Strichartz inequality for orthonormal functions associated with Hermite-Schrödinger operator
aa r X i v : . [ m a t h . F A ] F e b STRICHARTZ INEQUALITY FOR ORTHONORMAL FUNCTIONSASSOCIATED WITH HERMITE-SCHR ¨ODINGER OPERATOR
SHYAM SWARUP MONDAL AND JITENDRIYA SWAIN
Abstract.
The Strichartz inequality for the system of orthonormal functions forthe Hermite operator H = − ∆ + | x | on R n has been proved in [2], using theclassical Strichartz estimates for the free Schr¨odinger propagator for orthonormalsystems [4, 5] and the link between the Schr¨odinger kernel and the Mehler kernelassociated with the Hermite semigroup [19].In this article, we give an alternative proof of the above result in connectionwith the restriction theorem with respect to the Hermite transform with an opti-mal behavior of the constant in the limit of a large number of functions. As anapplication, we show the well-posedness results in Schatten spaces for the nonlinearHermite-Hartree equation. Introduction
Consider the free Schr¨odinger equation i∂ t u ( x, t ) = − ∆ u ( x, t ) x ∈ R n , t ∈ R (1.1) u ( x,
0) = f ( x ) . It is well known that e it ∆ f is the unique solution to the initial value problem (1.1).The following remarkable estimate for the solution to the initial value problem (1.1)is first obtained by Strichartz [13] in connection with Fourier restriction theory: Theorem 1.1. [13]
Let f ∈ L ( R n ) . If p, q ≥ satisfying ( p, q, n ) = (1 , ∞ , and p + nq = n, then e it ∆ f ∈ L pt L qx ( R × R n ) and satisfies the inequality Z R (cid:18)Z R n (cid:12)(cid:12)(cid:0) e it ∆ f (cid:1) ( x ) (cid:12)(cid:12) q dx (cid:19) pq dt ≤ C (cid:18)Z R n | f ( x ) | dx (cid:19) p . The above inequality have been substantially generalized for a system of orthonor-mal functions by Frank-Lewin-Lieb-Seiringer [4] and Frank-Sabin [5].
Date : February 16, 2021.2010
Mathematics Subject Classification.
Primary 35Q41, 47B10; Secondary 35P10, 35B65.
Key words and phrases.
Strichartz inequality, Schr¨odinger equations, Hermite operator, Or-thonormal functions, Hartree equation. heorem 1.2. [4, 5] Assume that p, q, n ≥ such that ≤ q < n + 1 n − and p + nq = n. For any (possibly infinite) system u j of orthonormal functions in L ( R n ) and anycoefficients ( n j ) ⊂ C , we have Z R Z R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j n j (cid:12)(cid:12)(cid:0) e it ∆ u j (cid:1) ( x ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx ! pq dt ≤ C pn,q X j | n j | qq +1 ! p ( q +1)2 q , where C n,q is a universal constant which only depends on n and q . Further extensions of Theorem 1.1 has been made for the Schr¨odinger’s equationof the form i ∂u∂t ( x, t ) + ∆ u ( x, t ) − V ( x ) u ( x, t ) = 0, for a suitable potential V byseveral authors [6, 7, 8]. We are interested in the case when V ( x ) = | x | . When V ( x ) = | x | , the initial value problem (1.1) turns out to an initial value problem forthe Schr¨odinger equation associated with the Hermite operator H = − ∆ + | x | : i∂ t u ( x, t ) = Hu ( x, t ) x ∈ R n , t ∈ R (1.2) u ( x,
0) = f ( x ) . If f ∈ L ( R n ), the solution of the initial value problem (1.2) is given by u ( x, t ) = e − itH f ( x ) . The Strichartz inequality in this case has been proved by Koch-Tataru[10] or Nandakumaran-Ratnakumar [14] resulting the following.
Theorem 1.3. [10, 14]
Let f ∈ L ( R n ) . If p, q ≥ satisfying ( p, q, n ) = (1 , ∞ , and p + nq = n, then k e − itH f k L pt L qx ( T × R n ) ≤ C k f k . This result has been extended for orthonormal family of functions by Bez et. al. in[2](Theorem 6.4) using the Strichartz inequality for system of orthonormal functionsin Theorem 1.2 and the link between the Schr¨odinger kernel and the Mehler kernelfor the Hermite semigroup [19].
Theorem 1.4. [2] (Strichartz inequality for orthonormal functions for Hermite op-erator) Let p, q, n ≥ such that ≤ q < n + 1 n − and p + nq = n. For any (possibly infinite) system ( u j ) of orthonormal functions in L ( R n ) and anycoefficients ( n j ) ⊂ C , we have Z π − π Z R n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j n j (cid:12)(cid:12)(cid:0) e − itH u j (cid:1) ( x ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q dx ! pq dt ≤ C pn,q X j | n j | qq +1 ! p ( q +1)2 q , (1.3) where C n,q is a universal constant which only depends on n and q . n this paper we give an alternative proof of Theorem 1.4 in connection with therestriction theorem associated with Hermite transform with an optimal behavior ofthe constant in the limit of a large number of functions. The idea of proving Theorem1.4 is motivated by the famous works of Strichartz [13] and Frank-Sabin [5], wherethe Fourier restriction theorem for some quadratic surfaces is linked to space-timedecay estimates for certain evolution equations.Let f ∈ L ( R n ). Define the Hermite transform of f byˆ f ( µ ) = Z R n f ( x ) φ µ ( x ) dx, µ ∈ N n where φ µ ’s are the n -dimensional Hermite functions (defined in section 2). If f ∈ L ( R n ) then { ˆ f ( µ ) } ∈ ℓ ( N n ) and satisfies the Plancherel formula k f k = X µ ∈ N n | ˆ f ( µ ) | , where N denotes the set of all non-negative integers. The inverse Hermite transformis given by f ( x ) = X µ ∈ N n ˆ f ( µ ) φ µ ( x ) . Given a discrete surface S in N n , we define the restriction operator ( R S f )( x ) := { ˆ f ( µ ) } µ ∈ S and the operator dual to R S (called the extension operator) as E S ( { ˆ f ( µ ) } ) := P µ ∈ S ˆ f ( µ ) φ µ ( x ) and ask the following question. For which exponents 1 ≤ p ≤ , the sequence of Hermite transforms of a function f ∈ L p ( R n ) belongs to ℓ ( S )?This question can be reframed to the boundedness of the operator E S from ℓ ( S ) to L p ′ ( R n ), where p ′ is the conjugate exponent of p by a duality argument. Since E S isbounded from ℓ ( S ) to L p ′ ( R n ) if and only if T S := E S ( E S ) ∗ is bounded from L p ( R n )to L p ′ ( R n ).Similar techniques have already been adopted to prove the restriction theorems forthe Fourier transform for certain quadratic surfaces of R n by Stein [17] and Strichartz[13] by introducing an analytic family of operators T z on a strip a ≤ Re z ≤ b . Theyprove the L − L boundedness of T z on the line Re z = b and L − L ∞ boundednesson the line Re z = a in their set up. Further using Stein’s interpolation theorem[18], they deduce the L p − L p ′ boundedness of T S = T c for some exponent p and some c ∈ ( a, b ). Note that H¨older’s inequality implies that L p − L p ′ boundedness of theoperator of T S is equivalent to the L − L boundedness of the operator W T S W for any W , W ∈ L p − p ( R n ). However, Frank and Sabin [5] considered the similarproblem for the system of orthonormal functions and show that the operator W T S W belongs to a Schatten class, where W , W ∈ L p − p ( R n ), which is more general result L p − L p ′ boundedness of T S . In this paper we consider discrete surface S ⊂ N n +10 with respect to countingmeasure and introduce an analytic family of operators ( T z ) defined on the strip ≤ Re z ≤ b in the complex plane such that T S = T c for some c ∈ ( a, b ). We showthat the operator W T S W belongs to a Schatten class for W , W ∈ L p − p ( R n × T ).Then applying the well known duality argument (see Lemma 2.1), we obtain theStrichartz inequality for orthonormal functions for Hermite operator (1.3).The schema of the paper apart from introduction is as follows: In Section 2,we discus the spectral theory of the Hermite operator and the kernel estimates forthe Hermite semigroup and obtain the duality principle in terms of Schatten spacebounds of the operator W e − itH ( e − itH ) ∗ W . In section 3, we prove Strichartz inequal-ity for 1 ≤ q < n +1 n − for the system of orthonormal functions associated with theHermite operator. Also we prove the optimality of Schatten exponent (2.8) andshow that (1.3) fails at the end point q = n +1 n − . Finally we consider the non-linearHermite-Hartree equation in Schatten spaces and prove the global well-posednessresults in section 4 as an application to inhomogeneous Strichatz inequality.2. Inequality for orthonormal functions and its dual
In this section we discus the spectral theory of the Hermite operator and esti-mate the kernel for the Hermite semigroup. We start with the definition of Hermitefunctions.2.1.
Hermite Operator and the Spectral theory:
Let N be the set of all non-negative integers. Let H k denote the Hermite polynomial on R , defined by H k ( x ) = ( − k d k dx k ( e − x ) e x , k ∈ N and h k denote the normalized Hermite functions on R defined by h k ( x ) = (2 k √ πk !) − H k ( x ) e − x , k ∈ N . The higher dimensional Hermite functions denoted by Φ α are obtained by takingtensor product of one dimensional Hermite functions. Thus for any multi-index α ∈ N n and x ∈ R n , we define Φ α ( x ) = Q nj =1 h α j ( x j ) . The family { Φ α } formsan orthonormal basis for L ( R n ). They are eigenfunctions of the Hermite operator H = − ∆ + | x | corresponding to eigenvalues (2 | α | + n ), where | α | = P nj =1 α j . Given f ∈ L ( R n ) we have the Hermite expansion f = X α ∈ N n ( f, Φ α ) Φ α = ∞ X k =0 X | α | = k ( f, Φ α ) Φ α = ∞ X k =0 P k f, where P k denotes the orthogonal projection of L ( R n ) onto the eigenspace spannedby { Φ α : | α | = k } .The solution to the initial value problem (1.2) is given by u ( x, t ) = e − itH f ( x ) = ∞ X k =0 e − it (2 k + n ) P k f ( x ) . (2.1) he solution (2.1) can be expressed as an integral operator with kernel K it ( x, y ) = X α ∈ N n e − i (2 | α | + n ) t Φ α ( x )Φ α ( y ) . Since K it ( x, y ) = K i ( t +2 π ) ( x, y ), the solution is periodic in t with period 2 π. For z = r + it, r > , t ∈ R , we consider the kernel K z ( x, y ) = ∞ X k =0 e − z (2 k + n ) X | α | = k Φ α ( x )Φ α ( y ) . An application of Meheler’s formula, we get K z ( x, y ) = π − n e n ( r + it ) (cid:0) − ω (cid:1) − n e −
12 1+ ω − ω ( | x | + | y | ) + ω − ω x · y = π − n ( − sinh 2 z ) n e ( coth 2 z ( | x | + | y | ) − x · y sinh 2 z ) . Letting r → K it ( x, y ) = e − iπn π − n ( − i sinh 2 t ) n e i ( − coth 2 t ( | x | + | y | )+ x · y sinh 2 t ) . (2.2)We refer to [14] for a detailed study on the kernel associated with the operator e − itH .2.2. Schatten class and the duality principle.
Let H be a complex and sepa-rable Hilbert space in which the inner product is denoted by h , i H . Let T : H → H be a compact operator and let T ∗ denotes the adjoint of T . For 1 ≤ r < ∞ , theSchatten space G r ( H ) is defined as the space of all compact operators T on H suchthat ∞ X n =1 ( s n ( T )) r < ∞ , where s n ( T ) denotes the singular values of T, i.e. the eigenvalues of | T | = √ T ∗ T counted according to multiplicity. For T ∈ G r ( H ), the Schatten r -norm is defined by k T k G r = ∞ X n =1 ( s n ( T )) r ! r . An operator belongs to the class G ( H ) is known as Trace class operator. Also,an operator belongs to G ( H ) is known as Hilbert-Schmidt operator. With thesenotations, we state the following duality principle which can be obtained with atrivial modification to the duality principle due to Frank-Sabin [5].
Lemma 2.1. (Duality principle) Let p, q ≥ and α ≥ . Let Af ( x, t ) = e − itH f ( x ) .Then the following statements are equivalent. There is a constant
C > such that (cid:13)(cid:13) W AA ∗ W (cid:13)(cid:13) G α ( L ( T × R n )) C k W k L q − qt L p − px ( T × R n ) (2.3) for all W ∈ L q − q t L p − p x ( T × R n ) , where the function W is interpreted as anoperator which acts by multiplication. (2) For any orthonormal system ( f j ) j ∈ J in H and any sequence ( n j ) j ∈ J ⊂ C , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j ∈ J n j | Af j | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L q ′ t L p ′ x ( T × R n ) ≤ C ′ X j ∈ J | n j | α ′ ! /α ′ , (2.4) where C ′ is a constant. Refinement of Theorem 1.4.
Since the solution to the initial value problemin (1.2) is periodic in the time variable with period 2 π , the solution space is consideredto be L pt L qx ( T × R n ). Let S be the discrete surface S = { ( µ, ν ) ∈ N n × N : ν =2 | µ | + n } with respect to the counting measure. Then for all f ∈ L ( S ) and for all( x, t ) ∈ R n × [ − π, π ], the extension operator can be written as E S f ( x, t ) = X µ,ν ∈ S ˆ f ( µ, ν ) φ µ ( x ) e − itν , (2.5)where ˆ f ( µ, ν ) = Z R n Z T f ( x, t ) φ µ ( x ) e iνt dxdt. Choosingˆ f ( µ, ν ) = (cid:26) ˆ u ( µ ) if ν = 2 | µ | + n, u : R n → C in (2.5), we get E S f ( x, t ) = X µ,ν ∈ S ˆ f ( µ, ν ) φ µ ( x ) e − itν = Z R n X µ φ µ ( x ) φ µ ( y ) e − it (2 | µ | + n ) ! u ( y ) dy = e − itH u ( x ) . From this observation together with Lemma 2.1, we will rather prove that is equiv-alent to Theorem 1.4.The inequality (1.3) can also be written in terms of the operator γ := X j n j | u j i h u j | (2.6) n L ( R n ) , where the Dirac’s notation | u ih v | stands for the rank-one operator f v, f i u . For such γ , let γ ( t ) := e − itH γ e itH = X j n j (cid:12)(cid:12) e − itH u j (cid:11) (cid:10) e − itH u j (cid:12)(cid:12) . Then the density of the operator γ ( t ) is given by ρ γ ( t ) := X j n j (cid:12)(cid:12) e − itH u j (cid:12)(cid:12) . (2.7)With these notations (1.3) can be rewritten as (cid:13)(cid:13) ρ γ ( t ) (cid:13)(cid:13) L pt L qx ( T × R n ) C n,q k γ k G qq +1 , (2.8)where k γ k G qq +1 = X j | n j | qq +1 ! q +12 q . By Lemma 2.1, (2.3) and (2.4) are equivalentto the following bound: for any γ ∈ G α ′ ( H ) we have (cid:13)(cid:13) ρ e itH γ e itH (cid:13)(cid:13) L q ′ t L p ′ x ( T × R n ) ≤ C k γ k G α ′ ( H ) , with C > γ . Dual Strichartz inequality.
We will not prove Theorem 1.4 directly but provean inequality that is dual to (1.3). For a trace-class operator γ and bounded function V of compact support Tr( V ( x ) γ ) = Z R n V ( x ) ρ γ ( x ) dx, (2.9)where V ( x ) is identified with the corresponding multiplication operator on L ( R n )and for a time-dependent potential V ( t, x ) ∈ L ∞ c ([ − π, π ] × R n ) , we have (cid:12)(cid:12)(cid:12)(cid:12) Tr (cid:18)Z [ − π,π ] e itH V ( t, x ) e − itH dt (cid:19) γ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z [ − π,π ] Tr (cid:0) V e − itH γe itH (cid:1) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ k V k L p ′ t L q ′ x ( T × R n ) (cid:13)(cid:13) ρ γ ( t ) (cid:13)(cid:13) L pt L qx ( T × R n ) , where p ′ and q ′ are the dual to p and q respectively. For 1 < q ≤ n , a dualityargument shows that Theorem 1.4 is equivalent to the following: Theorem 2.2. (Dual Strichartz inequality in Schatten spaces). Assume that p ′ , q ′ , n ≥ satisfy n < q ′ < ∞ and p ′ + nq ′ = 2 , we have (cid:13)(cid:13)(cid:13)(cid:13)Z [ − π,π ] e itH V ( t, x ) e − itH dt (cid:13)(cid:13)(cid:13)(cid:13) G q ′ ≤ C n,q k V k L p ′ t L q ′ x ( T × R n ) (2.10) emark 2.3. For q ′ = ∞ and p ′ = 1 we have the following bound (cid:13)(cid:13)(cid:13)(cid:13)Z [ − π,π ] e itH V e − itH dt (cid:13)(cid:13)(cid:13)(cid:13) ≤ C n,q k V k L t L ∞ x ( T × R n ) . Inhomogeneous case.
Using (2.10), we prove the inhomogeneous Strichartzinequality for the following system. Consider the equation given by i ˙ γ ( t ) = [ − H, γ ( t )] + iR ( t )(2.11) γ ( t ) = 0 , where R ( t ) is a self-adjoint operator on L ( R n ) and is bounded for almost every t .The solution of (2.11) can be written as γ ( t ) = Z tt e i ( t − s ) H R ( s ) e i ( s − t ) H ds. (2.12)We obtain the following inhomogeneous Strichartz inequality. Corollary 2.4. (Inhomogeneous Strichartz inequality) Assume that p, q, n ≥ sat-isfy ≤ q < n + 1 n − and p + nq = n and let γ ( t ) be given by (2.12). Then (cid:13)(cid:13) ρ γ ( t ) (cid:13)(cid:13) L pt L qx ( T × R n ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)Z π − π e isH | R ( s ) | e − isH ds (cid:13)(cid:13)(cid:13)(cid:13) G qq +1 for a constant C which is independent of t . Proof.
Using (2.9) and the fact that | Tr( AB ) | ≤ Tr( | A || B | ) for self-adjoint operators A and B , we have (cid:12)(cid:12)(cid:12)(cid:12)Z πt Z R n V ( t, x ) ρ γ ( t ) ( x ) dxdt (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z πt Z tt Tr (cid:0) e itH V ( t, x ) e − itH e isH R ( s ) e − isH (cid:1) dsdt (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z πt Z tt Tr (cid:0) e itH | V ( t, x ) | e − itH e isH | R ( s ) | e − isH (cid:1) dsdt = Tr (cid:18)(cid:18)Z πt e itH | V ( t, x ) | e − itH dt (cid:19) (cid:18)Z πt e isH | R ( s ) | e − isH ds (cid:19)(cid:19) . Applying H¨older’s inequality for traces and (2.10) for the term involving V ( t, x ), theproof is complete by the duality argument. (cid:3) . Proof of Theorem 1.4
We make use of the following proposition to prove (2.3). The proof of the Propo-sition is based on the similar idea used in the proof of Proposition 1 of [5] withappropriate modifications. Therefore we only state the proposition without proof.
Proposition 3.1.
Let ( T z ) be an analytic family of operators on T × R n in the senseof Stein defined on the strip − λ ≤ Re z ≤ for some λ > . Assume that we havethe following bounds k T is k L ( T × R n ) → L ( T × R n ) ≤ M e a | s | , k T − λ + is k L ( T × R n ) → L ∞ ( T × R n ) ≤ M e b | s | , (3.1) for all s ∈ R , for some a, b ≥ and for some M , M ≥ . Then, for all W , W ∈ L λ ( T × R n , C ) the operator W T − W belongs to G λ ( L ( T × R n )) and we havethe estimate k W T − W k G λ ( L T × R n ) ) ≤ M − λ M λ k W k L λ t L λ x ( T × R n ) k W k L λ t L λ x ( T × R n ) . (3.2)To define the analytic family of operators, we first consider the generalized func-tions G z ( µ, ν ) = 1Γ( z + 1) ( ν − (2 | µ | + n )) z + , where x z + = ( x z for x >
00 for x ≤ . For Schwartz class functions φ on N n +10 , we have h G z , φ i := z +1) P µ,ν ( ν − (2 | µ | + n )) z + φ ( µ, ν ) and lim z →− h G z , φ i = lim z →− z + 1) X µ,ν φ ( µ, ν )( ν − (2 | µ | + n )) z + = X ( µ,ν ) ∈ S φ ( µ, ν ) . We refer to [15] for the distributional calculus of ( ν − (2 | µ | + n )) z + . Thus G − = δ S . Define the analytic family of operators T z by T z g ( x, t ) = X µ,ν ˆ g ( µ, ν ) G z ( µ, ν ) φ µ ( x ) e − iνt . Then T z g ( x, t ) = Z R n ( K z ( x, y, · ) ∗ g ( y, · ))( t ) dy, (3.3)where K z ( x, y, t ) = X µ,ν φ µ ( x ) φ µ ( y ) G z ( µ, ν ) e − iνt . Using the definition of G z ( µ, ν ) wehave K z ( x, y, t ) = 1Γ( z + 1) X µ φ µ ( x ) φ µ ( y ) e − it (2 | µ | + n ) ∞ X k =0 k z + e − itk . (3.4) roposition 3.2. For − < λ ≤ Re z ≤ , < | t | < π we have ∞ X k =0 k z + e − itk = ie iz π Γ( z + 1) t − z − . Proof.
Let F ( ξ ) = Z ξ f ( u ) du , where f ( u ) = u z + e − itu . Let { ξ k } be a sequence of realnumbers such that ξ k +1 − ξ k = c and lim k →∞ ξ k = ∞ . Since F is twice differentiable in(0 , ∞ ) we have F ( ξ ) = F ( a ) + ( ξ − a ) F ′ ( a ) + ( ξ − a ) F ( ξ ) , where F ( ξ ) = (cid:26) F ′ ( ξ ) − F ′ ( a ) ξ − a if ξ = a,F ′′ ( a ) ξ = a for every a > . Thus F (cid:16) ξ k + c (cid:17) = F ( ξ k ) + c f ( ξ k ) + c F (cid:16) ξ k + c (cid:17) (3.5)and F (cid:16) ξ k + c (cid:17) = F ( ξ k +1 ) − c f ( ξ k +1 ) + c F (cid:16) ξ k + c (cid:17) . (3.6)Subtracting (3.6) from (3.5) and taking summation over k from 0 to N , we get, F ( ξ N +1 ) = c f ( ξ N +1 ) + c N X k =0 f ( ξ k ) , where ξ = 0 . Letting N → ∞ and using the property that lim N →∞ f ( ξ N ) = 0, we getlim N →∞ Z [0 ,ξ N +1 ] f ( u ) du = c ∞ X k =0 f ( ξ k ) . (3.7)Taking ξ k = k, k ∈ N ∪ { } in (3.7) we have Z ∞ u z + e − itu du = ∞ X k =0 k z + e − itk . (3.8)The computation of the Fourier transform of u z + (see P 170 of [15]) is essentiallygiven by Z ∞ u z e − itu du = ie iz π Γ( z + 1) t − z − . Therefore X k k z + e − ikt = ie iz π Γ( z + 1) t − z − . (cid:3) ow we prove (1.3) when p = q . Theorem 3.3. (Diagonal case) Let n ≥ and let S ⊂ R n × T be a discrete surface.Then k W T S W k G n +2 ( L ( T × R n )) C k W k L n +2 t,x ( T × R n ) k W k L n +2 t,x ( T × R n ) for all W , W with a constant C > independent of W , W . Proof.
Clearly ( T z ) is an analytic in the sense of Stein defined on the strip − λ ≤ Re z ≤ λ >
1. We show that the ( T z ) satisfies (3.1). When Re ( z ) = 0,we have k T is k L ( T × R n ) → L ( T × R n ) = k G is k L ∞ ( T × R n ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) is ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ce π | s | / . (3.9)An application of H¨olders inequality in (3.3) gives | T z g ( x, t ) | ≤ sup t ∈ T ,y ∈ N n | K z ( x, y, t ) |k g k L ( R n × T ) (3.10)for g ∈ L ( R n × T ). Thus for z = − λ + is , T z is bounded from L ( T × R n ) to L ∞ ( T × R n ) if and only if | K z ( x, y, t ) | is bounded for each x, y ∈ R n . But by (3.4),Proposition (3.2) and (2.2) we get | K z ( x, y, t ) | ∼ C | t | Re ( z +1+ n ) e iz π . (3.11)So for each x, y , | K z ( x, y, t ) | is bounded if and only if Re ( z ) = − n +22 . The conclusionof the theorem follows by choosing λ = n +22 by Proposition 3.1 (cid:3) Now we are in a position to prove Theorem 1.4: Under the assumption of Theorem2.2, we show that the operator V ∈ L p ′ t L q ′ x ( T × R n ) Z T e itH V ( t, x ) e − itH dt ∈ G q ′ is bounded. Using the fact that e itH xe − itH = cos 2 t · x − sin 2 t · p , where p = i ∇ and x is the operator of multiplication, we deduce that e itH V ( t, x ) e − itH = V ( t, cos 2 t · x − sin 2 t · p ) . Therefore (cid:13)(cid:13)(cid:13)(cid:13)Z T e itH V ( t, x ) e − itH dt (cid:13)(cid:13)(cid:13)(cid:13) ≤ Z T k V ( t, · ) k L ∞ ( R n ) dt. Thus (2.10) holds for ( p ′ , q ′ ) = (1 , ∞ ). Further by Theorem 3.3, Lemma 2.1 and The-orem 2.2, (2.10) holds for ( p ′ , q ′ ) = (cid:0) n , n (cid:1) . Applying the complex interpola-tion method [1](chapter 4), we prove Strichartz inequality (1.3) for all 1 < q ≤ n .The following theorem provides the full range exponents of Strichartz inequality forsystem of orthonormal functions. For the range 1 + n < q < n +1 n − , we prove the dualinequality (see Lemma 2.1 ) to (1.3). heorem 3.4. Let S be the discrete surface S = { ( µ, ν ) ∈ N n × N : ν = 2 | µ | + n } with respect to the counting measure. Then for all exponents p, q ≥ satisfying p + nq = 1 , q > n + 1 we have k W T S W k G q ( L ( T × R n )) ≤ C k W k L pt L qx ( T × R n ) k W k L pt L qx ( T × R n ) with C > independent of W , W .Proof. For ( µ, ν ) ∈ N n × N , the family of generalized functions G z coincides with theoperator T S when z = −
1. Notice that the operator T − λ + is is an integral operatorwith kernel K − λ + is ( x, x ′ , t − t ′ ) defined in (3.3). An application of Hardy-Littlewood-Sobolev inequality along with (3.9) and (3.11) yields (cid:13)(cid:13) W λ − is T − λ + is W λ − is (cid:13)(cid:13) G = Z T Z R n W ( t, x ) λ | K − λ + is ( x, t, x ′ , t ′ ) | W ( t ′ , x ′ ) λ dxdx ′ dtdt ′ ≤ C Z T Z R n W ( t, x ) λ W ( t ′ , x ′ ) λ | t − t ′ | n +2 − λ dxdx ′ dtdt ′ ≤ C e π | s | Z T Z T k W ( t ) k λ L λx ( R n ) k W ( t ′ ) k λ L x ( R n ) | t − t ′ | n +2 − λ dtdt ′ ≤ C e π | s | k W k λ L λ λ − dt L λ x ( T × R n ) k W k λ L λ λ − dt L λ x ( T × R n ) provided we have 0 ≤ n + 2 − λ < , that is ( n + 1) / < λ ≤ n/ . By Theorem2.9 of [16] we have k W T − W k G λ ( L T × R n ) ) ≤ C k W k L λ λ − dt L λ x ( T × R n ) k W k L λ λ − dt L λ x ( T × R n ) . (cid:3) Optimality of the Schatten exponent:
In this section, we use a semi-classical argument based on coherent states to show that the power p ( q +1)2 q on theright hand side in (1.3) is optimal. Proposition 3.5. (Optimality of the Schatten exponent). Assume that n, p, q ≥ p + nq = n. Then we have sup γ ∈G r (cid:13)(cid:13) ρ e − itH γ e itH (cid:13)(cid:13) L pt L qx ( T × R n ) k γ k G r = + ∞ for all r > qq +1 . roof. Depending on the positive parameters β, L and µ , we construct the family ofoperators γ = 1(2 π ) n Z Z R n × R n e − x − ξ µ | F x,ξ i h F x,ξ | dxdξ, where F x,ξ ( z ) = (2 πβ ) − n e − ( z − x )24 β e iξ · z . The functions F x,ξ are normalized and satisfy Z Z R n × R n dxdξ (2 π ) n | F x,ξ i h F x,ξ | = 1 . By Mehler’s formula we get e itH F x,ξ ( z ) = ( − πi sin 2 t ) − n (2 πβ ) − n Z R n e − i cot 2 t ( z + y ) + i sin(2 t ) z · y e − ( y − x )24 β e iξ · y dy. Therefore (cid:12)(cid:12) e itH F x,ξ ( z ) (cid:12)(cid:12) = (cid:18) βπ (4 β cos t + sin t ) (cid:19) n e − β ( z − x cos 2 t + ξ sin 2 t )24 β t +sin2 2 t and ρ γ ( t ) ( z ) := ρ e itH γ e − itH ( z ) = Z Z R n × R n dxdξ (2 π ) d e − x − ξ µ (cid:12)(cid:12) e itH F x,ξ ( z ) (cid:12)(cid:12) = (cid:18) πβµL (4 β + 2 βL ) cos t + (1 + 2 µβ ) sin t (cid:19) n e − βz ( β βL ) cos2 2 t +(1+2 µβ ) sin2 2 t . So k ρ γ ( t ) k qL qx ( R n ) = (cid:18) πq (cid:19) n (cid:0) µL (cid:1) nq (cid:18) β (4 β + 2 βL ) cos t + (1 + 2 µβ ) sin t (cid:19) n ( q − . Further using the fact that n ( q − p = 2 q , we have k ρ γ ( t ) k pL pt L qx ( T × R n ) = (cid:18) πq (cid:19) np q (cid:0) µL (cid:1) np Z [ − π,π ] β (4 β + 2 βL ) cos t + (1 + 2 µβ ) sin t dt = √ π (cid:18) πq (cid:19) np q (cid:0) µL (cid:1) np β p β + βL √ µβ . hus k ρ γ ( t ) k L pt L qx ( T × R n ) = A n,p (cid:0) µL (cid:1) n ( L ) − p µ − p (cid:0) βL + 1 (cid:1) p (cid:16) µβ + 2 (cid:17) p = A n,p (cid:0) µL (cid:1) n − p (cid:0) βL + 1 (cid:1) p (cid:16) µβ + 2 (cid:17) p . Using the fact that n (cid:16) q (cid:17) = n − p and choosing 1 /µ ≪ β ≪ L , we obtain k ρ γ ( t ) k L pt L qx ( T × R n ) ≈ A n,p (cid:0) µL (cid:1) n ( q ) 2 − p ≈ A n,p − p (cid:0) µL (cid:1) n ( q q ) ≈ A n,p − p N q q , where N = Z R n γ ( z, z ) dz = Z Z Z R n × R n × R n dxdξ (2 π ) n e − x − ξ µ | F xξ ( z ) | dz = Z Z R n × R n dxdξ (2 π ) n e − x − ξ µ = A n L n µ n . An application of Berezin-Lieb inequality gives thatTr γ r ≤ Z Z R n × R n dxdξ (2 π ) d e − rx − rξ µ = r − n N, where r ≥ N = ( µL ) n n . Therefore (cid:13)(cid:13) ρ e − itH γ e itH (cid:13)(cid:13) L pt L qx ( T × R n ) k γ k G r ≥ A n,p − p r − nr N ( q q − r ) . (cid:3) The end point:
We prove that the operator Z [ − π,π ] e itH V ( t, x ) e − itH dt is neverin the Schatten space G n +1 i.e. the inequality (1.3) fails when q = n +1 n − even if V hasa fast decay in both space and time. In this context we have the following theorem. Theorem 3.6. (The end point) Let = V ∈ L ∞ c ([ − π, π ] × R n ) be a non-negativefunction with non-negative Fourier transform (in both space and time). Then Tr (cid:18)Z [ − π,π ] e itH V ( t, x ) e − itH dt (cid:19) n +1 = + ∞ . (3.12) roof. Consider the operator M V := Z [ − π,π ] e itH V ( t, x ) e − itH dt. Then M V is an integral operator whose kernel is given by K ( x, z ) = Z [ − π,π ] Z R n V ( t, y ) K − t ( x, y ) K t ( y, z ) dydt = ( − πi ) − n Z [ − π,π ] Z R n V ( t, y )(sin 2 t ) − n e i cot 2 t ( x − z ) − iy ( x − z )sin 2 t dydt. Using the notations from [3], the rotated Fourier transform of the kernel is given by d M V ( p, q ) = Z Z R n × R n K ( x, z ) e − ixp e izq dxdz = ( − πi ) n Z [ − π,π ] Z R n V ( t, y )(cos 2 t ) n e i tan 2 t ( q − p ) e iy t ( q − p ) dydt = M ( p − q , p − q )(say).The remaining part of the proof follows by proceeding exactly as in the proof ofProposition 2 in [4]. (cid:3) Application of Strichartz estimates: global well-posedness forthe Hermite-Hartree equation in Schatten spaces
In this section we show the well-posedness results in the sprit of [12, 11] for a systemof infinitely many equations (without a trace class assumption) in Schatten spacesfor the Hermite-Hatree equation by applying our orthonormal Strichartz inequalitiesfor Hermite operator.
Theorem 4.1.
Let ≤ q < n − and p + nq = n and w ∈ L q ′ ( R n ) . Then, for any γ ∈ G q ( q +1) , there exists a unique γ ∈ C t ([0 , T ] , G q ( q +1) ) satisfying ρ γ ∈ L pt L qx ([0 , T ] × R n ) and i∂ t γ = [ H + w ⋆ ρ γ , γ ] ,γ | t =0 = γ . Proof.
Let
R > k γ k G q ( q +1) = R < ∞ . Let T = T ( R ) ( ≤ X T = n ( γ, ρ ) ∈ C t (cid:16) [0 , T ]; G q ( q +1) (cid:17) × L pt L qx ([0 , T ] × R n ) : k ( γ, ρ ) k X T ≤
10 max { , C Stri } R o , here the norm k ( γ, ρ ) k X T is defined by k ( γ, ρ ) k X T := k γ k C t G q ( q +1) + k ρ k L pt L qx ([0 ,T ] × R n ) . Consider the mapΦ ( γ, ρ )( t ) = e − itH γ e itH − i Z t e i ( s − t ) H [ w ⋆ ρ γ ( s ) , γ ( s )] e i ( t − s ) H ds Using the above map, we define the contraction map Φ byΦ( γ, ρ ) = (Φ ( γ, ρ ) , ρ [Φ ( γ, ρ )]) , where we have used the notation ρ [ γ ] = ρ γ . Now k Φ ( γ, ρ ) k C t G qq +1 R + 2 Z T k w ∗ ρ ( s ) k L ∞ x k γ ( s ) k S qq +1 ds R + 2 T /p ′ k w k L q ′ x k ρ k L pt L qx k γ k C t S qq +1 R + 8 T /p ′ k w k L q ′ x max (cid:0) , C (cid:1) R and from Corollary 2.4 we also have k ρ [Φ ( γ, ρ )] k L pt L qx ≤ C Stri R + 8 C Stri T /p ′ k w k L q ′ x max (cid:0) , C (cid:1) R . Choosing T ≤ C Stri T /p ′ k w k L q ′ x max (cid:0) , C (cid:1) R ≤ C Stri R and Φ maps X to itself. Thus Φ is a contraction mapping and has a unique fixedpoint on X which is a solution to the Hatree equation on [0 , T ] . (cid:3) Acknowledgments
The first author thanks the Ministry of Human Resource Development, India forthe research fellowship and Indian Institute of Technology Guwahati for the supportprovided during the period of this work.
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Shyam Swarup MondalDepartment of MathematicsIIT GuwahatiGuwahati, Assam, India.
Email address : [email protected] Jitendriya Swain,Department of MathematicsIIT GuwahatiGuwahati, Assam, India.
Email address : [email protected]@iitg.ac.in