Super-diffusivity in a shear flow model from perpetual homogenization.

Transcription

1 Super-diffusivity in a shear flow model from perpetual homogenization. Gérard Ben Arous and Houman Owhadi arxiv:math/15199v1 [math.pr] 24 May 21 February 1, 28 Abstract This paper is concerned with the asymptotic behavior solutions of stochastic differential equations dy t = dω t Γ(y t )dt, y = and d = 2. Γ is a 2 2 skew-symmetric matrix associated to a shear flow characterized by an infinite number of spatial scales Γ 12 = Γ 21 = h(x 1 ), with h(x 1 ) = n= γ nh n (x 1 /R n ) where h n are smooth functions of period 1, h n () =, γ n and R n grow exponentially fast with n. We can show that y t has an anomalous fast behavior (E[ y t 2 ] t 1+ν with ν > ) and obtain quantitative estimates on the anomaly using and developing the tools of homogenization. Contents 1 Introduction 2 2 The model 2 3 Main results 3 4 Proofs Notations and proof of the theorem Anomalous mean squared displacement: theorem Anomalous behavior from perpetual homogenization Distinction between effective and drift scales. Proof of the equation Influence of the intermediate scale on the effective scales: proof of the inequalities 41 and Control of the homogenization process over the effective scales: proof of the equations 45 and Control of the influence of the perturbation scale: proof of the equation Stochastic separation between scales: proof of the equation Stochastic mixing: proof of the proposition EPFL, gerard.benarous@epfl.ch Technion, owhadi@techunix.technion.ac.il 1

2 1 Introduction Turbulent incompressible flows are characterized by multiple scales of mixing length and convection rolls. It is heuristically known and expected that a diffusive transport in such media will be super-diffusive. The first known observation of this anomaly is attributed to Richardson [Ric26] who analyzed available experimental data on diffusion in air, varying on about 12 orders of magnitude. On that basis, he empirically conjectured that the diffusion coefficient D λ in turbulent air depends on the scale length λ of the measurement. The Richardson law, D λ λ 4 3 (1) was related to Kolmogorov-Obukhov turbulence spectrum, v λ 1 3, by Batchelor [Bat52]. The super-diffusive law of the root-mean-square relative displacement λ(t) of advected particles λ(t) (D λ(t) t) 1 2 t 3 2 (2) was derived by Obukhov [Obu41] from a dimensional analysis similar to the one that led Kolmogorov [Kol41] to the λ 1 3 velocity spectrum. More recently physicists and mathematicians have started to investigate on the superdiffusive phenomenon (from both heuristic and rigorous point of view) by using the tools of homogenization or renormalization; we refer to M. Avellaneda and A. Majda [AM9], J. Glimm and Al. [FGLP9], [FGL + 91], [GLPP92], J. Glimm and Q. Zhang [GZ92], Q. Zhang [Zha92], M.B. Isichenko and J. Kalda [IK91]. It is now well known that homogenization over a periodic or ergodic divergence free drift has the property to enhance the diffusion [FP94],[FP96], [KO]. It is also expected that several spatial scales of eddies should give rise to anomalous diffusion between proper time scales outside homogenization regime or when the bigger scale has not yet been homogenized. We refer to M. Avellaneda [Ave96]; A. Fannjiang [Fan99]; Rabi Bhattacharya [Bha99] (see also [BDG99] by Bhattacharya - Denker and Goswami); A. Fannjiang and T. Komorowski [FK1] and this panorama is certainly not complete. The purpose of this paper is to implement rigorously on a shear flow model the idea that the key to anomalous fast diffusion in turbulent flows is an unfinished homogenization process over a large number of scales of eddies without a sharp separation between them. We shall assume that the ratios between the spacial scales are bounded. The underlying phenomenon is similar to the one related to anomalous slow diffusion from perpetual homogenization on an infinite number of scales of gradient drifts [Owh1], [BO1], the main difference lies in the asymptotic behavior of the multi-scale effective diffusivities D(n) associated with n spacial scales, i.e. D(n) diverge towards or converge towards with exponential rate depending on the nature of the scales: eddies or obstacles. 2 The model Let us consider in dimension two a Brownian motion with a drift given by the divergence of a shear flow stream matrix, i.e. the solution of the stochastic differential equation: dy t = dω t Γ(y t )dt, y = (3) where Γ is a skew-symmetric 2 2 shear flow matrix. ( ) h(x1 ) Γ(x 1,x 2 ) = h(x 1 ) (4) 2

3 The function (x 1,x 2 ) h(x 1 ) is given by a sum of infinitely many periodic functions with (geometrically) increasing periods h(x 1 ) = n= γ n h n ( x 1 R n ) (5) Where h n are smooth functions of period 1, h n () = normalized by the choosing their variance equal to one: Var(h n ) = (h(x) R n and γ n grow exponentially fast with n, i.e. Where r n are integers, r = 1, R n = h(y)dy) 2 dx = 1 (6) n r k (7) k= We choose γ = 1 and ρ min = inf n N r n 2 and γ min = inf n N (γ n+1/γ n ) > 1 and ρ max = sup n N r n < (8) γ max = sup(γ n+1 /γ n ) < (9) n N It is assumed that the first derivates of the potentials h n are uniformly bounded. (Osc(h) stands for suph inf h) K = suposc(h n ) <, n N K 1 = sup h n < (1) n N For ρ min > γ max, Γ is a well defined Lipschitz stream matrix and the solution of the stochastic differential equation 3 exists; is unique up to sets of measure with respect to the Wiener measure and is a strong Markov continuous Feller process. We shall prove its anomalous behavior under that assumption. For ρ min < γ max, one can obtain similar results by introducing further uniform bounds on higher order derivates of the fluctuations h n. 3 Main results Our objective is to show that the solution 3 is abnormally fast and the asymptotic subdiffusivity will be characterized as an anomalous behavior of the variance at time t, i.e. E [y 2 t ] t1+ν as t. More precisely there exists a constant ρ (γ min,γ max,k,k 1 ) and a time t (γ min,γ max,r 1,K,K 1 ) such that Theorem 3.1. If ρ min > ρ and y t is a solution of 3 then for t > t with E [ y t.e 2 2 ] = t 1+ν(t) (11) ln γ min ln ρ max + ln γmax γ min C 1 ln t ν(t) ln γ max ln ρ min + ln γ min γ max + C 2 ln t (12) Where the constants C 1 and C 2 depends on ρ min,γ min,γ max,ρ max,k,k 1 3

4 We remark that if γ max = γ min = γ and ρ max = ρ min = ρ then ν(t) ln γ/ln ρ. The key of the fast asymptotic behavior of the variance of the solution of 3 is the geometric rate of divergence towards of the multi-scale effective matrices associated to a finite number of scales. More precisely, for k,p N, k p we shall write H k,p = p γ n h n (x/r n ) (13) n=k and Γ k,p the skew-symmetric matrix given by Γ k,p 1,2 (x 1,x 2 ) = H k,p (x 1 ). Let D(Γ,p ) be the effective diffusivity associated to homogenization of the periodic operator L Γ,p = 1/2 Γ,p. Then it is easy to see that ( ) D(Γ,p 1 ) = D(Γ,p (14) ) 22 and it shall be shown that Theorem 3.2. For 1 + 4(1 ǫ) ǫ = 4K 1 ( ρmin (γ min 1) ) 1 < 1 (15) p γk 2 D(Γ,p ) (1 + ǫ) k= p γk 2 (16) The super-diffusive behavior can be explained and controlled by a perpetual homogenization process taking place over the infinite number of scales,...,n,... The idea of the proof of theorem 3.1 is to distinguish, when one tries to estimate 11, the smaller scales which have already been homogenized (,...,n ef called effective scales), the bigger scales which have not had a visible influence on the diffusion (n dri,..., called drift scales because they will be replaced by a constant drift in the proof) and some intermediate scales that manifest their particular shapes in the behavior of the diffusion (n ef + 1,...,n dri 1 = n ef + n per called perturbation scales because they will enter in the proof as a perturbation of the homogenization process over the smaller scales). The number of effective scales of is fixed by the mean squared displacement of y t.e 1. Writing n ef (t) = inf{n : t R 2 n} one proves that E[(y t.e 2 ) 2 ] D(Γ,n ef(t) )t Assume for instance R n = ρ n and γ n = γ n then n ef (t) ln t/(2ln ρ) and E[(y t.e 2 ) 2 ] t 1+ln γ ln ρ We remark that the quantitative control is sharper than the one associated to a perpetual homogenization on a gradient drift [Owh1]; this is explained by the fact that the number of perturbation scales is limited to only one scale with a divergence free drift. Nevertheless the main difficulty is to control the influence of this intermediate scale and the core of that control is based on the following mixing stochastic inequality (we write TR d the torus of dimension d and side 1) Proposition 3.1. Let R > and f,g (H 1 (TR 1))2 such that f(y)dy = and G(y)dy =, let r > and t > E [ G(b t ) k= 1 f(rb s )ds ] f L 2 (T 1 R ) G L 2 (T 1 R )2r 1 (17) 4

5 Physical interpretation We want to emphasize that to obtain a super-diffusive behavior of the solution of 3 of the shape E[y 2 t ] t 1+ν with ν > it is necessary to assume the exponential rate of growth of the parameters γ n, this has a clear meaning when the flow is compared on a heuristic point of view to a real turbulent flow. The parameters γ n h n /R n represents the amplitude of the pulsations of the eddies of size R n. It is a well known characteristic of turbulence ([LL84] p. 129) that the amplitude of the pulsations increase with the scale, since for all scales h n K 1 one should have limγ n /R n to reflect that image. In our model, it is sufficient to assume to exponential rate of divergence of γ n to obtain a super-diffusive behavior. Let us also notice on a heuristic point of view (our model is not isotropic and does not depend on the time) that the energy dissipated per unit time and unit volume in the eddies of scale n is of order of ǫ n γ2 n Rn 4 K2 2 (18) So saying that the energy is dissipated mainly in the small eddies is equivalent to saying that γ n /R 2 n as n or if R n = ρ n and γ n = ρ αn, this equivalent to say that α < 2. The Kolmogorov-Obukhov s law is equivalent to say that K 2 <, for all n h n() = and γ n R 4 3 n (19) or if R n = ρ n and γ n = ρ αn, this is equivalent to say that α = 4 3. Overlapping ratios The super-diffusive behavior in the theorem 3.1 requires a minimal separation between scales, i.e. ρ min > ρ and this condition is necessary. Assume for instance R n = ρ n and γ n = γ n, then if h n (x 1 ) = h(x) γ p h(a p x) it is easy to see that for ρ = a, Γ is bounded and y t has a normal behavior by Norris s Aronson type estimates [Nor97]. Thus, as for a gradient drift [Owh1], it is easy to see on simple examples that when ρ ρ the solution of 3 may have a normal or a super-diffusive behavior depending on the value of ρ and the shapes of h n and ratios of normal behavior may be surrounded by ratios of anomalous behavior. This phenomenon is created by a strong overlap between spatial ratios between scales. Remark: fast separation between scales The feature that distinguishes a strong fast behavior from a weak one is the rate at which spatial scales do separate. Indeed one can follow the proof of the theorem 3.1, changing the condition ρ max < into R n = R n 1 [ρ nα /R n 1 ] (ρ,α > 1) and γ max = γ min = γ to obtain Theorem 3.3. For t > t (γ 2,R 2,K,K 1 ) C 1 tγ β(t) E [ y t.e 2 2 ] C 2 tγ β(t) (2) Where the constants C 1 and C 2 depends on ρ,γ,α,k 1 with β(t) = 2(2ln ρ) 1 α (ln t) 1 α (21) And as α 1 the behavior of the solution of 3 pass from weakly anomalous to strongly anomalous. 5

6 4 Proofs 4.1 Notations and proof of the theorem 3.2 In this subsection we shall prove the theorem 3.2 using the explicit formula of the effective diffusivity. We shall first introduce the basic notations that shall also be used to prove the theorem 3.1. Let J be smooth T1 2 periodic 2 2 skew-symmetric matrix such that J 12 (x 1,x 2 ) = j(x 1 ) and consider the periodic operator L J = 1/2 J. We call χ l the solution of the cell problem associated to L J, i.e. the T1 2 periodic solution of L J (χ l l.x) = with χ l () =. One easily obtains that [ x 1 χ l (x 1,x 2 ) = 2l 2 j(y)dy x 1 ] j(y)dy The solution of the cell problem allows to compute the effective diffusivity t ld(j)l = l χ l (x) 2 dx that leads to T 2 1 D(J) = For a R periodic function f we shall write Var(f) = 1/R ( ) Var(j) R (f(x) R Using the notation 13, from the equation 23 we obtain 14 with Now we shall prove by induction on p that (for ǫ by 15) (1 ǫ) (22) (23) f(y)dy) 2 dx (24) D(Γ,p ) 2,2 = 1 + 4Var(H p ) (25) p γk 2 Var(Hp ) (1 + ǫ) k= p γk 2 (26) Then the equation 16 of the theorem will follow by 25. The equation 26 is trivially true for p =. From the explicit formula of H p we shall show in the paragraph that Var(H p ) Var(H p 1 ) γp 2 2γ pk 1 rp 1 Var(H p 1 ) (27) Assuming 26 to be true at the rank p one obtains k= ( p Var(H p ) (1 + ǫ) 1 2γ p+1 (γ k /γ p+1 ) 2)1 2 k= (28) (1 + ǫ) 1 2 γp+1 (γ min 1) 1 and it is easy to see that the condition 15 implies that ǫ 2K 1 (1+ǫ) 1 2(ρ min (γ min 1)) 1, combining this with 27 and 28 one obtains that Var(H p+1 ) Var(H p ) γ 2 p+1 ǫγ2 p+1 (29) which proves the induction and henceforth the theorem. 6

7 From the equation Var(H p ) = one obtains with ( 1 (H p 1 (R p x) H p 1 (R p y)dy) + γ p (h p (x) h p (y)dy) ) 2 dx (3) Var(H p ) Var(H p 1 ) γ 2 p 2γ p E (31) E = Cov(S Rp H p 1,h p ) where Cov stands for the covariance: Cov(f,g) = (f(x) f(y)dy)(g(x) g(y)dy)dx and S R is the scaling operator S R f(x) = f(rx). We shall use the following mixing lemma whose proof is an easy exercise Lemma 4.1. Let (g,f) ( C 1 (T1 d ) 2 and R N then g(x)s R f(x)dx g(x)dx f(x)dx ( g /R) T d 1 By the lemma 4.1 and Cauchy-Schwartz inequality E h p R p 1 R p T d 1 Combining this with 31 one obtains 27. T d 1 T d 1 f dx (32) S Rp H p 1 (x) dx K 1 r p Var(H p 1 ) (33) 4.2 Anomalous mean squared displacement: theorem Anomalous behavior from perpetual homogenization Let y t be the solution of 3. Define n ef (t) = inf{n N : t R 2 n+1(γ n 1 /γ n+1 ) K 2 1 (1 γ max/ρ min ) 2 } (34) n ef (t) shall be the number of effective scales that have one can consider homogenized in the estimation of the mean squared displacement at the time t. Indeed, we shall show in the sub subsection that for ρ min > C 1,K1,γ min,γ max and t > C 2,K,γ min,γ 1,R 1 one has (1/4)tγ 2 n ef (t) 1 E[ (y t.e 2 ) 2] C 3,K1,γ min γ 2 n ef (t)+1 t (35) Combining this with the bounds 8 and 9 on R n and γ n one obtains easily the theorem Distinction between effective and drift scales. Proof of the equation 35 By the Ito formula one has y t.e 2 = ω t.e 2 + And by the independence of ω t.e 2 from ω t.e 1 one obtains 1 h(ω s.e 1 )ds (36) E [ (y t.e 2 ) 2] [ ( t = t + E 1 h(ω s.e 1 )ds ) ] 2 (37) 7

8 thus for all p N, using h = H p + H p+1, one easily obtains (writing H p = H,p ) E [ (y t.e 2 ) 2] t + 1 [ ( t 2 E 1 H p (ω s.e 1 )ds ) ] [ 2 ( t E 1 H p+1, (ω s.e 1 )ds ) ] 2 [ ( t t + 2E 1 H p (ω s.e 1 )ds ) ] [ 2 ( t + 2E 1 H p+1, (ω s.e 1 )ds ) ] (38) 2 Now we shall bound the larger scales 1 H p+1, as drift scales, i.e. bound them by a constant drift using 1 H p+1, K 1 (γ p+1 /R p+1 )(1 γ max /ρ min ) 1 E [ (y t.e 2 ) 2] t + 1 [ ( t 2 E 1 H p (ω s.e 1 )ds ) ] 2 t 2( K 1 (γ p+1 /R p+1 )(1 γ max /ρ min ) 1 ) 2 [ ( t t + 2E 1 H p (ω s.e 1 )ds ) ] 2 + 2t 2( K 1 (γ p+1 /R p+1 )(1 γ max /ρ min ) 1) 2 [ ( Write I p = E 1H p (ω s.e 1 )ds ) ] 2, since H p is periodic, for t large enough, I p should behave like tγp 2. Nevertheless since the ratios between the scales are bounded, to control the asymptotic lower bound of the mean squared displacement in 39 we shall need a quantitative control of I p that is sharp enough to show that the influence of the effective scales is not destroyed by the larger ones. This control is based on stochastic mixing inequalities and it shall be shown in the sub subsection that for ρ min > 8K 1 /(γ min 1) one has (39) I p t23 ( γ 2 p 1(1 1/γ min ) 2 + K K 1 γ p 1 (γ p /r p )(1 1/γ min ) 1) + t 2 K 2 1(γ 2 p/r 2 p) + t68γ p 1 γ p R p 1 K 2 (1 1/γ min ) 1 + 6K 2 (1 1/γ min ) 1 γ p 1 γ p (R 2 p/r p ) (4) + 16K 2 γ 2 p 1R 2 p 1 and I p tγ p 1 (γ p 1 16K 1 γ p /r p ) t68γ p 1 γ p R p 1 K 2 (1 1/γ min) 1 6K 2 (1 1/γ min ) 1 γ p 1 γ p (R 2 p/r p ) 8K 2 γ 2 p 1R 2 p 1(1 1/γ min ) 2 (41) Choosing p = n ef (t) given by 34 one obtains 35 from 4, 41 and 39 by straightforward computation under the assumption ρ min > C K,K 1,γ max,γ min Influence of the intermediate scale on the effective scales: proof of the inequalities 41 and 4 In this sub subsection we shall prove the inequalities 41 and 4 by distinguishing the scale p as a perturbation scale, i.e. controlling its influence on the homogenization process over the scales,...,p 1. More precisely, writing b t the Brownian motion ω t.e 1 one has with [ ( t I p = I p 1 + E 1 H p,p (b s )ds ) ] 2 + 2J p (42) [ ( t J p = E 1 H p,p (b s )ds )( 1 H p 1 (b s )ds )] (43) 8

9 We shall then control 42 by bounding 1 H p,p by a constant drift to obtain [ ( t E 1 H p,p (b s )ds ) ] 2 t 2 K1 2 γ2 p /R2 p (44) In the sub subsection we shall control the homogenization process over the scales,...,p 1 and use our estimates on D(Γ,p 1 ) to obtain that for ρ min > 8K 1 /(γ min 1) I p 1 tγ 2 p 12(1 1/γ min ) 2 + R 2 p 1γ 2 p 116K 2 (1 1/γ min ) 2 (45) I p 1 tγ 2 p 1 2(1 1/γ min) 2 R 2 p 1 γ2 p 1 8K2 (1 1/γ min) 2 (46) In the sub subsection we shall bound J p is an error term by mixing stochastic inequalities to obtain J p γ p 1 γ p (1 1/γ min ) 1( ) trp 1 34K 2 + (t/r p)8k 1 + Rp 2 r 1 p 3K2 (47) Combining 42, 44, 45, 46 and 47 one obtains 4 and Control of the homogenization process over the effective scales: proof of the equations 45 and 46 Writing for m p, κ m,p Rp = 1/R p H m,p (y)dy and κ p = κ,p one obtains by the Ito formula 1 H p 1 (b s )ds = 2 bt (H p 1 (y) κ p 1 )dy 2 (H p 1 (b s ) κ p 1 )db s (48) Using the periodicity of H p : x ( H p 1 (y) κ p 1) dy R p 1 K γ p 1 (1 1/γ min ) 1 (49) Now we shall show that for ρ min > 8K 1 /(γ min 1) and p N E[ (H p (b s ) κ p ) 2 ds] (1 1/γ min ) 2( tγp ) R2 p γ2 p 4K2 (5) tγp 2 2 R2 p γ2 p 4K2 (1 1/γ min) 2 Combining 48, 49 and 5 one obtains 45 and 46 by straightforward computation. The proof of 5 is based on standard homogenization theory: writing f(x) = x ( H p (y) κ p) 2 Var(H p )x (51) One obtains by Ito formula ( x and g(x) = 2 Rp ) f(y)dy (x/r p ) f(y)dy (52) E[ (H p (b s ) κ p ) 2 ds] = Var(H p )t + E[g(b t )] (53) Using the periodicity of g, one has g 4K 2 γ2 pr 2 p(1 1/γ min ) 2 Combing this with the estimate 26 on Var(H p ) one obtains 5 from 53. 9

10 4.2.5 Control of the influence of the perturbation scale: proof of the equation 47 From the equations 43, 48 and x 1 H p,p (ω s.e 1 )ds = 2 ( H p 1 (y) κ p 1) dy R p 1 γ p 1 K (1 1/γ min ) 1 (54) bt (H p,p (y) κ p,p )dy 2 (H p,p (b s ) κ p,p )db s (55) One obtains using by straightforward computation (using Cauchy-Schwartz inequality) that with and J p 2γ p 1 γ p R p 1 K 2 (1 1/γ min) 1 t + 4 J p,2 + 2 J p,3 (56) [ ] J p,2 = E (H p 1 (b s ) κ p 1 )(H p,p (b s ) κ p,p )ds [ J p,3 = E 1 H p 1 (b s )ds bt ] (H p,p (y) κ p,p )dy We shall show in the sub subsection that the ratio r p allows a stochastic separation between the scales,...,p 1 and p reflected by the following inequality Jp,2 γp 1 γ p K (1 1/γ min ) 1( ) 8K R p 1 t + 2tK1 /r p (59) Using the proposition 3.1 (that we shall prove in sub subsection 4.2.7) with G(x) = x (Hp,p (y) κ p,p )dy, r = r p and f(r p x) = H p 1 (x) k p 1 that J p,3 γp 1 γ p Rpr 2 p 1 15K(1 2 1/γ min ) 1 (6) Combining 56, 59 and 6 one obtains Stochastic separation between scales: proof of the equation 59 Writing for x R, g(x) = one obtains by Ito formula x (57) (58) (H p 1 (y) κ p 1 )(H p,p (y) κ p )dy (61) bs 2E[ g(y)dy] = J p,2 (62) Using the functional mixing lemma 4.1 one obtains easily that g(x) 2K γ p 1 (1 1/γ min ) 1( 4γ p K R p 1 + x K 1 (γ p /r p ) ) (63) Combining this with 62 one obtains 59. 1

11 4.2.7 Stochastic mixing: proof of the proposition 3.1 By the scaling law of the Brownian motion E [ G(b t /R) (1/R) 1 f(rb s /R)ds ] = RE [ G(b t R 2 ) R 2 1 f(rb s )ds ] and it is sufficient to prove the proposition assuming R = 1. Let us write I = E [ G(b t ) 1 f(rb s )ds ] (64) We shall prove the proposition 3.1 by expanding 64 on the the Fourier decompositions of f and G (written f k and G k ) and controlling trigonometric functions. Write for k,m Z J k,m = By straightforward computation J k,m = f(x) = k Z f k e ik2πx (65) E [ e ikr2πbs e im2πbt] ds (66) e (2π)2 (kr+m) 2 2 s (2π) 2 m2 2 (t s) m2 (2π)2 = e t1 ( (kr) 2 e (2π) krm)t (2π) 2 ( (kr)2 2 + krm) (in last fraction of the above equation, if the denominator is equal to, we consider it as a limit to obtain the exact value t). Now I = k,m Z 2 J km ik2πf k G m (68) Thus since f = G =, by Cauchy-Schwartz inequality I f k ( (2π) 2 m 2 G m 2) 1 2 J 1 2 k (69) k Z with m Z J k = 1 ( 1 e (2π)2( (kr) 2 m 2 m Z ( k2 r 2 2 +krm)t 2 + krm)(2π) 2 ) 2e (2π) 2 m 2 t Using (1 e tx )/x 3t for x > and the fact that the minimum of k 2 r/2 + km is reached for m kr/2 and we obtain J k ( e (2π)2 m 2 t 2 4t k 2 r ( 1 ) 2) m 2 krm(2π) 2 m Z m Z (67) (7) 4/r 2 (71) Observing that G 2 L 2 (T 1 1 ) = (2π)2 m Z m2 G m 2 one obtains from 69 and 71 I G L 2 (T 1 1 )(2/r) k Z f k (72) Which, using Cauchy-Schwartz inequality leads to I G L 2 (T1 1) f L 2 (T1 1 )2/r (73) 11

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