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Weighted estimates for selfadjoint operators and weak dispersion

Piero D’Ancona

Dipartimento di Matenatica SAPIENZA - Universit` a di Roma www.mat.uniroma1.it/people/dancona

XXIX Convegno Nazionale di Analisi Armonica In honour of Guido Weiss Bardonecchia, 15–19 giugno 2009

Joint work (in progress) with

• Federico Cacciafesta (PhD student, Sapienza, Roma)

The motivation stems from joint work with

• Reinhard Racke (Konstanz)

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Weighted estimates for fractional operators

Our goal (more on this later): prove

• a weighted L2 estimate for a fractional power of an electromagnetic

Laplacian H = (i∇ − A)2 + V with standard weights w(x) = x−s: x−sHαf L2 ≤ Cx−s(−∆)αf L2, α = 1 4

• with precise bounds on C in terms of A, V

Notice that for α = 1

2 this is a reverse Riesz transform estimate

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Natural approach: use Stein-Weiss interpolation for the analytic family x−sHz(−∆)−zxs and reduce the problem to pure imaginary powers Hiy Unfortunately the kernel of Hiy is not smooth in general, thus standard singular integrals theory does not apply

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Hebisch in 1990 proved Lp and weak (1,1) estimates for g(−∆ + V (x)), V ≥ 0 smooth and decaying at infinity, g(s) smooth and bounded Thus kernel smoothness can be relaxed and Lp estimates still hold. How much can it be relaxed? Several works, culminating in Auscher and Martell 2007 (AIM) with a very powerful maximal lemma (good-λ inequality)

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The results

Assumption (H) H is a non negative selfadjoint operator on L2(Rn) satisfying

• a gaussian estimate

0 ≤ e−tH(x, y) ≤ K0 · t− n

2 e− |x−y|2 d1t ,

d1 > 0

• a finite speed of propagation property

|x − y| > d2t = ⇒ cos(t √ H)(x, y) = 0, d2 > 0. The relevance of these conditions was noticed in Sikora and Wright 2000 We can assume d1 = d2 = 1 in (H) (just rescale H → λH)

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Example

An electromagnetic laplacian H = (i∇ − A(x))2 + V (x) satisfies (H) provided

• A ∈ L2

loc(Rn; Rn)

• V real valued of Kato class with negative part V− = max{−V , 0}

small enough, in the sense that V−K < πn/2/Γ(n/2 − 1) The gaussian estimate follows from D’A-Pierfelice 2005 in the case A ≡ 0, combined with the pointwise diamagnetic inequalty of Barry Simon: for any test function φ, |et[(∇−iA(x))2−V ]φ| ≤ et(∆−V )|φ|.

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Recall that V is in the Kato class when sup

x lim r↓0

• |x−y|<r

|V (y)| |x − y|n−2 dy, (n ≥ 3) while the Kato norm is defined by V K = sup

x

• |V (y)|

|x − y|n−2 dy (n ≥ 3) (log |x − y| when n = 2)

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Condition (H) is quite general and satisfied by ’reasonable’ operators:

Example

Any elliptic selfadjoint operator H = H∗ = −

• i,j

∂iaij(x)∂j with aij = aji ∈ C ∞, 0 < λ ≤ [aij] ≤ Λ, satisfies (H) The smoothness of the coefficients aij can be substantially relaxed (e.g. for the heat kernel estimate aij ∈ L∞ is sufficient)

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Extended goal:

• weighted Lp estimates for operators g(

√ H) and general weights

• good control of the norm of g(

√ H) The smoothness of g(s) will be measured via the following norm (where φ a fixed test function supported in [1/2, 2]) µσ(g) = sup

λ>0

ξσ+ n

2 F[φSλg]L1

µσ is ’close’ to gW σ+ n

2 ,∞ and is dominated by standard Sobolev or

Besov norms: µσ(g) ≤ c(n)g

B

σ+ n+1 2 2,1

≤ c(n, ǫ)gHσ+ n+1

2 +ǫ

ǫ > 0.

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Main result:

Theorem

Let w be a weight function in Ap for some 1 < p < ∞, H an operator satisfying Assumption (H), and g a bounded function on R such that µσ(g) is finite for some σ > 0. Then, for all 1 < q < ∞, g( √ H)f Lq(w) ≤ C(1 + µσ(g) + g2

L∞)f Lq(w)

where C = c(n, σ, p, q, w)K 1+2p2 .

Remark

C depends on w through wAp = supQ

• −
• Q w

−

• Q w 1−p′p/p′

and on the norm of the uncentered maximal function in Lp(w)

Remark

We also improve Hebisch’ original estimates

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Application: imaginary powers

Consider g(s) = s2iy, y ∈ R, so that g( √ H) = Hiy We have gL∞ = 1 and µσ(g) ≤ C(1 + |y|)σ+ n

2

[proof: chain rule when s + n/2 ∈ 2N0, interpolation for s + n/2 ∈ R+] Thus the norm of Hiy : Lp(w) → Lp(w) is bounded by C(1 + |y|)

n 2 +ǫ

Compare with the optimal Lp and weak (1, 1) estimates of Sikora and Wright 2000 for L = − ∂iaij∂j LiyLp→Lp ≤ C(1 + |y|)

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Application: fractional powers

Consider the special case H = (i∇ − A)2 + V , n ≥ 3

Lemma

Let w ∈ Ar, 1 < r < n/2. Then Hf Lr(w) ≤ C(1 + |A|2 − i∇ · A + V L

n 2 + ALn) · ∆f Lr(w)

[proof: H¨

• lder inequality + Muckenhoupt-Wheeden 1974]

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By Stein-Weiss interpolation on the family w Hz(−∆)−zw −1 and Theorem 1, we obtain

Corollary

For 0 ≤ θ ≤ 1, 1 < q <

n 2θ, we have the estimate

Hθf Lq(w) ≤ C(−∆)θf Lq(w) provided w ∈ Ar for some r > qθ, with C ≃ (1 + |A|2 − i∇ · A + V L

n 2 + ALn)θ

Still in progress — can certainly be optimized

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Motivation: dispersive equations

Dispersive equations characterized by

• conserved quantities (typically of L2 type),
• spreading over increasing volumes (as t ր ∞)

Thus the size of the solution, measured in suitable norms, decreases Decay can be measured in a variety of norms; the strongest estimates are the pointwise estimates

Three examples

Schr¨

• dinger

iut + ∆u = 0: u = eit∆f = c

• eiφ(t,x,ξ)

f dξ, φ(t, x, ξ) = t|ξ|2 + x · ξ |eit∆f | ≤ C |t|

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Wave u ≡ ∂2

t u − ∆u = 0:

u = eit|D|f = c

• eiφ(t,x,ξ)

f dξ, φ(t, x, ξ) = t|ξ| + x · ξ |eit|D|f | ≤ C |t|

n−1 2 f ˙

B

n+1 2 1,1

Klein-Gordon u + u ≡ ∂2

t u − ∆u + u = 0:

u = eitDf = c

• eiφ(t,x,ξ)

f dξ, φ(t, x, ξ) = tξ + x · ξ |eitDf | ≤ C |t|

n 2 f

B

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Heuristically: larger ’space’ available for dispersion = ⇒ better decay. Decay improves

• in higher space dimension
• on manifolds with negative curvature

Decay is worse, or absent

• on manifolds with positve curvature
• in compact situations (loss of derivatives)

Standard consequence of pointwise estimates: Strichartz estimates eit∆f Lp

t Lq x ≤ Cf L2,

2 p + n q = n 2 Mixture of PDE and harmonic analysis techniques (Kato, Yajima, Ginibre, Velo, Bourgain, Tao et al.) Applications: decay estimates are the basic tools for local and global existence, scattering, regularity etc. of nonlinear equations

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Waveguides

We investigate dispersive equations on a waveguide, i.e. a domain Ω = Rn

x × ω,

ω ⊆ Rm

y

Waveguides model physically interesting structures, e.g.

• a nanotube: ω compact, m = 2, n = 1
• a layer: Ω = Rn × (a, b)

Intermediate between the compact and non-compact situations How does the shape of Ω influence the dispersive properties? Guess: The decay should be determined by the dimension n of the unbounded component, e.g. for the wave equation ∼ t− n−1

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Natural approach: expand the solution u(t, x, y) in eigenfunctions on ω u =

• j

u(j)(t, x)φj(y), −∆yφj = λ2

j φj

E.g. for the wave equation utt − ∆xu − ∆yu = 0 = ⇒ u(j)

tt − ∆xu(j) + λ2 j u(j) = 0,

j ≥ 1 Surprise: we get a family of Klein-Gordon equations, with faster decay |u(j)| ∼ t− n

2

Summing over j, this gives a better than expected decay for the original equation (Lesky and Racke 2003; Metcalf, Sogge, Stewart 2005) A fairly complete and ’elementary’ theory can be developed for flat waveguides, leading to satisfying results for the corresponding nonlinear problems

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A more realistic situation

Question:

what happens if the waveguide is not flat?

First difficulty

Ω is not a product! We can assume it has an ’approximate’ product structure: Ω ⊆ Rn

x × Rm y ,

with sections Ωx = {y : (x, y) ∈ Ω} of ’similar shape’, such that a function v(x, y) on Ω can be expanded as v(x, y) =

• j

v (j)(x)φj(x, y) where φj(x, y), for each x, is an eigenfunction of ∆y on Ωx

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Under suitable spectral assumptions we obtain a family of perturbed equations for the components u(j)(t, x): iu(j)

t

− ∆xu(j) + Vj(x)u(j) = 0 Note that the Vj increase polynomially in j (like the eigenvalues of ∆) Several results exist concerning pointwise and Strichartz estimates for perturbed equations The natural approach should be then

• prove estimates for each u(j)

|u(j)(t, x)| ≤ C(Vj) tα u(j)(0, x)X

• sum the estimates over j

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Second difficulty

We need to know how the constants depend on Vj; this will reflect in a loss of derivatives in the remaining variables y (as expected in compact situations) Unfortunately, all existing proofs of pointwise estimates rely on Fredholm theory = ⇒ no quantitative estimates are available! This is still an open (and interesting) problem

Workaround

Use smoothing estimates, which can be proved by concrete multiplier methods, to deduce Strichartz estimates

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By the multiplier method one can prove, under suitable assumptions on V , that the solution of iut − ∆u + V (x)u = 0 satisfies a smoothing estimate x−s(−∆)1/4uL2L2 ≤ C(V )H1/4u(0)L2 where H = −∆ + V and C(V ) is explicit. The weight x−s can be replaced by much more general weights

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Third difficulty, and motivation

We proved a smoothing estimate in the form x−s(−∆)1/4uL2L2 ≤ C(V )H1/4u(0)L2 but to deduce Strichartz we need an estimate like x−suL2L2 ≤ C(V )u(0)L2 (1) Clearly it would be sufficient to prove a weighted estimate x−sH1/4gL2 ≤ Cx−s(−∆)1/4gL2 with precise dependence of C on the perturbation V , so that x−sH1/4uL2L2 ≤ C(V )H1/4u(0)L2 which implies (1). This is precisely the content of the Corollary.

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Tools for Theorem 1

First ingredient: a good-λ inequality of Auscher and Martell

• Mf denotes the uncentered maximal function
• cq is the norm of the weak (q, q) estimate for M
• As,1 is the class of weights such that ∃C: for every cube Q and

every E ⊆ Q w(E) w(Q) ≤ C |E| |Q| 1

s

The following statement is slightly simplified (and the constants C0, C1 have been computed explicitly)

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Lemma (Auscher–Martell 2007)

Let F, G be positive measurable functions on Rn, 1 < q ≤ ∞, a ≥ 1, 1 ≤ s < ∞, w ∈ As,1. Assume that for every ball B there exist GB, HB positive functions such that F ≤ GB + HB a.e. on B, HBLq(B) ≤ a(MF(x) + G(y)) · |B|

1 q

for every x, y ∈ B, (2) GBL1(B) ≤ G(x) · |B| for every x ∈ B. (3) Then for all λ > 0, γ < 1, K ≥ 2n+2a, we have (C0 = 26(n+q)(c1 + cq)) w{MF > Kλ, G ≤ γλ} ≤ C0wAs,1 · γ K + aq K q 1

s

· w{MF > λ}. (4) As a consequence, if F is L1 and 1 ≤ p < q/s, MFLp(w) ≤ C1GLp(w), C1 =

• (8C0wAs,1 + 2n+3)ap

s 1−ps/q .

(5)

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The choice of the functions is the following (for arbitrarily large ν and a given f ):

• F(x) = |g(

√ H)f |ν

• G(x) = c(n, σ)νK ν

0 (1 + µ + g2 L∞)ν · M(|f |ν)(x)

• GB = 2ν|g(

√ H)(1 − ψr( √ H))f |ν

• HB = 2ν|g(

√ H)ψr( √ H)f |ν where ψ(s) is a fixed test function, ψr(s) = ψ(rs) is adapted to the radius r of the ball B

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Second ingredient: kernel estimates, Hebisch–style. The function HB, corresponding to low frequencies (smoothing operator) is estimated using

Lemma

Assume H satisfies (H). Let ψ ∈ C ∞

c (R), and for r > 0 write

ψr(s) = ψ(rs). Then, for all m ≥ 0, |ψr( √ H)(x, y)| ≤ C(n, m, ψ)K 2

0 ·

x − y r −m r −n. (6)

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The function GB, corresponding to high frequencies (singular part) is estimated via a dyadic decomposition (φ(s) = ψ(2s) − ψ(s)) g( √ H)(1 − ψr( √ H)) =

• k≤0

gk+lg r( √ H), gj(s) = g(s)φ(2js). The terms of the series are estimated using the following

Lemma

Assume H satisfies (H), and g has support in [−R, R]. Then we have for all a ≥ 0 g( √ H)xa ≤ c(n, a, R) · K0ξa+n/2 gL1 (7) where c(n, a, R) is independent of the operator H. Here we used the notation, for an operator A with integral kernel A(x, y), Axa = max

• sup

x

• x − ya|A(x, y)|dy,

sup

y

• x − ya|A(x, y)|dx
• ;

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The finite speed of propagation was used in the proof of the previous Lemma, to obtain an inequality of the form cos(ξ √ H)f L1 ξn/2

Q

f L2(Q). where f is any test function and the unit cubes Q form an almost disjoint cover of Rn

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