Convolution estimates and the Gross-Pitaevskii hierarchy
aa r X i v : . [ m a t h . A P ] D ec CONVOLUTION ESTIMATES AND THE GROSS-PITAEVSKIIHIERARCHY
WILLIAM BECKNER
Abstract.
Extensions to higher-dimensions are given for a convolution estimate used byKlainerman and Machedon in their study of uniqueness of solutions for the Gross-Pitaevskiihierarchy. Such estimates determine more general forms of Stein-Weiss integrals involvingrestriction to smooth submanifolds.
Analysis of the Gross-Pitaevskii hierarchy has led to the development and application offunctional analytic mappings for the rigorous description of many-body interactions in quan-tum dynamics. In their formative and influential paper on uniqueness of solutions for theGross-Pitaevskii hierarchy, Klainerman and Machedon determine uniform bounds for a three-dimensional convolution integral. The idea of their argument rests on an extension of theclassical convolution for Riesz potentials Z S | w − g | λ | y | µ dσ where S is a smooth submanifold in R n , w ∈ R m and the objective is to bound the size of theintegral by an inverse power of | w | under suitable conditions on λ and µ . Such an estimate canbe viewed as a step in the larger and dual program for understanding how smoothness controlsrestriction to a non-linear sub-variety (see [1]). Two natural extensions to higher dimensionsare suggested here: | w | Z R n ×···× R n δ h τ + X ′ | x k | − | x n | i δ (cid:16) w − X x k (cid:17) Y | x k | − ( n − dx · · · dx n (1) | w | n − Z R n × R n × R n δ (cid:2) τ + | z | + | x | − | y | (cid:3) δ ( w − x − y − z ) (cid:2) | z | | x | | y | (cid:3) − ( n − dx dy dz (2)with the objective being to determine uniform bounds in terms of the variables τ > w ∈ R n with n > τ = 1:Λ n ( w ) = | w | Z R n ×···× R n δ h X ′ | x k | − | x n | i δ (cid:16) w − X x k (cid:17) Y | x k | − ( n − dx · · · dx n ∆ n ( w ) = | w | n − Z R n × R n × R n δ (cid:2) | z | + | x | − | y | (cid:3) δ ( w − x − y − z ) (cid:2) | x | | y | | z | (cid:3) − ( n − dx dy dz One observes that the first expression is an extension of the classical convolution form( g ∗ f ∗ · · · ∗ f n )( w ) , g ∈ L ( R n ) , f k ∈ L n/ ( n − ( R n )which is uniformly continuous and in the class C ( R n ) using the Riemann-Lebesgue lemma.Here the convolution for Lebesgue classes is replaced by Riesz potentials, but the multivariableintegration is constrained to be on a hyperbolic surface invariant under action by the indefiniteorthogonal group. Theorem 1. Λ n ( w ) is bounded for n ≥ ; ∆ n ( w ) is bounded for n ≥ . The argument for the proof of Theorem 1 will be developed in several steps and will bereduced to the second statement when the dimension is at least four. Note that Λ = ∆ , andthis is the case determined by Klainerman and Machedon. Proof.
Step 1: for n = 2, Λ ( w ) is unbounded. This case is instructive and will identify themethod used later in the proof of the second part.Λ ( w ) = | w | Z R δ (cid:0) | w − y | − | y | (cid:1) | w − y | | y | dy = | w | Z ∞ Z π/ − π/ δ (cid:2) | w | − rw cos θ (cid:3) √ r − dr dθ (since cos θ must be positive and | y | > | w | Z √ − u p (1 + | w | ) − | w | u du = | w | | w | Z √ u √ − u √ − βu du , β = 4 | w | (1 + | w | ) ≤ π | w | | w | F (cid:18) ,
12 ; 1; β (cid:19) = 2 | w | | w | K ( p β )where F denotes the hypergeometric function and K the complete elliptic integral. Λ ( w ) = ∞ for any w on the unit sphere | w | = 1. Observe that for β ≃ | w | ≃ ( w ) ≃ − ln (cid:16)p − β/ (cid:17) A similar calculation will now give:
Lemma.
For n = 2 and < α < ,α ( w ) = | w | α Z R δ (cid:0) | w − y | − | y | (cid:1) | w − y | − α | y | − α dy is uniformly bounded in w . Observe that by dilation symmetry this result is equivalent to uniform boundedness with τ > w ∈ R for | w | α Z R δ (cid:0) τ + | w − y | − | y | (cid:1) | w − y | − α | y | − α dy . ONVOLUTION ESTIMATES AND THE GROSS-PITAEVSKII HIERARCHY 3
Step 2: let n ≥
4; then using the second delta function for the variable x n − Λ n ( w ) = | w | Z R n ×···× R n δ (cid:20) n − X | x k | + (cid:12)(cid:12)(cid:12) w − n − X x k − x n − − x n (cid:12)(cid:12)(cid:12) + | x n − | − | x n | (cid:21) × n − Y k =1 | x k | − ( n − (cid:12)(cid:12)(cid:12) w − n − X x k − x n − − x n (cid:12)(cid:12)(cid:12) − ( n − (cid:16) | x n − | | x n | (cid:17) − ( n − dx · · · dx n − dx n − dx n = | w | Z R n × · · · × R n | {z } ( n −
3) copies n − Y | x k | − ( n − (cid:12)(cid:12)(cid:12) w − n − X x k (cid:12)(cid:12)(cid:12) − ( n − × "(cid:12)(cid:12)(cid:12) w − n − X x k (cid:12)(cid:12)(cid:12) n − Z R n × R n δ h X | x k | + (cid:12)(cid:12)(cid:12) w − X x k − x − y (cid:12)(cid:12)(cid:12) + | x | − | y | i × h(cid:12)(cid:12)(cid:12) w − X x k − x − y (cid:12)(cid:12)(cid:12) | x | | y | i − ( n − dx dy (cid:21) dx . . . dx n − ≤ c n | w | Z R n ×···× R n n − Y | x k | − ( n − (cid:12)(cid:12)(cid:12) w − n − X x k (cid:12)(cid:12)(cid:12) − ( n − dx . . . dx n − where c n = sup τ,v | v | n − Z R n × R n δ h τ + | v − x − y | + | x | − | y | ih | v − x − y | | x | | y | i − ( n − dx dy = sup w ∆ n ( w )where in the earlier expression, τ = 1 + P n − | x k | and v = w − P n − x k . ThenΛ n ( w ) ≤ c n | w | Z R n ×···× R n Y | x k | − ( n − (cid:12)(cid:12)(cid:12) w − X x k (cid:12)(cid:12)(cid:12) − ( n − dx . . . dx n − Using the following notation for the Fourier transform and its action on Riesz potentials( F f )( x ) = Z R n e πixy f ( y ) dy F h | x | − λ i ( ξ ) = π − n/ λ Γ( n − λ )Γ( λ ) | ξ | − ( n − λ ) | w | Z R n ×···× R n n − Y | x k | − ( n − (cid:12)(cid:12)(cid:12) w − n − X x k (cid:12)(cid:12)(cid:12) − ( n − dx · · · dx n − = π [( n − − / (cid:20) Γ (cid:16) n − (cid:17)(cid:21) − ( n − h Γ (cid:16) n − (cid:17)i − Hence Λ n ( w ) is bounded for n ≥ n ( w ) is bounded for n ≥ n −
1) is the unique uniform inverse power whereone can preserve dilation invariance and obtain a reduction of this type that connects bounds forintegrals of the form Λ n , ∆ n . Perhaps this circumstance reflects a larger underlying symmetryin addition to the correspondence with the property that the convolution of n functions in L n/ ( n − ( R n ) will be uniformly continuous. WILLIAM BECKNER
Step 3: consider n = 2∆ ( w ) = | w | Z R × R δ h | w − x | + | x − y | − | y | ih | w − x | | x − y | | y | i − dx dy = | w | Z R dx Z R dy δ h | w − x | + | x | − | x | | y | cos θ i | w − x | − | y | − h | y | − − | w − x | i − / = 2 | w | Z R dx | w − x | − Z √ − u h (1 + | w − x | + | x | ) − | x | u (1 + | w − x | ) i − / du = | w | Z R | w − x | − (cid:20)h | w − x | + | x | i − Z √ u √ − u p − β ( x ) u du (cid:21) dx where β ( x ) = 4 | x | (1 + | w − x | )(1 + | w − x | + | x | ) ≤ . Since for 0 < u < √ − u p − β ( x ) ≤ − uβ ( x )∆ ( w ) ≤ (cid:20)Z u − / (1 − u ) − / du (cid:21) | w | Z R | w − x | − h | w − x | + | x | i − / (cid:12)(cid:12)(cid:12) | w − x | −| x | (cid:12)(cid:12)(cid:12) − / dx Set w = | w | ξ , dilate by | w | and choose ξ as the x direction∆ ( w ) ≤ c Z R (cid:0) | x − | + | x | (cid:1) − / h h ( x − / + | x | ii − / (cid:12)(cid:12)(cid:12) x − (cid:16)
1+ 1 | w | (cid:17)(cid:12)(cid:12)(cid:12) − / dx dx Rearrange in the variable x using Z R f ( x, y ) g ( x, y ) h ( x, y ) dx dy ≤ Z R f ( x, y ) g ( x, y ) h ( x, y ) dx dy where f ( x, y ) is the equimeasurable symmetric decreasing rearrangement of | f ( x, y ) | in thevariable x ∈ R . Then ∆ ( w ) ≤ c Z R | x | − (1 + 4 | x | ) − / | x | − / dx which is a convergent integral as one sees by using polar coordinates. Hence ∆ ( w ) is uniformlybounded. The option to directly use rearrangement depends on the choice of the inverse power,e.g., the value ( n − Step 4: by using simple radial coordinate estimates, one can obtain for n > c denotes ageneric constant) ∆ n ( w ) ≤ c ∆ ( ¯ w )where ¯ w ∈ R with | w | = | ¯ w | . ONVOLUTION ESTIMATES AND THE GROSS-PITAEVSKII HIERARCHY 5
Observe that the integrands for both expressions treated here, Λ n ( w ) and ∆ n ( w ), are func-tions only of lengths and polar angles so that facilitates the simplicity of the argument.∆ n ( w ) = | w | n − Z R n × R n δ h | w − x | + | x | − x · y i | w − x | − n − | y | − n − × h | y | − − | w − x | i − ( n − dx dy = σ ( S n − | w | n − Z R n | w − x | − ( n − (2 | x | ) n − (cid:0) | w − x | + | x | (cid:1) − ( n − × Z u − / (1 − u ) ( n − / (1 − β ( x ) u ) − ( n − / du dx with β ( x ) as before. Since 0 ≤ β ( x ) ≤ ≤ u ≤ − u ) ( n − / (1 − β ( x ) u ) − ( n − / ≤ (1 − u ) − / (1 − β ( x ) u ) − / In the integral over R n . first make the change of variables z = w − x , and then dilate z by | w | and integrate out the non-polar angle variables.∆ n ( w ) ≤ n − (cid:2) σ ( S n − ) (cid:3) | w | Z ∞ Z π (cid:20) | w | | z − ξ | | z | w [ | z − ξ | + | z | ] (cid:21) n − × (cid:2) | w | (cid:0) | z − ξ | + | z | (cid:1)(cid:3) − d | z | (sin θ ) n − dθ Z u − / (1 − u ) − / (1 − β ( | w | ( z − ξ )) u ) − / du Using (cid:20) | w | | z − ξ | | z | | w | ( | z − ξ | + | z | ) (cid:21) ≤ , ∆ n ( w ) ≤ (cid:2) σ ( S n − ) (cid:3) | w | Z ∞ Z π − π | z | (cid:2) | w | ( | z − ξ | + | z | ) (cid:3) − Z u − / (1 − u ) − / (1 − β ( | w | ( z − ξ )) u ) − / du | z | d | z | dθ Now since we can take ξ as defining the polar angle for the coordinate system, and | z − ξ | onlydepends on this angle and the length | z | , z and ξ can be repositioned as vectors in R so that∆ n ( w ) ≤ (cid:2) σ ( S n − ) (cid:3) | w | Z R | z | (cid:2) | w | (cid:0) | z − ξ | + | z | (cid:1)(cid:3) − Z u − / (1 − u ) − / (cid:0) − β (cid:0) | w | ( z − ξ ) (cid:1) u (cid:1) − / du dz Reversing the previous coordinate changes of dilation and translation∆ n ( w ) ≤ (cid:2) σ ( S n − ) (cid:3) | w | Z R | w − x | − (cid:2) | x | + | w − x | (cid:3) − / Z u − / (1 − u ) − / (1 − β ( x ) u ) − / du dx which then gives the required controlsup w ∈ R n ∆ n ( w ) ≤ (cid:2) σ ( S n − ) (cid:3) sup w ∈ R ∆ ( w ) WILLIAM BECKNER and hence the uniform bound for ∆ ( w ) gives a uniform bound for ∆ n ( w ), n >
2. Thiscompletes the proof of Theorem 1.As noted above, the inverse power | x | − ( n − has a special role for the convolution estimatesdiscussed here; still the two-dimensional result from the Lemma is suggestive that useful boundsmight be obtained for inverse powers close to α = n −
1. Consider for τ > w ∈ R n | w | ρ Z R n ×···× R n δ h τ + X ′ | x k | − | x n | i δ h w − X x k i Π | x k | − α dx . . . dx n (3)For dilation invariance, p = 2 + n ( α − n + 1) with the further requirement of positivity for thepossibility of boundedness; that means ( n − ≥ α > ( n − − /n so that asymptotically a ≃ n − | w | σ Z R n × R n δ (cid:2) τ + | x | − | y | (cid:3) δ ( w − x − y ) | x | − α | y | − α dx dy (4)Here σ = 2 α + 2 − n for dilation invariance. For both forms, it suffices to show uniform boundsfor τ = 1, and in two dimensions they are the same and already proved in the argument for theLemma. Theorem 2.
For n ≥ , σ = 2 α + 2 − n and ( n − / < α < ( n − n,α ( w ) = | w | σ Z R n × R n δ (cid:2) | x | − | y | (cid:3) δ ( w − x − y ) | x | − α | y | − α dx dy is uniformly bounded for w ∈ R n .Proof. Let n ≥ n,α ( w ) = | w | σ Z R n δ (cid:0) | w − y | − | y | (cid:1) | w − y | − α | y | − α dy = 2 π ( n − / Γ(( n − / | w | σ Z ∞ Z δ (cid:0) | w | − | w | ru (cid:1) ( r − − α/ r n − α − (1 − u ) ( n − / dr du = π ( n − / α − n Γ(( n − / (cid:20) | w | | w | (cid:21) α − n +1 Z (1 − u ) ( n − / u α − ( n − / − (cid:0) − β (2) u (cid:1) − α/w du where β ( w ) = | w | (1+ | w | ) ≤ − β ( w ) u ) − α/ ≤ (1 − u ) − α/ . ThenΘ n,α ( w ) ≤ π ( n − / α − n Γ(( n − / (cid:20) | w | | w | (cid:21) α − n +1 Z (1 − u ) ( n − − α ) / − u α − ( n − / − du = π ( n − / α − n Γ(( n − − α ) /
2) Γ((2 α − n + 1) / n − /
2) Γ( α/ (cid:20) | w | | w | (cid:21) α − n +1 Hence Θ n,α ( w ) is bounded for ( n − / < α < n −
1. Note that Λ ,α = Θ ,α . Theorem 3.
For n ≥ , ρ = 2 + n ( α − n + 1) and n − > α > n − − /n Λ n,α ( w ) = | w | ρ Z R n ×···× R n δ h X ′ | x k | − | x n | i δ (cid:16) w − X x k (cid:17) Π | x k | − α dx . . . dx n is uniformly bounded for w ∈ R n . ONVOLUTION ESTIMATES AND THE GROSS-PITAEVSKII HIERARCHY 7
Proof.
The argument here rests on the boundedness of Θ n,α following the method of Step 2 inthe proof of Theorem 1. For n ≥ x n − andwrite Λ n,α ( w ) = | w | ρ Z R n ×···× R n | {z } ( n −
2) copies n − Π | x k | − α (cid:12)(cid:12) w − n − X x k (cid:12)(cid:12) − σ × (cid:20)(cid:12)(cid:12) w − n − X x k (cid:12)(cid:12) σ Z R n δ h X | x k | + (cid:12)(cid:12) w − X x k − y (cid:12)(cid:12) − | y | i × h(cid:12)(cid:12) w − X x k − y (cid:12)(cid:12) | y | i − α dy (cid:21) dx . . . dx n − ≤ c n | w | ρ Z R n ×···× R n n − Π | x k | − α (cid:12)(cid:12) w − n − X x k (cid:12)(cid:12) − σ dx . . . dx n − where c n = sup τ,v | v | σ Z R n δ (cid:2) τ + | v − y | − | y | (cid:3) (cid:2) | v − g | | y | (cid:3) − α dy = sup w Θ n,α ( w )and in the earlier expression, τ = 1 + P n − | x k | and v = w − P n − x k . ThenΛ n,α ( w ) ≤ c n | w | ρ Z R n ×···× R n Π | x k | − α (cid:12)(cid:12) w − X x k (cid:12)(cid:12) − σ dx . . . dx n − with the integral | w | ρ Z R n ×···× R n Π | x k | − α (cid:12)(cid:12) w − X x k (cid:12)(cid:12) − σ dx . . . dx n − being constant in w so that Λ n,α ( w ) is bounded for n ≥ n,α ( w ) is bounded for n ≥ Z S | w − y | λ | y | µ dv which then the convolution algebra for Riesz potentials allows an extended multilinear result.For completeness, an outline is given for non-uniform Riesz potentials. Theorem 4.
For n ≥ , σ = α + λ + 2 − n , α + λ > n − and < α < n − n,α,λ ( w ) = | w | σ Z R n × R n δ (cid:2) | x | − | y | (cid:3) δ ( w − x − y ) | x | − α | y | − λ dx dy (5) is uniformly bounded for w ∈ R n .Proof. Let n ≥
2; observe that since | y | ≥
1, there is no upper bound for λ in this computation.Θ n,α,λ ( w ) = | w | σ Z n R δ (cid:2) | w − y | − | y | (cid:3) | w − y | − α | y | − λ dy = 2 π ( n − / Γ(( n − / | w | σ Z ∞ Z δ (cid:0) | w | − | w | ru (cid:1) ( r − − α/ r n − λ − (1 + u ) ( n − / dr du = 2 α + λ − n π ( n − / Γ(( n − / (cid:20) | w | | w | (cid:21) α + λ − n +1 Z (1 − u ) ( n − / u ( α + λ − n +1) / − (1 − β ( w ) u ) − α/ du WILLIAM BECKNER where β ( w ) = | w | (1+ | w | ) ≤ − β ( w ) u ) − α/ ≤ (1 − u ) − α/ . ThenΘ n,α,λ ( w ) = 2 α + λ − n π ( n − / Γ(( n − / (cid:20) | w | | w | (cid:21) α + λ − n +1 Z u ( α + λ − n +1) / − (1 − u ) ( n − − α ) / − du = 2 α + λ − n π ( n − / Γ(( α + λ − n + 1) /
2) Γ(( n − − α ) / n − /
2) Γ( λ/ (cid:20) | w | | w | (cid:21) α + λ − n +1 Hence Θ n,α,λ ( w ) is bounded for λ >
0, 0 < α < n − α + λ > n − (cid:3) Theorem 5.
For n ≥ , consider real-valued exponents < α k < n , k = 1 , . . . , n − and λ > with α = P α k and ρ = 2 + α + λ − n ( n − so that < ρ < n . Further assume one exponent α i together with λ satisfies: < α i < n − and n − < α i + λ < n − ; relabel this a i as α n − . Then Λ n,α,λ = | w | ρ Z R n × R n δ (cid:20) X ′ | x k | − | x n | (cid:21) δ (cid:16) w − X x k (cid:17) Π | x k | − α k | x n | − λ dx . . . dx n is uniformly bounded for w ∈ R n .Proof. As in Theorem 3, the argument here rests on the uniform boundedness of | w | σ Z R n × R n δ (cid:2) τ + | x | − | y | (cid:3) δ ( w − x − y ) | x | − α n − | y | − λ dx dy for τ > w ∈ R n which is determined by Theorem 4. For n ≥ x n − and write with σ = α n − + λ + 2 − n Λ n,α,λ = | w | ρ Z R n ×···× R n | {z } ( n −
2) copies n − Π | x k | − α k (cid:12)(cid:12)(cid:12) w − n − X x k (cid:12)(cid:12)(cid:12) − σ × (cid:20)(cid:12)(cid:12)(cid:12) w − n − X x k (cid:12)(cid:12)(cid:12) σ Z R n δ (cid:20) X | x k | + (cid:12)(cid:12)(cid:12) w − X x k − y (cid:12)(cid:12)(cid:12) − | y | (cid:21) × (cid:12)(cid:12)(cid:12) w − X x k − y (cid:12)(cid:12)(cid:12) − α n − | y | − λ (cid:21) dx . . . dx n − ≤ c n,α,λ | w | ρ Z R n ×···× R n n − Π | x k | − α k (cid:12)(cid:12)(cid:12) w − n − X x k (cid:12)(cid:12)(cid:12) − σ dx . . . dx n − where c n,α,λ = sup τ,v | v | σ Z R n δ (cid:2) τ + | v − y | − | y | (cid:3) | v − y | − α n − | y | − λ and in the earlier expression τ = 1 + P n − | x k | and v = w − P n − x k . ThenΛ n,α,λ ( w ) ≤ c n,α,λ | w | ρ Z R n ×···× R n Π | x k | − α k (cid:12)(cid:12)(cid:12) w − X x k (cid:12)(cid:12)(cid:12) − σ dx . . . dx n − with the integral | w | ρ Z R n ×···× R n Π | x k | − α k (cid:12)(cid:12)(cid:12) w − X x k (cid:12)(cid:12)(cid:12) − σ dx . . . dx n − being constant in w so that Λ n,α,λ ( w ) is bounded in w for n ≥ n,α,λ ( w ) is bounded for n ≥ α n − and λ . (cid:3) ONVOLUTION ESTIMATES AND THE GROSS-PITAEVSKII HIERARCHY 9
Implicit in the formulation of the problems treated here is the continuing development ofnew forms that characterize control by smoothness for size. As an example and a consequenceof the principal estimate obtained here, bounds for new Stein-Weiss integrals with a kerneldetermined by restriction to a smooth submanifold can be shown.
Theorem 6.
Define K ( w, v ) = Z R n ×···× R n Π | x k | − ( n − h(cid:12)(cid:12) w − X x k (cid:12)(cid:12) (cid:12)(cid:12) v − X x k (cid:12)(cid:12)i − ( n − × δ h X ′ | x k | − | x n | i dx . . . dx n , n ≥ K n,α ( w, v ) = Z R n ×···× R n Π | x k | − α h(cid:12)(cid:12) w − X x k (cid:12)(cid:12) (cid:12)(cid:12) v − X x k (cid:12)(cid:12)i − λ × δ h X ′ | x k | − | x n | i dx . . . dx n , n ≥ λ = n ( n − α + 1) / − , n − > α > n − − /nH n,α ( w, v ) = Z R n × R n | x | − α | y | − α (cid:2) | w − x − y | | v − x − y | (cid:3) − (3 n/ − − α ) × δ (cid:0) | x | − | y | (cid:1) dx dy , n ≥ , ( n − / < α < n − J ( w, v ) = Z R n × R n × R n (cid:2) | x | | y | | z | (cid:3) − ( n − (cid:12)(cid:12) w − x − y − z | | v − x − y − z | (cid:3) − ( n +1) / × δ (cid:2) | x | + | z | − | y | (cid:3) dx dy dz , n ≥ then for non-negative f ∈ L ( R n ) and T representing the above kernels Z R n × R n f ( w ) T ( w, v ) f ( v ) dw dv ≤ c Z R n | f | dx (6) Proof.
Apply Pitt’s inequality and the uniform bounds obtained above for Λ n , Λ n,α , Θ n,α and∆ n . Here c is a generic constant. (cid:3) Practical application for such convolution-type estimates has proved to be efficient by re-placing the Riesz potentials with the Fourier transform of Bessel potentials ([2], [3]); advantageis achieved by removing local singularities while gaining integrability on the potential side andimproving the range of application as “smoothing operators”; still the lack of homogeneity lim-its determination of precise dependence on parameters in computing best size estimates. But aswith exact model calculations, the role of Riesz potentials can result in “very elegant and usefulformulae” that underline intrinsic geometric structure, capture essential features of symmetryand uncertainty, and provide insight to precise lower-order effects.
References [1] W. Beckner,
Multilinear embedding estimates for the fractional Laplacian , Math. Res. Lett. (in press).[2] T. Chen and N. Pavlovic,
On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies ,Discr. Contin. Dyn. Syst. (2010), 715–739.[3] K. Kirkpatrick, B. Schlein and G. Staffilani, Derivation of the two-dimensional nonlinear Schr¨odinger equa-tion from many body quantum mechanics , Amer. J. Math. (2011), 91–130.[4] S. Klainerman and M. Machedon,
On the uniqueness of solutions to the Gross-Pitaevskii hierarchy , Comm.Math. Phys. (2008), 169–185. [5] E.M. Stein,
Singular integrals and differentiability properties of functions , Princeton University Press, 1970.
Department of Mathematics, The University of Texas at Austin, 1 University Station C1200,Austin TX 78712-0257 USA
E-mail address ::