0,y_2,ξ_2)=⟨ y_2,ξ_2⟩

and

    b(0,y_2,ξ_0)=1.

It is a classical result (see Zworski <cit.> Part 3, particularly Chapters 10-11) that

    W^⋆(x_1)W(x_1)=  and   W(x_1)W^⋆(x_1)=.

Let 

    v(x_1,x_2)=W(x_1)u(x_1,·)

 then

    u=W^⋆v.

So to obtain Theorem <ref> we need to to obtain a bound

    W^⋆v_L^p≲ h^-δ(n,p,k)u_L^2.

First we see that v is a strong joint quasimode of hD_x_1 and h^k+1D^k+1_x_2. 


Suppose W(x_1) and v(x_1,x_2) are given by (<ref>) and  (<ref>) respectively. Then v is a  strong joint  quasimode of order h of the operators hD_x_1 and h^k+1D^k+1_x_2. 




Note that

    (hD_x_1)^Mv   =hD_x_1(W(x_1)u(x_1,·))
       =W(x_1)(hD_x_1-a(x,hD_x_2)^Mu.

Therefore since W(x_1) is unitary

    (hD_x_1)^Mv_L^2_x_2≲(hD_x_1-a(x,hD_x_2))^Mu_L^2_x_2.

Using the fact that v is localised to an O(1) region gives us

    (hD_x_2)^Mv_L^2≲ h^Mu_L^2.

So v is certainly a strong quasimode of order h of hD_x_1. 

Recall that the factorisation of p_2(x,ξ) ensures that u is a quasimode of 

    (hD_x_1-q(x,hD_x_2))u=0.

We have assumed that we are localised about the point (0,0) where both p_1(0,0)=0=p_2(0,0), which implies a(0,0)=0=q(0,0). Recall  that W^⋆(x_1) is the propagator for the time evolution equation

    (hD_x_1-a(x,hD_x_2))W^⋆(x_1)=0

where x_1 acts as the time variable.  Define the classical system

    ẋ_2(x_1)=∂_ξ_2 a(x_1,x_2,ξ_2)
    
    x_2(0)=x_2
    ξ̇_2(x_1)=-∂_x_2a(x_1,x_2,ξ_2)
    ξ_2(0)=ξ_2

and let

    q̃(x,ξ_2)   =q(x_1,x_2(x_1),ξ_2(x_1))
    ã(x,ξ_2)   =a(x,x_2(x_1),ξ_2(x_1)).

 Egorov's theorem tells us that

    q̃(x,hD_x_2)   =W(x_1)q(x,hD_x_2)W^⋆(x_1)+hr_1(x,hD)
       and
    ã(x,hD_x_2)   =W(x_1)a(x,hD_x_2)
    )W^⋆(x_1)+hr_2(x,hD),

where both r_i(x,hD):L^2→ L^2 are bounded independent of h. So if we let

    p̃_2(x,ξ)=ξ_1+ã(x,ξ_2)-q̃(x,ξ_2)

then

    (hD_x_1)^M_1(p̃_2(x,hD))^M_2v   =(hD_x_1)^M_1(p̃_2(x,hD))^M_2-1W(hD_x_1-q(x,hD_x_2)u
           +h(hD_x_1)^M_1p_2(x,hD)^M_2-1r_1(x,hD)W(x_1)u
         ⋮
       =∑_j=0^M_2h^j(hD_x_1)^M_1R_j(x,hD)W(hD_x_1-q(x,hD_x_2))^M_2-ju
       =∑_j=0^M_2h^j(hD_x_1)^M_1-1R_j(x,hD)W(hD_x_1-a(x,hD_x_2))(hD_x_1-q(x,hD_x_2))^M_2-ju
           +∑_j=0^M_2h^j+1(hD_x_1)^M_1-1R_j,1(x,hD)W(hD_x_1-q(x,hD_x_2))^M_2-ju
         ⋮
       =∑_m=0^M_1∑_j=0^M_2h^j+mR_j,m(x,hD)(hD_x_1-a(x,hD_x_2))^M_1-m(hD_x_1-q(x,hD_x_2)^M_2-ju

where all the pseudodifferential operators R_j,m(x,hD) are bounded L^2→ L^2 independent of h. Therefore

    (hD_x_1)^M_1(p̃_2(x,hD))^M_2v_L^2≲ h^M_1+M_2u_L^2

and v is a strong joint quasimode of order h  of p̃_2(x,ξ) and hD_x_1. Since a(0,0)=0=q(0,0) we have that p̃_2(0,0)=0.  Consider 

    ∂^r/∂ξ_2^rp̃_2(x,ξ_2)=∂^r(a-q)/∂ξ_2(x_1,x_2(t),ξ_2(t))(∂ξ_2(x_1)/∂ξ_2)^r+R(x_1,x_2,ξ_2)

where R(x_1,x_2,ξ_2) is a sum of terms all of which have a factor of

    ∂^γx_2(x_1)/∂ξ^γ_2|_(x,ξ)=(0,0)

with γ≤r or a factor of

    ∂^βξ_2(x_1)/∂ξ_2^β|_(x,ξ)=(0,0)

for 2≤β≤ r. Now

    x_2(x_1)=x_2+O(|x_1|)  and  ξ_2=ξ_2+O(|x_1|).
 
So at x_1=0 all of the factors of the form (<ref>) and (<ref>) are zero and

    ∂ξ_2(x_1)/∂ξ_2|_x_1=0=1.

Since the derivatives of a and q agree up to the kth derivative (but not at the k+1st derivative) we have that

    ∂^r/∂ξ_2^rp̃_2(x,ξ_2)|_(0,0)=
    0   1≤ r≤ k
    
    c≠ 0    r=k+1.
 
That is the order of contact is preserved. We can therefore write

    p̃_2(x,ξ)=ξ_1-ξ_2^k+1g(x,ξ_2)

for some g(x,ξ_2) with the property that g(0,0)≠ 0. Therefore

    p̃_2(x,hD)=hD_x_1-(h^k+1D_x_1^k+1)G(x,hD_x_2)

where

    G(x,ξ_2)=g(x,ξ_2)+O(h).

That is G(x,hD_x_2) is invertible. Now by Lemma <ref> with e_1(x,hD)= and e_2(x,hD)=-,  v is a strong joint quasimode of order h  of hD_x_1 and h^k+1D_x_2^k+1G(x,hD_x_2). A second application of Lemma <ref> with e_1(x,hD)=0 and e_2(x,hD)=G^-1(x,hD_x_2) gives that v is a strong joint quasimode of order h of hD_x_1 and h^k+1D_x_2^k+1. 





§ PROOF OF THEOREM <REF>

We are now in a position to prove Theorem <ref>. We have that

    u=W^⋆(x_1)v

where v is a  strong joint quasimode of order h of both hD_x_1 and h^k+1D^k+1. Therefore we expect the support of [v] to sit in a h× h^1/k+1 about ξ=0. To incorporate the property  that v is a hD_x_1 quasimode we decompose v with a continuous wavelet transform in x_1. That is,

    v(x_1,x_2)=1/C_f∫1/|a|^5/2X_v(a,b,x_2)f(a^-1(x_1-b))dadb

where

    X_v(a,b,x_2)=1/|a|^1/2∫ f(a^-1(y_1-b))v(y_1,x_2)dy_1

and f is a wavelet satisfying

    ∫ f(τ)dτ=0  and   C_f=∫|f̂(ξ)|^2/|ξ|dξ<∞.

Since v is an order h quasimode of hD_x_1 we would expect the major contributions to (<ref>) to come from the region where |a|∼ 1, and indeed (in Theorem <ref>) we find this is the case. 

When we incorporate this decomposition we obtain

    W^⋆(x_1)v   =1/C_f2π h∫ e^i/h( ϕ(x_1,x_2,ξ_2)-y_2ξ_2)b(x_1,y_2,ξ_2)1/|a|^5/2X_v(a,b,x_2)f(a^-1(x_1-b))dadbdξ_2dy_2
       =1/C_f2π h∫ e^i/hϕ(x_1,x_2,ξ_2)b(x_1,y_2,ξ_2)f(a^-1(x_1-b))[X_v](a,b,ξ_2)dξ_2dadb

where 

    [X_v](a,b,ξ_2)=[X(a,b,·)](ξ_2)=1/(2π h)^1/2∫ e^-i/hx_2ξ_2X_v(a,b,x_2)dx_2.

We re-write (<ref>) as

    W^⋆(x_1)v=1/C_f∫1/|a|^5/2W_a^⋆(x_1)[[X_v](a,·,·)]da

where

    W_a^⋆(x_1)G=1/(2π h)^1/2∫ e^i/hϕ(x_1,x_2,ξ_2)b(x_1,y_x,ξ_2)f(a^-1(x_1-b))G(ξ_2,b)dξ_2db.

Now we want to include the property that v is also a quasimode of h^k+1D^k+1. Since we expect this restriction to require that [v] is supported mainly in the region |ξ_2|≤ h^1/k+1 we decompose W_a^⋆ dyadically into regions with |ξ_2|∼ 2^jh^1/k+1. To that end choose χ_0:→ supported in [-2,2]  and χ supported in [1/2,3/2] so that

    1=χ_0(h^-1/k+1|ξ_2|)+∑_j=1^Jχ(2^-jh^-1/k+1|ξ_2|)  where   2^Jh^1/k+1=1.

Then let

    W_a,j^⋆(x_1)G=1/(2π h)^1/2∫ e^i/hϕ(x_1,x_2,ξ_2)b(x_1,y_x,ξ_2)f(x_1-b/a)χ(|ξ_2|/2^jh^1/k+1)G(ξ_2,b)dξ_2db


    W_a,0^⋆(x_1)G=1/(2π h)^1/2∫ e^i/hϕ(x_1,x_2,ξ_2)b(x_1,x_2,ξ_2)f(x_1-b/a)χ_0(|ξ_2|/h^k+1)G(ξ_2,b)dξ_2db.

We will proceed by proving L^2→ L^p estimates for each W_a,j^⋆, (in Theorem <ref>), and then L^2 bounds for 

    [X_v^j]=χ(2^-jh^-1/1+k)[X_v^j],

(in Propositions <ref> and <ref>). Finally in Theorem <ref> we combine this information to obtain the estimates on W(x_1)^⋆v necessary to prove Theorem <ref>. 



Suppose W_a,j^⋆(x_1) and W_a,0^⋆(x_1) are given by (<ref>) and (<ref>). Then for a≤ 2^-2jh^1-2/k+1,

    W_a,j(·)G_L^p≲ a^1/2+1/ph^(-1+1/k+1)(1/2-1/p)2^j(1/2-1/p)G_L^2.

For a≥2^-2jh^1-2/k+1,

    W_a,j(·)G_L^p≲ a^1/2 2^μ(p,j)h^-δ(p,k)G_L^2

where 

    δ(p,k)   =(1/2-2/p)-1/k+1(1/2-3/p)   6≤ p≤∞
    1/4-1/2p   2≤ p≤ 6,
       and
    μ(p,j)   =
    j(1/2-3/p)   6≤p≤∞
    
    0     2≤ p≤ 6.




Here we use a well developed technique to obtain the L^p estimates. Indeed the general idea dates back Tomas-Stein's <cit.> treatment of L^2→ L^p extension (dual restriction) results. There are three important steps.


  * Thinking again as x_1 as the `time' variable we aim to compute a Strichartz L^p_x_1L^p_x_2 estimate. To that end we compute W_a,j^⋆(x_1)W_a,j(z_1) and find estimates of the form

    W_a,j^⋆(x_1)W_a,j(z_1)_L^1→ L^∞   ≲ h^-γ_∞Q_∞(|x_1-z_1|,h,a)
    W_a,j^⋆(x_1)W_a,j(z_1)_L^2→ L^2   ≲ h^-γ_2Q_2(|x_1-z_1|,h,a)

where the Q_i(|x_1-z_1|,h,a) captures decay as |x_1-z_1| increases.

  * As in the Keel-Tao <cit.> treatment of abstract Strichartz estimates we interpolate to obtain 

    W_a,j^⋆(x_1)W_a,j(z_1)_L^p'→ L^p≲ h^-γ_pQ_p(|x_1-z_1|,h,a).


  * Finally we resolve the |x_1-z_1| integral using Young's inequality, or Hardy-Littlewood-Sobolev's inequality in the borderline case where Q_p(|x_1-z_1|,h,a)^p/2 just fails to be integrable.


First note that for any x_1

    W_a,j(x_1)_L^2_x_2→ L^2_x_2≲W_a(x_1)_L^2_x_2→ L^2_x_2≲ 1.

Then using the support properties of f and the fact that v localised in an O(1) region we see that

    W_a,j(x_1)_L^2_x_2L^2_b→ L^2_x_1L^2_x_2≲ a^1/2.

So

    W_a,j^⋆(x_1)W_a,j(z_1)_L^2→ L^2≲ a.

Therefore we need only focus on the L^1→ L^∞ estimate. Computing W_a,j^⋆W_a,j we have that,

    W_a,j^⋆(x_1)W_a,j(z_1)g=W_a,j^⋆W_a,j(x_1,z_1)g=∫ K(x_1,x_2,z_1,z_2)g(z_1,z_2)dz_1dz_2

where for 1≤j≤ J

    K_j(x_1,x_2,z_1,z_2)=1/2π h∫ e^i/h(ϕ(x_1,x_2,ξ_2)-ϕ(z_1,z_2,ξ_2))f(a^-1(x_1-b))f(a^-1(z_1-b))
    ×χ^2(2^-jh^-1/k+1|ξ_2|)B(x_1,x_2,z_1,z_2,ξ_2)dξ_2db

and

    K_0(x_1,x_2,z_1,z_2)=1/2π h∫ e^i/h(ϕ(x_1,x_2,ξ_2)-ϕ(z_1,z_2,ξ_2))f(a^-1(x_1-b))f(a^-1(z_1-b))
    ×χ^2_0(h^-1/k+1ξ_2)B(x_1,x_2,z_1,z_2,ξ_2)dξ_2db.

Note that for the kernel to be non-zero we require that |x_1-z_1|≤Ca for some sufficiently large constant. Also since v is localised to an O(1) region we must  have |x_1-z_1|≤ 1. We estimate the b integral using the support properties of f. To compute the ξ_2 integral we want to appeal to the stationary phase lemma (it is here that we use the curvature assumption on p_1(x,ξ)). As it <cit.> and <cit.> we have that

    ϕ(x_1,x_2,ξ_2)-ϕ(z_1,z_2,ξ_2)=⟨ x_2-z_2,ξ_2+z_1F(z_1,x_2,z_2,ξ_2)⟩
    
    +(x_1-z_1)a(0,x_2,ξ_2)+O(|x_1-z_1|^2).

The curvature assumption guarantees that |∂^2_ξ_2ξ_2a(0,x_2,ξ_2)|>c>0 so

    |∂^2_ξ_2ξ_2(ϕ(x_1,x_2,ξ_2)-ϕ(z_1,z_2,ξ_2))|≥c|x_1-z_1|.

Therefore, had the symbol had been smooth in ξ_2, the stationary phase lemma would tell us that greatest contribution would come from a h^1/2|x_1-z_1|^-1/2 region about the critical point. However, the cut off in ξ_2 is not smooth. In fact for |x_1-z_1|≤ 2^-2jh^1-2/k+1 we get a better estimate by just using the support properties of χ (or χ_0). When |x_1-z_1|≥2^-2jh^1-2/k+1 the regularity of the cut off is equal or better than the natural regularity introduced in the proof of the stationary phase lemma. Therefore

    |K_j(x_1,x_2,z_1,z_2)|≤
    a2^jh^-1+1/k+1   |x_1-z_1|≤2^-2jh^1-2/k+1
    
    ah^-1/2|x_1-z_1|^-1/2   |x_1-z_1|≥2^-2jh^1-2/k+1

and

    W_a,j^⋆W_a,j(x_1,z_1)_L^1→ L^∞≲
    a2^jh^-1+1/k+1   |x_1-z_1|≤2^-2jh^1-2/k+1
    
    ah^-1/2|x_1-z_1|^-1/2   |x_1-z_1|≥2^-2jh^1-2/k+1.

Since we have the L^2→ L^2 bound of

    W_a,j^⋆W_a,j(x_1,z_1)_L^2→ L^2≲ a

we can interpolate to get

    W_a,j^⋆W_a,j(x_1,z_1)_L^p'→ L^p≲
    a2^2j(1/2-1/p)h^2(-1+1/k+1)(1/2-1/p)   |x_1-z_1|≤ 2^-2jh^1-2/k+1
    
    ah^-(1/2-1/p)|x_1-z_1|^-(1/2-1/p)   |x_1-z_2|≥ 2^-2j h^1-2/k+1.

We are now in a position to examine our two cases


  Case 1 a≤ 2^-2jh^1-2/k+1

  Case 2 a ≥2^-2jh^1-2/k+1.


In Case 1 the restriction that |x_1-z_1|≤ Ca means that we are always in the situation where

    W_a,j^⋆W_a,j(x_1,z_1)_L^p'→ L^p≲ a2^2j(1/2-1/p)h^2(-1+1/k+1)(1/2-1/p).

So

    W_a,j^⋆W_a,j(·,·)_L^p'→ L^p   ≲ a2^2j(1/2-1/p)h^2(-1+1/k+1)(1/2-1/p)(∫_0^adt)^2/p
       ≲ a^1+2/p2^2j(1/2-1/p)h^2(-1+1/k+1)(1/2-1/p).

In Case 2 (for p≠ 6) Young's inequality gives

    W_a,j^⋆W_a,j(·,·)_L^p'→ L^p≲ a2^2j(1/2-1/p)h^2(-1+1/k+1)(1/2-1/p)(∫_0^2^-2jh^1-2/k+1dt)^2/p
    
    +ah^-(1/2-1/p)(∫_2^-2jh^1-2/k+1^a|t|^-p/2(1/2-1/p)dt)^2/p

and so

    W_a,j^⋆W_a,j(·,·)_L^p'→ L^p≲ 2^2μ(p,j)h^-2δ(p,j).

Using Hardy-Littlewood-Sobolev's inequality to resolve the the case where p=6 we obtain

    W_a,j^⋆W_a,j(·,·)_L^6/5→ L^6≲ 2^2μ(6,j)h^-2δ(6,j).






Finally we can put this information together with the estimates from Proposition <ref> and Proposition <ref> (whose proofs we defer to the end of this section). 


If W(x_1) and v are given by (<ref>) and (<ref>) respectively,

    W^⋆(·)v_L^p≲ h^-δ(p,k)u_L^2.





Clearly

    W^⋆v_L^p≲∑_j=0^JW^⋆_jv_L^p

where

    W^⋆_jv=1/C_f∫1/|a|^5/2W_a,j^⋆[[X_v](a,b,·)]da.

We divide the a integral into three parts

    W^⋆_jv=T_j,1[[X_v]]+T_j,2[[X_v]]+T_j,3[[X_v]]

with

    T_j,1[[X_v]]   =1/C_f∫_|a|≤ 2^-2jh^1-2/k+11/|a|^5/2W_a,j[[X_v](a,·,·)]da,
    
    T_j,2[[X_v]]   =1/C_f∫_2^-2jh^1-2/k+1≤|a|≤ 11/|a|^5/2W_a,j[[X_v](a,·,·)]da,
       and
    
    T_j,3[[X_v]]   =1/C_f∫_|a|≥11/|a|^5/2W_a,j[[X_v](a,·,·)]da.

Note that the localisation  applied to W_a,j^⋆ means that it is enough to consider T_j,i acting on [X_v^j] given by

    [X_v^j](a,b,ξ_2)=χ(2^-jh^-1/k+1|ξ_2|)[X_v](a,b,ξ_2)   j≥ 1
    [X_v^0](a,b,ξ_2)=χ_0(h^-1/k+1ξ_2)[X_v](a,b,ξ_2)   j=0.

Let's treat T_j,1 first. Notice that since k≥1, 2^-2jh^1-2/k+1≤1 so we can use the result of Proposition <ref>, namely for |a|≤1,

    [X_v^j](a,·,·)_L^2_ξ_2L^2_b≲ a^3/22^-jM
 
for any natural number M. 
Then

    T_j,1[[X^j_v]]_L^p   ≲∫_|a|≤2^-2jh^1-2/k+1|a|^-2+1/ph^(-1+1/k+1)(1/2-1/p)2^j(1/2-1/p)[X^j_v](a,·,·)_L^2_ξ_2L^2_b
       ≲∫_|a|≤ 2^-2jh^1-2/k+1|a|^-1/2+1/ph^(-1+1/k+1)(1/2-1/p)2^j(1/2-1/p-M)u_L^2
       ≲ h^-1/2(k+1)+1/p(2-3/k+1)2^j(-1/2-2/p-M)u_L^2

which is better than (<ref>). Now considering T_j,2 we have

    T_j,2[[X^j_v]]_L^p   ≲∫_2^-2jh^1-2/k+1≤|a|≤ 1a^-2 2^μ(p,j)h^-δ(p,k)[X^j_v](a,·,·)_L^2_ξ_2L^2_bda
       ≲∫_2^-2jh^1-2/k+1≤|a|≤ 1a^-1/2 2^μ(p,j)-jMh^-δ(p,k)u_L^2
       ≲ 2^μ(p,j)-Mjh^-δ(p,k)u_L^2.

Finally to deal with T_j,3 we use the result of Proposition <ref> which states that for |a|≥1 

    [X_v^j](a,·,·)_L^2_ξ_2L^2_b≲ 2^-jM
 
for any natural number M. This gives us

    T_j,3[[X^j_v]]_L^p   ≲∫_1≤|a|a^-22^μ(p,j)h^-δ(p,k)[X^j_v](a,·,·)_L^2_ξ_2L^2_b
       ≲∫_1≤|a|a^-22^μ(p,j)-jMh^-δ(p,k)u_L^2
       ≲ 2^μ(p,j)-jMh^-δ(p,k)u_L^2.

Therefore by making M large enough

    W^⋆_j(·)v_L^p≲ h^-δ(p,k)2^-2ju_L^2

and so summing

    W^⋆(·)v_L^p≲ h^-δ(p,k)u_L^2.




Below we provide the proofs of Propositions <ref> and <ref>.


Suppose v is a  strong  joint quasimode of order hu_L^2 of hD_x_1 and h^k+1D^k+1_x_2 and  [X_v^j](a,b,ξ_2)
is given by

    [X_v^j](a,b,ξ_2)=χ(2^-jh^-1/k+1ξ_2)[X_v(a,b,·)](ξ_2)   1≤ j≤ J,


    [X_v^0](a,b,ξ_2)=χ_0(h^-1/k+1ξ_2)[X_v(a,b,·)](ξ_2).

 Then for all |a|≤1

    [X_v^j](a,·,·)_L^2_bL^2_ξ_2≲ 2^-jMa^3/2u_L^2

for any  M∈.



The semiclassical Fourier transform preserves L^2 norms so for fixed a and b

    [X_v^0](a,b,·)_L^2_ξ_2≲[X_v](a,b,·)_L^2_ξ_2=X_v(a,b,·)_L^2_x_2.

Now

    X_v(a,b,x_2)=1/|a|^1/2⟨ f(a^-1(x_1-b),v(x_1,x_2)⟩_x_1.

Since f has integral zero it can be written as the derivative of a function g which is also compactly supported. That is

    f(x_1)=iD_x_1g(x_1)⇒ f(a^-1(x_1-b))=aiD_x_1g(a^-1(x_1-b)).

Therefore

    X_v(a,b,x_2)   =ia/|a|^1/2⟨ D_x_1g(a^-1(x_1-b)),v(x_1,x_2)⟩_x_1
       =-ia/|a|^1/2⟨ g(a^-1(x_1-b),D_x_1v(x_1,x_2)⟩_x_1.

So

    |X_v(a,b,x_2)|≲ ag_L^2D_x_1v(x_1,x_2)_L^2_x_1(B_a(b))

and

    X_v(a,b,·)_L^2_x_2≲ aD_x_1v(x_1,x_2)_L^2_x_1(B_a(b))L^2_x_2.

Therefore

    [X_v^0](a,b,·)_L^2_ξ_2≲ aD_x_1v(x_1,x_2)_L^2_x_1(B_a(b))L^2_x_2.

Computing the b integral we have

    [X_v^0](a,·,·)_L^2_bL^2_ξ_2   ≲ a^3/2D_x_1v_L^2
       ≲ a^3/2u_L^2.

Where in the final step we have used the fact that v is an O_L^2(hu_L^2) quasimode of hD_x_1. 


When j≥1, we first write

    [X_v^j(a,b,·)](ξ_2)=1/ξ_2^(k+1)Mξ_2^(k+1)M[X_v^j(a,b,·)](ξ_2)

and see that

    [X_v^j](a,b,·)_L^2_ξ_2   ≲ h^-M2^-j(k+1)Mξ_2^(k+1)M[X_v](a,b,·)_L^2_ξ_2
       =h^-M2^-j(k+1)M(h^k+1D_x_2^k+1)^MX_v(a,b,·)_L^2_x_2.

Now

    (h^k+1D^k+1_x_2)^MX_v^j(a,b,x_2)=(h^k+1D^k+1_x_2)^M1/|a|^1/2⟨ f(a^-1(x_1-b),v(x_1,x_2)⟩_x_1.

Following the same manipulations as the j=0 case

    (h^k+1D^k+1_x_2)^MX_v^j(a,b,x_2)   =-ia/|a|^1/2⟨ g(a^-1(x_1-b),(h^k+1D^k+1_x_2)^MD_x_1v(x_1,x_2)⟩_x_1,
    |(h^k+1D^k+1_x_2)^MX_v^j(a,b,x_2)|   ≲ a (h^k+1D^k+1_x_2)^MD_x_1v(x_1,x_2)_L^2_x_1(B_a(b))

and

    (h^k+1D^k+1_x_2)^MX_v^j(a,b,·)_L^2_x_2≲ a (h^k+1D^k+1_x_2)^MD_x_1v_L^2_x_1(B_a(b))L^2_x_2.

Therefore

    ξ_2^(k+1)M[X_v^j](a,b,·)_L^2_ξ_2≲ a(h^k+1D^k+1_x_2)^MD_x_1v_L^2_x_1(B_a(b))L^2_x_2.

and so

    [X_v^j](a,b,·)_L^2_ξ_2≲ ah^-M2^j(k+1)M(h^k+1D^k+1_x_2)^MD_x_1v_L^2_x_1(B_a(b))L^2_x_2.

Integrating in b gives us

    [X_v^j](a,·,·)_L^2_ξ_2L^2_b≲ a^3/2(h^k+1D^k+1_x_2)^MD_x_1v_L^2.

Finally since v is a strong joint quasimode of order h of hD_x_1 and h^k+1D_x_2^k+1 we obtain

    [X_v^j](a,·,·)_L^2_ξ_2L^2_b≲ a^3/22^-j(k+1)Mu_L^2.
 




Suppose v is a strong joint quasimode of order h of hD_x_1 and h^k+1D^k+1_x_2 and [X_v^j] is as in Proposition <ref>. Then for |a|≥1,

    [X_v](a,·,·)_L^2_ξ_2L^2_b≲ 2^-jMu_L^2

for any  M∈.



For j=0 we again use the fact that the semiclassical Fourier transform preserves L^2 norms to write,

    [X_v^0](a,b,·)_L^2_ξ_2≲[X_v](a,b,·)_L^2_ξ_2=X_v(a,b,·)_L^2_x_2.

Referring to the definition, (<ref>), of X_v(a,b,x_2) we have

    X_v(a,b,x_2)=1/|a|^1/2⟨ f(a^-1(x_1-b)),v(x_1,x_2)⟩_x_1.

Remember that v(x_1,x_2) is localised to an O(1) region. Therefore

    |X_v(a,b,x_2)|≲1/|a|^1/2v_L^2_x_1

and

    [X_v]_L^2_ξ_2=X_v(a,b,·)_L^2_x_2≲1/|a|^1/2v_L^2.

Now once |b|≫ |a| the support of f(a^-1(x_1-b)) no longer overlaps with the support of v so

    [X_v](a,·,·)_L^2_ξ_2L^2_b≲v_L^2≲u_L^2.

If j≥ 1, ξ_2 is supported away from zero on the support of [X_v^j]. So again writing

    [X_v^j(a,b,·)](ξ_2)=1/ξ_2^(k+1)Mξ_2^(k+1)M[X_v^j(a,b,·)](ξ_2)

we see that

    [X_v^j](a,b,·)_L^2_ξ_2   ≲ h^-M2^-j(k+1)Mξ_2^(k+1)M[X_v](a,b,·)_L^2_ξ_2
       =h^-M2^-j(k+1)M(h^k+1D_x_2^k+1)^MX_v(a,b,·)_L^2_x_2.

As in the j=0 case we write

    X_v(a,b,x_2)=1/|a|^1/2⟨ f(a^-1(x_1-b)),v(x_1,x_2)⟩_x_1

so

    |(h^k+1D^k+1_x_2)^MX_v(a,b,x_2)|   ≲1/|a|^1/2|⟨ f(a^-1(x_1-b)),(h^k+1D^k+1_x_2)^Mv(x_1,x_2)⟩_x_1|
       ≲1/|a|^1/2(h^k+1D^k+1)^Mv_L^2_x_1.

Since v is a strong O_L^2(hu_L^2) quasimode of h^k+1D_x_2^k+1

    (h^k+1D^k+1_x_2)^MX_v(a,b,·)_L^2_x_2≲h^M/|a|^1/2u_L^2

and

    [X_v^j](a,b,·)_L^2_ξ_2≲2^-j(k+1)M/|a|^1/2u_L^2.

Since X_v(a,b,x_2)=0 for b≫ a

    [X_v^j](a,·,·)_L^2_ξ_2≲ 2^-j(k+1)Mu_L^2.





§ APPENDIX A: SEMICLASSICAL ANALYSIS



Throughout this paper we have used  some  standard results from semiclassical analysis. For the readers convenience we record the results in this appendix  and direct them to <cit.> and <cit.> for the proofs and further discussion. In this paper we always use the left quantisation of semiclassical pseudodifferential operators. That is given a symbol p(x,ξ) we define the operator

    Op_h(p)u=p(x,hD)u=1/(2π h)^n∫ e^i/h⟨ x-y,ξ⟩p(x,ξ)u(y)dξ dy



Let p(x,hD), q(x,hD) be left-quantised semiclassical pseudodifferential operators with symbols p(x,ξ) and q(x,ξ) respectively. The the symbol of p(x,hD)∘q(x,hD) is given by

    p(x,ξ)# q(x,ξ)=e^ih⟨D_ξ,D_y⟩p(x,ξ)q(y,η)|_x=y,ξ=η
    
      =∑_kh^k/k!(⟨D_ξ,D_y⟩/i)^kp(x,ξ)q(y,η)|_x=y,ξ=η.

  
  
  
  Let p(x,hD), q(x,hD) be left-quantised semiclassical pseudodifferential operators. Then
  
    [p(x,hD),q(x,hD)]=hOp_h({p,q})+h^2r(x,hD)

  where {p,q} is the Poisson bracket of p and q and the remainder operator r(x,hD) satifies

    r(x,hD)_L^2→L^2≲ 1.




Let p(x,hD) be a left-quantised, semiclassical pseudodifferential operator with symbol p(x,ξ) such that |p(x,ξ)|>c>0. Then there exists an inverse operator (p(x,hD))^-1 with

    (p(x,hD))^-1_L^2→ L^2≲ 1.





Suppose W(x_1) has the property that

    hD_x_1W(x_1)=-W(x_1)a(x,hD_x_2)+O_L^2(h^∞)

then W(x_1) has the parametrix representation 

    W(x_1)g=1/2π h∫ e^i/h(x_2ξ_2-ϕ(x_1,y_2,ξ_2))b(x_1,y_2,ξ_2)g(y_2)dξ_2dy_2

where

    ∂_x_1ϕ+a(x,∂_y_2ϕ,ξ_2)=0  ϕ(0,y_2,ξ_2)=y_2ξ_2


    b(0,y_2,ξ_2)=1.




We provide only a sketch of this standard result. Using the parametrix (<ref>) we have that

    hD_x_1W(x_1)g=1/2π h∫ e^i/h(x_2ξ_2-ϕ(x_1,y_2,ξ_2))(∂_x_1ϕ b+h∂_x_1b)(x_1,y_2,ξ_2)g(y_2)dξ_2dy_2

and

    -Wa(x,hD_x_2)g=-1/(2π h)∫ e^i/h(x_2ξ_2-ϕ(x_1,y_2,ξ_2)+(y_2-z_2)η_2)b(x_1,y_2,ξ_2)a(x_1,y_2,η_2)g(z_2)dξ_2dy_2dη_2dz.

Now computing the (y_2,η_2) integral via the method of stationary phase we find that there is a non-degenerate critical point where

    y_2-z_2=0  and   -∂_y_2ϕ+η_2=0.

So

    -Wa(x,hD_x_2)g=-1/2π h∫ e^i/h(x_2ξ_2-ϕ(x_2,z_2,ξ_2))(b(x_1,z_2,ξ_2)a(x_1,z_2,∂_y_2ϕ)+hr(x_1,z_2,ξ_2))g(z_2)dξ_2dz

where r(x_1,z_2,ξ_2) is determined by the lower order terms in the stationary phase expansion. Therefore if ϕ satisfies (<ref>),

    (hD_x_1W(x_1)+W(x_1)a(x,hD_x_2))g=1/2π h∫ e^i/h(x_2ξ_2-ϕ(x_2,z_2,ξ_2))h(∂_x_1b+r)(x_1,z_2,ξ_2)g(z_2)dξ_2dz_2.

That is the error is already O(h). To continue improving we write

    b(x_1,y_2,ξ_2)=∑_k=0^∞h^kb_k(x_1,y_2,ξ_2)

and successively solve transport equations to achieve an O(h^∞) error. 




§ ACKNOWLEDGEMENTS

The author would like to acknowledgement the comments and suggestions of the reviewer.



