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) Piero D’Ancona Weighted estimates 2 / 30
Weighted estimates for fractional operators Our goal (more on this later): prove • a weighted L 2 estimate for a fractional power of an electromagnetic Laplacian H = ( i ∇ − A ) 2 + V with standard weights w ( x ) = � x � − s : α = 1 �� x � − s H α f � L 2 ≤ C �� x � − s ( − ∆) α f � L 2 , 4 • with precise bounds on C in terms of A , V Notice that for α = 1 2 this is a reverse Riesz transform estimate Piero D’Ancona Weighted estimates 3 / 30
Natural approach: use Stein-Weiss interpolation for the analytic family � x � − s H z ( − ∆) − z � x � s and reduce the problem to pure imaginary powers H iy Unfortunately the kernel of H iy is not smooth in general, thus standard singular integrals theory does not apply Piero D’Ancona Weighted estimates 4 / 30
Hebisch in 1990 proved L p 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 L p 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) Piero D’Ancona Weighted estimates 5 / 30
The results Assumption (H) H is a non negative selfadjoint operator on L 2 ( R n ) satisfying • a gaussian estimate 2 e − | x − y | 2 0 ≤ e − tH ( x , y ) ≤ K 0 · t − n d 1 t , d 1 > 0 • a finite speed of propagation property √ | x − y | > d 2 t = ⇒ cos( t H )( x , y ) = 0 , d 2 > 0 . The relevance of these conditions was noticed in Sikora and Wright 2000 We can assume d 1 = d 2 = 1 in (H) (just rescale H → λ H ) Piero D’Ancona Weighted estimates 6 / 30
Example An electromagnetic laplacian H = ( i ∇ − A ( x )) 2 + V ( x ) satisfies (H) provided • A ∈ L 2 loc ( R n ; R n ) • V real valued of Kato class with negative part V − = max {− V , 0 } small enough, in the sense that � V − � K < π n / 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 , | e t [( ∇− iA ( x )) 2 − V ] φ | ≤ e t (∆ − V ) | φ | . Piero D’Ancona Weighted estimates 7 / 30
Recall that V is in the Kato class when � | V ( y ) | sup x lim | x − y | n − 2 dy , ( n ≥ 3) r ↓ 0 | x − y | < r while the Kato norm is defined by � | V ( y ) | � V � K = sup | x − y | n − 2 dy ( n ≥ 3) x (log | x − y | when n = 2) Piero D’Ancona Weighted estimates 8 / 30
Condition (H) is quite general and satisfied by ’reasonable’ operators: Example Any elliptic selfadjoint operator � H = H ∗ = − ∂ i a ij ( x ) ∂ j i , j with a ij = a ji ∈ C ∞ , 0 < λ ≤ [ a ij ] ≤ Λ, satisfies (H) The smoothness of the coefficients a ij can be substantially relaxed (e.g. for the heat kernel estimate a ij ∈ L ∞ is sufficient) Piero D’Ancona Weighted estimates 9 / 30
Extended goal: √ • weighted L p 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]) �� ξ � σ + n 2 F [ φ S λ g ] � L 1 µ σ ( g ) = sup λ> 0 µ σ is ’close’ to � g � W σ + n 2 , ∞ and is dominated by standard Sobolev or Besov norms: µ σ ( g ) ≤ c ( n ) � g � ≤ c ( n , ǫ ) � g � H σ + n +1 ǫ > 0 . σ + n +1 + ǫ 2 2 B 2 , 1 Piero D’Ancona Weighted estimates 10 / 30
Main result: Theorem Let w be a weight function in A p for some 1 < p < ∞ , H an operator satisfying Assumption (H), and g a bounded function on R such that µ σ ( g ) is finite for some σ > 0 . Then, for all 1 < q < ∞ , √ H ) f � L q ( w ) ≤ C (1 + µ σ ( g ) + � g � 2 � g ( L ∞ ) � f � L q ( w ) where C = c ( n , σ, p , q , w ) K 1+2 p 2 . 0 Remark � � � Q w 1 − p ′ � p / p ′ � � C depends on w through � w � A p = sup Q Q w and on − − the norm of the uncentered maximal function in L p ( w ) Remark We also improve Hebisch’ original estimates Piero D’Ancona Weighted estimates 11 / 30
Application: imaginary powers Consider g ( s ) = s 2 iy , y ∈ R , so that √ H ) = H iy g ( We have � g � L ∞ = 1 and µ σ ( g ) ≤ C (1 + | y | ) σ + n 2 [proof: chain rule when s + n / 2 ∈ 2 N 0 , interpolation for s + n / 2 ∈ R + ] Thus the norm of H iy : L p ( w ) → L p ( w ) is bounded by C (1 + | y | ) n 2 + ǫ Compare with the optimal L p and weak (1 , 1) estimates of Sikora and Wright 2000 for L = − � ∂ i a ij ∂ j n � L iy � L p → L p ≤ C (1 + | y | ) 2 Piero D’Ancona Weighted estimates 12 / 30
Application: fractional powers Consider the special case H = ( i ∇ − A ) 2 + V , n ≥ 3 Lemma Let w ∈ A r , 1 < r < n / 2 . Then � Hf � L r ( w ) ≤ C (1 + �| A | 2 − i ∇ · A + V � L 2 + � A � L n ) · � ∆ f � L r ( w ) n [proof: H¨ older inequality + Muckenhoupt-Wheeden 1974] Piero D’Ancona Weighted estimates 13 / 30
By Stein-Weiss interpolation on the family w H z ( − ∆) − z w − 1 and Theorem 1, we obtain Corollary n For 0 ≤ θ ≤ 1 , 1 < q < 2 θ , we have the estimate � H θ f � L q ( w ) ≤ C � ( − ∆) θ f � L q ( w ) provided w ∈ A r for some r > q θ , with C ≃ (1 + �| A | 2 − i ∇ · A + V � L 2 + � A � L n ) θ n Still in progress — can certainly be optimized Piero D’Ancona Weighted estimates 14 / 30
Motivation: dispersive equations Dispersive equations characterized by • conserved quantities (typically of L 2 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¨ odinger iu t + ∆ u = 0: � φ ( t , x , ξ ) = t | ξ | 2 + x · ξ e i φ ( t , x ,ξ ) � u = e it ∆ f = c f d ξ, | e it ∆ f | ≤ C 2 � f � L 1 n | t | Piero D’Ancona Weighted estimates 15 / 30
Wave � u ≡ ∂ 2 t u − ∆ u = 0: � u = e it | D | f = c e i φ ( t , x ,ξ ) � f d ξ, φ ( t , x , ξ ) = t | ξ | + x · ξ C | e it | D | f | ≤ 2 � f � ˙ n +1 n − 1 2 | t | B 1 , 1 Klein-Gordon � u + u ≡ ∂ 2 t u − ∆ u + u = 0: � u = e it � D � f = c e i φ ( t , x ,ξ ) � f d ξ, φ ( t , x , ξ ) = t � ξ � + x · ξ | e it � D � f | ≤ C 2 � f � n n 2 +1 | t | B 1 , 1 Piero D’Ancona Weighted estimates 16 / 30
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 p + n 2 q = n � e it ∆ f � L p x ≤ C � f � L 2 , t L q 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 Piero D’Ancona Weighted estimates 17 / 30
Waveguides We investigate dispersive equations on a waveguide, i.e. a domain Ω = R n ω ⊆ R m x × ω, y Waveguides model physically interesting structures, e.g. • a nanotube: ω compact, m = 2, n = 1 • a layer: Ω = R n × ( 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 2 Piero D’Ancona Weighted estimates 18 / 30
Natural approach: expand the solution u ( t , x , y ) in eigenfunctions on ω � u ( j ) ( t , x ) φ j ( y ) , − ∆ y φ j = λ 2 u = j φ j j E.g. for the wave equation tt − ∆ x u ( j ) + λ 2 j u ( j ) = 0 , u ( j ) u tt − ∆ x u − ∆ y u = 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 Piero D’Ancona Weighted estimates 19 / 30
Recommend
More recommend
