Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: The 2D case

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Ann. I. H. Poincaré – AN 31 (2014) 1267–1288

www.elsevier.com/locate/anihpc

Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: The 2D case

Jean Bourgain

, Aynur Bulut

School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, United States Received 4 May 2013; received in revised form 30 August 2013; accepted 26 September 2013

Available online 9 October 2013

Abstract

Our first purpose is to extend the results from[14]on the radial defocusing NLS on the disc inR²to arbitrary smooth (defocusing) nonlinearities and show the existence of a well-defined flow on the support of the Gibbs measure (which is the natural extension of the classical flow for smooth data). We follow a similar approach as in[8]exploiting certain additional a priori space–time bounds that are provided by the invariance of the Gibbs measure.

Next, we consider the radial focusing equation with cubic nonlinearity (the mass-subcritical case was studied in[15]) where the Gibbs measure is subject to anL²-norm restriction. A phase transition is established. For sufficiently smallL²-norm, the Gibbs measure is absolutely continuous with respect to the free measure, and moreover we have a well-defined dynamics. For sufficiently largeL²-norm cutoff, the Gibbs measure concentrates on delta functions centered at 0. This phenomenon is similar to the one observed in the work of Lebowitz, Rose, and Speer[13]on the torus.

©2013 Elsevier Masson SAS. All rights reserved.

1. Introduction

The purpose of this work is to establish global well-posedness results for the initial value problems associated to the defocusing (−) and focusing (+) nonlinear Schrödinger equation,

iu_t+u∓ |u|^αu=0,

u|t=0=φ (1)

withα∈2Nin the defocusing case, posed on the two-dimensional unit ballB₂⊂R², and withα=_d⁴ in the focusing case, posed on thed-dimensional unit ballB_d⊂R^d,d2. In both cases, we prescribe Dirichlet boundary conditions u(t )=0 on∂B_dfor allt∈R.

In order to obtain results globally in time we will appeal to a probabilistic viewpoint, invoking the construction of an invariant Gibbs measure developed in the setting of nonlinear dispersive equations in the works[3–5]. To motivate our discussion below, let us first recall that a Hamiltonian system of the form

* Corresponding author.

E-mail addresses:bourgain@math.ias.edu(J. Bourgain),abulut@math.ias.edu(A. Bulut).

0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2013.09.002

d dt

p_i q_i

i=1,...,n

=

−∂H /∂q_i

∂H /∂p_i

i=1,...,n

(2) withH=H (p1, p2, . . . , pn, q1, q2, . . . , qn)is subject to the following invariance property: theGibbs measure

e^{−H (p,q)}dL²ⁿ(p, q) satisfies

A

e^{−H (p,q)}dL²ⁿ(p, q)=

{(p(t),q(t)):(p(0),q(0))∈A}

e⁻H (p(t ),q(t ))dL²ⁿ

p(t ), q(t )

(3)

for every measurable set A⊂R²ⁿ andt∈R, whereL²ⁿ denotes Lebesgue measure and we use the abbreviation p=(p1, . . . , pn),q=(q1, . . . , qn).

The relevance of this observation to our present study is that the equation in(1)is of the formiut=∂H /∂u, with conserved Hamiltonian

H (φ)=1 2

B

|∇φ|²dx± 1 α+2

B

|φ|^α+2dx.

In order to access the invariance(3)of the Gibbs measure in this infinite dimensional setting, we shall consider a sequence of finite-dimensional projections of the problem(1), namely

iu_t+u∓P_N

|u|^αu

=0,

u|t=0=PNφ (4)

for every integerN1, whereP_Ndenotes the frequency truncation operator defined via the relation PN

n∈N

anen(x)

=

{n∈N:znN}

anen(x),

with(e_n)the sequence of radial eigenfunctions and(z²_n)the sequence of associated eigenvalues of−with vanishing Dirichlet boundary conditions.

Solutionsu_Nto(4)exist globally in time and can be represented as u_N(t, x)=

{n∈N:znN}

u_n(t)e_n(x),

and the equation may be written in the form(2)with p_i=Re

u_n(t)

, q_i=Im u_n(t)

.

The Hamiltonian associated to the finite-dimensional projected problem(4)is then H_N(φ)=1

2z_nN

z²_nφ(n)²± 1 α+2

B

P_Nφ(x)^α+2dx.

Furthermore, the flow map φ_N→u_N(t)

leaves invariant the Gibbs measureμ_Gcorresponding to(4)defined by dμ_G=e^−HN(φ)dφ=e^∓

α+21 P_Nφ ^α+2

Lα+2

x dμ^{(N )}_F , (5)

withμ^{(N )}_F denoting the free probability measure induced by the mapping ω→ 1

π{n∈N:z_nN}

gn(ω) zn

e_n, ω∈Ω

where(g_n)is a sequence of normalized independent Gaussian random variables on a probability space(Ω, p,M).

As noted in[3,6], when writing(5)one must take care to ensure (i) theμ_F-a.s. existence of the norm P_Nφ _Lα+2 x

and (ii) the integrability of the densitye^∓

α+21 PNφ ^α+2

Lα+2

x with respect to the measureμF. In both the defocusing and focusing cases, the first condition is satisfied as a consequence of estimates on the eigenfunctions. On the other hand, while the second condition is trivial in the defocusing setting, it is not satisfied in general when focusing interactions are present.

In the setting of focusing periodic NLS on the one-dimensional torus, the non-integrability of the density was overcome in the work of Lebowitz, Rose, and Speer[13]by restriction to a ball in the conservedL²_xnorm. In particular, one fixesρ >0 and considers

dμ_G=e

1

α+2 P_Nφ ^α+2

Lα+2 x χ_{{ PNφ}

L2

x<ρ}(φ) dμ^{(N )}_F

which is again invariant under the evolution and can be normalized to a well-defined measure for allα4 (the case α=4 requiresρsufficiently small).

1.1. Main results of the present work

In recent works[8,9](see also[7]), we addressed the Cauchy problems corresponding to(1) for the defocusing nonlinear wave and Schrödinger equations on the unit ball ofR³, establishing global well-posedness results for radial solutions with random initial data according to the support of the Gibbs measure (almost surely in the randomization) (see also[11]).

We say that functions u, u_N :I ×B_d →C are solutions of (1), (4), respectively, if they belong to the class C_t(I;H_x^σ(B_d))for someσ <¹₂ and satisfy the associated integral equations

u(t )=φ±i t

0

e^{i(t−τ )}u(τ )^αu(τ )

dτ, t∈I, (6)

and

uN(t)=PNφ±i t

0

e^{i(t−τ )}PNu(τ )^αu(τ )

dτ, t∈I. (7)

To proceed with our discussion, recall that we consider the sequence of finite-dimensional projections(4). Our estimates will typically be uniform in the truncation parameterN. To accommodate this, we will often make use of the probability measureμ_F induced by the mapping

ω→φ^(ω):= 1 πn∈N

gn(ω) z_n e_n.

Note that with this notation, one hasμ^{(N )}_F =PN[μF]. Moreover, for eachN1, the support of the Gibbs measure μGcorresponds to the set

PNφ^(ω): ω∈Ω .

With this probabilistic framework in mind, our first main result, concerning the defocusing problem, takes the following form:

Theorem 1.1.Fix α∈2N. With the above notations, for N∈N,ω∈Ω, letu_N denote the solution to(4) in the defocusing case on the two-dimensional unit ball with initial dataPNφ=PNφ^(ω). Then almost surely inΩ, for every 0< T <∞there existsu_∗∈Ct([0, T );H_x^s(B2)),s < ¹₂such thatuN converges tou_∗inCt([0, T );H_x^s(B2)).

We remark thatTheorem 1.1was announced in[7]. The restriction on the nonlinearity toα∈2Nis by no means essential, and serves only to simplify the estimates on the nonlinearity, avoiding technicalities due to fractional powers.

The caseα <4 was treated in[14].

The proof ofTheorem 1.1further develops the method of[8]and consists of an analysis of convergence properties of solutions to the truncated equations(4). In order to perform this analysis, we will make use of three key ingredients:

(i) A detailed study of embedding properties associated to the Fourier restriction spacesX^s,b (seeLemma 2.3and Lemma 2.5).

(ii) A probabilistic estimate demonstrating how the randomization procedure leads to additionalL^pxL^q_t control, al- most surely in the probability space (seeProposition 3.2).

(iii) A bilinear estimate of the nonlinearity enabling one to estimate interactions of high and low frequencies, allowing for a paraproduct-type analysis in the present setting (seeProposition 4.1).

The embeddings established inLemma 2.3andLemma 2.5use frequency decomposition techniques, exploiting the product structure inherent in theL⁴_t norm and the Plancherel identity. A technical tool used to estimate the fre- quency interactions at this stage are arithmetic estimates for the counting of lattice points on circles (see in particular Lemma 2.1).

On the other hand, the probabilisticL^p_xL^q_t bounds ofProposition 3.2make essential use of the fact thatu_N is a solution of (4). More precisely, the improvement in integrability follows from the invariance of the Gibbs measure and bounds for functions belonging to its support.

Turning to the bilinear estimateProposition 4.1, theX^s,b norm of certain products in the Duhamel formula are estimated byX^s,bandL²_tH_x^γ norms (γ >0 small) of its factors – this involves appropriate high and low frequency localizations. The proof of this proposition is in the flavor of similar estimates in theR^d setting, with the additional component that the usual convolution identities are replaced with estimates on the correlation of eigenfunctions.

To conclude the proof of Theorem 1.1, the ingredients (i), (ii) and (iii) are combined in order to show that the approximate solutionsu_N-almost surely converge in the spaceX^s,bvia a bootstrap-type argument. We refer the reader to Section5for the full details of the argument.

Our second main result, treating the focusing problem, is as follows.

Theorem 1.2. Set α=2. For each N ∈N,ω∈Ω letuN denote the solution to (4) in the focusing case on the two-dimensional unit ball with initial dataPNφ=PNφ^(ω)and subject to an appropriateL²-norm restriction. Then for every0< T <∞, there exists almost surelyu_∗∈C_t([0, T );H_x^s(B₂)),s < ¹₂, such thatu_N converges tou_∗ in C_t([0, T );H_x^s(B₂)).

The main additional issue in the proof ofTheorem 1.2is to show theμ_F-integrability of the map φ→e

α+21 φ ^α+2

Lα+2 x χ_{{ PNφ}

L2 x<ρ}(φ)

provided thatρ >0 is chosen sufficiently small. This result is stated inProposition 6.1, and ensures a bound on the L²_x-truncated Gibbs measures

dμ^{(N )}_G =e

1 α+2 φ ^α+2

Lα+2 x χ_{{ PNφ}

L2

x<ρ}(φ) dμ^{(N )}_F .

Once we have the invariant measure at our disposal, the convergence of the solutions of the truncated equations follows from the same argument as in the defocusing case forα=2, leading to a well-defined dynamics on the support of the modified Gibbs measure.

For subscritical nonlinearityα <2, the corresponding result was established in[15](with arbitraryL²-truncation).

InRemark 6.4in Section6, we will also comment on what happens for largerL²-norm restrictionρ.

1.2. Outline of the paper

The remainder of this paper is structured as follows: in Section2we establish our notation and recall the definitions of the function spaces which will be used in the remainder of the paper. Section3is then devoted to the proof of a probabilistic estimate for solutions corresponding to initial data in the support of the Gibbs measure. In Section4, we establish a key bilinear estimate on the nonlinearity, while the proof ofTheorem 1.1is contained in Section 5.

We conclude by establishingTheorem 1.2in Section6.

2. Preliminaries 2.1. Notation

Let us now establish some brief notational conventions. Unless otherwise indicated, we will use the conventions n∈N,m∈Z, while capital lettersK,N andMshall denote dyadic integers of the form 2^k,k0. Throughout our arguments we will frequently make use of a dyadic decomposition in frequency, writing

f (x)=

n

f (n)e _n(x)=

N1n∼N

f (n)e _n(x),

where for each dyadic integerN, the conditionn∼N is characterized byNn <2N (likewise, we saym∼Mif M|m|2M). We shall also use the notationx =(1+ |x|²)^1/2.

For eachn∈N, letz_n∈R\ {0}be such thatz²_nis thenth eigenvalue of the Dirichlet Laplacian onB₂, and recall thatz_nsatisfies

zn=π

n−1 4

+O

1 n

. (8)

Following the usual convention, we will often refer to(zn)as the sequence of frequencies for functions defined on the ballB2. Moreover, letendenote thenth radial eigenfunction, corresponding to the eigenvaluez²_n. One then has

e_{n L}^p

x 1, p∈ [2,4), e_{n L}^p

x log(2+n)^1/4, p=4,

en L^px n^12−p2, p∈(4,∞). (9)

We now state a basic probabilistic estimate for Gaussian random variables. In particular, if(gn)is a sequence of independent (normalized) complex Gaussians, then we have

n

α_ng_n(ω) L^q(dω)

√

q

n

|α_n|² 1/2

. (10)

Moreover, if X(ω) is a Gaussian process with values in some normed space (E, · ), of finite expectation Eω[ X ], it follows that

e^{c(E[ X X ])2}< C

and hence Pω

X > tEω

X

e^−ct2, t >1. (11)

The results and analysis in this section appear basically in[14]and are repeated here in a form suitable for our presentation and in the interest of being self-contained.

2.2. Arithmetic estimates

As usual in the study of nonlinear Schrödinger equations on bounded domains (e.g. the case of tori treated in[3, 4]) an essential component of our analysis will rely upon arithmetical bounds for the sequence of frequencies. In particular, we shall use the following:

Lemma 2.1.There existsc >0such that for everyR > R₁1and all boxesQ⊂R²of sizeR₁, we have (n1, n2)∈Z²: n²₁+n²₂=R², (n1, n2)∈Qexp

c logR₁ log logR₁

. (12)

Proof. LetR₁< R be given. Suppose first thatR₁andRsatisfyR^1/3R₁. Note that factorization in the Gaussian integersZ+iZimplies the bound

(n₁, n₂)∈Z²: n²₁+n²₂=R²expGaussian prime factors ofR²<exp logR

log logR. (13)

Since logR∼logR₁, the inequality(12)now follows from(13).

On the other hand, suppose thatR₁R^1/3. It then follows by Jarnick’s theorem for lattice points on circles that the left-hand side of(12)is at most 2, which gives the claim in this case. 2

Lemma 2.2.Letz²_nbe thenth eigenvalue of the Dirichlet Laplacian onB₂and letQ⊂R²be a box of sizeR₁. Then for any∈R₊,

(n₁, n₂)∈Z²: z²_n

1+z_n²

2−<1, (n1, n₂)∈Qexp

c logR₁ log logR₁

.

Proof. According to(8),z²_n=π²(n−¹₄)²+O(1). Therefore the equation|z_n²

1+z²_n

2−|<1 implies π²

n₁−1

4 2

+π²

n₂−1 4

2

−

< O(1), (4n1−1)²+(4n2−1)²−16

π²

< O(1)

and we can applyLemma 2.1settingn₁=4n1−1,n₂=4n2−1. 2 2.3. Description of theX^s,bspaces

FixI= [0, T )with 0< T <¹₂, and letX^s,b(I )denote the class of functionsf:I×B→Crepresentable as f (t, x)=

n,m

fn,men(x)e(mt), (t, x)∈I×B (14)

for which the norm f _s,b:=inf

n,m

z_n^2s

z²_n−m2b

|f_n,m|² 1/2

is finite, with the infimum taken over all representations(14), cf.[1,2]. Throughout the remainder of the paper, we will assume 0< T <¹₂, unless otherwise indicated.

We now give two lemmas expressing some embeddings of the spaceX^s,b which will be essential components of our analysis below. Similar estimates appear already in[14](see in particular[14, Proposition 4.1]).

Lemma 2.3.Let ¹₄< b <1and2p <4be given. Then, lettingPIf =

zn∈If (n)e n, we have for >0,f ∈S and intervalsI⊂R,

P_If _L^p

xL⁴_t

|I| PIf 0,b forb >¹₂,

|I|^1−2b+ P_If _0,b forb <¹₂. (15)

Proof. We begin by establishing the first inequality in(15), for which we shall compute the norm directly.

Fix >0 and write P_If (t, x)=

m∈Z zn∈I

f (m, n)e _n(x)e(mt)=

m

zn∈I

f m+

z²_n , n

e_n(x)e z²_nt

e(mt).

Performing a dyadic decomposition into intervalsm∼M, we obtain P_If _L^p

xL⁴_t

M

f_{M L}^p

xL⁴_t

with f_M=

m∼M zn∈I

f m+

z²_n , n

e_n(x)e z²_nt

e(mt).

We then have f_{M L}^p

xL⁴_t

m∼M

zn,z_n∈I

|z²_n+(z_n)²−|<1

f m+

z²_n, n f m+

(z_n)² , n

e_n(x)e_n(x)e^{it 1/2}

L^p/2_x L²_t

m∼M

z_n,z_n∈I

|z²_n+(z_n)²−|<1

f m+

z²_n, n f m+

(z_n)² , n

e_n(x)e_n(x) ²1/2

^1/2

L^p/2x

, (16)

where in obtaining the last inequality we have used the Plancherel identity in thet variable.

Using the Cauchy–Schwarz inequality andLemma 2.2, (16)

m∼M

sup

zn,z_n∈I

|z²_n+(z_n)²−|<1

1 1/4

zn∈I

f m+

z²_n, n²en(x)^{2 1/2}

L^p/2x

m∼M

|I|

zn∈I

f m+

z²_n, n²e_n(x)²

L^p_x

1/2

m∼M

|I|

zn∈I

f m+

z²_n, n²1/2

where to obtain the last inequality we used the eigenfunction estimate(9).

Invoking the Cauchy–Schwarz inequality once more, (16)|I|M¹²

m∼Mzn∼I

f m+

z²_n, n²1/2

= |I|M¹²

zn∼I m−z²_n∼M

f (m, n)²1/2

|I|M^12−b P_If _0,b. (17)

On the other hand to obtain the second inequality in(15), a similar calculation yields f_{M L}^p

xL⁴_t

zn∈I

m∼M

f m+

z²_n , n

e(mt ) L⁴_t

e_n(x)

L^p_x

M¹⁴

zn∈I

m−z²_n∼M

f (m, n)²1/2

|I|^1/2M^14−b P_If _0,b. (18)

Thus from(17),(18) f_{M L}^p

xL⁴_t min

|I|M^12−b,|I|¹²M^14−b

P_If _0,b and summation inMgives the desired estimates. 2

Remark 2.4.The estimates obtained inLemma 2.3also allow us to conclude f _L^p

xL⁴_t C f _,b forb1/2 andp4 and

f _L^p

xL⁴_t C f _1−2b+,b for 1/4< b <1/2.

Indeed, appealing to the decompositionf=

NP_{[N,2N )}f withN1 dyadic and applying(15)on each interval I= [N,2N )(with˜=/2), we obtain

f _L^p

xL⁴_t

N

N^/2 P_{[N,2N )}f _0,b

N

N^−/2 f _,b f _,b

forb1/2. An identical calculation with an application of the second inequality in(15)in place of the first inequality of(15)gives the claim for 1/4< b <1/2.

As a consequence of Lemma 2.3, we have the following estimates on the nonlinear term of the Duhamel for- mula(7).

Lemma 2.5.For each intervalI⊂Rand everyb >¹₂, >0, there existsC=C(b, ) >0such that

P_I t

0

e^{i(t−τ )}f (τ ) dτ

0,b

C|I|^2b−1+ f

L

4 3+

x L

4 t3

. (19)

Moreover, the inequality

t

0

e^{i(t−τ )}f (τ ) dτ 0,b

C(√

− )^2b−1+f

L

4 3+

x L

4 t3

(20) also holds for allf ∈S.

Proof. We begin by showing(19), invoking the representation f (t, x)=

m,n

f (m, n)e n(x)e(mt) and observing that the left-hand side of(19)is

m∈Z

zn∈I

t

0

e^{i(t−τ )z2n}f (m, n)e _n(x)e^imτdτ 0,b

=

m∈Z zn∈I

f (m, n)e _n(x)·e(mt)−e(z²_nt ) i(m−z²_n)

0,b

m∈Z zn∈I

|f (m, n) |² m−z²_n^2(1−b)

1/2

+

z_n∈I

m

f (m, n) m−z²_n

²1/2

m∈Z z_n∈I

|f (m, n) |² m−z²_n^2(1−b)

1/2

= P_If _0,−(1−b) (21)

where we have used Cauchy–Schwarz to obtain the second inequality.

By duality arguments followed by Hölder’s inequality combined withLemma 2.3, we then have PIf 0,−(1−b)=sup

PIf (t, x)PIg(t, x) dt dx

: g∈L², PIg 0,1−b1

f

L

4 3+

x L

4 t3

PIg

L

4+3 x1+3L⁴_t

|I|^{1−2(1−b)+} f

L

4 3+

x L

4 t3

P_Ig _0,1−b |I|^2b−1+ f

L

4 3+

x L

4 t3

. (22)

The inequality(19)now follows by combining(21)and(22).

The proof of(20)proceeds in a similar manner. Arguing as above and usingRemark 2.4, we obtain

m∈Z n∈N

t

0

e^{i(t−τ )z2n}f (m, n)e _n(x)e^imτdτ

−(2b−1+),b

f _{−(2b−1+),−(1−b)}

sup

f

L

4 3+

x L

4 t3

g

L

4+3 1+3 x L⁴_t

: g _{2b−1+,1−b}1

f

L

4 3+

x L

4 3 t

g 2b−1+,1−b

f

L

4 3+

x L

4 3 t

from which the desired inequality(20)follows immediately. 2 3. Probabilistic estimates

In this section, we establish a collection of essential probabilistic estimates which will enable us to obtain long-time control over solutions to(4). We remark that these estimates are uniform in the truncation parameterN; this uniformity is important in the convergence proof of the next section.

We begin by establishing an almost sure bound on initial data belonging to the support of the Gibbs measureμ_G. Lemma 3.1.Fixs <¹₂. Then we have the bound

μ_F φ: N

1 2−s

0 P_N0φ _Hx^s > λ

exp

−λ^c

(23) for allN₀1sufficiently large, whereφ=φ^(ω)=

n∈Ng_n(ω) zn e_n.

Proof. Fixq12 to be determined. The Tchebyshev and Minkowski inequalities together with the estimate(10)then imply that the left-hand side of(23)is bounded by

N⁽

1 2−s)q₁ 0

λ^q1 PN₀φ ^q1

L^q_ω¹(dμ_F;H_x^s)N⁽

1 2−s)q₁ 0

λ^q1

∞ n=N0

g_n(ω) z⁽¹_n^−s)en

q₁

L²_x(L^q_ω¹(dμ_F))

N

1 2−s 0

√q₁ λ

q1 _∞

n=N₀

e_n(x) ²

L²_x

z²⁽¹_n ^−s) q₁/2

N

1 2−s 0

√q1

λ

q_{1 ∞}

n=N0

z⁻_n^2(1−s) q1/2

√

q1

λ q1

, (24)

where the summation in the third line is bounded by N₀^−1+2s for N₀ sufficiently large, as a consequence of the asymptotic representation (8) for the sequence of eigenvalues (z_n). Optimizing(24)in q₁, we obtain the desired estimate(23). 2

In particular, we note thatLemma 3.1includes a description of the decay present when considering the restriction of φto high frequencies. The next proposition is an essential ingredient and combines this type of probabilistic estimate with the invariance of the Gibbs measure μG under the finite-dimensional evolution to obtain certain space–time bounds of large deviation type.

Proposition 3.2.LetT >0be given. Then for every0σ <¹₂,2p <_σ² andq <∞, EμF(√

− )^σu_φ

L^pxL^q_t

< C(σ, p, q, T )

whereu_N=u^(φ)_N is the solution of (4)corresponding to initial dataP_Nφand theL^q_t norm is taken on the interval [0, T ). In fact, there is the stronger distributional inequality

μ_F φ: (√

− )^σu_N

L^p_xL^q_t > λ

exp

−λ^c

, λ >0, N1, for somec >0.

Proof. Fixλ₁>0 to be determined later in the argument. Then, denotingu=u^(φ)_N , we have μF

φ: (√

− )^σu

L^p_xL^q_t > λ

e^2+1αλ21+αμG

φ: (√

− )^σu

L^p_xL^q_t > λ +μ_F

φ: (√

− )^σu

L^p_xL^q_t > λ

∩

φ: φ _L2+α x > λ₁

=:(I)+(II). (25)

To estimate term (I), we observe that for everyq₁max{p, q}the Tchebyshev inequality and Minkowski inequal- ity for integrals allow us to bound

μ_G φ: (√

− )^σu

L^p_xL^q_t > λ by

1

λ^q1 (√

− )^σu^q1

L^pxL^q_t dμG(φ) T^1/q

λ q1

(√

− )^σφ

L^p_x^q1

L^q1(dμG)

T

√q₁ λ

q1

znN

|e_n(x)|² z²⁽¹_n ^{−σ )}

1/2 ^q1

L^px

(26) where we use the invariance of the Gibbs measureμ_Gunder the truncated evolution(4). Using Minkowski’s inequality (sincep2), we then have

(26) √

q1

λ

q1

znN

e_n(x) ²

L^px

z²⁽¹n ^{−σ )}

q1/2

. (27)

Now, recalling the eigenfunction bounds(9), we have the bound

n∈N

n^1−4p

z_n^{2(1−σ )}C₀+C₁

nN₀

n^{−(1−2σ )−p4} <∞, (28)

where we have used the hypothesisp < ²_σ.

Combining(28)with(27)gives μ_G

φ:(√

− )^σu

L^p_xL^q_t > λ

√q₁ λ

q₁

(29) where the implicit constant is independent ofN. Optimizing(29)inq₁then implies the bound

(I)exp 1

2+αλ²₁^+α

exp

−λ²/(2e)

. (30)

To estimate term (II), we fixq₂>2+αand again apply the Tchebyshev and Minkowski inequalities to bound the term

μ_F

φ: φ _L2+α x > λ₁ by

1 λ^q₁²

φ ^q2

L²x^+α

dμ_F(φ) 1 λ^q₁²

φ _L^q2(dμ_F)^q2

L²_x^+α

√q₂ λ₁

q₂

n

|e_n(x)|² z²_n

1/2 ^q2

L²x^+α

√q2

λ₁

q2

n

e_n(x) ²

L^2+α_x

z²_n

q2/2

.

Appealing to the eigenfunction bounds(9)and the asymptotic representation(8)forzn, we estimate

n∈N

n^1−2+α4

z²_n C₀+

nN0

n⁻¹⁻²+⁴α <∞

forN₀sufficiently large. As a consequence we obtain μF

φ: φ _L2+α x > λ1

√

q2

λ₁ q2

.

Minimizing the right-hand side over admissible values ofq2we get the bound (II)exp

−λ²₁/(2e)

. (31)

Combining(25)with(30)and(31)and optimizing inλ1then gives μ_F

φ: (√

− )^σu

L^p_xL^q_t > λ

exp

−cλ²+⁴α as desired. This completes the proof ofProposition 3.2. 2

We will also require a slight refinement ofProposition 3.2which is a consequence of similar arguments, and allows for more precise estimates.

Proposition 3.3.LetT,σ,p,q and(u_N)be as inProposition3.2. Then for allM, N1withM < N, one has the distributional inequality

μ^{(N )}_F

φ_N: (√

− )^σ(u_N−P_Mu_N)

L^p_xL^q_t > λ

< e^{−(θ λ)c} where we have setθ=T^−1qM^p2−σ.