On the L p norm of spectral clusters for compact manifolds with boundary

c

2007 by Institut Mittag-Leffler. All rights reserved

On the L ^p norm of spectral clusters for compact manifolds with boundary

by

Hart F. Smith

University of Washington Seattle, WA, U.S.A.

Christopher D. Sogge

Johns Hopkins University Baltimore, MD, U.S.A.

1. Introduction

LetM be a compact 2-dimensional manifold with boundary, and let P be an elliptic, second-order differential operator onM, self-adjoint with respect to a density dµ, and with vanishing zero-order term, so that in local coordinates

(P f)(x) =%(x)⁻¹

n

X

j,k=1

∂_j(%(x)g^jk(x)∂_kf(x)), dµ=%(x)dx. (1.1)

We take the g^jk’s to be positive, so that the Dirichlet eigenvalues ofP can be written as {−λ²_j}^∞_j=0.

Let χ_λ be the projection of L²(dµ) onto the subspace spanned by the Dirichlet eigenfunctions for whichλ_j∈[λ, λ+1]. In the case whereM is compact without boundary of dimensionn>2, and the coefficients ofP areC^∞functions, Sogge [14] established the following bounds:

kχλfkL^q(M)6Cλ((n−1)/2)(1/2−1/q)kfkL²(M), 26q6qn, (1.2) kχλfkL^q(M)6Cλn(1/2−1/q)−1/2kfkL²(M), qn6q6∞. (1.3) Furthermore, the exponent ofλis sharp on every such manifold (see, e.g., [15]). In the case of a sphere, the examples which prove sharpness are in fact eigenfunctions. For (1.2), the appropriate example is an eigenfunction which concentrates in aλ^−1/2-diameter tube about a geodesic. For (1.3), the example is a zonal eigenfunction ofL² norm λ^(n−1)/2 which takes on values comparable toλon aλ⁻¹-diameter ball about each of the north and

The authors were supported by the National Science Foundation, Grants DMS-0140499, DMS- 0099642, and DMS-0354668.

south poles. Approximate spectral clusters with similar properties can be constructed in the interior of any smooth manifold, showing that for spectral clusters (though not necessarily eigenfunctions) the exponents in (1.2) and (1.3) are also lower bounds on manifolds with boundary.

In [13], the authors showed that, on a manifold of dimension n>2 for which the boundary is everywhere strictly geodesically concave (such as the complement inRⁿ of a strictly convex set), the estimates (1.2) and (1.3) both hold.

On the other hand, Grieser [5] observed that in the unit disk{x:|x|61} there are eigenfunctions of the Laplacian, for Dirichlet as well as for Neumann boundary conditions, of eigenvalue−λ²that concentrate within aλ^−2/3-neighborhood of the boundary. These are the classical Rayleigh whispering gallery modes (see [10] and [9]). The Fourier–

Airy calculus of Melrose and Taylor allows one to construct an approximate spectral cluster with similar localization properties near any boundary point ofM at which the boundary is strictly convex (the gliding case). Consequently, ifM is of dimension 2 and the boundary has a point of strict convexity with respect to the metric g (for instance, any smoothly bounded planar domain endowed with the standard Laplacian and either Dirichlet or Neumann conditions), then the following bounds cannot be improved upon:

kχλfk_Lq(M)6Cλ(2/3)(1/2−1/q)kfk_L2(M), 26q68, (1.4) kχλfk_Lq(M)6Cλ2(1/2−1/q)−1/2kfk_L2(M), 86q6∞. (1.5) In this paper we show that the estimates (1.4) and (1.5) hold on any 2-dimensional compact manifold with boundary, forP as above and either Dirichlet or Neumann con- ditions assumed. Estimate (1.4) follows by interpolation of the trivial caseq=2 with the caseq=6, so we restrict attention toq>6 for (1.4). Forq>6, the above estimates are an immediate consequence of the following theorem (see for example [8] or [11]).

Theorem 1.1. Suppose that usolves the Cauchy problem on R×M,

∂_t²u(t, x) =P u(t, x), u(0, x) =f(x), ∂tu(0, x) = 0, (1.6) and satisfies either Dirichlet conditions:

u(t, x) = 0 if x∈∂M ,

or Neumann conditions, where N_x is a unit normal field with respect to g:

N_x·∇_xu(t, x) = 0 if x∈∂M. Then,the following bounds hold for 66q68:

kuk_L^q_xL2

t(M×[−1,1])6Ckfk_Hγ(q)(M), γ(q) =2 3

1 2−1

q

,

and the following bounds hold for 86q6∞:

kuk_L^q_xL2

t(M×[−1,1])6Ckfk_Hδ(q)(M), δ(q) = 2 1

2−1 q

−1 2.

In the statement of the theorem, the space H^s(M) refers to the Sobolev space of ordersonM determined, respectively, by Dirichlet or Neumann eigenfunctions.

Our approach to proving Theorem 1.1 is to work in geodesic normal coordinates near∂M, and to extend both the operatorP and the solutionuacross the boundary, to obtainuas a solution to a wave equation on an open set, but for an operator with coef- ficients of Lipschitz regularity. We then adapt a frequency dependent scaling argument, originally developed to handle Lipschitz metrics, to metrics with the particular type of codimension-1 singularities that the extendedP will have.

We remark that, for operators of the type (1.1) with %and the g^jk’s of Lipschitz regularity, the estimate (1.4) is known on the range 26q66, as established by the first author in [12], along with a weaker version of (1.5) having larger exponent if q <∞.

It is not currently known what the sharp exponents are for general Lipschitz P, since the known counterexamples satisfy the estimates (1.5). The estimates for q=∞ were established for eigenfunctions recently by Grieser [6], while the sup-norm estimates for spectral clusters were obtained by the second author in [16].

For q=∞, the squarefunction estimate of Theorem 2.1 below was shown in [12] to hold for operators P with Lipschitz coefficients, which in particular implies the q=∞

case of Theorem 1.1 for P on a manifold with boundary. Our proof here of the case q <∞, however, depends crucially on the fact that if u is appropriately microlocalized away from directions tangent to∂M, then better squarefunction estimates hold than do for directions near to tangent. In other words, we exploit the fact that the more highly localized eigenfunctions considered in [5] are associated only to gliding directions along

∂M, not directions transverse to∂M.

A historical curiosity is that the critical L²!L⁸ bounds for χλ have an analog in Euclidean space which seems to be the first restriction theorem for the Fourier transform.

To explain this, we first notice that, by duality, ourL²!L⁸bounds are equivalent to the statement thatχλ:L^8/7!L² with norm O(λ^1/4). The Euclidean analog would say that ifχλ:L^8/7(R²)!L²(R²) denotes the projection onto Fourier frequencies |ξ|∈[λ, λ+1], then this operator also has norm O(λ^1/4). An easy scaling argument shows then that the latter result is equivalent to the following Fourier restriction theorem for the circle

Z 2π

0

|f(cosˆ θ,sinθ)|²dθ ^1/2

6Ckfk_L8/7(R²), f∈C₀^∞(R²).

Motivated by a question of DeLeeuw [4], Stein [17] proved this by a now standard T T^∗ argument, using the fact that convolution with cdθ maps L^8/7(R²)!L⁸(R²) by the Hardy–Littlewood–Sobolev theorem, as |cdθ|6C|x|^−1/2. Since this argument does not use the oscillations ofcdθ, one can strengthen the above restriction theorem to show that, forj>1, one has the uniform bounds

Z 2^−j

0

|fˆ(cosθ,sinθ)|²dθ ^1/2

6C2^−j/8kfk_L8/7(R²), f∈C₀^∞(R²). (1.7) By the Knapp example, there is no small angle improvement for the critical

L^6/5(R²)−!L²(S¹)

restriction theorem of Stein–Tomas. A key step for us is that in the setting of compact manifolds with boundary, we also get the sameO(2^−j/8) improvement in ourL⁸-estimates when microlocalized to regions of phase space that correspond to bicharacteristics that are of angle comparable to 2^−j from tangency to the boundary.

In higher dimensions the natural analog of (1.4)–(1.5) would say that kχλfkL^q(M)6Cλ(2/3+(n−2)/2)(1/2−1/q)kfkL²(M), 26q66n+4

3n−4, (1.8)

kχλfk_Lq(M)6Cλn(1/2−1/q)−1/2kfk_L2(M), 6n+4

3n−46q6∞. (1.9) By higher dimensional versions of the Rayleigh whispering gallery modes, this would be sharp if true. At present we are unable to prove this estimate but, as we shall indicate in the final section, we can prove the bounds in (1.9) for the smaller range of exponents q>4 ifn>4, andq>5 ifn=3. We hope to return to the problem of proving sharp results in higher dimensions in a future work.

Notation. We use the following notation. The symbol a.b means that a6Cb, where C is a constant that depends only on globally fixed parameters (or on N, α andβ, in case of inequalities involving general integers).

For convenience, we will letx3 serve as substitute for the time variablet. We use d=(d1, d2, d3) to denote the gradient operator, andD=−id.

2. Dyadic localization arguments

The estimates of Theorem 1.1 hold if u is supported away from ∂M, by the results of [8]. Consequently, by finite propagation velocity and the use of a smooth partition of unity, we may assume that, forT small, the solutionu(t, x) in Theorem 1.1 is for|t|6T

supported in a suitably small coordinate patch centered on the boundary. Note that if we establish Theorem 1.1 on the set|t|6T for some smallT, it then holds for T=1 by energy conservation.

We work in boundary normal coordinates for the Riemannian metric gjkthat is dual to g^jk of (1.1). Thus,x2>0 will define the manifold M, andx1is a coordinate function on∂M which we choose so that∂x₁ is of unit length along∂M. In these coordinates,

g₂₂(x₁, x₂) = 1, g₁₁(x₁,0) = 1 and g₁₂(x₁, x₂) = g₂₁(x₁, x₂) = 0. (2.1) The metric g^jk forP is the inverse to gjk, and the same equalities hold for it.

We now extend the coefficient g¹¹and%in an even manner across the boundary, so that

g¹¹(x1,−x2) = g¹¹(x1, x2) and %(x1,−x2) =%(x1, x2). (2.2) The extended functions are then piecewise smooth, and of Lipschitz regularity across x₂=0. Because g is diagonal, the operatorP is preserved under the reflectionx₂7!−x₂. After multiplying %(x) by a constant, and rescaling variables if necessary, we may assume that on the ball|x|<1 the function%(x) isC^N(R²₊)-close to the function 1, and g^jk(x) isC^N(R²₊)-close to the Euclidean metric, whereN is suitably large, andc0 will be taken suitably small:

k%−1k_CN(R²₊)6c₀ and kg^jk−δ^jkk_CN(R²₊)6c₀. (2.3) We may then extend %and g^jk globally, preserving conditions (2.1)–(2.3), so thatP is defined globally onR²and such that

%(x) = 1 and g^jk(x) =δ^jk for|x|>³4. (2.4) We then extend the initial data f and the solutionuto be odd inx2 (respectively, even in x2 in case of Neumann conditions). This extension map is seen to map the Dirichlet (respectively, Neumann) Sobolev spaceH²(Rⁿ₊) toH²(Rⁿ), and henceH^δ(Rⁿ₊) to H^δ(Rⁿ) for 06δ62. The extended solution u thus solves the extended equation

∂²_tu=P u on R×R², with the extended initial data f. The result of Theorem 1.1 is therefore a direct consequence of the following one.

Theorem 2.1. Suppose that the operator P takes the form (1.1),and that %and g satisfy conditions (2.1)–(2.4)above. Let usolve the Cauchy problem on R×R²:

∂_t²u(t, x) =P u(t, x), u(0, x) =f(x), ∂tu(0, x) =g(x). (2.5)

Then the following bounds hold for 66q68:

kuk_L^q_xL2

t(R²×[−1,1]).(kfk_Hγ(q)+kgk_Hγ(q)−1), γ(q) =2 3

1 2−1

q

,

and the following bounds hold for 86q6∞:

kuk_L^q_xL²

t(R²×[−1,1]).(kfk_Hδ(q)+kgk_Hδ(q)−1), δ(q) = 2 1

2−1 q

−1 2.

We begin by reducing matters to compactly supported u satisfying an inhomoge- neous equation. Henceforth, we will use the notationx₃=t. Letφ(x) be a smooth even function onR³, equal to 1 for|x|6³2, and vanishing for|x|>2. We may then write

3

X

j=1

D_j(a^jj(x)D_j(φu)(x)) =

3

X

j=1

D_jF_j(x), where

a³³(x) =%(x) and a^jj(x) =−%(x)g^jj(x) forj= 1,2.

We express this equation concisely asDAD(φu)=DF, and observe that, for 06δ62, kφuk_Hδ(R³)+kFk_Hδ(R³).kfk_Hδ+kgk_H^δ−1.

This is a consequence of energy estimates, which hold separately onR³₊andR³₋, together with the fact thatDAD(φu) is compactly supported and has integral 0, so may be written asDF.

We may thus assume that u(x) is supported in the ball |x|62, and need to show that

kuk_LqL².kuk_Hγ(q)+kFk_Hγ(q), 66q68, (2.6) kuk_LqL².kuk_Hδ(q)+kFk_Hδ(q), 86q6∞, (2.7) whereDADu=DF.

Next let Γ(ξ) be a multiplier of order 0, supported in the set ξ:¹₄|ξ3|6|(ξ1, ξ2)|64|ξ3|}, which equals 1 on the set

ξ:¹₂|ξ3|6|(ξ1, ξ2)|62|ξ3|}.

The operatorDAD is elliptic on the support of 1−Γ, and we may write DAD(1−Γ(D))u= (1−Γ(D))DF−D[A,Γ(D)]Du.

As a consequence of the Coifman–Meyer commutator theorem [2] (see also [20, Proposi- tion 3.6.B]) the operator [A,Γ(D)] mapsH^δ−1!H^δ for 06δ61. Hence, the right-hand side of the above belongs to H^δ−1, and, by Sobolev embedding and elliptic regularity (see, for example, [20, Theorem 2.2.B], which applies in the Sobolev setting), we have

k(1−Γ(D))uk_LqL².k(1−Γ(D))uk_Hδ(q)+1.kuk_Hδ(q)+kFk_Hδ(q).

Indeed, there is an extra ¹₂ derivative inδ(q)+1 beyond the Sobolev indexn(1/2−1/q), so this holds for all 26q6∞. Sinceγ(q)>δ(q) forq68, this implies that (2.6) and (2.7) hold forureplaced by (1−Γ(D))uon the left-hand side.

It thus remains to establish (2.6) and (2.7) with u replaced on the left by Γ(D)u.

We take a Littlewood–Paley decomposition inξto write Γ(D)u=

∞

X

k=1

Γk(D)u=

∞

X

k=1

uk,

with ˆu_k supported in a region where|(ξ1, ξ₂)|≈|ξ3|and|ξ|≈2^k. Since these regions have finite overlap in theξ₃ axis, we have

kΓ(D)uk_LqL².kukk_Lq`<sup>2</sup><sub>k</sub>L<sup>2</sup>.kukk<sub>`</sub>2 kL^qL², where we useq>2 at the last step.

Now let Ak denote the matrix of coefficients obtained by truncating the frequencies ofaⁱⁱ(x) to|ξ|6c2^k for a fixed smallc. We then haveDAkDuk=DFk, where

F_k= Γ_k(D)F+[A,Γ_k(D)]Du+(A_k−A)Du_k. (2.8) Note that the inhomogeneity Fk is now localized in frequency to |ξ3|≈|ξ|≈2^k, by the frequency localizations ofAk anduk.

We claim that, for 06δ61,

∞

X

k=1

2^2kδkFkk²_L2.kuk²_Hδ+kFk²_Hδ.

This follows by orthogonality for the first term on the right of (2.8), and the last term is handled by the boundkA−AkkL^∞.2^−k. The middle term is handled by the Coifman–

Meyer commutator theorem, which yields thatP∞

k=1ε_k[A,Γ_k(D)] maps H^δ−1!H^δ for all sequencesε_k=±1.

We thus are reduced to establishing uniform estimates for each dyadically localized pieceu_k. We thus fix a frequency scale λ=2^k for the rest of this paper. We then need to prove the following estimates, where we now setDA_λDu_λ=F_λ,

kuλkL^qL²(R³).λ^γ(q)(kuλkL²(R³)+λ⁻¹kFλkL²(R³)), 66q68, ku_λk_LqL²(R³).λ^δ(q)(ku_λk_L2(R³)+λ⁻¹kF_λk_L2(R³)), 86q6∞.

Since we are usingx₃ orthogonality to make this reduction, we must control the norms of theu_λ’s globally. However, sinceuis supported in the ball of radius 2, it is easy to see that the norm ofu_λover|x|>3 is bounded byλ⁻¹kuk_L2, so in fact it suffices to establish the above estimate with the left-hand side norm taken over the cube of sidelength 3.

If we let v_λ denote the localization ofu_λ to frequencies where|ξ2|>¹₈|ξ3|, then the squarefunction estimates hold forv_λas on an open manifold:

kvλk_LqL²(R³).λ^δ(q)(kuλk_L2(R³)+kFλk_L2(R³)), 66q6∞.

This will follow as a consequence of the techniques that we use to handle the part ofu_λ with frequencies localized to angles≈1 from theξ₃ axis.

Consequently, we will assume that supp(ˆuλ)⊆

ξ:|ξ1| ∈₁

2λ,2λ

,|ξ2|610¹λand|ξ3| ∈₁

2λ,2λ .

On this region, the operator DA_λD is hyperbolic with respect to the x₁ direction. We can thus takep(x, ξ⁰), a positive elliptic symbol in ξ⁰=(ξ₂, ξ₃), so that

a¹¹_λ (x)(ξ₁²−p(x, ξ⁰)²) =

3

X

j=1

a^jj_λ(x)ξ_j² if |ξ2|6¹9λ and |ξ3| ∈₁

3λ,3λ ,

and such that

p(x, ξ⁰) =|ξ⁰| if ξ⁰∈/

−¹₈λ,¹₈λ

×1 4λ,4λ

.

We also smoothly setp(x, ξ⁰)=1 nearξ⁰=0. Thus, p(x, ξ⁰), d_xp(x, ξ⁰)∈S_1,1¹ ,

andp(x, ξ⁰) differs from|ξ⁰|by a symbol supported in the dyadic shell|ξ⁰|≈λ.

Next, let pλ(x⁰, ξ) be obtained by truncating the symbol p(x, ξ⁰) to x⁰-frequencies less thancλ, where cis a small constant. Then, uniformly overλ,

p_λ(x, ξ⁰)−p(x, ξ⁰)∈S_1,1⁰ and supp(p_λ−p)⊂ {ξ⁰:|ξ⁰| ≈λ}.

Furthermore, the symbol-composition rule holds forpλ to first order. Consequently, we can write

(D₁+p_λ(x, D⁰))(D₁−p_λ(x, D⁰))u_λ=F_λ⁰, where

kF_λ⁰k_L2(R³).λkuλk_L2(R³)+kFλk_L2(R³).

The functionu_λcan be written as the sum of four pieces with disjoint Fourier transforms, according to the possible signs of ξ₁ and ξ₃. We restrict attention to the piece u⁺_λ, supported whereξ₁>0 and ξ₃>0. Estimates for the other pieces will follow similarly.

Since p_λ is x⁰-frequency localized, F_λ⁰ also splits into four disjoint pieces. The symbol ξ₁+p(x, ξ⁰) is elliptic on the regionξ₁>0, hence we may write

D1u⁺_λ−pλ(x, D⁰)u⁺_λ=F_λ⁰⁰, where

kF_λ⁰⁰k_L2(R³).kuλk_L2(R³)+λ⁻¹kFλk_L2(R³). Finally, we have that

pλ(x, D⁰)−pλ(x, D⁰)^∗∈Op(S⁰_1,1),

and is dyadically supported inξ⁰. We have thus reduced the proof of Theorem 2.1 to the proof of the following result.

Theorem 2.2. Suppose that the x⁰-Fourier transform of uλ satisfies the support condition

supp(ˆuλ)⊆

ξ⁰:|ξ2|610¹λand ξ3∈₁

2λ,2λ , and that

D1uλ−Pλuλ=Fλ,

where Pλ=¹₂(pλ(x, D⁰)+pλ(x, D⁰)^∗). Then, for S=[0,1]×R²,

ku_λk_LqL²(S).λ^γ(q)(ku_λk_L∞L²(S)+kF_λk_L2(S)), 66q68, kuλk_LqL²(S).λ^δ(q)(kuλk_L∞L²(S)+kFλk_L2(S)), 86q6∞.

(2.9)

The use of the L^∞L² norm ofu_λ is allowed by Duhamel and energy bounds. Here, as in what follows, we are using the shorthand mixed-norm notation thatL^pL^q=L^p_x

1L^q_x0. 3. The angular localization

In this section we take a further decomposition ofuλ, by decomposing its Fourier trans- form dyadically in theξ2 variable. The reductions of the previous section required only the fact that the coefficientsa^jj(x) were Lipschitz functions. The reduction to estimates for angular pieces depends on the fact that the singularities of a^jj(x), and hence the points where thex2-derivatives ofpλ(x, ξ⁰) are large, occur only atx2=0. Consequently, various error terms that arise in this further reduction will be highly concentrated at x2=0, which we express through weightedL² estimates.

We will take a dyadic decomposition of theξ₂variable, from scaleξ₂≈λ^2/3toξ₂≈λ.

Thus, for 16j <N_λ=¹₃log₂λ, letβ_j(ξ⁰)=β_j(ξ₂, ξ₃) denote a smooth cutoff satisfying supp(βj)⊂[2^−j−2λ,2^−j+1λ]×₁

4λ,4λ ,

andβN_λ supported in [−λ^2/3, λ^2/3]×₁

4λ,4λ

, such that, withβ_−j(ξ2, ξ3)=βj(−ξ2, ξ3),

Nλ

X

j=1

β_j(ξ⁰)+

−1

X

j=1−Nλ

β_j(ξ⁰) = 1 if |ξ2|6¹₈λ and ξ₃∈₁

2λ,2λ .

Let

uj(x) =βj(D⁰)uλ(x).

If we define

θ_j= 2^−|j|,

thenuj has frequencies localized toξ2≈±θjξ3, or|ξ2|.λ^−1/3ξ3in casej=Nλ.

On the microlocal support ofu_j, the bicharacteristic equation for the principal sym- bolξ1−pλ(x, ξ⁰) satisfiesdx2/dx1≈±θj, respectively asj >0 orj <0. A bicharacteristic curve passing through the microlocal support ofuj will satisfy this condition on an in- terval ofx1-length less thanεθj, ifεis a small constant. It is thus natural that we will have good estimates foruj on slabs of widthεθj in thex1variable, and it turns out that this is sufficient to prove Theorem 2.2.

In proving estimates foruj, it is convenient to work with the symbolpj obtained by truncatingp(x, ξ⁰) tox⁰-frequencies less thancθ^−1/2_j λ^1/2. This finer truncation than that ofpλis chosen so that, after rescaling the space byθj, the rescaled symbolpj(θjx,·) will bex⁰-frequency truncated atµ^1/2, whereµ=θjλis the frequency scale of the rescaled so- lutionuj(θjx). This square root truncation is consistent with the wave packet techniques we use, and is standard in the construction of parametrices for rough metrics.

The energy of the induced error term (Pλ−Pj)uwill be large at x2=0, but decays away from x2=0 at a rate that is integrable along bicharacteristic curves that traverse the boundary at angleθj. This error term can thus be considered as a bounded driving force, and we call this termGj below.

In the next two sections we will establish the following result.

Theorem 3.1. Let S_j,k denote the slab x₁∈[kεθ_j,(k+1)εθ_j] for 06k6ε⁻¹2^|j|. Then, if

D1uj−Pjuj=Fj+Gj,

it holds uniformly over j, k, and 66q6∞, that

kujk_LqL²(Sj,k).λ^δ(q)θ^1/2−3/q_j kujk_L^∞_L2(Sj,k)+kFjk_L1L²(Sj,k)

+λ^1/4θ^1/4_j khλ^1/2θ^−1/2_j x2i⁻¹ujk_L2(Sj,k)

+λ^−1/4θ_j^−1/4khλ^1/2θ^−1/2_j x2i²Gjk_L2(Sj,k)

.

For j=Nλ, it holds that

kujk_LqL²(S_j,k).λ^δ(q)θ_j^1/2−3/q(kujk_L∞L²(S_j,k)+kFj+Gjk_L1L²(S_j,k)).

The gain of the factorθ^1/2−3/q_j reflects the fact that, forq >6, there is an improvement in the squarefunction estimates if the solution is localized to a small conic set in frequency.

The terms G_j arise naturally in both the linearization step of Lemma 4.4 and the paradifferential smoothing (6.2). They reflect the fact that the singularities ofd²a^jj(x) are localized tox₂=0. The weightedL²bound onu_j is a characteristic energy estimate.

If θ_j≈1, then the weightedL² bound on G_j dominates theL¹_x

2L²_x

1,x₃ norm ofG_j, and exchangingx₁ and x₂ we could treat G_j and F_j the same. In this case the bound onu_j would be dominated by the L^∞_x

2L²_x

1,x₃ norm. For smallθ_j’s, however, we cannot usex₂ as our “time” variable, and we are forced to work with the weighted L² norms.

These weighted norms can be thought of as an energy norm along the bicharacteristic flow at angleθj. Precisely, if one replaced x2 by θj(x1−c) in the weight, then the weighted L²norms ofuj andGj would behave like theL^∞L²andL¹L² norms, respectively. The crossing pointc differs, however, for different bicharacteristics.

The proof of Theorem 3.1 is contained in §4 and§5. In§6 we establish the appro- priate bounds on the norms occuring on the right side if, as above,uj=βj(D⁰)uλ, while Fj and Gj are defined in (6.1)–(6.2) below.

To state the bounds required, letcj,kdenote the term occuring inside the parentheses on the right-hand side of Theorem 3.1. In§6, we show that, ifD1uλ−Pλuλ=Fλ, then we have a uniform summability condition

X

j

c²_j,k(j).kuλk²_L∞L²(S)+kFλk²_L2(S), (3.1)

wherek(j) denotes any sequence of values forksuch that the slabsS_j,k(j)are nested, in that forj >0 we haveS_j+1,k(j+1)⊂S_j,k(j)(with the analogous condition forj <0.)

In the remainder of this section we show how Theorem 2.2 follows from Theorem 3.1 together with the bound (3.1).

We first remark that, if q is a fixed index with q6=8, the bounds of Theorem 2.2 hold (with constant depending onq) under the weaker assumption that the cj,k’s are

uniformly bounded by the right-hand side of (3.1). To see this, we sum over the 2^jε⁻¹ slabs and write

kujkL^qL²(S)6 ^2jε−1

X

k=1

kujk^q_LqL²(S_j,k)

1/q

.λ^δ(q)θ^1/2−4/q_j kcj,kk`<sup>∞</sup><sub>j</sub> `^∞_k

.λ^δ(q)θ^1/2−4/q_j (ku_λk_L∞L²(S)+kF_λk_L2(S)).

The values ofθj=2^−|j| vary dyadically fromλ^−1/3 to 1. Forq >8 we can sum overj to obtain

kuλk_LqL²(S).λ^δ(q)(kuλk_L^∞_L2(S)+kFλk_L2(S)), and for 66q <8 the sum yields

kuλkL^qL²(S).λδ(q)−(1/3)(1/2−4/q)(kuλkL^∞L²(S)+kFλkL²(S)).

The above exponent of λequals γ(q), yielding the desired bound. The geometric sum, however, increases asq!8, and yields a logarithmic loss inλatq=8.

To obtain the bound atq=8, and hence uniform bounds overqin Theorem 2.2, we use the following worst-case branching argument. We consider terms withj >0 here, the negative terms being controlled by the same argument.

LetS_1,k(1) denote the slab at scale ε2⁻¹ that maximizes kuλk_L8L²(S1,k). Since the decomposition ofuλintouj’s is a Littlewood–Paley decomposition in theξ2variable, we have

kuλk²_L8L²(S_1,k(1)).

^Nλ

X

j=1

|uj|^{2 1/2}

2

L⁸L²(S_1,k(1))

.

By the Minkowski inequality,

^Nλ

X

j=1

|uj|^{2 1/2}

2

L⁸L²(S_1,k(1))

6ku1k²_L8L²(S_1,k(1))+

X

S2,k⊂S_1,k(1)

^Nλ

X

j=2

|uj|^{2 1/2}

8

L⁸L²(S_2,k)

^2/8

6ku1k²_L8L²(S_1,k(1))+2^2/8

^Nλ

X

j=2

|uj|² 1/2

2

L⁸L²(S_2,k(2))

,

where k(2) is chosen to maximize k(P∞

j=2|u_j|²)^1/2k_L8(S_2,k) among the two slabs S_2,k contained inS_1,k(1). Repeating this procedure yields a nested sequence such that

ε^1/4kuλk²_L8L²(S)6ku1k²_L8L²(S_1,k(1))+2^2/8ku2k²_L8L²(S_2,k(2))+2^4/8ku3k²_L8L²(S_3,k(3))+...

.λ^2δ(8)(c²_1,k(1)+c²_2,k(2)+c²_3,k(3)+...),

where the last inequality holds by Theorem 3.1 sinceθ_j^1/2−3/8=2^−j/8. The case q=8 of Theorem 2.2 follows by (3.1).

4. The wave packet transform

The purpose of this section and the next one is to establish Theorem 3.1. We assume for these two sections that we have fixedλandθ_j, and considerj >0 so thatξ₂>0 (except for the termj=N_λ, where|ξ₂|6λ^2/3).

We will rescale the space byθj. Thus, we work with the function u(x) =uj(θjx),

which, forj6=Nλ, is supported in the set ξ:ξ2∈₁

4θjµ,2θjµ

andξ3∈₁

4µ,4µ , where

µ=θjλ

is now the frequency scale foru(x). Forj=N_λ, we have|ξ2|6µ^1/2and θ_Nλ=µ^−1/2. Letq(x, ξ⁰) denote the rescaled symbol

q(x, ξ⁰) =θjpj(θjx, θ⁻¹_j ξ⁰),

which is truncated to x⁰-frequencies less than cµ^1/2. For|ξ⁰|≈µ, the symbol q satisfies the estimates

|∂_x^β∂_ξ^α0q(x, ξ⁰)|.

µ^1−|α|, if|β|= 0,

c0(1+µ^(|β|−1)/2θjhµ^1/2x2i^−N)µ^1−|α|, if|β|>1. (4.1) This follows from (6.32).

In the remainder of this section and the next one, we will drop the index j. The quantities θ and µ are the two relevant parameters for our purposes. After rescaling the estimates of Theorem 3.1, and translating S_j,k in x₁ to x₁=0, we are reduced to establishing the following result. Here,S denotes the (x₁, x⁰) slab [0, ε]×R².

Theorem 4.1. Suppose that u(ξ)ˆ is supported in the set ξ:ξ2∈1

4θµ,2θµ

and ξ3∈1

4µ,4µ , respectively

ξ:|ξ2|6µ^1/2 and ξ3∈₁

4µ,4µ in case θ=µ^−1/2. Suppose that usatisfies D1u−q(x, D⁰)u=F+G

on the slabS, where q satisfies (4.1),and is truncated to x⁰-frequencies less than cµ^1/2. Then the following bounds hold, uniformly over θ, µ,and 66q6∞:

kukL^qL²(S).µ^δ(q)θ^1/2−3/q kukL^∞L²(S)+kFkL¹L²(S)+µ^1/4θ^1/2khµ^1/2x₂i⁻¹ukL²(S)

+µ^−1/4θ_j^−1/2khµ^1/2x₂i²GkL²(S)

, and forθ=µ^−1/2

kukL^qL²(S).µ^δ(q)θ^1/2−3/q(kukL^∞L²(S)+kF+GkL¹L²(S)).

We introduce at this point the phase-space transform which will be used to establish Theorem 4.1. This transform is essentially the C´ordoba–Fefferman wave packet trans- form [3]. The precise form here is a simple modification of the Fourier–Bros–Iagolnitzer transform used by Tataru in [18] and [19] to establish Strichartz estimates for low reg- ularity metrics; the difference is that in our applications we use a Schwartz function with compactly supported Fourier transform, instead of the Gaussian, as the fundamen- tal wave packet. This is useful in that it strictly localizes the frequency support of the transformed functions. Our transform will act on thex⁰=(x2, x3) variables.

We use the notion of the previous sections: x=(x1, x2, x3)=(x1, x⁰), where x3 de- notes the variablet.

Fix a real, radial Schwartz functiong(z⁰)∈S(R²), with kgk_L2(R²)=(2π)⁻¹, and as- sume that its Fourier transform ˆg(ζ⁰) is supported in the ball{ζ⁰:|ζ⁰|6c}. Forµ>1, we defineTµ:S⁰(R²)!C^∞(R⁴) by the rule

(T_µf)(x⁰, ξ⁰) =µ^1/2 Z

e^{−ihξ0,y0−x0i}g(µ^1/2(y⁰−x⁰))f(y⁰)dy⁰. A simple calculation shows that

f(y⁰) =µ^1/2 Z

e^{ihξ0,y0−x0i}g(µ^1/2(y⁰−x⁰))(Tµf)(x⁰, ξ⁰)dx⁰dξ⁰, so thatT_µ^∗Tµ=I. In particular,

kTµfk_L2(R⁴)=kfk_L2(R²). (4.2) It will be useful to note that this holds in a more general setting.

Lemma4.2. Suppose that gx⁰,ξ⁰(y⁰)is a family of Schwartz functions onR², depend- ing on the parameters x⁰ and ξ⁰, with uniform bounds over x⁰ and ξ⁰ on each Schwartz norm of g. Then the operator

(Tµf)(x⁰, ξ⁰) =µ^1/2 Z

e^{−ihξ0,y0−x0i}gx⁰,ξ⁰(µ^1/2(y⁰−x⁰))f(y⁰)dy⁰ satisfies the bound

kTµfkL²(R⁴).kfkL²(R²). Proof. Tµ is bounded if and only if T_µ^∗ is bounded. Since

kT_µ^∗Fk²_L26kTµT_µ^∗FkL²kFkL²,

it suffices to see thatTµT_µ^∗ is bounded onL²(dy⁰dξ⁰).

The operatorT_µT_µ^∗ is an integral operator with kernel K(y⁰, η⁰;x⁰, ξ⁰) =µe^{ihη0,y0i−ihξ0,x0i}

Z

e^{ihξ0−η0,z0i}g_y⁰_,η⁰(µ^1/2(z⁰−y⁰))g_x⁰_,ξ⁰(µ^1/2(z⁰−x⁰))dz⁰. A simple integration by parts argument shows that

|K(y⁰, η⁰;x⁰, ξ⁰)|.(1+µ^−1/2|η⁰−ξ⁰|+µ^1/2|y⁰−x⁰|)^−N,

with constants depending only on uniform bounds for a finite collection of seminorms of g_x⁰_,ξ⁰ depending on N. The L²(R⁴) boundedness of T_µT_µ^∗ then follows by Schur’s lemma.

A corollary of Lemma 4.2 is that, forN positive or negative,

khµ^1/2x2i^NTµfk_L2(R⁴).khµ^1/2x2i^Nfk_L2(R²), (4.3) by consideringg_x⁰(y)=hµ^1/2x₂i^Nhµ^1/2x₂−y2i^−Ng(y⁰).

The next two lemmas relate the conjugation of q(x, D⁰), by the wave packet trans- form, to the Hamiltonian flow underq. These results are analogous to [18, Theorem 1], in that the error term is one order better where the metric has two bounded derivatives.

In our case the second derivatives are large along the boundaryx₂=0, which leads to larger errors there. A key fact for our paper is that the errors are suitably integrable along the Hamiltonian flow ofq.

Lemma 4.3. Let q(x, ξ⁰)satisfy the estimates (4.1). Suppose that |ξ⁰|≈µ. Then,if q(y, D_y⁰)^∗ acts on the y⁰ variable, and y₁=x₁,we can write

(q(y, D_y⁰)^∗−idξ⁰q(x, ξ⁰)·dx⁰+idx⁰q(x, ξ⁰)·dξ⁰)[e^{ihξ0,y0−x0i}g(µ^1/2(y⁰−x⁰))]

=e^{ihξ0,y0−x0i}gx,ξ⁰(µ^1/2(y⁰−x⁰)), whereg_x,ξ⁰(·)denotes a family of Schwartz functions onR²depending on the parameters xand ξ⁰,each of which has Fourier transform supported in the ball of radius 2c. If k · k denotes any of the Schwartz seminorms, we have

kgx,ξ⁰k.1+c0µ^1/2θhµ^1/2x2i⁻³, where c₀ is the small constant of (2.3).

Proof. LettingFdenote the Fourier transform with respect toy⁰, we write F(q(y, D_y⁰)^∗−idξ⁰q(x, ξ⁰)·dx⁰+id_x⁰q(x, ξ⁰)·dξ⁰)[e^{ihξ0,y0−x0i}g(µ^1/2(y⁰−x⁰))](η⁰)

=e^−ihη0,x0iµ⁻¹gd_x,ξ⁰(µ^−1/2(η⁰−ξ⁰)),

wheregd_x,ξ⁰(η⁰) is equal to Z

e^−ihη0,y0i[q(x+µ^−1/2y⁰, ξ⁰+µ^1/2η⁰)−q(x, ξ⁰)−dx⁰,ξ⁰q(x, ξ⁰)·(µ^−1/2y⁰, µ^1/2η⁰)]g(y⁰)dy⁰

= Z 1

0

(1−σ) Z

e^−ihη0,y0i∂_σ²(q(x+σµ^−1/2y⁰, ξ⁰+σµ^1/2η⁰))g(y⁰)dy⁰

dσ.

The spectral restriction onqandgimplies that this vanishes for|η⁰|>2c. Consequently, it suffices to establishC^∞bounds inη⁰for the term in brackets, uniformly overσ∈[0,1] and

|η⁰|62c. Since the effect of differentiating the integrand with respect toη⁰ is innocuous, as the rapid decrease ing(y⁰) counters any polynomial in y⁰, we content ourselves with establishing uniform pointwise bounds on the term in brackets. Note that

|ξ⁰+σµ^1/2η⁰| ≈µ.

The effect of∂_σ² is to bring out factors ofµ^±1/2, and to differentiateqtwice. Ifqis differentiated at most once inx⁰, then the bounds

|∂x⁰∂_ξ⁰q(x, ξ⁰)|.1 and |∂_ξ²0q(x, ξ⁰)|.µ⁻¹ for|ξ⁰| ≈µ,

yield bounds of size 1 on the term. If q is differentiated twice in x⁰, then by (4.1) we have the bounds, for|ξ⁰|≈µ,

µ⁻¹|∂_x²0q(x+σµ^−1/2y⁰, ξ⁰)|.c0+c0µ^1/2θhµ^1/2x2+σy2i⁻³ .1+c0µ^1/2θhµ^1/2x2i⁻³hy2i³.

The rapid decrease ofg(y⁰) absorbs the term hy2i³, leading to the desired bounds.

We now take the wave packet transform of the solutionu(x) with respect to thex⁰ variables, and introduce the notation ˜u(x, ξ⁰)=(Tµu)(x, ξ⁰). The functionsFe(x, ξ⁰) and G(x, ξe ⁰) in the next lemma, though, include terms in addition to the transforms of F andGof Theorem 4.1. LetSedenote the (x1, x⁰, ξ⁰) slab [0, ε]×R⁴=S×R²_ξ0.

Lemma4.4. Under the above conditions, we may write

(d₁−d_ξ⁰q(x, ξ⁰)·d_x⁰+d_x⁰q(x, ξ⁰)·d_ξ⁰)˜u(x, ξ⁰) =Fe(x, ξ⁰)+G(x, ξe ⁰), where

kFek_L1L²(S)e +µ^−1/4θ^−1/2khµ^1/2x2i²Gke _L2(S)e .kukL^∞L²(S)+kFkL¹L²(S)

+µ^1/4θ^1/2khµ^1/2x2i⁻¹uk_L2(S)

+µ^−1/4θ^−1/2khµ^1/2x2i²Gk_L2(S). (4.4)