Random weighted Sobolev inequalities on R d and application to Hermite functions
aa r X i v : . [ m a t h . A P ] D ec RANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d ANDAPPLICATION TO HERMITE FUNCTIONS by Aur´elien Poiret, Didier Robert & Laurent Thomann
Abstract . —
We extend a randomisation method, introduced by Shiffman-Zelditch and developed byBurq-Lebeau on compact manifolds for the Laplace operator, to the case of R d with the harmonic oscillator.We construct measures, thanks to probability laws which satisfy the concentration of measure property,on the support of which we prove optimal weighted Sobolev estimates on R d . This construction relies onaccurate estimates on the spectral function in a non-compact configuration space. As an application, weshow that there exists a basis of Hermite functions with good decay properties in L ∞ ( R d ), when d ≥
1. Introduction and results1.1. Introduction. —
During the last years, several papers have shown that some basic resultsconcerning P.D.E. and Sobolev spaces can be strikingly improved using randomization techniques. Inparticular Burq-Lebeau developed in [ ] a randomisation method based on the Laplace operator ona compact Riemannian manifold, and showed that almost surely, a function enjoys better Sobolevestimates than expected, using ideas of Shiffman-Zelditch [ ]. This approach depends heavily onspectral properties of the operator one considers. In this paper we are interested in estimates inSobolev spaces based on the harmonic oscillator in L ( R d ) H = − ∆ + | x | = d X j =1 ( − ∂ j + x j ) . We get optimal stochastic weighted Sobolev estimates on R d using the Burq-Lebeau method. Indeedwe show that there is a unified setting for these results, including the case of compact manifolds.We also make the following extension: In [ ], the construction of the measures relied on Gaussianrandom variables, while in our work we consider general random variable which satisfy concentration Mathematics Subject Classification . —
Key words and phrases . —
Harmonic oscillator, spectral analysis, concentration of measure, Hermite functions.D. R. was partly supported by the grant “NOSEVOL” ANR-2011-BS01019 01.L.T. was partly supported by the grant “HANDDY” ANR-10-JCJC 0109.
AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN of measure estimates (including discrete random variables, see Section 2). However, we obtain theoptimal estimates only in the case of the Gaussians.We will see that the extension from a compact manifold to an operator on R d with discrete spectrumis not trivial because of the complex behaviour of the spectral function on a non-compact configurationspace.In our forthcoming paper [ ], we will give some applications to the well-posedness of nonlinearSchr¨odinger equations with Sobolev regularity below the optimal deterministic index.Most of the results stated here can be extended to more general Schr¨odinger Hamiltonians −△ + V ( x )with confining potentials V . This will be detailed in [ ].Let d ≥
2. We want to define probability measures on finite dimensional subspaces E h ⊂ L ( R d ),based on spectral projections with respect to H . We denote by { ϕ j , j ≥ } an orthonormal basisof L ( R d ) of eigenvectors of H (the Hermite functions), and we denote by { λ j , j ≥ } the nondecreasing sequence of eigenvalues (each is repeated according to its multiplicity): Hϕ j = λ j ϕ j .For h >
0, we define the interval I h = [ a h h , b h h [ and we assume that a h and b h satisfy, for some a, b, D > , δ ∈ [0 , h → a h = a, lim h → b h = b, < a ≤ b and b h − a h ≥ Dh δ , with any D > δ < D ≥ δ = 1. This condition ensures that N h , thenumber (with multiplicities) of eigenvalues of H in I h tends to infinity when h →
0. Indeed, we cancheck that N h ∼ ch − d ( b h − a h ), in particular lim h → N h = + ∞ , since d ≥
2. In the sequel, we writeΛ h = { j ≥ , λ j ∈ I h } and E h = span { ϕ j , j ∈ Λ h } , so that N h = h = dim E h . Finally, we denoteby S h = (cid:8) u ∈ E h : k u k L ( R d ) = 1 (cid:9) the unit sphere of E h .In the sequel, we will consider sequences ( γ n ) n ∈ N so that there exists K > | γ n | ≤ K N h X j ∈ Λ h | γ j | , ∀ n ∈ Λ h , ∀ h ∈ ]0 , . This condition means that on each level of energy λ n , n ∈ Λ h , one coefficient | γ k | cannot be muchlarger than the others. Sometimes, in order to prove lower bound estimates, we will need the strongercondition ( K > K N h X j ∈ Λ h | γ j | ≤ | γ n | ≤ K N h X j ∈ Λ h | γ j | , ∀ n ∈ Λ h , ∀ h ∈ ]0 , . This so-called “squeezing” condition means that on each level of energy λ n , n ∈ Λ h , the coefficients | γ k | have almost the same size. For instance (1.2) or (1.3) hold if there exists ( d h ) h ∈ ]0 , so that γ n = d h for all n ∈ Λ h . ANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d Consider a probability space (Ω , F , P ) and let { X n , n ≥ } be independent standard complexGaussians N C (0 , γ n ) n ∈ N satisfies (1.2), we define the random vector in E h v γ ( ω ) := v γ,h ( ω ) = X j ∈ Λ h γ j X j ( ω ) ϕ j . We define a probability measure P γ,h on S h by: for all measurable and bounded function f : S h −→ R Z S h f ( u )d P γ,h ( u ) = Z Ω f v γ ( ω ) k v γ ( ω ) k L ( R d ) ! d P ( ω ) . We can check that in the isotropic case ( γ j = √ N h for all j ∈ Λ h ), P γ,h is the uniform probabilityon S h (see Appendix C).Finally, let us recall the definition of harmonic Sobolev spaces for s ≥ p ≥ W s,p = W s,p ( R d ) = (cid:8) u ∈ L p ( R d ) , H s/ u ∈ L p ( R d ) (cid:9) , H s = H s ( R d ) = W s, . The natural norms are denoted by k u k W s,p and up to equivalence of norms we have (see [ , Lemma 2.4])for 1 < p < + ∞ k u k W s,p = k H s/ u k L p ≡ k ( − ∆) s/ u k L p + kh x i s u k L p . Estimates for frequency localised functions. —
Our first result gives properties of theelements on the support of P γ,h , which are high frequency localised functions. Namely Theorem 1.1 . —
Let d ≥ . Assume that ≤ δ < / in (1.1) and that condition (1.3) holds. Thenthere exist < C < C , c > and h > such that for all h ∈ ]0 , h ] . P γ,h h u ∈ S h : C | log h | / ≤ k u k W d/ , ∞ ( R d ) ≤ C | log h | / i ≥ − h c . Moreover the estimate from above is satisfied for any δ ≥ with D large enough. It is clear that under condition (1.3), there exist 0 < C < C , so that for all u ∈ S h , and s ≥ C h − s/ ≤ k u k H s ( R d ) ≤ C h − s/ , since all elements of S h oscillate with frequency h − / . Thus Theorem 1.1 shows a gain of d/ L ∞ , and this induces a gain of d derivatives compared to the usual deterministic Sobolevembeddings. This can be compared with the results of [ ] where the authors obtain a gain of d/ ] as well) do not dependon the length of the interval of the frequency localisation I h (see (1.1)), but only on the size of thefrequencies. This is a consequence of the randomisation, and from the bound (3.15). AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN
We will see in Theorem 4.1 that the upper bound in Theorem 1.1 holds for any 0 ≤ δ ≤ X which satisfy the concentration of measure property. However, toprove the lower bound (see Corollary 4.8), we have to restrict to the case of Gaussians: in the generalcase, under Assumption 1, we do not reach the factor | ln h | / . Following the approach of [
18, 2 ],we first prove estimates of k u k W d/ , ∞ ( R d ) with large r and uniform constants (see Theorem 4.12), andwhich are essentially optimal for general random variables (see Theorem 4.13).The condition δ < / ], Feng and Zelditch prove similar estimates forthe mean and median for the L ∞ -norm of random holomorphic fields. Global Sobolev estimates. —
Using a dyadic Littlewood-Paley decomposition, we nowgive general estimates in Sobolev spaces; we refer to Subsection 4.1 for more details. For s ∈ R and p, q ∈ [1 , + ∞ ], we define the harmonic Besov space by(1.5) B sp,q ( R d ) = n u = X n ≥ u n : X n ≥ nqs/ k u n k qL p ( R d ) < + ∞ o , where the u n have frequencies of size ∼ n . The space B sp,q ( R d ) is a Banach space with the normin ℓ q ( N ) of { ns/ k u n k L p ( R d ) } n ≥ .We assume that γ satisfies (1.2) and X n ≥ | γ | Λ n < + ∞ where | γ | n := X k : λ k ∈ [2 k , k +1 [ | γ k | . Then we set v γ ( ω ) = + ∞ X j =0 γ j X j ( ω ) ϕ j , so that almost surely v γ ∈ B , ( R d ) and its probability law defines a measure µ γ in B , ( R d ). Noticethat we have H s ( R d ) ⊂ B , ( R d ) ⊂ L ( R d ) , ∀ s > . We have the following result
Theorem 1.2 . —
For every ( s, r ) ∈ R such that r ≥ and s = d ( − r ) there exists c > suchthat for all K > we have (1.6) µ γ h u ∈ B , ( R d ) : k u k W s,r ( R d ) ≥ K k u k B , ( R d ) i ≤ e − c K . In particular µ γ -almost all functions in B , ( R d ) are in W s,r ( R d ) . ANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d If γ satisfies (1.2) and the (weaker) condition X n ≥ | γ | n < + ∞ , then µ γ defines a probability measureon L ( R d ) and we can prove the estimate(1.7) µ γ h u ∈ L ( R d ) : k u k W s,r ( R d ) ≥ K k u k L ( R d ) i ≤ e − c K , with s = d ( − r ) when r < + ∞ and s < d/ r = + ∞ . From this result it is easy todeduce space-time estimates (Strichartz) for the linear flow e − itH u , which can be used to study thenonlinear problem. This will be pursued in [ ]. An application to Hermite functions. —
Similarly to [ ], the previous results give someinformation on Hilbertian bases. We prove that there exists a basis of Hermite functions with gooddecay properties. Theorem 1.3 . —
Let d ≥ . Then there exists a Hilbertian basis of L ( R d ) of eigenfunctions of theharmonic oscillator H denoted by ( ϕ n ) n ≥ such that k ϕ n k L ( R d ) = 1 and so that for some M > andall n ≥ , (1.8) k ϕ n k L ∞ ( R d ) ≤ M λ − d n (1 + log λ n ) / . We refer to Theorem 5.1 for a more quantitative result, and where we prove that for a naturalprobability measure, almost all Hermite basis satisfies the property of Theorem 1.3 (see also Corol-lary 4.14). For the proof of this result, we need the finest randomisation with δ = 1 and D = 2 in (1.1),so that P γ,h is a probability measure on each eigenspace.The result of Theorem 1.3 does not hold true in dimension d = 1. Indeed, in this case one can provethe optimal bound (see [ ])(1.9) k ϕ n k L ∞ ( R ) ≤ Cn − / . Let us compare (1.10) with the general known bounds on Hermite functions. We have Hϕ n = λ n ϕ n ,with λ n ∼ cn /d , therefore (1.10) can be rewritten(1.10) k ϕ n k L ∞ ( R d ) ≤ M n − / (1 + log n ) / . For a general basis with d ≥
2, Koch and Tataru [ ] (see also [ ]) prove that k ϕ n k L ∞ ( R d ) ≤ Cλ d − n , which shows that (1.8) induces a gain of d − ϕ n ) n ≥ which satisfy the conclusion of the Theorem. Forinstance, the basis ( ϕ ⊺ n ) n ≥ obtained by tensorisation of the 1D basis does not realise (1.10) becauseof (1.9) which implies the optimal bound k ϕ ⊺ n k L ∞ ( R d ) ≤ Cλ − / n . Observe also that the basis of radial Hermite functions does not satisfy (1.10) in dimension d ≥
2. Asin [ , Th´eor`eme 8], it is likely that the log term in (1.10) can not be avoided. AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN
Notations . —
In this paper c, C > denote constants the value of which may change from line to line.These constants will always be universal, or uniformly bounded with respect to the other parameters.We denote by H = − ∆+ | x | = P dj =1 ( − ∂ j + x j ) the harmonic oscillator on R d , and for s ≥ we definethe Sobolev space H s by the norm k u k H s = k H s/ u k L ( R d ) ≈ k u k H s ( R d ) + kh x i s u k L ( R d ) . More generally,we define the spaces W s,p by the norm k u k W s,p = k H s/ u k L p ( R d ) . We write L r,s ( R d ) = L r ( R d , h x i s d x ) ,and its norm k u k r,s . The rest of the paper is organised as follows. In Section 2 we describe the general probabilistic settingand we prove large deviation estimates on Hilbert spaces. In Section 3 we state crucial estimates onthe spectral function of the harmonic oscillator. Section 4 is devoted to the proof of weighted Sobolevestimates and of the mains results. In Section 5 we prove Theorem 1.3.
Acknowledgements . —
The authors thank Nicolas Burq for discussions on this subject and for hissuggestion to introduce conditions (1.2) - (1.3) .
2. A general setting for probabilistic smoothing estimates
Our aim in this section is to unify several probabilistic approaches to improve smoothing estimatesestablished for dispersive equations. This setting is inspired by papers of Burq-Lebeau [ ], Burq-Tzvetkov [
3, 4 ] and their collaborators.
Definition 2.1 . — We say that a family of Borelian probability measures ( ν N , R N ) N ≥ satisfies theconcentration of measure property if there exist constants c, C > N ∈ N such thatfor all Lipschitz and convex function F : R N −→ R (2.1) ν N h X ∈ R N : (cid:12)(cid:12) F ( X ) − E ( F ( X )) (cid:12)(cid:12) ≥ r i ≤ c e − Cr k F k Lip , ∀ r > , where k F k Lip is the best constant so that | F ( X ) − F ( Y ) | ≤ k F k Lip k X − Y k ℓ .For a comprehensive study of these phenomena, we refer to the book of Ledoux [ ]. Notice thatone of the main features of (2.1) is that the bound is independent of the dimension of space, whichenables to take N large.Typically, in our applications, F will be a norm in R N .Let us give some significative examples of such measures. • If ( ν N , R N ) N ≥ is a family of probability measures which satisfies a Log-Sobolev estimate withconstant C ⋆ >
0, then (2.1) is satisfied for all Lipschitz function F : R N −→ R (see [ , Th´eor`eme 7.4.1, ANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d page 123]). Recall that a probability measure ν N on R N satisfies a Log-Sobolev estimate if there exists C > N ≥ f ∈ C b ( R N )(2.2) Z R N f ln (cid:0) f E ( f ) (cid:1) d ν N ( x ) ≤ C Z R N |∇ f | d ν N ( x ) , E ( f ) = Z R N f d ν N ( x ) . Such a property is usually difficult to check. See [ ] for more details. Notice that the convexity of F is not needed. • A probability measure of the form d ν N ( x ) = c α,N exp (cid:0) − P Nj =1 | x j | α (cid:1) d x , x ∈ R N , satisfies (2.1) ifand only if α ≥ , page 109]). • Assume that ν is a measure on R with bounded support, then ν N = ν ⊗ N satisfies the concentrationof measure property. This is the Talagrand theorem [ ] (see also [ ] for an introduction to the topic). Assumption 1 . —
Consider a probability space (Ω , F , P ) and let { X n , n ≥ } be a sequence ofindependent, identically distributed, real or complex random values. In the sequel we can assume thatthey are real with the identification C ≈ R . Moreover, we assume that for all n ≥ ,(i) Denote by ν law of the X n . We assume that the family ( ν ⊗ N , R N ) N ≥ satisfies the concentrationof measure property in the sense of Definition 2.1.(ii) The r.v. X n is centred: E ( X n ) = 0 .(iii) The r.v. X n is normalized: E ( X n ) = 1 . Under Assumption 1, for all n ≥
1, and ε > E (e εX n ) < + ∞ . Indeed, by Definition 2.1 with F ( X ) = X n E (e εX n ) = Z + ∞ ν ( e CεX n > λ )d λ = 1 + Z + ∞ ν (cid:0) | X n | > r ln λε (cid:1) d λ ≤ Z + ∞ λ − εC d λ < + ∞ . Next, with the inequality s | x | ≤ εx / s / (2 ε ), we obtain that for all s ∈ R , E (e sX n ) ≤ C e Cs whichin turn implies (see [ , Proposition 46]) that there exists C > s ∈ R (2.4) E (e sX n ) ≤ e Cs . Remark 2.2 . — Condition (2.4) is weaker that (2.2): a family of independent centred r.v. { X n , n ≥ } which satisfies (2.4) does not necessarily satisfy (2.1) for all Lipschitz function F . Indeed, using Kol-mogorov estimate, one can prove (see [ ]) that condition (2.1) is equivalent to(2.5) Z R d e sF dν ≤ e Cs k F k Lip , ∀ s ∈ R , for all Lipschitz function F with ν -mean 0.We conclude with the elementary property AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN
Lemma 2.3 . —
Assume that { X n } satisfies (2.4) and that { α j , ≤ j ≤ N } are real numbers suchthat X ≤ j ≤ N α j ≤ . Then X := X ≤ j ≤ N α j X j satisfies (2.4) with the same constant C .Proof . — It is a direct application of (2.5) with F ( X ) = N X j =1 α j X j . In this sub-section K is a separable complex Hilbertspace and K is a self-adjoint, positive operator on K with a compact resolvent. We denote by { ϕ j , j ≥ } an orthonormal basis of eigenvectors of K , Kϕ j = λ j ϕ j , and { λ j , j ≥ } is the nondecreasing sequence of eigenvalues of K (each is repeated according to its multiplicity). Then we geta natural scale of Sobolev spaces associated with K defined for s ≥ K s = Dom( K s/ ).Now we want to introduce probability measures on these spaces and on some finite dimensionalspaces of K .Let us describe in our setting the randomization technique deeply used by Burq-Tzvetkov in [ ].Let γ = { γ j } j ≥ a sequence of complex numbers such that X j ≥ λ sj | γ j | < + ∞ .Consider a probability space (Ω , F , P ) and let { X n , n ≥ } be independent, identically distributedrandom variables which satisfy Assumption 1.We denote by v γ = X j ≥ γ j ϕ j ∈ K s , and we define the random vector v γ ( ω ) = X j ≥ γ j X j ( ω ) ϕ j . Wehave E ( k v γ k K ) < + ∞ , therefore v γ ∈ K s , a.s. We define the measure µ γ on K s as the law of therandom vector v γ . The Kakutani theorem. —
The following proposition gives some properties of the mea-sures µ γ (see [ ] for more details). Proposition 2.4 . —
Assume that all random variables X j have the same law ν .(i) If the support of ν is R and if γ j = 0 for all j ≥ then the support of µ γ is K s .(ii) If for some ε > we have v γ / ∈ K s + ε then µ γ ( K s + ε ) = 0 .(iii) Assume that we are in the particular case where d ν ( x ) = c α e −| x | α d x with α ≥ . Let γ = { γ j } and β = { β j } be two complex sequences and assume that (2.6) X j ≥ (cid:12)(cid:12)(cid:12)(cid:12) γ j β j (cid:12)(cid:12)(cid:12)(cid:12) α/ − ! = + ∞ . Then the measures µ γ and µ β are mutually singular, i.e there exists a measurable set A ⊂ H s suchthat µ γ ( A ) = 1 and µ β ( A ) = 0 . We give the proof of ( iii ) in Appendix A.We shall see now that condition (1.2) (resp. (1.3)) can be perturbed so that Proposition 2.4 givesus an infinite number of mutually singular measures on K s . ANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d Lemma 2.5 . —
Let γ satisfying (1.2) (resp. (1.3)) and δ = { δ n } n ≥ such that | δ n | ≤ ε | γ n | for every n ≥ n . Then for every ε ∈ [0 , √ − , the sequence γ + δ satisfies (1.2) (resp. (1.3)) (with newconstants). We do not give the details of the proof. From this Lemma and Proposition 2.4 we get an infinitenumber of measures µ γ with γ satisfying (1.2) (resp. (1.3)). Let ε j be any sequence such that X j ≥ ε j = + ∞ and lim sup ε j < √ − ε ⊗ γ the sequence ε j γ j . Then µ γ and µ γ + ε · γ aremutually singular. Measures on the sphere S h . — Now we consider finite dimensional subspaces E h of K defined by spectral localizations depending on a small parameter 0 < h ≤ h − is a measure ofenergy for the quantum Hamiltonian K ). In the sequel, we use the notations I h = [ a h h , b h h [, N h , Λ h and E h introduced in Section 1.1, and we assume that (1.1) is satisfied. Observe that E h is the spectralsubspace of K in the interval I h : E h = Π h K where Π h is the orthogonal projection on K . For simplicity,we sometimes denote by N = N h , Λ = Λ h , . . . , with implicit dependence in h . Our goal is to finduniform estimates in h ∈ ]0 , h [ for a small constant h > E h (2.7) v γ ( ω ) := v γ,h ( ω ) = X j ∈ Λ γ j X j ( ω ) ϕ j , and assume that (1.3) is satisfied. In the sequel we denote by | γ | = X n ∈ Λ γ n .Now we consider probabilities on the unit sphere S h of the subspaces E h . The random vector v γ in (2.7) defines a probability measure ν γ,h on E h . Then we can define a probability measure P γ,h on S h as the image of by v v k v k . Namely, we have for every Borel and bounded function f on S h ,(2.8) Z S h f ( w ) P γ,h (d w ) = Z E h f (cid:18) v k v k K (cid:19) ν γ,h (d v ) = Z Ω f (cid:18) v γ ( ω ) k v γ ( ω ) k K (cid:19) P (d ω ) . Remark that we have k v γ ( ω ) k K = X j ∈ Λ | γ j | | X j ( ω ) | and E ( k v γ k K ) = X j ∈ Λ | γ j | = | γ | . Let us detail two particular cases of interest: • If | γ n | = √ N for all j ∈ Λ and if X n follows the complex normal law N C (0 ,
1) then P γ,h is theuniform probability on S h considered in [ ]. This follows from (2.8) and property of Gaussian laws. AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN • Assume that for all n ∈ N , P ( X n = 1) = P ( X n = −
1) = 1 /
2, then P γ,h is a convex sum of 2 N Dirac measures. Indeed we have k v γ ( ω ) k K = P j ∈ Λ | γ j | = | γ | . Denote by ( ε ( k ) ) ≤ k ≤ N all thesequences so that ε ( k ) j = ± ≤ j ≤ N , and setΦ k = 1 | γ | X j ∈ Λ γ j ε ( k ) j ϕ j , ≤ k ≤ N . Then P γ,h = 12 N N X k =1 δ Φ k . To get an optimal lower bound for L ∞ estimates we shall need a stronger normal concentrationestimate than estimate given in (2.1). Hence we make the following assumptions: Assumption 2 . —
We assume that(i) The random variables X j are standard independent Gaussians N C (0 , .(ii) The sequence γ satisfies (1.3). Let L be a linear form on E h , and denote by e L = X j ∈ Λ h | L ( ϕ j ) | . The main result of this section isthe following Theorem 2.6 . —
Let L be a linear form on E h . Suppose that (1.2) holds and that Assumption 1 issatisfied. Then there exist C , c > so that (2.9) P γ,h h u ∈ S h : | L ( u ) | ≥ t i ≤ C e − c NeL t , ∀ t ≥ , ∀ h ∈ ]0 , h ] , Moreover, if (1.3) holds, there exist C , c > and ε , h > so that (2.10) C e − c NeL t ≤ P γ,h h u ∈ S h : | L ( u ) | ≥ t i , ∀ t ∈ (cid:2) , ε √ e L √ N (cid:3) , ∀ h ∈ ]0 , h ] . Furthermore, if Assumption 2 is satisfied, there exist C , C , c , c , ε , h > so that (2.11) C e − c NeL t ≤ P γ,h h u ∈ S h : | L ( u ) | ≥ t i ≤ C e − c NeL t , ∀ t ∈ [0 , ε √ e L ] , ∀ h ∈ ]0 , h ] . Since P γ,h is supported by S h , the bounds in the previous result don’t depend on | γ | Λ . Therestriction on t ≥ | L ( u ) | ≤ √ e L , ∀ u ∈ S h . In the applications we give, there is some embedding K s → C ( M ), for s > M is a metric space. We have E ⊆ T s ∈ R K s , thus we can consider the Dirac evaluation linear form δ x ( v ) = v ( x ). In this case we have e L = X j ∈ Λ | ϕ j ( x ) | = e x , which is usually called the spectral functionof K in the interval I . ANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d For example, one can consider the Laplace-Beltrami operator on compact Riemannian manifolds,namely K = −△ and K s = H s ( M ) are the usual Sobolev spaces: this is the framework of [ ]. InSection 3 we will apply the result of Theorem 2.6 to the Harmonic oscillator K = −△ + | x | on R d .In this latter case K s is the weighted Sobolev space K s = (cid:8) u ∈ H s ( R d ) , | x | s u ∈ L ( R d ) (cid:9) , s ≥ . Remark 2.7 . — In the particular case where P γ,h is the uniform probability on S h , we have theexplicit computation P γ,h h u ∈ S h : | L ( u ) | ≥ t i = Φ (cid:18) t √ e L (cid:19) , where(2.12) Φ( t ) = I [0 , ( t )(1 − t ) N − , and (2.11) follows directly. For a proof of (2.12), see [ ] or in Appendix C of this paper for analternative argument.For the proof of Theorem 2.6 we will need the following result. Proposition 2.8 . —
Assume that γ satisfies (1.2) . Let L be a linear form on E h . Then we have thelarge deviation estimate P h ω ∈ Ω : | L ( v γ ) | ≥ t i ≤ − κ NeL | γ | t , where κ = κ K . As a consequence, if ν γ,h denotes the probability law of v γ , then ν γ,h h w ∈ E h : | L ( w ) | ≥ t i ≤ − κ NeL | γ | t . Proof . — We have L ( v γ ) = X j ∈ Λ γ n X n ( ω ) L ( ϕ n ) . It is enough to assume that L ( v γ ) is real and to estimate P (cid:2) ω ∈ Ω : L ( v γ ) ≥ t (cid:3) . Using the Markovinequality, we have for all s > P (cid:2) L ( v γ ) ≥ t (cid:3) ≤ e − st E (e sL ( v γ ) ) , and thanks to (1.2) we have X j ∈ Λ | γ j L ( ϕ j ) | ≤ K | γ | N X j ∈ Λ | L ( ϕ j ) | . Using Lemma 2.3 we get P (cid:2) L ( v γ ) ≥ t (cid:3) ≤ e − st e κ K eLN | γ | s , and with the choice s = κ tNK e L | γ | we obtain P (cid:2) L ( v γ ) ≥ t (cid:3) ≤ e − κ N | γ | eL t . It will be useful to show that k v γ ( ω ) k K is close to its expectation for large N . AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN
Lemma 2.9 . —
Let γ satisfying the squeezing condition (1.3). Then then exists c > (dependingonly on K and K ) such that for every ε > P h ω ∈ Ω : (cid:12)(cid:12) k v γ ( ω ) k K − | γ | (cid:12)(cid:12) > ε i ≤ − εc N | γ | . Proof . — It is enough to consider the real case, so we assume that γ n and X n are real and { X n , n ≥ } have a common law ν . We also assume that | γ | = 1.We have k v γ ( ω ) k K = X j ∈ Λ | γ j | X j ( ω ) := M N ( ω ) . From large number law, k v γ ( ω ) k K converges to 1 a.s. To estimate the tail we use the Cramer-Chernofflarge deviation principle (see e.g. [ , §
5, Chapter IV]). This applies because from (2.3) we knowthat f ( s ) := E (e sX ) is C in ] − ∞ , s [ for some s > g ( s ) = log( f ( s )) which is well defined for s < s . Now, since the X j are i.i.d., for t, s ≥ P (cid:2) M N > t (cid:3) = P (cid:2) e sNM N > e sNt (cid:3) ≤ E (e sNM N )e − sNt = Y j ∈ Λ e − ( Ns | γ j | t − g ( Ns | γ j | )) . Next, apply the Taylor formula to g at 0: g (0) = 0, g ′ (0) = 1 so tτ − g ( τ ) = ( t − τ + O ( τ ), hencethere exists s > ≤ τ ≤ s , tτ − g ( τ ) ≥ ( t − τ . Then, with t = 1 + ε , and since N | γ j | ≤ K we get P (cid:2) M N > ε (cid:3) ≤ Y j ∈ Λ e − εNs | γ j | / = e − εsN/ , provided s > ε > N ≥
1. The same computation appliedto − M N gives as well P (cid:2) M N < − ε (cid:3) ≤ e − εc N . Proof of (2.9). — By homogeneity, we can assume that | γ | Λ = 1. Denote by(2.13) A = (cid:8) ω ∈ Ω : (cid:12)(cid:12) k v γ ( ω ) k K − (cid:12)(cid:12) ≤ / (cid:9) . By the Cauchy-Schwarz inequality, for all u ∈ S h , we obtain | L ( u ) | ≤ e / L . Thus in the sequel we canassume that t ≤ e / L . Then, from Proposition 2.8 and Lemma 2.9 we have(2.14) P γ,h h u ∈ S h : | L ( u ) | ≥ t i = P (cid:2) ω ∈ Ω : | L ( v ( ω )) | ≥ t k v ( ω ) k L (cid:3) = P (cid:2) ( | L ( v ( ω )) | ≥ t k v ( ω ) k L ) ∩ A (cid:3) + P (cid:2) ( | L ( v ( ω )) | ≥ t k v ( ω ) k L ) ∩ A c (cid:3) . Therefore P γ,h h u ∈ S h : | L ( u ) | ≥ t i ≤ P (cid:2) | L ( v ( ω )) | ≥ t/ (cid:3) + P ( A c ) ≤ C e − c NeL t + 2e − c N ≤ C e − c NeL t , ANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d which implies (2.9).We now turn to the proof of (2.10). We will need the following result Lemma 2.10 . —
We suppose that γ satisfies (1.3) and that Assumption 1 is satisfied. Then thereexist C > , c > , h > , ε > such that P h ω ∈ Ω : | L ( v γ ( ω )) | ≥ t i ≥ C e − c NeL | γ | t , ∀ t ∈ (cid:2) , ε √ e L | γ | Λ √ N (cid:3) , ∀ h ∈ ]0 , h ] . Proof . — Let us first recall the Paley-Zygmund inequality (1) : Let Z ∈ L (Ω) be a r.v such that Z ≥ < λ < P (cid:0) Z > λ k Z k (cid:1) ≥ (cid:16) (1 − λ ) k Z k k Z k (cid:17) . We apply (2.15) to the random variable Z = | Y N | , with Y N = √ N √ e L | γ | Λ L ( v γ ) = √ N √ e L | γ | Λ X j ∈ Λ γ j X j L ( ϕ j ) , and λ = 1 /
2. By (1.3), we have c ≤ k Y N k ≤ C uniformly in N ≥
1. Next, recall the Khinchininequality (see e.g. [ , Lemma 4.2] for a proof) : there exists C > k ≥ a n ) ∈ ℓ ( N ) k X n ∈ Λ X n ( ω ) a n k L k P ≤ C √ k (cid:16) X n ∈ Λ | a n | (cid:17) . Therefore, there exists C > k Y N k ≤ C . As a result, there exist η > ε > N ≥ P ( | Y N | > η ) > ε , which implies the result. Proof of (2.10). — We assume that | γ | Λ = 1, and consider the set A defined in (2.13). Then by (2.14)and the inequality P ( B ∩ A ) ≥ P ( B ) − P ( A c ) we get P γ,h h u ∈ S h : | L ( u ) | ≥ t i ≥ P (cid:2) ( | L ( v ( ω )) | ≥ t k v ( ω ) k L ) ∩ A i ≥ P (cid:2) | L ( v ( ω )) | ≥ t/ (cid:3) − P ( A c ) ≥ C e − c NeL t − − c N , where in the last line we used Lemma 2.10 and Lemma 2.9. This yields the result if t ≤ ε √ e L √ N with ε > Lemma 2.11 . —
We suppose that Assumption 2 is satisfied. Then there exist C > , c > , h > , ε > such that P h ω ∈ Ω : | L ( v γ ( ω )) | ≥ t i ≥ C e − c NeL | γ | t , ∀ t ≥ , ∀ h ∈ ]0 , h ] . (1) We thank Philippe Sosoe for this suggestion. AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN
Proof . — Denote by γ ⊗ L ( ϕ ) the vector ( γ ⊗ L ( ϕ )) j = γ j L ( ϕ j ). Observe that, thanks to (1.3), K | γ | e L N ≤ | γ ⊗ L ( ϕ ) | = X j ∈ Λ h γ j | L ( ϕ j ) | ≤ K | γ | e L N .
Then, using the rotation invariance of the Gaussian law and the previous line, we get P h ω ∈ Ω : | L ( v γ ( ω )) | ≥ t i = P h (cid:12)(cid:12)(cid:12) h γ ⊗ L ( ϕ ) | γ ⊗ L ( ϕ ) | , X i (cid:12)(cid:12)(cid:12) ≥ t | γ ⊗ L ( ϕ ) | i = 1 √ π Z | s |≥ t | γ ⊗ L ( ϕ ) | e − s / d s ≥ C e − cNeL | γ | t . The estimate (2.11) then follows from Lemma 2.11 and with the same argument as for Lemma 2.10.
Concentration phenomenon. —
We now state a concentration property for P γ,h , inheritedfrom Assumption 1 and condition (1.3). See [ ] for more details on this topic. Proposition 2.12 . —
Suppose that the i.i.d. random variables X j satisfy Assumption 1 and supposethat condition (1.3) is satisfied. Then there exist constants K > , κ > (depending only on C ⋆ ) suchthat for every Lipschitz function F : S h −→ R satisfying | F ( u ) − F ( v ) | ≤ k F k Lip k u − v k L ( R d ) , ∀ u, v ∈ S h , we have (2.16) P γ,h h u ∈ S h : | F − M F | > r i ≤ K e − κNr k F k Lip , ∀ r > , h ∈ ]0 , , where M F is a median for F . Recall that a median M F for F is defined by P γ,h (cid:2) u ∈ S h : F ≥ M F (cid:3) ≥ , P γ,h (cid:2) u ∈ S h : F ≤ M F (cid:3) ≥ . In Proposition 2.12, the distance in L can be replaced with the geodesic distance d S on S h , sincewe can check that k u − v k L ( R d ) ≤ d S ( u, v ) = 2 arcsin (cid:0) k u − v k L ( R d ) (cid:1) ≤ π k u − v k L ( R d ) . When P γ,h is the uniform probability on S h , Proposition 2.12 is proved in [ , Proposition 2.10],and the proof can be adapted in the general case (see Appendix D). The factor N in the exponentialof r.h.s of (2.16) will be crucial in our application. ANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d
3. Some spectral estimates for the harmonic oscillator
Our goal here is to apply the general setting of Section 2 to the harmonic oscillator in R d . Thisway we shall get probabilistic estimates analogous to results proved in [ ] for the Laplace operator ina compact Riemannian manifold.In the following, we consider the Hamiltonian H = −△ + V ( x ) with V ( x ) = | x | , x ∈ R d for d ≥
2. For this model, all the necessary spectral estimates are already known. More general confiningpotentials V shall be considered in the forthcoming paper [ ].A first and basic ingredient in probabilistic approaches of weighted Sobolev spaces is a good knowl-edge concerning the asymptotic behavior of eigenvalues and eigenfunctions of H . The eigenvalues ofthis operator are the (cid:8) j + · · · + j d )+ d, j ∈ N d (cid:9) , and we can order them in a non decreasing sequence { λ j , j ∈ N } , repeated according to their multiplicities. We denote by { ϕ j , j ∈ N } an orthonormalbasis in L ( R d ) of eigenfunctions (the Hermite functions), so that Hϕ j = λ j ϕ j . The spectral functionis then defined as π H ( λ ; x, y ) = X λ j ≤ λ ϕ j ( x ) ϕ j ( y ) (recall that this definition does not depend on thechoice of { ϕ j , j ∈ N } ). When the energy λ is localized in I ⊆ R + we denote by Π H ( I ) the spectralprojector of H on I . The range E H ( I ) of Π H ( I ) is spanned by { ϕ j ; λ j ∈ I } and Π H ( I ) has an integralkernel given by π H ( I ; x, y ) = X [ j : λ j ∈ I ] ϕ j ( x ) ϕ j ( y ) . We will also use the notation E H ( λ ) = E H ([0 , λ ]), N H ( λ ) = dim[ E H ( λ )]. We begin with some general interpolation results which willbe needed in the sequel. In R d , the spectral function π H ( λ ; x, x ) is fast decreasing for | x | → + ∞ soit is natural to work with weighted L p norms. We denote by h x i s = (1 + | x | ) s/ and introduce thefollowing Lebesgue space with weight L p,s ( R d ) = (cid:26) u, Lebesgue measurable : Z | u ( x ) | p h x i s d x < + ∞ (cid:27) = L p ( R d , h x i s d x ) , endowed with its natural norm, which we denote by k u k p,s . For p = ∞ , we set k u k ∞ ,s = sup x ∈ R d h x i s | u ( x ) | .The following interpolation inequalities hold true. Let 1 ≤ p ≤ p ≤ p ≤ + ∞ and κ ∈ ]0 ,
1[ suchthat p = κp + − κp . Then for p < + ∞ we have(3.1) k u k L p,s ( R d ) ≤ ( k u k L p ,s ( R d ) ) − κ ( k u k L p ,s ( R d ) ) κ , with s = p − pp − p s + p − pp − p s . In the case p = + ∞ , we have(3.2) k u k L p,s ( R d ) ≤ (sup R d h x i s | u ( x ) | ) − p /p ( k u k L p ,s ( R d ) ) p /p , with s = ( p − p ) s + s . AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN
We recall here some more or less standardproperties stated in [ ]. To begin with, we state a ”soft” Sobolev inequality. Lemma 3.1 . —
For all u ∈ E H ( I )(3.3) | u ( x ) | ≤ ( π H ( I ; x, x )) / k u k L ( R d ) . Proof . — We have u ( x ) = Π u ( x ) = Z R d π H ( I ; x, y ) u ( y )d y. Using the Cauchy-Schwarz inequality(3.4) | u ( x ) | ≤ (cid:18)Z R d | π H ( I ; x, y ) | d y (cid:19) / k u k L ( R d ) . Now we use that Π H ( I ) is an orthonormal projector.(3.5) π H ( I ; x, y ) = Z R d π H ( I ; x, z ) π H ( I ; z, y )d z and π H ( I ; x, y ) = π H ( I ; y, x ) . Finally, from (3.4) and (3.5) with y = x we get (3.3).The next result gives a bound on π H . Lemma 3.2 . —
The following bound holds true (3.6) π H ( λ ; x, x ) ≤ Cλ d/ exp (cid:18) − c | x | λ (cid:19) , ∀ x ∈ R d , λ ≥ . Proof . — Let K ( t ; x, y ) be the heat kernel of e − tH . It is given by the following Mehler formula (2) (3.7) K ( t ; x, y ) = (2 π sinh 2 t ) − d/ exp (cid:18) − tanh t | x + y | − | x − y | t (cid:19) . So we have(3.8) K ( t ; x, x ) = Z R e − tµ d π H ( µ ; x, x ) = (2 π sinh 2 t ) − d/ exp( −| x | tanh t ) . We set t = λ − , integrate in µ on [0 , λ ] and get π H ( λ ; x, x ) ≤ e K ( λ − ; x, x ) . Assuming λ ≥ λ , λ large enough, we easily see that (3.6) is a consequence of (3.8).Let u ∈ E H ( λ ). From (3.3) and (3.6) we get | u ( x ) | ≤ Cλ d/ exp (cid:18) − c | x | λ (cid:19) k u k L ( R d ) , where c, C > x ∈ R d nor λ ≥ (2) The Mehler formula can also be obtained from the Fourier transform computation of the Weyl symbol of e − tH (see[ , Exercise IV-2]). ANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d Remark 3.3 . — From (3.6), we can deduce that for every θ > C θ > π H ( λ ; x, x ) ≤ C θ λ ( d + θ ) / h x i − θ , which by (3.3) implies with the semiclassical parameter h = λ − h x i θ/ h ( d + θ ) / | u ( x ) | ≤ C θ k u k L ( R d ) , ∀ u ∈ E H ( h − ) . We can easily see that this uniform estimate is true for u ∈ E ( I h ) where I h = [ ah , bh ] with a < b . Forsmaller energy intervals we can get much better estimates, as we will see in Lemma 3.5. Remark 3.4 . — Let us compare the previous results with the case of a compact Riemannian mani-fold M , and when H = −△ is the Laplace operator. We have the uniform H¨ormander estimate [ ]:(3.9) π H ( λ ; x, x ) = c d ( x ) λ d/ + O ( λ ( d − / ) , where 0 < c d ( x ) is a continuous function on M . Thus from (3.4) and (3.9) we get for some con-stant C S > k u k L ∞ ( M ) ≤ C S λ d/ k u k L ( M ) , ∀ u ∈ E ( λ ) . Let us emphasis here that it results form the uniform Weyl law (3.9) that π H ( λ ; x, x ) has an upperbound and a lower bound of order λ d/ . For confining potentials like V the behavior of π H ( λ ; x, x )is much more complicated because of the turning points: (cid:8) | x | = λ (cid:9) . This behavior was analyzedin [ ]. From the Weyl law for the harmonicoscillator we have N H ( λ ) = c d λ d + O ( λ d − ) , c d > , we deduce that if (1.1) is satisfied with δ = 1 then we have(3.10) αh − d ( b h − a h ) ≤ N h ≤ βh − d ( b h − a h ) , α > , β > . The main result of this section is the following lemma. It is a consequence of the work of Thangavelu[ , Lemma 3.2.2, p. 70] on Hermite functions. This was proved later Karadzhov [ ] with a differentmethod. It could also be deduced from much more general results by Koch, Tataru and Zworski [ ] and it is also related, after rescaling, with results obtained by Ivrii [ , Theorem 4.5.4]. Lemma 3.5 . —
Let d ≥ and assume that | µ | ≤ c , ≤ p ≤ + ∞ and θ ≥ . Then there exists C > so that for all λ ≥ k π H ( λ + µ ; x, x ) − π H ( λ ; x, x ) k L p, ( p − θ ( R d ) ≤ Cλ α , with α = d (1 + p ) − θ (1 − p ) . AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN
Proof . — Recall the following estimates proved in [ , Theorem 4]: For d ≥ x ∈ R (3.11) | π H ( λ + µ ; x, x ) − π H ( λ ; x, x ) | ≤ Cλ d/ − , λ ≥ , | µ | ≤ . and for every ε > N ≥ C ε ,N such that(3.12) π H ( λ ; x, x ) ≤ C ε ,N | x | − N , for | x | ≥ (1 + ε ) λ. From (3.11) we get that for every C > C > | π H ( λ + µ ; x, x ) − π H ( λ ; x, x ) | ≤ C (1 + | µ | ) λ d/ − , λ ≥ , | µ | ≤ C λ. Then from (3.13) and (3.12) we get that for every θ ≥ C such that(3.14) | π H ( λ + µ ; x, x ) − π H ( λ ; x, x ) | ≤ C (1 + | µ | ) λ d/ − θ/ h x i − θ , λ ≥ , | µ | ≤ C λ. Therefore, by (3.12), to get the result of Lemma 3.5, it is enough to integrate the previous inequalityon | x | ≤ c λ / .From (3.14), we easily get an accurate estimate for the spectral function e x = π H ( b h h ; x, x ) − π H ( a h h ; x, x ) . Lemma 3.6 . —
Assume that (1.1) is satisfied with < δ ≤ . For any θ ≥ there exists C > suchthat (3.15) h x i θ e x ≤ CN h h ( d − θ ) / . Using (3.3) and interpolation inequalities we get Sobolev type inequalities for u ∈ E h , θ ≥ p ≥ k u k L ∞ ,θ/ ( R d ) ≤ C (cid:16) N h h ( d − θ ) / (cid:17) / k u k L ( R d ) , which in turn implies, by (3.1)(3.17) k u k L p,θ ( p/ − ( R d ) ≤ C (cid:16) N h h ( d − θ ) / (cid:17) − p k u k L ( R d ) . By (3.10), the previous inequality can be written as k u k L p,θ ( p/ − ( R d ) ≤ C ( b h − a h ) − p h − ( d + θ )( − p ) k u k L ( R d ) , ∀ p ∈ [2 , + ∞ ] , ∀ θ ∈ [0 , d ] . Remark 3.7 . — For similar bounds for eigenfunctions or quasimodes, we refer to [ ].
4. Probabilistic weighted Sobolev estimates
We apply here the general probabilistic setting of Section 2 when K = H is the harmonic oscillator, K = L ( R d ) and { ϕ j , j ∈ N } an orthonormal basis of Hermite functions. Recall that S h is the unitsphere of the complex Hilbert space E h , identified with C N or R N , and that P γ,h is the probabilityon S h defined as in Section 2. ANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d We divide this section in two parts: in the first part, under Assumption 1, we establish upperbounds and in the second part we obtain lower bounds, but only in the case of Gaussian randomvariables (Assumption 2), and under the condition 0 ≤ δ < / We suppose here that Assumption 1, (1.2) and (1.1) with 0 ≤ δ ≤ Theorem 4.1 . —
There exist h ∈ ]0 , , c > and C > such that if c = d (1 + d/ , we have (4.1) P γ,h h u ∈ S h : h − d − θ k u k L ∞ ,θ/ ( R d ) > Λ i ≤ Ch − c e − c Λ , ∀ Λ > , ∀ h ∈ ]0 , h ] . Proof . — We adapt here the argument of [ ]. To begin with, by (3.15) and (2.9), there exists c > θ ∈ [0 , d ], every x ∈ R d , and every Λ > P γ,h h u ∈ S h : h x i θ h − d − θ | u ( x ) | > Λ i ≤ e − c Λ . Now, we will need a covering argument. Our configuration space is not compact but using (3.12) wehave, for every u ∈ S h , | u ( x ) | ≤ C N | x | − N , for | x | ≥ (1 + ε ) h − / . So choosing
R > u inside the box B R h = { x ∈ R d , | x | ∞ ≤ Rh − / } . We divide B R h in small boxes of side with length τ small enough. We use the gradientestimate |∇ x u ( x ) | ≤ Ch − / − d/ , ∀ u ∈ S h , and (4.2) at the center of each small box to get the result.For x, x ′ ∈ R d we have |h x i θ/ u ( x ) − h x ′ i θ/ u ( x ′ ) | ≤ C ( h x i θ/ | u ( x ) − u ( x ′ ) | + h x i θ/ | x − x ′ || u ( x ′ ) | ) . Let { Q τ } τ ∈ A be a covering of B R h with small boxes Q τ with center x τ and side length τ small enough.Then for every x ∈ Q τ we have(4.3) h ( θ − d ) / |h x i θ/ u ( x ) − h x τ i θ/ u ( x τ ) | ≤ Cτ h − / − d/ . We choose(4.4) τ ≈ ε Λ2 C h / d/ and h ε > | x | ∞ > Rh − / ⇒ h ( θ − d ) / h x i θ/ | u ( x ) | ≤ ε Λ2 , ∀ h ∈ ]0 , h ε ] . Then using (4.2), (4.3), (4.4) and (4.5) we get(4.6) P γ,h h u ∈ S h : h − d − θ k u k L ∞ ,θ/ ( R d ) > Λ i ≤ A e − c (1 − ε ) Λ , ∀ Λ > , ∀ h ∈ ]0 , h ε ] . Using now that A ≈ Ch − c with c = d (1 + d/
4) we get (4.1) from (4.6).We can deduce probabilistic estimates for the derivatives as well. Recall that the Sobolev spaces W s,p ( R d ) are defined in (1.4). AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN
Corollary 4.2 . —
For any multi index α, β ∈ N d there exists ˜ c such that P γ,h h u ∈ S h : h | α | + | β | − d k x α ∂ βx u k L ∞ ( R d ) > Λ i ≤ Ch − c e − ˜ c Λ , ∀ Λ > , ∀ h ∈ ]0 , h ] . In particular we have, for every s > , P γ,h h u ∈ S h : h s − d k u k W s, ∞ ( R d ) > Λ i ≤ Ch − c e − ˜ c Λ , ∀ Λ > , ∀ h ∈ ]0 , h ] . Proof . — We apply (4.1) using that from the spectral localization of u ∈ E h we have k x α ∂ βx u k L ( R d ) ≤ Ch − | α | + | β | k u k L ( R d ) , k H s u k L ( R d ) ≤ Ch − s/ k u k L ( R d ) . The following corollary shows that we get a probabilistic Sobolev estimate improving the deter-ministic one (3.16) with probability close to one as h →
0. The improvement is ”almost” of order N / h ≈ (cid:0) ( b h − a h ) h − d (cid:1) / . Choosing Λ = √− K log h for K >
Corollary 4.3 . —
Let c , c > be the constants given by Theorem 4.1. Then for every K > c c wehave P γ,h h u ∈ S h : k u k L ∞ ,θ/ ( R d ) > Kh d − θ | log h | / i ≤ h Kc − c , ∀ h ∈ ]0 , h ] , ∀ θ ∈ [0 , d ] . P γ,h h u ∈ S h : k u k W s, ∞ ( R d ) > Kh d − s | log h | / i ≤ h Kc − c , ∀ h ∈ ]0 , h ] , ∀ s ≥ . Let us give now an application to a probabilistic Sobolev embedding for the Harmonic oscillator.We shall use a Littlewood-Paley decomposition with h j = 2 − j . Let θ a C ∞ real function on R such that θ ( t ) = 0 for t ≤ a , θ ( t ) = 1 for t ≤ b/ < a < b/
2. Define ψ − ( t ) = 1 − θ ( t ), ψ j ( t ) = θ ( h j t ) − θ ( h j +1 t ) for j ∈ N . Notice that the support of ψ j is in [ ah j , bh j ].For every distribution u ∈ S ′ ( R d ) we have the Littlewood-Paley decomposition u = X j ≥− u j , with u j = X k ∈ N ψ j ( λ k ) h u, ϕ k i ϕ k and we have u j ∈ E h j .The Besov spaces for the Harmonic are naturally defined as follows: if p, r ∈ [1 , ∞ ] and s ∈ R , u ∈ B sp,r if and only if k u k B sp,r := X j ≥− jsr/ k u j k rL p ( R d ) /r < + ∞ . We shall use here the spaces B s , ∞ . For every s > B s , ∞ ⊆ L ( R d ) ⊆ B , ∞ . ANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d Another scale of spaces is defined as G m = (cid:8) u ∈ S ′ ( R d ) : X j ≥ j m k u j k L ( R d ) < + ∞ (cid:9) , m ≥ . Then for every s > m ≥ B s , ∞ ⊆ G m ⊆ L ( R d ).It is not difficult to see that G m can be compared with the domain in L ( R d ) of the operator log s H .This domain is denoted by H s log , the norm being the graph norm. For every s > / H m + s log ⊂ G m ⊂ H m log . Notice that we do not need that the energy localizations ψ j are smooth and we can define the samespaces with ψ ( t ) = I [1 , ( t ) so that the energy intervals [2 j , j +1 [ are disjoint.Let us now define probabilities on G m as we did for Sobolev spaces H s . Let γ j be a sequence ofcomplex numbers satisfying (1.2) and such that(4.7) X j ≥ j m | γ | Λ j < + ∞ , where Λ j = Λ h j and v γ = X j ≥ γ j ϕ j , v γ ( ω ) = X j ≥ γ j X j ( ω ) ϕ j , so that v γ is a.s in G m and its probability law defines a measure µ mγ in G m . This measure satisfies alsothe following properties as in Proposition 2.4.(i) If the support of ν is R and if γ j = 0 for all j ≥ µ mγ is G m .(ii) If u γ ∈ G m and v γ / ∈ G s where s > m then µ mγ ( G s ) = 0. In particular µ mγ ( H s ) = 0 for every s > iii ) in Proposition 2.4 we can construct singular measures µ mγ and µ mβ .Now we can state the following corollary of Theorem 4.1. Corollary 4.4 . —
Suppose that γ satisfies (1.2) with a < b and (4.7) with m = 1 / . Then for themeasure µ / γ almost all functions in the space G / are in the space C [ d/ H where C ℓH ( R d ) = n u ∈ C ℓ ( R d ) : k x α ∂ βx u k L ∞ ( R d ) < + ∞ , ∀ | α | + | β | ≤ ℓ o . In particular if v γ ∈ H s , s > and if v γ / ∈ H s , s > s , then we have µ / γ ( B σ , ∞ ) = 1 for every σ > and we have an a.s embedding of the Besov space B σ , ∞ in C [ d/ H .Proof . — Let u = X n ≥− u n ∈ G / with u n ∈ E h n . For κ > B κn = n v ∈ E h n : k x α ∂ βx v k L ∞ ( R d ) ≤ κ √ n k v k L ( R d ) , ∀ | α | + | β | ≤ [ d/ o . AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN
We have, using Corollary 4.2 ν γ,n ( B κn ) ≥ − e − n ( c κ − c ) . So if B κ = (cid:8) u ∈ G / : u ∈ E h , u n ∈ B κn , ∀ n ≥ (cid:9) , then we have µ / γ ( B κ ) ≥ Y n ≥ (cid:0) − e − n ( c κ − c ) (cid:1) ≥ − ε ( κ )with lim κ → + ∞ ε ( κ ) = 0. More precisely we have ε ( κ ) ≈ e − cκ for some c > u ∈ B κ we have k x α ∂ βx u k L ∞ ( R d ) ≤ X n ≥− k x α ∂ βx u n k L ∞ ( R d ) ≤ κ X n ≥− √ n k u n k L ( R d ) := κ k u k G / . So the corollary is proved.
Remark 4.5 . — In the last corollary, for every s > γ such that µ / γ ( H s ) = 0. Sothe smoothing property is a probabilistic effect similar to the Khinchin inequality.From the proof we get a more quantitative statement. There exists c > µ / γ h k u k W d/ , ∞ ≥ κ k u k G / i ≤ e − cκ . Remark 4.6 . — The proof of the corollary depends on the squeezing assumption (1.2) on γ . Forexample if (1.2) is satisfied for b h − a h ≈ h then we can consider the energy decomposition in intervals[2 n, n + 1)[ instead of the dyadic decomposition. So when applying Theorem 4.1 with h of order n we get h − c e − c Λ = e c log n − c Λ .Then taking Λ = κ √ log n with κ large enough, in the construction of B κn we have to replace √ n by √ log n . In the conclusion the space G / is replaced by ˜ G / where˜ G m = (cid:8) u ∈ S ′ ( R d ) : X j ≥ log m j k u j k L ( R d ) < + ∞ (cid:9) , u j := X j ≤ λ n < j +1) h u, ϕ j i ϕ j . Here we suppose that thestronger Assumption 2 and (1.1) with δ < / k u k L ∞ ,θ/ ( R d ) .The spectral condition δ < / ≤ h − / .A first step is to get two sides weighted L r estimates for large r which is a probabilistic improvementof (3.17). Denote by(4.8) β r,θ = d − θ − r ) . Theorem 4.7 . —
Assume that θ ∈ [0 , d ] , and denote by M r a median of k u k L r,θ ( r/ − . Then thereexist < C < C , K > , c > , h > such that for all r ∈ [2 , K | log h | ] and h ∈ ]0 , h ] such that (4.9) P γ,h h u ∈ S h : (cid:12)(cid:12)(cid:12) k u k L r,θ ( r/ − − M r (cid:12)(cid:12)(cid:12) > Λ i ≤ (cid:0) − c N /rh h − β r,θ Λ (cid:1) . ANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d and where C √ rh d − θ (1 − r ) ≤ M r ≤ C √ rh d − θ (1 − r ) , ∀ r ∈ [2 , K log N ] . This result shows that k u k L r,θ ( r/ − has a Gaussian concentration around its median.From (4.9) we deduce that for every κ ∈ ]0 , K >
0, there exist 0 < C < C , c > h > r ∈ [2 , K | log h | κ ], h ∈ ]0 , h ] and Λ > P γ,h h u ∈ S h : C √ rh d − θ (1 − r ) ≤ k u k L r,θ ( r/ − ≤ C √ rh d − θ (1 − r ) i ≥ − e − c | log h | − κ . As a consequence of Theorem 4.7, for every θ ∈ [0 , d ] we get a two sides weighted L ∞ estimateshowing that Theorem 4.1 and its corollary are sharp. Corollary 4.8 . —
After a slight modification of the constants in Theorem 4.7, if necessary, we getthat for all θ ∈ [0 , d ] and h ∈ ]0 , h ](4.10) P γ,h h u ∈ S h : C | log h | / h ( d − θ ) / ≤ k u k L ∞ ,θ/ ≤ C | log h | / h ( d − θ ) / i ≥ − h c . To prove these results we have to adapt to the unbounded configuration space R d the proofs of [ ,Theorems 4 and 5] which hold for compact manifolds. The concentration result stated in Proposi-tion 2.12 will prove useful. Proof of Theorem 4.7 . — Denote by F r ( u ) = k u k L r,θ ( r/ − and by M r its median. Thanks to (3.17)we have the Lipschitz estimate | F r ( u ) − F r ( v ) | ≤ C (cid:16) N h h d − θ (cid:17) − r k u − v k L ( R d ) , ∀ u, v ∈ S h . Therefore, by (2.16) and (4.8), we have for some c > P γ,h h u ∈ S h : | F r ( u ) − M r | > Λ i ≤ (cid:0) − c N /rh h − β r,θ Λ (cid:1) . The next step is to estimate M r . Denote by A rr = E h ( F rr ) the moment of order r and compute, with s = θ ( r/ − A rr = E h (cid:18)Z R d h x i s | u ( x ) | r d x (cid:19) = r Z R d h x i s (cid:16) Z + ∞ s r − P γ,h h u ∈ S h : | u ( x ) | > s i d s (cid:17) d x. Thus by (2.11) we get C r Z R d h x i s (cid:16) Z ε √ e x s r − e − c Nex s d s (cid:17) d x ≤ A rr ≤ C r Z R d h x i s (cid:16) Z + ∞ s r − e − c Nex s d s (cid:17) d x. Performing the change of variables t = c j Ne x s we obtain that there exist C , C > C r ( c N ) − r/ (cid:18)Z R d h x i s e r/ x d x (cid:19) Z εN t r/ − e − t d t ≤ A rr ≤ C r ( c N ) − r/ (cid:18)Z R d h x i s e r/ x d x (cid:19) Γ( r/ , AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN with ε = c ε . We need to estimate the term R εN t r/ − e − t d t from below. Using the elementaryestimate Z + ∞ T t r/ − e − t d t ≤ T r/ e − T Γ( r/ , T ≥ , we get that there exists ε > N large and r ≤ ε N log N then we have Z εN t r/ − e − t d t ≥ Γ( r/ . So we get the expected lower bound, ∀ r ∈ [1 , ε N log N ],e − r/ C − r (cid:18)Z R d h x i s e r/ x d x (cid:19) N − r/ Γ( r/ ≤ A rr ≤ C rN − r/ (cid:18)Z R d h x i s e r/ x d x (cid:19) Γ( r/ . and where Γ( r/
2) can be estimated thanks to the Stirling formula: there exist 0 < C < C such that( C r ) r/ ≤ Γ( r/ ≤ ( C r ) r/ , ∀ r ≥ . Now we need the following lemma which will be proven in Appendix B. The upper bound can beseen as an application of Lemma 3.5 with λ = h − and µ = ( b h − a h ) h − ∼ N h h d − . Lemma 4.9 . —
Assume that θ > − d/ ( p − . Then there exist < C < C and h > such that C N h h β p,θ ≤ (cid:18)Z R d h x i θ ( p − e px d x (cid:19) /p ≤ C N h h β p,θ , for every p ∈ [1 , ∞ [ and h ∈ ]0 , h ] where β r,θ = d − θ (1 − r ) . From this lemma we get(4.13) C p rh β r,θ ≤ A r ≤ C p rh β r,θ , ∀ r ≥ , h ∈ ]0 , h ] . Now we have to compare A r and the median M r . We have |A r − M r | r = (cid:12)(cid:12) k F r k L r ( S h ) − kM r k L r ( S h ) (cid:12)(cid:12) r ≤ k F r − M r k rL r ( S h ) = r Z ∞ s r − P γ,h (cid:2) | F r − M r | > s (cid:3) d s. Then using the large deviation estimate (4.11) we get |A r − M r | ≤ CN − /r p rh β r,θ , ∀ r ≥ . Choosing r ≤ K log N , ( K <
1) and N large, from (4.13) we obtain(4.14) C p rh β r,θ ≤ M r ≤ C p rh β r,θ , ∀ r ∈ [2 , K log N ]and the proof of Theorem 4.7 follows using (4.14) and (4.11). Remark 4.10 . — The upper-bound in Lemma 4.9 is true for δ = 1. This is proved in Appendix B.Now let us prove Corollary 4.8. ANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d Proof of Corollary 4.8 . — For simplicity we assume that θ = d . Using (4.1) it is enough to prove thatthere exist C > h > c > P γ,h h u ∈ S h : k u k L ∞ ,d/ ≤ C | log h | / i ≤ h c , ∀ h ∈ ]0 , h ] . Let u ∈ S h , then by (3.2) we have the interpolation inequality k u k L r,d ( r/ − ( R d ) ≤ k u k − /rL ∞ ,d/ . So we get P γ,h h u ∈ S h : k u k L ∞ ,d/ ≤ C | log h | / i ≤ P γ,h (cid:20) u ∈ S h : k u k L r,d ( r/ − ≤ (cid:16) C | log h | / (cid:17) − /r (cid:21) , and choosing r = r h = ε | log h | we obtain P γ,h h u ∈ S h : k u k L ∞ ,d/ ≤ C | log h | / i ≤ P γ,h " u ∈ S h : k u k L rh,d ( rh/ − ≤ (cid:18) C √ ε r / h (cid:19) − /r h . Then choosing h > C √ ε small enough and Λ = c | log h | / we can conclude that (4.15) is satisfiedusing (4.9). Remark 4.11 . — Concerning the mean M ∞ of F ∞ ( u ) := k u k L ∞ ,d/ it results from Corollary 4.8,(4.1) and (3.16) that we have the two sides estimates C | log h | / ≤ M ∞ ≤ C | log h | / , ∀ h ∈ ]0 , h ] . It is not difficult to adapt the proof of (4.9) and (4.10) for the Sobolev norms k u k W s,p ( R d ) . It is enoughto remark that considering L s u ( x ) = H s/ u ( x ) we have e L s = X j ∈ Λ λ sj ϕ j ( x ) . But for j ∈ Λ, λ j is of order h − hence there exists C > C − h − s e x ≤ e L s ≤ Ch − s e x . Using this property we easily get the next result, which in particular implies Theorem 1.1. Let M r,s be the median of u
7→ k u k W s,r ( R d ) , and recall the definition (4.8). Then Theorem 4.12 . —
Let s ≥ . There exist < C < C , K > , c > , h > such that for all r ∈ [2 , K | log h | ] and h ∈ ]0 , h ](4.16) P γ,h h u ∈ S h : (cid:12)(cid:12)(cid:12) k u k W s,r ( R d ) − M r,s (cid:12)(cid:12)(cid:12) > Λ i ≤ (cid:0) − c N /rh h − β r, + s Λ (cid:1) . where C √ rh βr, − s ≤ M r,s ≤ C √ rh βr, − s , ∀ r ∈ [2 , K log N ] . AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN
In particular, for every κ ∈ ]0 , , K > , there exist C > , C > , c > such that for every r ∈ [2 , K | log h | κ ] we have P γ,h h u ∈ S h : C √ rh d (1 − r ) h − s ≤ k u k W s,r ( R d ) ≤ C √ rh d (1 − r ) h − s i ≥ − e − c | log h | − κ , For r = + ∞ we have for all h ∈ ]0 , h ] P γ,h h u ∈ S h : C | log h | / h d − s ≤ k u k W s, ∞ ( R d ) ≤ C | log h | / h d − s i ≥ − h c . Namely, k u k W s,r ( R d ) ≈ h − s/ k u k L r, ( R d ) + k u k L r,s ( R d ) , and h − s/ k u k L r, ( R d ) ∼ h d (1 − r ) h − s , k u k L r,s ( R d ) ∼ h d (1 − r ) h − s r . Under Assumption 1, we prove a weaker version ofTheorem 4.12.
Theorem 4.13 . —
Suppose that Assumption 1 is satisfied. Let s ≥ , κ ∈ ]0 , , K > . There exist < C < C , K > , c > , h > such that for all r ∈ [2 , K | log h | κ ] and h ∈ ]0 , h ] P γ,h h u ∈ S h : C h d (1 − r ) h − s ≤ k u k W s,r ( R d ) ≤ C √ rh d (1 − r ) h − s i ≥ − e − c | log h | − κ , . For r = + ∞ we have for all h ∈ ]0 , h ] P γ,h h u ∈ S h : C h d − s ≤ k u k W s, ∞ ( R d ) ≤ C | log h | / h d − s i ≥ − h c . Therefore, we have optimal constants in the control of the W s,r ( R d ) norms when r < + ∞ and forgeneral random variables which satisfy the concentration property, but when r = + ∞ we lose thefactor | log h | / in the lower bound. Proof . — We can follow the main lines of the proof of Theorem 4.12. Here compared to (4.12) we get A rr ≥ C rN − r/ (cid:18)Z R d h x i s e r/ x d x (cid:19) Z ε t r/ − e − t d t ≥ CN − r/ (cid:18)Z R d h x i s e r/ x d x (cid:19) ε r/ , and this explains the loss of the factor √ r . L p -Sobolev estimates. — Here we extend the L ∞ - random estimatesobtained before to the L r -spaces for any real r ≥
2, and we prove Theorem 1.2. Let us recall thedefinition (1.5) of the Besov spaces, where we use the notations of Subsection 4.1 for the dyadicLittlewood-Paley decomposition.
ANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d Proof of Theorem 1.2 . — Recall that for every σ > m we can choose γ such that µ γ ( H σ ) = 0.Denote by F r,s ( u ) = k u k W s,r . The Lipschitz norm of F r,s satisfies k F r,s k Lip ≤ Ch − s + d ( − r ) N − r h . Let us denote by M r,s the median of F r,s on the sphere S h for the probability P γ,h and by A rr,s themean of F rr,s . From Proposition 2.12 we have, for some 0 < c < c ,(4.17) P γ,h h u ∈ S h : | F r,s − M r,s | > K i ≤ exp (cid:16) − c N K k F r,s k Lip (cid:17) ≤ exp (cid:16) − c N /r K (cid:17) . With the same computations as for (4.14) we get(4.18) A r,s ≈ √ r and |A r,s − M r,s | . √ rN − /r . These formulas are obtained from (2.9) applied to the linear form L s u := H s u ( x ) noticing that e L s = X j ∈ Λ h | H s ϕ j ( x ) | ≈ h − s e x . Then taking c > ν γ,h h v ∈ E h : k v k W s,r ≥ K k v k L ( R d ) i ≤ exp (cid:16) − c N /r K (cid:17) , ∀ K ≥ . Then from (4.19) we proceed as for the proof of Corollary 4.4. For simplicity we consider here theusual Littlewood-Paley decomposition. Then we have N /r ≈ nd/r . So the end of the proof followsby considering B κn = (cid:8) v ∈ E n : k v k W s,r ≤ K k v k L ( R d ) (cid:9) . So for a fixed r ≥ c > µ γ Y n ≥ B Kn ≥ − e − c K . Using the isometry u H − m/ u between B s , and B m + s , for all real m ≥
0, we can get the followingcorollary to Theorem 1.2.
Corollary 4.14 . —
Let m ≥ and assume that γ satisfies (1.2) and X n ≥ nm | γ | n < + ∞ . Then for s = d ( − r ) + m and r ≥ , we have µ γ h u ∈ B m , : k u k W m + s,r ≥ K k u k B m , i ≤ e − c K . AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN
5. Application to Hermite functions
We turn to the proof of Theorem 1.3 and we can follow the main lines of [ , Section 3]. We usehere the upper bounds estimates of Section 4.1 in their full strength. Firstly, we assume that for all j ∈ Λ h , γ j = N − / h and that X j ∼ N C (0 , P h := P γ,h is the uniform probability on S h . Weset h k = 1 /k with k ∈ N ∗ , and a h k = 2 + dh k , b h k = 2 + (2 + d ) h k . Then (1.1) is satisfied with δ = 1 and D = 2. In particular, each interval I h k = h a h k h k , b h k h k h = [2 k + d, k + d + 2[only contains the eigenvalue λ k = 2 k + d with multiplicity N h k ∼ ck d − , and E h k is the correspondingeigenspace of the harmonic oscillator H . We can identify the space of the orthonormal basis of E h k with the unitary group U ( N h k ) and we endow U ( N h k ) with its Haar probability measure ρ k . Thenthe space B of the Hilbertian bases of eigenfunctions of H in L ( R d ) can be identified with B = × k ∈ N U ( N h k ) , which can be endowed with the measure d ρ = ⊗ k ∈ N d ρ k . Denote by B = ( ϕ k,ℓ ) k ∈ N , ℓ ∈ J ,N hk K ∈ B a typical orthonormal basis of L ( R d ) so that for all k ∈ N ,( ϕ k,ℓ ) ℓ ∈ J ,N hk K ∈ U ( N h k ) is an orthonormal basis of E h k .Then the main result of the section is the following, which implies Theorem 1.3. Theorem 5.1 . —
Let d ≥ . Then, if M > is large enough, there exist c, C > so that for all r > ρ h B = ( ϕ k,ℓ ) k ∈ N , ℓ ∈ J ,N hk K ∈ B : ∃ k, ℓ ; k ϕ k,ℓ k W d/ , ∞ ( R d ) ≥ M (log k ) / + r i ≤ C e − cr . We will need the following result
Proposition 5.2 . —
Let d ≥ . Then, if M > is large enough, there exist c, C > so that for all r > and k ≥ ρ k h B k = ( ψ ℓ ) ℓ ∈ J ,N hk K ∈ U ( N h k ) : ∃ ℓ ∈ J , N h k K ; k ψ ℓ k W d/ , ∞ ( R d ) ≥ M (log k ) / + r i ≤ Ck − e − cr . Proof . — The proof is similar to the proof of [ , Proposition 3.2]. We observe that for any ℓ ∈ J , N h k K ,the measure ρ k is the image measure of P h k under the map U ( N h k ) ∋ B k = ( ψ ℓ ) ℓ ∈ J ,N hk K ψ ℓ ∈ S h k . ANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d Then we use that S h k ⊂ E h k is an eigenspace and by Theorem 4.1 we obtain that for all ℓ ∈ J , N h k K ρ k h B k = ( ψ ℓ ) ℓ ∈ J ,N hk K ∈ U ( N h k ) : k ψ ℓ k W d/ , ∞ ( R d ) ≥ M (log k ) / + r i = P h k h u ∈ S h k : k u k W d/ , ∞ ( R d ) ≥ M (log k ) / + r i = P h k h u ∈ S h k : k d/ k u k L ∞ , ( R d ) ≥ M (log k ) / + r i ≤ Ck c − M c e − c r , where c , c > Ck c e − c r , with c = c − M c + d − Proof of Theorem 5.1 . — We set F k,r = (cid:8) B k = ( ψ ℓ ) ℓ ∈ J ,N hk K ∈ U ( N h k ) : ∀ ℓ ∈ J , N h k K ; k ψ ℓ k W d/ , ∞ ( R d ) ≤ M (log k ) / + r (cid:9) , and F r = ∩ k ≥ F k,r . Then for all r > ρ ( F cr ) ≤ X k ≥ ρ k ( F ck,r ) ≤ C X k ≥ k − e − cr = C ′ e − cr , and this completes the proof.We have the following consequence of the previous results. Corollary 5.3 . —
For ρ -almost all orthonormal basis ( ϕ k,ℓ ) k ∈ N , ℓ ∈ J ,N hk K of eigenfunctions of H wehave k ϕ k,ℓ k L ∞ ( R d ) ≤ ( M + 1) k − d (1 + log k ) / , ∀ k ∈ N , ∀ ℓ ∈ J , N h k K . Proof . — Apply (5.1) with r = (log k ) / and denote, for k ≥
2, Ω k the eventΩ k = (cid:8) B = ( ϕ k,ℓ ) , ∃ ℓ ∈ J , N h k K , k ϕ k,ℓ k L ∞ ( R d ) ≥ ( M + 1) k − d/ (log k ) / (cid:9) . We have ρ (Ω k ) ≤ Ck . Therefore from the Borel-Cantelli Lemma we have ρ [lim sup Ω k ] = 0 and thisgives the corollary. Appendix AProof of Proposition 2.4 ( iii ) Proof . — Denote by f γ ( x ) = c α γ e − ( | x | γ ) α , γ >
0. We have, with obvious identifications, µ γ = ⊗ j ≥ ( f γ j d x ) . Denote by π j = Z R (cid:18) f γ j f β j (cid:19) / f β j d x. AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN
According to the main result of [ ] the measures µ γ and µ β are mutually singular if the infinite product Q j ≥ π j is divergent. From elementary computations we get π j = (cid:18) γ j β j (cid:19) α/ + 12 (cid:18) β j γ j (cid:19) α/ ! − /α . • If π j has not 1 as limit then the product is divergent. • If π j has 1 as limit then the infinite product is divergent if X j ≥ ( π − αj −
1) = + ∞ . So, using that12 ( x + 1 x ) = 1 + 12 (1 − x ) + O (1 − x ) , we see that the infinite product is divergent if (2.6) is satisfied. Appendix B L p weighted spectral estimates for the Harmonic oscillator Our goal here is to give a self-contained proof of Lemma 4.9. It could be proved using the semi-classical functional calculus for pseudo-differential operators [ ], but for the harmonic oscillator it ispossible to use the exact Mehler formula and elementary properties of Hermite functions to get theresult. B.1. A functional calculus with parameter for the Harmonic oscillator. —
The startingpoint is the inverse Fourier transform f ( H ) = 12 π Z e itH ˆ f ( t )d t, where f is in the Schwartz space S ( R ).We want estimates for the integral kernel K f ( x, y ) of f ( H ). To do that it is convenient to firstcompute the Weyl symbol W f ( H ) ( x, ξ ) of f ( H ) and use that K f ( x, y ) = (2 π ) − d Z R d W f ( H ) (cid:0) x + y , ξ (cid:1) e i ( x − y ) · ξ d ξ. For basic properties about the Weyl calculus see for example [ ]. The unitary operator e itH has anexplicit Weyl symbol w ( t, x, ξ ) :(B.1) w ( t, x, ξ ) = 1(cos t ) d e i tan t ( | x | + | ξ | ) , for | t | < π . Formula (B.1) can be easily proved from the Mehler formula (3.7) and also directly (see [ , Exer-cise IV]).Let us introduce a cutoff χ ∈ C ∞ ( R ), χ ( t ) = 1 for | t | < ε , χ ( t ) = 0 for | t | > ε with 0 < ε < π/ R f = 12 π Z e itH (1 − χ ( t )) ˆ f ( t )d t ANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d and ˜ W f ( x, ξ ) = 12 π Z R w ( t, x, ξ ) χ ( t ) ˆ f ( t )d t. We apply these formulas to give estimates with f h ( s ) = f ( hs ) where h > KR f h ( x, y ) of the operator R f h . Lemma B.1 . —
There exists M > such that for every M ≥ there exists C M > such that (B.2) | KR f h ( x, y ) | ≤ C M h M (cid:0) h x ih y i (cid:1) − M − d − k f k M + M , ∀ h ∈ ]0 , , ∀ x, y ∈ R d , where k f k m = sup j + k ≤ m,t ∈ R | t j d k d t k ˆ f ( t ) | .Proof . — Denote by ˆ g h ( t ) = (1 − χ ( t )) ˆ f ( th ). So we have R f,h = g h ( H ) and for every M, M ′ ≥ | µ M g h ( µ ) | ≤ Z | t |≥ ε (cid:12)(cid:12) d M d t M ˆ g h ( t ) (cid:12)(cid:12) d t ≤ C M,M ′ ( f ) h M ′ . So we have | KR f h ( x, y ) | = (cid:12)(cid:12) X j g h ( λ j ) ϕ j ( x ) ϕ j ( y ) (cid:12)(cid:12) ≤ Ch M (cid:0) X j λ − Mj | ϕ j ( x ) | (cid:1) / (cid:0) X j λ − Mj | ϕ j ( y ) | (cid:1) / . Recall the Sobolev estimate in the harmonic spaces: for every s > d + r there exists C = C sr suchthat h x i r | u ( x ) | ≤ C k u k H s , ∀ u ∈ H s ( R d ) . So we get, for s > d + r , | KR f h ( x, y ) | ≤ Ch M ( h x ih y i ) − r X j λ s − Mj . Using that λ j ≈ j /d and choosing r = M + d + 1 we get (B.2).Our aim is to estimate the kernel of f (cid:16) H − νλµ (cid:17) for large λ , | µ | ≥ Dλ − δ where D > δ < / ν is fixed in an interval [ ν , ν ], where 0 < ν < ν . All our estimates will be uniformin ν , so for convenience we shall take ν = 1.Denote by g λ,µ ( s ) = f (cid:16) s − λµ (cid:17) so we have ˆ g λ,µ ( t ) = µ e − itλ ˆ f ( µt ). We consider the dilated Weylsymbol: W λ,µ ( x, ξ ) = ˜ W g λ,µ ( √ λx, √ λξ ). Then we have(B.3) W λ,µ ( x, ξ ) = µ π Z R e iλ Φ( t,x,ξ ) χ ( t )cos( t ) d ˆ f ( µt )d t, with the phase Φ( t, x, ξ ) = tan t ( | x | + | ξ | ) − t . AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN
Lemma B.2 . —
Assume that δ < . Then for every N, M ≥ we have W λ,µ ( x, ξ ) = X j (1 − δ )+ k (2 − δ )
First remark that when θ ≥
0, the upper-bound is a direct conse-quence of (3.12) and (3.13) and this holds true for δ = 1. The bound (3.13) being a rather difficultresult, we shall prove by the same method the estimate from above and from below for δ < / f be a non negative C ∞ function in ] − C , C [ with a compact support, such that f = 1 in [ − C , C ]. We choose two cutofffunctions f ± with f + as above and f − such that supp( f − ) ⊆ ] C , C [, f − = 1 in [2 C , C /
2] where C < C /
4. If K ± ,h ( x, y ) is the Schwartz kernel of f ± (cid:16) H − h − µ (cid:17) ( h = λ is now a small parameter, ν ∈ [ ν , ν ]). We have K − ,h ( x, x ) ≤ e x ≤ K + ,h ( x, x ) . So we have to prove(B.6) C N h h β p,θ ≤ (cid:18)Z R d h x i θ ( p − K ± ,h ( x, x ) p d x (cid:19) /p ≤ C N h h β p,θ . Recall that K ± ,h ( x, x ) = (2 π ) − d Z R d W ± ,h ( x, ξ )d ξ where W ± ,h ( x, ξ ) is the Weyl symbol of the operator f ± (cid:16) H − h − µ (cid:17) . So using (B.4) it is not difficult tosee that it is enough to consider only the principal term given by the following formula K f,h ( x, x ) = (2 π ) − d Z R d f (cid:18) | x | + | ξ | − h − µ (cid:19) d ξ. We shall detail now the lower-bound; the upper-bound is proved in the same way. Denote by K − ( x ) = K f,h ( x, x ) and s = θ ( p − x = h − / y , ξ = h − / η Z R d h x i s K − ( x ) p d x = (2 π ) − dp Z R dx h x i s Z R dξ f − (cid:18) | x | + | ξ | − h − µ (cid:19) d ξ ! p d x = (2 π ) − dp h − (1+ p ) d/ Z R dy h h − / y i s Z R dη f − (cid:18) | y | + | η | − hµ (cid:19) d η ! p d y Using the property of the support of f − we obtain Z R dη f − (cid:18) | y | + | η | − hµ (cid:19) d η & hµ, and that | y | ≤ f − . Next, Z | y |≤ h h − / y i s d y = h d/ Z | x |≤ h − / h x i s d x ∼ Ch d/ , if s < − d,C | ln h | h d/ , if s = − d,Ch − s/ , if s > − d. Finally, we get (B.6) using that µ ≈ h δ − so µh ≈ h δ ≈ h d N h . AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN
Appendix CProof of (2.12)To begin with, we identify the complex sphere of C N with the real sphere S N − = (cid:8) w ∈ R N : w + · · · + w N = 1 (cid:9) ⊂ R N . Denote by P N the uniform probability measure on S N − and by µ N the Gaussian measure on R N ofdensity d µ = 1(2 π ) N exp (cid:0) − N X j =1 x j (cid:1) d x . . . d x N . It is easy to check that P N is the image measureof µ N by the map G : R N −→ S N − ( x , . . . , x N ) qP Nj =1 x j ( x , . . . , x N ) . Indeed, µ N ◦ G − is a probability measure on S N − which is invariant by the isometries of S N − ,therefore P N = µ N ◦ G − . For t ∈ [0 , t ) = P N (cid:0)p w + w > t (cid:1) , thenΦ( t ) = 1(2 π ) N Z I x + x >t P Nj =1 x j e − P Nj =1 x j d x . . . d x N = 1(2 π ) N Z I x + x > t − t P Nj =3 x j e − P Nj =1 x j d x . . . d x N . We make a spherical change of variables ( x , . . . , x N ) rσ and the polar change of variables( x , x ) = ( r cos θ, r sin θ ). Denote by s = t/ √ − t , thus there exists C N so thatΦ( t ) = C N Z r N − e − ( ρ + r ) I ρ>sr ρ d r d ρ = C N Z + ∞ r N − e − (1+ s ) r d r. Now, by the change of variables r ′ = (1 + s ) / r , there exists C N so thatΦ( t ) = C N (1 + s ) − ( N − = C N (1 − t ) N − , and C N = Φ(0) = 1. Appendix DProof of Proposition 2.12
For simplicity we assume that the random variables, the γ j and the space E h are real, and weidentify E h with R N , endowed with its natural Euclidean norm | y | . We also consider the γ -dependentnorm | y | γ = 1 N X ≤ j ≤ N y j γ j , y = ( y , · · · , y N ) . ANDOM WEIGHTED SOBOLEV INEQUALITIES ON R d Condition (1.3) means that we have 1 C | y | ≤ | y | γ ≤ C | y | . We define a probability measure ν γ in R N as the pull forward of the measure ν in R N by the mapping ϕ : ( x , · · · , x N )
7→ √ N ( γ x , · · · , γ N x N ). Notice that ν γ satisfies the concentration property ofDefinition 2.1.Now we follow the proof of of [ , Proposition 2.10]. Let F be a Lipschitz function on the sphere S h ,and by homogeneity, it is enough to assume that F is 1-Lipschitz. For x ∈ R N , G ( x ) = | x | (cid:0) F ( x | x | ) − F ( x ) (cid:1) , where x is a fixed point on S h . Thus G satisfies(D.1) | G ( x ) − G ( y ) | ≤ π + 1) | x − y | , and(D.2) F ( x | x | ) − F ( y | y | ) = G ( x | x | ) − G ( y | y | ) . By [ , Corollary 1.5], it is enough to prove that P γ,h ⊗ P γ,h h u, v ∈ S h : | F ( u ) − F ( u ) | ≥ r i = ν γ ⊗ ν γ h x, y ∈ R N : (cid:12)(cid:12)(cid:12) F ( x | x | ) − F ( y | y | ) (cid:12)(cid:12)(cid:12) ≥ r i ≤ C e − cNr . Denote by M γ the median of x
7→ | x | with respect to ν γ . Then by (D.2) ν γ ⊗ ν γ h x, y ∈ R N : (cid:12)(cid:12)(cid:12) F ( x | x | ) − F ( y | y | ) (cid:12)(cid:12)(cid:12) ≥ r i = ν γ ⊗ ν γ h x, y ∈ R N : (cid:12)(cid:12)(cid:12) G ( x | x | ) − G ( y | y | ) (cid:12)(cid:12)(cid:12) ≥ r i ≤ ν γ ⊗ ν γ h x, y ∈ R N : (cid:12)(cid:12)(cid:12) G ( x M γ ) − G ( y M γ ) (cid:12)(cid:12)(cid:12) ≥ r i + 2 ν γ h x ∈ R N : (cid:12)(cid:12)(cid:12) G ( x | x | ) − G ( x M γ ) (cid:12)(cid:12)(cid:12) ≥ r i . By (D.1) we have (cid:12)(cid:12)(cid:12) G ( x | x | ) − G ( x M γ ) (cid:12)(cid:12)(cid:12) ≤ π + 1) (cid:12)(cid:12)(cid:12)(cid:12) | x | M γ − (cid:12)(cid:12)(cid:12)(cid:12) , which implies from (2.1) that ν γ h x ∈ R N : (cid:12)(cid:12)(cid:12) G ( x | x | ) − G ( x M γ ) (cid:12)(cid:12)(cid:12) ≥ r i ≤ C e − c M γ r . Similarly, by (D.1) and (2.1) ν γ h x, y ∈ R N : (cid:12)(cid:12)(cid:12) G ( x M γ ) − G ( y M γ ) (cid:12)(cid:12)(cid:12) ≥ r i ≤ C e − c M γ r . To conclude the proof of the proposition we use the following AUR´ELIEN POIRET, DIDIER ROBERT & LAURENT THOMANN
Lemma D.1 . — In R N , denote by M γ ( N ) the median of x
7→ | x | with respect to ν γ , and by A γ ( N ) its expectation. Then there exist C, C , C > such that for all N ≥ |M γ ( N ) − A γ ( N ) | ≤ C and C √ N ≤ M γ ( N ) ≤ C √ N .
Proof . — Here we use the notation | x | N := ( x + · · · + x N ) / . By definition of M γ , we have for all t > P γ,h h | x | N ≥ M γ i ≤ e − t ( M γ −A γ ) E h e t ( | x | N −A γ ) i ≤ e − t ( M γ −A γ ) e ct . Then, we choose t = ( M γ − A γ ) / (2 c ) and get for some C > |M γ − A γ | ≤ C . This was the firstclaim.Next, by Cauchy-Schwarz, we obtain A γ ( N ) ≤ R R N | x | N d ν ( x ) = N . Now we prove that there exists C > N ≥ A γ ( N ) ≥ C √ N . Indeed, A γ ( N + 1) − A γ ( N ) = Z R N +1 x N +1 | x | N + | x | N +1 d ν ( x ) ≥ Z R N +1 x N +1 | x | N +1 d ν ( x ) = A γ ( N )2( N + 1) . This implies that for all N ≥ A γ ( N + 1) ≥ (1 − N + 1) ) − A γ ( N ) ≥ (1 + 12( N + 1) ) A γ ( N ) , and then A γ ( N ) ≥ P N A γ (1), whereln P N = N X k =2 ln(1 + 12 k ) = 12 ln N + O (1) , which yields the result. References [1] C. An´e, S. Blach`ere, D. Chafa¨ı, P. Foug`eres, I. Gentil, F. Malrieu, R. Cyril, and G. Scheffer. Sur lesin´egalit´es de Sobolev logarithmiques. Panoramas et Synth`eses, 10. Soci´et´e Math´ematique de France, Paris,2000.[2] N. Burq and G. Lebeau. Injections de Sobolev probabilistes et applications.
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Aur´elien Poiret , Laboratoire de Math´ematiques, UMR 8628 du CNRS. Universit´e Paris Sud, 91405 Orsay Cedex,France • E-mail : [email protected]
Didier Robert , Laboratoire de Math´ematiques J. Leray, UMR 6629 du CNRS, Universit´e de Nantes, 2, rue de laHoussini`ere, 44322 Nantes Cedex 03, France • E-mail : [email protected]