Thresholds for Existence of dispersion management solitons for general nonlinearities
aa r X i v : . [ m a t h . A P ] A ug THRESHOLDS FOR EXISTENCE OF DISPERSION MANAGEMENTSOLITONS FOR GENERAL NONLINEARITIES
MI-RAN CHOI, DIRK HUNDERTMARK, YOUNG-RAN LEE
Abstract.
We prove a threshold phenomenon for the existence of solitary solutions ofthe dispersion management equation for positive and zero average dispersion for a largeclass of nonlinearities. These solutions are found as minimizers of nonlinear and nonlocalvariational problems which are invariant under a large non–compact group. There existsa threshold such that minimizers exist when the power of the solitons is bigger than thethreshold. Our proof of existence of minimizers is rather direct and avoids the use of Lions’concentration compactness argument. The existence of dispersion managed solitons isshown under very mild conditions on the dispersion profile and the nonlinear polarizationof optical active medium, which cover all physically relevant cases for the dispersion profileand a large class of nonlinear polarizations, for example, they are allowed to change sign.
Contents
1. Introduction 11.1. The variational problems 11.2. The connection with nonlinear optics 52. Nonlinear estimates 82.1. Fractional Bilinear Estimates 82.2. Splitting the nonlocal nonlinearity 143. Strict subadditivity of the ground state energy 174. The existence proof 215. Threshold phenomena 326. Nonexistence 37Appendix A. Tightness and strong convergence in L Introduction
The variational problems.
We show the existence of minimizers for a family ofnonlocal and nonlinear variational problems E d av λ := inf (cid:8) H ( f ) : k f k = λ (cid:9) , (1.1)where λ >
0, the average dispersion d av ≥ k f k = R R | f | dx , the Hamiltonian takes theform H ( f ) := d av k f ′ k − N ( f ) , (1.2) Date : July 19, 2018, version continuous-general-nonlinearity-7-3.tex.c (cid:13) and the nonlocal nonlinearity is given by N ( f ) := Z Z R V ( | T r f ( x ) | ) dxψ ( r ) dr. (1.3)Here V : [0 , ∞ ) → R is a suitable nonlinear potential and T r = e ir∂ x is the solutionoperator of the free Schr¨odinger equation in one dimension. The function ψ is assumed tobe in suitable L p -spaces.If d av > f with additionally f ∈ H ( R ), the usual Sobolev space of square integrable functions whose distributionalderivative f ′ is also square integrable. One can recover our formulation (1.1) by setting k f ′ k := ∞ if f ∈ L \ H .Our interest in these variational problems stems from the fact that the minimizers of(1.1) are the building blocks for (quasi-)periodic breather type solutions, the so-calleddispersion management solitons, of the dispersion managed nonlinear Schr¨odinger equation.The function ψ is the density of a probability measure related to the local dispersionprofile d , see the discussion in Section 1.2, especially Lemma 1.8 and formula (1.10).The dispersion management solitons have attracted a lot of interest in the development ofultrafast longhaul optical data transmission fibers. So far, it has mainly been studied fora Kerr-type nonlinearity, i.e., the special case where V ( a ) = a . The purpose of this workis to extend our previous existence results from [12] to a large class of nonlinearities V and also to positive average dispersion. We address the connection of the above variationalproblems with nonlinear optics later in Section 1.2.The standard approach to show the existence of a minimizer of (1.1) is to identify it asthe strong limit of a suitable minimizing sequence, that is, a sequence ( f n ) n ∈ N ⊂ L ( R ),with k f n k = λ and E d av λ = lim n →∞ H ( f n ). The catch is that the above variational problemis invariant under translations of L ( R ) if d av > d av = 0. This invariance under a large non-compact groupof transformations leads to a loss of compactness since minimizing sequences can easilyconverge weakly to zero. The usual strategy to compensate for such a loss of compactnessis Lions’ concentration compactness method. In a previous paper, [12], we used an alterna-tive approach, which directly showed that modulo the natural symmetries of the problem,minimizing sequences stay compact. The tools were very much tailored to the special typeof Kerr nonlinearity. This paper extends our approach from [12] to a much more generalsetting. This extension is by no means straightforward, see Section 2 and Remark 1.5.Our main assumptions on the nonlinear potential V : R + → R are A V is continuous on R + = [0 , ∞ ) and differentiable on (0 , ∞ ) with V (0) = 0. Thereexist 2 ≤ γ ≤ γ < ∞ such that | V ′ ( a ) | . a γ − + a γ − for all a > . A V is continuous on R + and differentiable on (0 , ∞ ) with V (0) = 0. There exists γ > V ′ ( a ) a ≥ γ V ( a ) for all a > . A
3) There exists a > V ( a ) > f . g , if there exists a finite constant C > f ≤ Cg .These three assumptions above are our main requirements on the nonlinear potential.For our existence results, depending on whether d av = 0 or d av >
0, we will have to posesome additional restrictions on the range of γ ≤ γ . For example, if d av >
0, we will needthat 2 < γ ≤ γ <
10 and we will also have some additional L p conditions on ψ to ensure HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 3 the existence of minimizers for (1.1). We will see that these conditions yield a thresholdphenomenon, that is, f ∈ H ( R ), respectively f ∈ L ( R ), if d av = 0, with k f k = λ and H ( f ) = E d av λ exists at least for large enough power λ . In order to guarantee the existenceof solutions for arbitrarily small λ >
0, we need to strengthen assumption A3 to A
4) If d av > ε > < κ < V ( a ) & a κ for all 0 < a ≤ ε. If d av = 0 we assume that there exists ε > V ( a ) > < a ≤ ε . Remarks 1.1. (i) An integration shows that A1 implies | V ( a ) | . a γ + a γ . Much more important for us is the fact that A1 allows us to control the nonlocal nonlinearity N under splitting , see Lemma 2.14 and the discussion in Section 2.2.(ii) Examples of nonlinearities obeying assumptions A1 through A3 are given by V ( a ) = J X j =1 c j a s j with c j >
0, 2 < s j <
10 if d av >
0, respectively, 2 < s j < d av = 0, and J ∈ N ,but our assumptions also allow nonlinear potentials which can become negative, forexample, for d av > V ( a ) = − a + a for a ≥ . It certainly fulfills A1 . Since V ′ ( a ) a = − a + 8 a = 4( − a + a ) + 4 a ≥ V ( a ) , it also obeys A2 . Moreover, V ( a ) > a , so A3 holds.Similarly, if d av = 0 we can allow for V ( a ) = − a + a for a ≥ . If we did not assume A3 , then the nonlinearities could also be strictly negative for all a >
0, for example, V ( a ) = − a − a obeys A1 and because of V ′ ( a ) a = − a − a = 6( − a − a ) ≥ V ( a )also A2 , but then the critical threshold λ d av cr given in Theorems 1.2 and 1.4 wouldbe infinite. The threshold is finite once there exists f ∈ H ( R ) with N ( f ) >
0, seeTheorem 5.1.5.1.v.Our first main result is
Theorem 1.2 (Thresholds for existence for positive average dispersion) . Assume d av > , V obeys assumptions A1 through A3 for some < γ ≤ γ < , and ψ ∈ L α δ has compactsupport for some δ > , where α δ := α δ ( γ ) := max { , − γ + δ } . Then (i) There exists a threshold ≤ λ d av cr < ∞ such that E d av λ = 0 for < λ < λ d av cr and −∞ < E d av λ < for λ > λ d av cr . (ii) If < λ < λ d av cr , then no minimizer for the constrained minimization problem (1.1) exists. If γ ≥ , then λ d av cr > . (iii) If λ > λ d av cr , then any minimizing sequence for (1.1) is up to translations relativelycompact in L ( R ) , in particular, there exists a minimizer for (1.1) . This minimizer isalso a weak solution of the dispersion management equation (1.12) for some Lagrangemultiplier ω < E d av λ /λ < . M.-R. CHOI, D. HUNDERTMARK, Y.-R. LEE (iv) If V obeys in addition A4 , then λ d av cr = 0 . Remark 1.3. If γ < − γ < α δ = 1 for all small enough δ >
0. So if2 < γ ≤ γ < ψ ∈ L with compact support for Theorem 1.2 to hold.If 6 ≤ γ <
10 the density ψ has to have compact support and to be in L p with p slightlylarger than − γ . With the bound (1.9) this translates into some slightly more restrictiveintegrability bound on 1 /d , which still covers all physically relevant local dispersion profiles.We have a similar existence result in the case of d av = 0 under slightly different L p assumptions on the density ψ . Theorem 1.4 (Threshold for existence for zero average dispersion) . Assume d av = 0 and V obeys assumptions A1 through A3 with < γ ≤ γ < , and that the density ψ hascompact support and ψ ∈ L − γ + δ for some δ > . (i) There exists a threshold ≤ λ < ∞ such that E λ = 0 for < λ < λ and −∞
6, we can allow for the largest possible class oflocal dispersion profiles d , they only have to change sign finitely many times and theirzero set has to have Lebesgue measure zero.Even in the case of a Kerr nonlinearity, where V ( a ) ∼ a , i.e., γ = γ = 4, the abovetwo theorems strongly improve our result in [12] in terms of scales of L p spaces: In [12],we needed in addition that ψ ∈ L , whereas now with γ = 4, one sees that ψ ∈ L isenough for strictly positive average dispersion and for vanishing average dispersion weonly need L δ for arbitrarily small δ > d = [0 , − [1 , and extended to more general dispersion profiles in [10]. In the more HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 5 general setting discussed in this paper the smoothness and decay of solitary solutionsis an open problem.Concerning the question whether the range of exponents in Theorems 1.2 and 1.4 isoptimal, we note the following
Theorem 1.6.
Let V be a pure power law nonlinearity given by V ( a ) = ca γ for a ≥ andsome coefficients c, γ > . (i) Assume further that d av > and ψ is a probability density with compact support which isstrictly positive in a possibly one sided neighborhood of zero. Then H ( f ) is unboundedfrom below on H for any fixed k f k = λ > , if γ > . If γ = 10 , then H ( f ) isunbounded from below for any fixed k f k = λ > as long as c is large enough. (ii) If d av = 0 and γ > , then H ( f ) is unbounded from below on L for any fixed k f k = λ > , if ψ is a probability density with compact support which is strictly positive in apossibly one sided neighborhood of zero.If d av = 0 and γ = 6 , then no minimizers exist in the model case ψ = [0 , . Remark 1.7.
As the lower bound (1.11) from Remark 1.9.ii shows, the assumption of thefirst part of Theorem 1.6 is fulfilled if the dispersion profile d is bounded from above closeto zero, which includes all physically relevant dispersion profiles.The strategy of the proofs of our Existence Theorems 1.2 and 1.4 is as follows: Themain observation which shows that E d av λ < equivalent to gaining compactness is donein Theorem 4.1. The necessary space–time bounds which prevent splitting of minimizingsequences as soon as E d av λ < E d av λ = 0 and E d av λ < The connection with nonlinear optics.
Our main motivation for studying (1.1)comes from the fact that the minimizer of the variational problem is related to breather-typesolutions of the dispersion managed nonlinear Schr¨odinger equation i∂ t u = − d ( t ) ∂ x u − p ( | u | ) u, (1.4)where the dispersion d ( t ) is parametrically modulated and P ( u ) = p ( | u | ) u is the nonlinearinteraction due to the polarizability of the glass fiber cable. In nonlinear optics (1.4)describes the evolution of a pulse in a frame moving with the group velocity of the signalthrough a glass fiber cable, see [25]. As a warning : with our choice of notation the variable t denotes the position along the glass fiber cable and x the (retarded) time. Hence d ( t ) is not varying in time but denotes indeed a dispersion varying along the optical cable. Forphysical reasons it would not be a strong restriction to assume that d is piecewise constant,but we will not make this assumption in this paper. By symmetry, one assumes that P is odd and P (0) = 0 can always be enforced by adding a constant term. Most often onemakes a Taylor series expansion, keeping just the lowest order nontrivial term leads to P ( u ) ∼ | u | u , the Kerr nonlinearity, but we will not make this approximation. M.-R. CHOI, D. HUNDERTMARK, Y.-R. LEE
The dispersion management idea, i.e., the possibility to periodically manage the disper-sion by putting alternating sections with positive and negative dispersion together in anoptical glass-fiber cable to compensate for dispersion of the signal was predicted by Lin,Kogelnik, and Cohen already in 1980, see [17], and then implemented by Chraplyvy andTkach for which they received the Marconi prize in 2009. See the reviews [26, 27] and thereferences cited in [12] for a discussion of the dispersion management technique.The periodic modulation of the dispersion can be modeled by the ansatz d ( t ) = ε − d ( t/ε ) + d av . (1.5)Here d av ≥ d its mean zero part which we assume to haveperiod L . For small ε , equation (1.5) describes a fast strongly varying dispersion whichcorresponds to the regime of strong dispersion management.A technical complication is the fact that (1.4) is a non-autonomous equation. We seekto rewrite (1.4) into a more convenient form in order to find breather type solutions. Let D ( t ) = R t d ( r ) dr and note that as long as d is locally integrable and has period L withmean zero, D is also periodic with period L . Furthermore, T r = e ir∂ x is a unitary operatorand thus the unitary family t T D ( t/ε ) is periodic with period εL . Making the ansatz u ( t, x ) = ( T D ( t/ε ) v ( t, · ))( x ) in (1.4), a short calculation shows i∂ t v = − d av ∂ x v − T − D ( t/ε ) (cid:2) P ( T D ( t/ε ) v ) (cid:3) which is equivalent to (1.4) and still a non-autonomous equation.For small ε , that is, in the regime of strong dispersion management, T D ( t/ε ) is fast oscil-lating in the variable t , hence the solution v is expected to evolve on two widely separatedtime-scales, a slowly evolving part v slow and a fast, oscillating part with a small amplitude.Analogously to Kapitza’s treatment of the unstable pendulum which is stabilized by fastoscillations of the pivot, see [15], the effective equation for the slow part v slow was derivedby Gabitov and Turitsyn [7, 8] for the special case of a Kerr nonlinearity. It is given byintegrating the fast oscillating term containing T D ( t/ε ) over one period in t , i∂ t v slow = − d av ∂ x v slow − εL Z εL T − D ( r/ε ) (cid:2) P ( T D ( r/ε ) v slow ) (cid:3) dr = − d av ∂ x v slow − L Z L T − D ( r ) (cid:2) P ( T D ( r ) v slow ) (cid:3) dr. (1.6)This averaging procedure leading to (1.6) was rigorously justified in [29] for suitable dis-persion profiles d in the case of a Kerr nonlinearity. The averaged equation is autonomousand stationary solutions of (1.6) can be found by making the ansatz v slow ( t, x ) = e − iωt f ( x ) . (1.7)Before doing so, it turns out to be advantageous to rewrite the nonlocal nonlinear termin (1.6): Define a measure µ ( B ) by setting µ ( B ) := L R L B ( D ( r )) dr for any Lebesguemeasurable set B ⊂ R . Since µ ( B ) ≥ µ ( R ) = L R L R ( D ( r )) dr = L R L dr = 1, onesees that µ is a probability measure. Since µ is the image measure of normalized Lebesguemeasure on [0 , L ] under D , we can rewrite (1.6) as i∂ t v slow = − d av ∂ x v slow − Z R T − r (cid:2) P ( T r v slow ) (cid:3) µ ( dr ) . (1.8)One can easily check that the simplest case of dispersion management, L = 2, d = [0 , − [1 , , which is the case most studied in the literature, corresponds to the measure µ having HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 7 density [0 , , the uniform distribution on [0 , µ in the following Lemma 1.8 (Lemma 1.4 in [12]) . Assume that the dispersion profile d is locally integrable.Then the following holds. (i) The probability measure µ has compact support. (ii) If the set { d = 0 } has zero Lebesgue measure, then µ is absolutely continuous withrespect to Lebesgue measure. (iii) If furthermore d changes sign finitely many times on [0 , L ] and is bounded away fromzero then µ has a bounded density ψ . (iv) Moreover, if d changes sign finitely many times on [0 , L ] and for some p > Z L | d ( s ) | − p ds < ∞ , then µ has a density ψ ∈ L p . More precisely, we have the bound k ψ k L p . (cid:16) Z L | d ( s ) | − p ds (cid:17) p (1.9) where the implicit constant depends only on the number of sign changes of d and theperiod L . Remarks 1.9. (i) As explained in [12], the bound (1.9) is quite natural and sharp. Ittranslates L p restrictions on ψ into integrability conditions on d − p . The extreme case p = ∞ yields that ψ is bounded once d is bounded away from zero, and the case p = 1poses the weak additional restriction that the set where d is zero has Lebesgue measurezero.(ii) Without working too hard, one can derive a formula for the density ψ of the probabilitymeasure µ . We give the short argument from [12], for the reader’s convenience, sinceit has an important consequence for Theorem 1.6: We assume that the measure of theset { d = 0 } is zero, otherwise µ will have a Dirac point mass component, and that d changes sign only finitely many times on [0 , L ]. Let { A j } be a collection of disjointhalf–open intervals with ∪ j A j = [0 , L ) such that, on each A j , the dispersion profile d has a fixed sign and so D is strictly monotone. Then by the definition of µ , Z F ( r ) µ ( dr ) = X j L Z A j F ( D ( s )) ds = X j L Z A j F ( D ( s )) | D ′ ( s ) || D ′ ( s ) | − ds = X j L Z D ( A j ) F ( r ) 1 | d ( D − ( r )) | dr = Z supp ( µ ) F ( r ) 1 L X s ∈ D − ( { r } ) | d ( s ) | − dr. In the third equality we used a change of variables r = D ( s ) and that in each A j there isa unique s j ∈ A j such that D ( s j ) = r and for the last equality we set D − ( { r } ) = { s ∈ [0 , L ) | D ( s ) = r } , the set of pre-images of r within [0 , L ). Thus we have the formula ψ ( r ) = 1 L X s ∈ D − ( { r } ) | d ( s ) | − (1.10)for the density ψ of µ in terms of the dispersion profile d . Since D ( r ) = R r d ( s ) ds and d is locally integrable, D is continuous and we can use (1.10) to get a lowerbound for all r in the support of ψ close enough to zero, as long as d does not behave M.-R. CHOI, D. HUNDERTMARK, Y.-R. LEE too wildly: If d is bounded close to zero there exists r > m := L − inf {| d ( D ( r )) | − : 0 ≤ r ≤ r } > ψ ≥ m [0 ,r ] or ψ ≥ m [ − r , (1.11)which one of the above two lower bounds holds depends on the sign of d close to zero.Coming back to our discussion of the dispersion management equation, plugging (1.7)into (1.8), we see that f should solve ωf = − d av f ′′ − Z R T − r (cid:2) P ( T r f ) (cid:3) µ ( dr ) , (1.12)which is a nonlocal nonlinear eigenvalue equation for f . Testing (1.12) with suitable testfunctions h one gets the weak formulation ω h h, f i = d av h h ′ , f ′ i − h h, Z R T − r (cid:2) P ( T r f ) (cid:3) µ ( dr ) i where h h , h i is the scalar product on L ( R ) given by R R h ( x ) h ( x ) dx . Exchanging inte-grals, a formal calculation, using the unicity of T r , yields h h, Z R T − r (cid:2) P ( T r f ) (cid:3) µ ( dr ) i = Z R h T r h, P ( T r f ) i µ ( dr )and one arrives at the weak formulation of (1.12) in the form ω h h, f i = d av h h ′ , f ′ i − Z R h T r h, P ( T r f ) i µ ( dr ) , (1.13)supposed to hold for any h in the Sobolev space H ( R ).Using the formula from Lemma 4.9 for the derivative of the nonlocal nonlinearity N ( f )from (1.3), one sees that (1.13) is the weak form of the Euler-Lagrange equation associatedto the energy H ( f ) given in (1.2), as long as V ′ ( | T r f | )sgn( T r f ) = P ( T r f ). This is the caseif V ′ ( a ) = p ( a ) a = P ( a ) for all a > , i.e., V is the antiderivative of the polarizability P , V ( a ) := Z a P ( s ) ds. In this case it is, up to some technicalities, clear that any minimizer of the associatedconstrained minimization problem (1.1) will be a weak solution of (1.12) for some choiceof Lagrange multiplier ω , as long as the variational problem (1.1) admits minimizers. Inparticular, combining Theorems 1.2 and 1.4 with Lemma 1.8 one sees that (1.12) has anon trivial weak solution f under the condition that the assumptions A1 – A3 hold, at leastfor large enough power λ = k f k , or for arbitrary power, if additionally A4 holds for theantiderivative of P and that the dispersion profile d changes signs finitely many times and1 /d obeys some mild integrability conditions given by the right hand side of (1.9). Thisallows for a large class of dispersion profiles d , covering all physically relevant cases.2. Nonlinear estimates
Fractional Bilinear Estimates.
In this paper, the nonlocal nonlinearity is not apure power, thus the multilinear estimates from [12] cannot be used anymore. First, wegather the estimates which will be used in the proof of fat–tail Propositions 4.3 and 4.4,which are crucial for the existence proof in this paper. The core of the argument will besuitable splitting bounds on the nonlocal nonlinearity N ( f ) from (1.3) given in Proposition HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 9 V , one needs certain fractionallinear bounds on the building blocks from Definition 2.5.Since T r = e ir∂ x is the solution operator for the free Schr¨odinger equation in dimension one,we can express T r f for any nice f , for example, in the Schwartz class, as follows: T r f ( x ) = 1 √ πir Z R e i | x − y | r f ( y ) dy (2.1)= 1 √ π Z R e ixη e − irη ˆ f ( η ) dη, (2.2)where ˆ f is the Fourier transform of f given byˆ f ( η ) = 1 √ π Z R e − ixη f ( x ) dx. As a first step, we note that, for ψ in suitable L p spaces, certain space time norms of T r f are bounded. Lemma 2.1.
Let f ∈ L ( R ) , ≤ q ≤ and ψ ∈ L − q ( R ) . Then k T r f k L q ( R ,dxψdr ) . k ψ k L − q ( R ) k f k . (2.3) Proof.
Using the H¨older inequality, we get
Z Z R | T r f | q dxψdr = Z Z R (cid:16) | T r f | − q )4 ψ (cid:17) (cid:16) | T r f | q − (cid:17) dxdr ≤ (cid:18)Z Z R | T r f | ψ − q dxdr (cid:19) − q (cid:18)Z Z R | T r f | dxdr (cid:19) q − . Since T r is unitary on L ( R ), Z Z R | T r f | ψ − q dxdr = k f k Z R ψ − q dr and the Strichartz inequality [9, 23], needed here only in one dimension, gives Z Z R | T r f | dxdr ≤ S k f k and so (2.3) follows. Remark 2.2.
The sharp value of the constant in the one-dimensional Strichartz inequalityis known to be S = 12 − / , the two dimensional sharp constant is known, too, and it isalso known that Gaussians are the only maximizers in the Strichartz inequality in one andtwo space dimensions, see [6] and [14]. In recent years there has been a renewed interestin establishing existence of maximizers for certain space time norms of solutions of linearevolution equations, like the Schr¨odinger or wave equation, see, for example, [2, 3, 5, 13].To take advantage of the fact that an interaction term containing the product of twoterms of the form T r f and T r f is typically small if the functions ˆ f and ˆ f have separatedsupports, we need Lemma 2.3 (Fractional bilinear estimate) . Let ≤ p < and f , f ∈ L ( R ) whoseFourier transforms have separated supports, say s = dist(supp ˆ f , supp ˆ f ) > . Then k T r f T r f k L p ( R , dxdr ) . s (3 − p ) /p k f kk f k . (2.4) Remark 2.4.
The bound (2.4) is a well-known bilinear estimate for p = 2, see [1]. Forreaders’ convenience, we give a proof of (2.4) for any 2 ≤ p <
3. As the proof shows, (2.4)holds also for p = 3, without any support condition on ˆ f and ˆ f . Proof.
Using (2.2), we get T r f ( x ) T r f ( x ) = 12 π Z Z R e ix ( η + η ) − ir ( η + η ) ˆ f ( η ) ˆ f ( η ) dη dη . Doing the change of variables a = η + η , b = η + η , with Jacobian J = ∂ ( a,b ) ∂ ( η ,η ) = 2( η − η )and introducing F ( a, b ) := 1 | J | ˆ f ( η ( a, b )) ˆ f ( η ( a, b )) [0 , ∞ ) ( b )one sees T r f ( x ) T r f ( x ) = 12 π Z Z R e ixa − irb F ( a, b ) dadb, that is, up to sign in one of the variables, T r f ( x ) T r f ( x ) is the space-time Fourier transformof F . Since p ≥
2, one can apply the Hausdorff-Young inequality, see, e.g., [16], whichreduces to Plancherel’s identity for p = 2, to get k T r f T r f k L p ( R × R ,dxdr ) ≤ k F k L p ′ ( R ,dadb ) with p ′ the dual index to p . Undoing the above change of variables, one sees k F k L p ′ ( R ,dadb ) = 2 − /p (cid:18)Z Z R | η − η | p ′ − | ˆ f ( η ) ˆ f ( η ) | p ′ dη dη (cid:19) /p ′ . (2.5)If p = p ′ = 2, we use | η − η | ≥ s on the support of the product ˆ f ˆ f to get k F k L ( R ,dadb ) . √ s k ˆ f kk ˆ f k which concludes the proof for p = 2, since the Fourier transform is an isometry on L .Since 3 / < p ′ <
2, one can use the Hardy-Littlewood-Sobolev inequality to see(2.5) ≤ s − /p ′ Z Z R | ˆ f ( η ) | p ′ | ˆ f ( η ) | p ′ | η − η | − p ′ dη dη ! p ′ . s (3 − p ) /p k ˆ f kk ˆ f k which yields (2.4) for 2 < p < Definition 2.5.
For any γ ≥
2, define M γψ ( f , f ) := Z Z R | T r f || T r f | ( | T r f | + | T r f | ) γ − dxψdr. Remark 2.6.
At first M γψ ( f , f ) is defined only when f , f are Schwartz functions. UsingProposition 2.7 below one sees that for all γ ≥ ψ ∈ L one can extend M γψ ( f , f ) toall of H , and even to all of L if 2 ≤ γ ≤ ψ in certain L p spaces, by density of theSchwartz functions. HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 11
Proposition 2.7. (i)
Let ≤ γ ≤ and ψ ∈ L − γ . Then M γψ ( f , f ) . k f kk f k ( k f k + k f k ) γ − (2.6) where the implicit constant depends only on γ and the L − γ norm of ψ . (ii) Let ≤ γ < ∞ and ψ ∈ L . Then M γψ ( f , f ) . k f kk f k ( k f k H + k f k H ) γ − (2.7) where the implicit constant depends only on γ and the L norm of ψ .Proof. Using H¨older’s inequality for 3 functions with exponents γ , γ , and γ/ ( γ −
2) one has M γψ ( f , f ) ≤ k T r f k L γ ( R ,dxψdr ) k T r f k L γ ( R ,dxψdr ) k| T r f | + | T r f |k γ − L γ ( R ,dxψdr ) . Applying the triangle inequality and Lemma 2.1 completes the proof of (2.6).Similarly, using H¨older’s inequality with exponents 2 ,
2, and ∞ shows M γψ ( f , f ) ≤ k T r f k L ( R ,dxψdr ) k T r f k L ( R ,dxψdr ) sup r ∈ R ( k T r f k L ∞ + k T r f k L ∞ ) γ − k ψ k γ − L ≤ k ψ k γL k f kk f k sup r ∈ R ( k T r f k L ∞ + k T r f k L ∞ ) γ − (2.8)where we also used Lemma 2.1. Using the simple bound k h k L ∞ ≤ k h kk h ′ k , (2.9)whose proof we postpone to the end of this proof, and the fact that the derivative and T r commute and T r is unitary on L ( R ), one getssup r ∈ R k T r f k L ∞ ≤ ( k f kk f ′ k ) / ≤ k f k H and similar for T r f . Thus the second factor in (2.8) is bounded bysup r ∈ R ( k T r f k L ∞ + k T r f k L ∞ ) γ − ≤ ( k f k H + k f k H ) γ − (2.10)which finishes the proof of (2.7).It remains to give an argument for (2.9). This is well–known, but we give the short prooffor convenience of the reader. It is enough to assume that h ∈ C ∞ ( R ). Then | h ( x ) | = Z x −∞ h ( s ) h ′ ( s )) ds = − Z ∞ x h ( s ) h ′ ( s )) ds. So | h ( x ) | ≤ Z R (cid:12)(cid:12) h ( s ) h ′ ( s ) (cid:12)(cid:12) ds ≤ k h kk h ′ k using the Cauchy-Schwarz inequality. Proposition 2.8.
Let s = dist(supp ˆ f , supp ˆ f ) > .If < γ < , τ > and ψ ∈ L β ( γ,τ ) , then M γψ ( f , f ) . s − α ( γ,τ ) k f kk f k ( k f k + k f k ) γ − , where α ( γ, τ ) := min { γ − τ , − γ τ } and β ( γ, τ ) := − γ − α ( γ,τ ) . Remark 2.9.
Note that β ( γ, τ ) is only slightly bigger than − γ since α ( γ, τ ) > τ → ∞ and that it is increasing in γ . So we loose only an epsilon, by choosing τ large enough, with respect to the bound from Proposition 2.7. Proof.
Let 0 < α < to be chosen later and write M γψ ( f , f ) = Z Z R (cid:8) ( | T r f || T r f | ) − α ψ (cid:9) {| T r f || T r f |} α (cid:8) ( | T r f | + | T r f | ) γ − (cid:9) dxdr. Now use H¨older’s inequality for 3 functions with exponents p , α , and, γ − , where1 p = 1 − α − γ −
26 = 8 − γ − α M γψ ( f , f ) ≤ (cid:18)Z Z R | T r f T r f | − α )8 − γ − α ψ − γ − α dxdr (cid:19) − γ − α k T r f T r f k αL ( R ,dxdr ) k| T r f | + | T r f |k γ − L ( R ,dxdr ) . Up to a constant, the third factor is bounded by ( k f k + k f k ) γ − , using the triangle andStrichartz inequalities. Using Lemma 2.2, the second factor is bounded by k T r f T r f k αL ( R ,dxdr ) . s − α k f k α k f k α . For the first factor, we note that with the help of the Cauchy-Schwarz inequality one gets
Z Z R | T r f T r f | − α )8 − γ − α ψ − γ − α dxdr ≤ (cid:18)Z Z R | T r f | − α )8 − γ − α ψ − γ − α dxdr (cid:19) / (cid:18)Z Z R | T r f | − α )8 − γ − α ψ − γ − α dxdr (cid:19) / . In order to use Lemma 2.1 for this, we need to have 2 ≤ q ≤ q = − α )8 − γ − α . Thisis equivalent to 6 α < − γ , 6 α ≤ γ − α ≤ − γ .Moreover, we need ψ − γ − α ∈ L − q = L − γ − α )6(6 − γ − α ) hence ψ ∈ L − γ − α . Now we come to the choice of α : In order to guarantee that 0 < α <
1, 6 α < − γ ,6 α ≤ γ −
2, and 2 α ≤ − γ , we take any τ > α := α ( γ, τ ). Then one checks that α obeys the above bounds to finish the proof. Lemma 2.10 (Duality) . Define e ψ ( s ) := 1(2 | s | ) − γ ψ (cid:0) − s (cid:1) for s = 0 . Then M γψ ( f , f ) = M γ e ψ ( ˇ f , ˇ f ) (2.11) where ˇ f is the inverse Fourier transform of f . Remark 2.11.
Of course, the definition of e ψ depends on γ , but we drop this dependence inour notation, for simplicity. For 2 ≤ γ ≤
6, Proposition 2.7 yields a natural a priori boundon M γψ ( f , f ) which depends on the L − γ norm of ψ . It is an easy exercise to check that k ψ k L − γ = k e ψ k L − γ , so Proposition 2.7 and the duality expressed in (2.11) are consistent. HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 13
Proof.
Without loss of generality, assume that f and f are Schwartz functions for thecalculations below. Defining u j ( r, x ) := ( T r f j )( x ) and ˇ u j ( r, x ) := ( T r ˇ f j )( x ), j = 1 ,
2, usingthe explicit form of the free time evolution (2.1) for u j ( r, x ), and expanding the square, onesees u j ( r, x ) = 1 √ ir e i x r ˇ u j (cid:16) − r , − x r (cid:17) which is often called pseudo-conformal invariance of the free Schr¨odinger evolution. Then M γψ ( f , f )= Z Z R (cid:12)(cid:12)(cid:12) ˇ u (cid:16) − r , − x r (cid:17)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ˇ u (cid:16) − r , − x r (cid:17)(cid:12)(cid:12)(cid:12) (cid:16)(cid:12)(cid:12)(cid:12) ˇ u (cid:16) − r , − x r (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˇ u (cid:16) − r , − x r (cid:17)(cid:12)(cid:12)(cid:12)(cid:17) γ − (2 | r | ) γ/ dxψ ( r ) dr. (2.12)Doing first the change of variables x = − ry , dx = 2 | r | dy and then r = − / (4 s ) with dr = (2 | s | ) − ds , yields(2.12) = Z Z R | ˇ u ( s, y ) | | ˇ u ( s, y ) | ( | ˇ u ( s, y ) | + | ˇ u ( s, y ) | ) γ − (2 | s | ) − γ dyψ ( − s ) ds which completes the proof.This duality is a convenient tool in the proof of the analogue of Proposition 2.8 whenthe functions f and f have separated supports. Proposition 2.12.
Let s = dist(supp f , supp f ) > . If < γ < , τ > and ψ ∈ L β ( γ,τ ) ( | r | α ( γ,τ ) β ( γ,τ ) dr ) , then M γψ ( f , f ) . s − α ( γ,τ ) k f kk f k ( k f k + k f k ) γ − . (2.13) Proof.
Given the duality expressed in Lemma 2.10 this is now simple: We have M γψ ( f , f ) = M γ e ψ ( ˇ f , ˇ f )and note that the assumption on the separation of the supports of f and f means, of course,that ˇ f and ˇ f have separated Fourier support, so Proposition 2.8 applies to M γ e ψ ( ˇ f , ˇ f ) aslong as e ψ is in the correct L p space. A short calculation shows k e ψ k pL p ( dr ) = Z R (2 | r | ) p (6 − γ )2 − | ψ ( r ) | p dr and (2.13) follows by choosing p = β ( γ, τ ).To handle the cases with 6 ≤ γ <
10 for positive average dispersion, we need a fractionalbilinear estimate for M γψ in H as follows. Proposition 2.13 ( H bilinear estimate) . Let γ ≥ and ψ ∈ L ( R ) with compact support.Then for any f , f ∈ H ( R ) with s = dist(supp f , supp f ) > , M γψ ( f , f ) . s − k f k H k f k H ( k f k H + k f k H ) γ − , where the implicit constant depends only on the support and the L norm of ψ .Proof. From the definition of M γψ ( f , f ) one sees M γψ ( f , f ) ≤ k T r f T r f k L ( R ,dxψdr ) sup r ∈ R ( k T r f k L ∞ + k T r f k L ∞ ) γ − . (2.14)We use (2.10) to bound the second factor in (2.14) by ( k f k H + k f k H ) γ − . To bound the first factor, we use the positive operators P ≤ L and P > L from Lemma B.4for suitably chosen L >
0. Although they are not projection operators, we think of P ≤ L as‘projecting’ onto frequencies localized to . L and P > L as ‘projecting’ onto large frequencies & L . At the same time, the supports of P ≤ L f and P > L f will still be essentially separated.See Lemma B.2 and B.4 in Appendix B for the properties of P ≤ L and P > L which we willneed.Since P ≤ L + P > L = on L ( R ) by Lemma B.2, we can use the triangle inequality and thelinearity of T r to split k T r f T r f k L ( R , dxψdr ) ≤ k T r P > L f T r f k L ( R , dxψdr ) + k T r P ≤ L f T r P > L f k L ( R , dxψdr ) + k T r P ≤ L f T r P ≤ L f k L ( R , dxψdr ) . (2.15)The Cauchy–Schwarz inequality and Lemma 2.1 yield k T r P > L f T r f k L ( R , dxψdr ) ≤ k T r P > L f k L ( R , dxψdr ) k T r f k L ( R , dxψdr ) . k P > L f kk f k . L − k f k H k f k , where we also use (B.20) in the last bound. Note that the implicit constant from Lemma2.1 depends only on the L norm of ψ . Switching the roles of f and f , using in additionthat P ≤ L ≤ , shows k T r P ≤ L f T r P > L f k L ( R , dxψdr ) . L − k f kk f k H . To bound the last term of the right hand side in (2.15), we use the bound (B.22) to get k T r P ≤ L f T r P ≤ L f k L ( R , dxψdr ) ≤ k ψ k L sup | r |≤ R k T r P ≤ L f T r P ≤ L f k L ( R ,dx ) ≤ k ψ k L A R L e L − B ,R s k f kk f k , with R > ψ ⊂ [ − R, R ] and the constants A R and B ,R fromLemma B.4. Therefore k T r f T r f k L ( R , dxψdr ) . h L e L − B ,R s + L − i k f k H k f k H for any L ≥
0. Choosing 2 L = B ,R s , we get k T r f T r f k L ( R , dxψdr ) . s − k f k H k f k H , and using this in (2.14) proves Proposition 2.13.2.2. Splitting the nonlocal nonlinearity.
For the nonlinear potential V : R + → R ourassumption A1 guarantees a simple bound which is central for our existence proofs. Lemma 2.14.
Assume that V obeys A1 . Then | V ( a ) | . a γ + a γ (2.16) for all a ≥ . Moreover, | V ( | z + w | ) − V ( | z | ) | . | w | (cid:0) ( | z | + | w | ) γ − + ( | z | + | w | ) γ − (cid:1) (2.17) and | V ( | z + w | ) − V ( | z | ) − V ( | w | ) | . | z || w | (cid:0) ( | z | + | w | ) γ − + ( | z | + | w | ) γ − (cid:1) (2.18) for all z, w ∈ C . HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 15
Proof.
As already observed in Remark 1.1.i, (2.16) follows from integrating the bound for V ′ . For the second claim, let c = min {| z | , | z + w |} and d = max {| z | , | z + w |} ≤ | z | + | w | .Then d − c = || z + w | − | z || ≤ | w | and using the triangle inequality and A1 , we have | V ( | z + w | ) − V ( | z | ) | ≤ d Z c | V ′ ( a ) | da . ( d γ − + d γ − )( d − c ) ≤ (( | z | + | w | ) γ − + ( | z | + | w | ) γ − ) | w | . For the last claim note that since V (0) = 0, we have V ( | z + w | ) − V ( | z | ) − V ( | w | ) = 0 if atleast one of z and w equals zero. So assume z, w = 0 in the following. Then V ( | z + w | ) − V ( | z | ) − V ( | w | ) = (cid:20) | z | + | w | V ( | z + w | ) − | z | V ( | z | ) (cid:21) | z | + (cid:20) | z | + | w | V ( | z + w | ) − | w | V ( | w | ) (cid:21) | w | . (2.19)Moreover,1 | z | + | w | V ( | z + w | ) − | z | V ( | z | ) = 1 | z | + | w | ( V ( | z + w | ) − V ( | z | )) − | w | ( | z | + | w | ) | z | V ( | z | ) . (2.20)Using (2.16) we have | w | ( | z | + | w | ) | z | | V ( | z | ) | . | w || z | + | w | ( | z | γ − + | z | γ − ) ≤ | w | (( | z | + | w | ) γ − + ( | z | + | w | ) γ − ) , which together with (2.17) in (2.20) gives (cid:12)(cid:12)(cid:12)(cid:12) | z | + | w | V ( | z + w | ) − | z | V ( | z | ) (cid:12)(cid:12)(cid:12)(cid:12) . (( | z | + | w | ) γ − + ( | z | + | w | ) γ − ) | w | and a similar inequality holds when we switch z and w , so (2.19) and (2.20) imply (2.18).Recall that the nonlocal nonlinearity is given by N ( f ) = Z Z R V ( | T r f ( x ) | ) dxψdr. Proposition 2.15 (Boundedness) . Assume that V obeys assumption A1 . Furthermore, for j = 1 , choose κ j with ( γ j − + ≤ κ j ≤ γ j − and assume that ψ ∈ L − γ κ ∩ L − γ κ .Then for all f ∈ H ( R ) | N ( f ) | . k f ′ k κ k f k γ − κ + k f ′ k κ k f k γ − κ , (2.21) where the implicit constant depends only on the L − γ κ and L − γ κ norms of ψ . More-over, if ≤ γ ≤ γ ≤ and κ = κ = 0 , then the above bound extends to all f ∈ L ( R ) . Remark 2.16.
As the condition in Proposition 2.15 indicates, we need κ j > γ j > j = 1 ,
2. If 2 ≤ γ ≤ γ ≤
6, we can bound N ( f ) solely in terms of the L norm of f . This is not possible anymore if some exponent γ j is bigger than 6. In this caseone has to use an L ∞ bound and (2.9) to extract some excess power and for this one hasto pay the price that the bound then contains the L norm of the derivative of f , but thisallows to go beyond the exponent 6 in the existence results for d av > Here ( x ) + = max { x, } is the positive part of x ∈ R . Note also that the choice κ j = γ j − ψ ∈ L the nonlinearity is bounded by | N ( f ) | . k f ′ k γ − k f k γ + k f ′ k γ − k f k γ . (2.22)Moreover, using k f ′ kk f k ≤ k f k H , (2.22) also gives the bound | N ( f ) | . k f k (cid:16) k f k γ − H + k f k γ − H (cid:17) (2.23)where the implicit constant only depends on the L norm of ψ . Proof of Proposition 2.15:
Take an arbitrary f ∈ H ( R ). As in the proof of Proposition2.13 we can use (2.9) to get sup r ∈ R k T r f k L ∞ ≤ k f ′ kk f k . Thus, for any γ ≥ κ ≥ γ − κ >
0, we have
Z Z R | T r f ( x ) | γ dxψdr ≤ sup r ∈ R k T r f k κL ∞ Z Z R | T r f ( x ) | γ − κ dxψdr ≤ k f ′ k κ k f k κ Z Z R | T r f ( x ) | γ − κ dxψdr. If, in addition, 2 ≤ γ − κ ≤ ψ ∈ L − γ + κ ( R ), then we can use Lemma 2.1 to see Z Z R | T r f ( x ) | γ − κ dxψdr . k f k γ − κ , where the implicit constant depends only on the L − γ + κ norm of ψ . Thus, Z Z R | T r f ( x ) | γ dxψdr . k f ′ k κ k f k γ − κ for all ( γ − + ≤ κ ≤ γ −
2. With the bound (2.16) and the definition of N ( f ) this proves(2.21) under the assumption that ψ ∈ L − γ κ ∩ L − γ κ . Proposition 2.17 (Splitting N ) . Assume that V obeys assumption A1 . (i) If ≤ γ ≤ γ ≤ and ψ ∈ L ∩ L − γ , then | N ( f + f ) − N ( f ) − N ( f ) | . k f kk f k (cid:0) k f k + k f k (cid:1) . (2.24)(ii) If < γ ≤ γ < and τ > , then with α ( γ , τ ) and β ( γ , τ ) as in Proposition 2.8, | N ( f + f ) − N ( f ) − N ( f ) | . s − min { α ( γ ,τ ) ,α ( γ ,τ ) } k f kk f k (cid:0) k f k + k f k (cid:1) (2.25) if ψ ∈ L ∩ L β ( γ ,τ ) and s = dist(supp ˆ f , supp ˆ f ) > , or ψ ∈ L β ( γ ,τ ) has compactsupport and s = dist(supp f , supp f ) > . (iii) If ≤ γ ≤ γ < ∞ and ψ ∈ L , then | N ( f + f ) − N ( f ) − N ( f ) | . k f kk f k (cid:16) k f k γ − H + k f k γ − H (cid:17) . (2.26)(iv) If ≤ γ ≤ γ < ∞ and ψ ∈ L has compact support, then | N ( f + f ) − N ( f ) − N ( f ) | . s − k f k H k f k H (cid:16) k f k γ − H + k f k γ − H (cid:17) (2.27) with s = dist(supp f , supp f ) > . HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 17
Proof.
Because of Lemma 2.14 and the Definition 2.5 of M γψ , we have | N ( f + f ) − N ( f ) − N ( f ) |≤ Z Z R (cid:12)(cid:12)(cid:12) V ( | T r f ( x ) + T r f ( x ) | ) − V ( | T r f ( x ) | ) − V ( | T r f ( x ) | ) (cid:12)(cid:12)(cid:12) dxψdr . M γ ψ ( f , f ) + M γ ψ ( f , f ) . (2.28)So (2.26) follows from Proposition 2.7 and (2.27) follows from Proposition 2.13, noting alsothat ( a + b ) γ − + ( a + b ) γ − . a γ − + b γ − , for all a, b ≥
0. Similarly, (2.24) follows from Proposition 2.7 as long as ψ ∈ L − γ ∩ L − γ .Since we also assume ψ ∈ L for convenience, this condition reduces to ψ ∈ L ∩ L − γ .For the proof of (2.25), we first assume s = dist(supp ˆ f , supp ˆ f ) >
0. Then Proposition2.8 shows M γψ ( f , f ) . s − α ( γ,τ ) k f kk f k ( k f k + k f k ) γ − for any 2 < γ < τ >
1, as long as ψ ∈ L β ( γ,τ ) .Thus (2.25) follows from (2.28) as long as ψ ∈ L β ( γ ,τ ) ∩ L β ( γ ,τ ) . Noting1 < β ( γ , τ ) ≤ β ( γ , τ ) and L ∩ L β ( γ ,τ ) ⊂ L β ( γ ,τ ) ∩ L β ( γ ,τ ) finishes the proof of (2.25) when ˆ f and ˆ f have separated supports.If s = dist(supp f , supp f ) >
0, we make the simple observation that for any compactlysupported ψ one has ψ ∈ L p ⇒ ψ ∈ L p ( | r | a dr ) ∩ L for any weight | r | a with a ≥ p ≥
1. With this observation, the above proofs carry overto the case that the functions f and f have separated supports, using now Proposition2.12 instead of Proposition 2.8.3. Strict subadditivity of the ground state energy
Recall that for d av ≥ H ( f ) = d av k f ′ k − N ( f )and E d av λ = inf (cid:8) H ( f ) : k f k = λ (cid:9) , where, if f ∈ L \ H , we set k f ′ k = ∞ , so the infimum in the definition of E d av λ is over all f ∈ H with fixed L norm if d av > α δ = max { , − γ + δ } for δ ≥ Lemma 3.1.
Assume that V obeys assumption A1 . (i) If d av = 0 , < γ ≤ γ ≤ and ψ ∈ L ∩ L − γ , then for every λ > −∞ < E λ ≤ , in particular, the variational problem (1.1) is well–posed. (ii) If d av > , < γ ≤ γ < and ψ ∈ L ∩ L α δ for some δ > , then the energyfunctional H ( f ) is coercive in k f ′ k for fixed k f k , that is, lim k f ′ k→∞ H ( f ) = ∞ (3.1) for fixed k f k = λ > . Also −∞ < E d av λ ≤ and thus the variational problem (1.1) is well–posed and any minimizing sequence ( f n ) n ⊂ H ( R ) for E d av λ is bounded in the H norm. (iii) If V obeys assumption A4 , then E d av λ < for any λ > and d av ≥ .Proof. If 2 < γ ≤ γ ≤ κ = κ = 0 in Proposition 2.15 to see that for any ψ ∈ L ∩ L − γ one has | N ( f ) | . k f k γ + k f k γ . Thus for d av = 0 we have E λ = − sup k f k = λ N ( f ) & − ( λ γ + λ γ ) > −∞ . To get a finite lower bound for E d av λ for d av > < γ ≤ γ <
10 we have to do alittle bit of numerology first: If α δ = 1, simply set κ j := γ j −
2. Since ψ ∈ L the conditionson ψ from Proposition 2.15 are clearly satisfied. Note that if α δ = 1, then necessarily γ <
6, thus also γ ≤ γ <
6, and hence( γ j − + = 0 ≤ κ j = γ j − . This shows that the condition on κ j from Proposition 2.15 are fulfilled and it also showsthat κ j ≤ γ − < α δ >
1, pick β j > − γ j − β j = α δ >
1. Setting κ j := (4 − β j ) + we certainlyhave 0 ≤ κ j <
4. Also, since 10 − γ j − β j >
0, we have β j < − γ j and this implies κ j = (4 − β j ) + ≥ ( γ j − + . Also, since − γ j − β j >
1, we have κ j < γ j − . So again, the conditions on κ j from Proposition 2.15 are fulfilled. So from (2.21) we get | N ( f ) | . k f ′ k κ k f k γ − κ + k f ′ k κ k f k γ − κ and there exists a constant C > L − γ κ and L − γ κ norms of ψ such that H ( f ) ≥ d av k f ′ k − C (cid:16) k f ′ k κ k f k γ − κ + k f ′ k κ k f k γ − κ (cid:17) . (3.2)Since κ j = (4 − β j ) + , for α δ >
1, we have 6 − γ j + κ j ≥ − γ j − β j , so1 < − γ j + κ j ≤ − γ j − β j = α δ and by interpolating, or simply H¨older’s inequality, the constant C in (3.2) can be made todepend only on the L and L α δ norms of ψ . HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 19
If we fix k f k = λ >
0, then we can rewrite (3.2) with k f ′ k = t as H ( f ) ≥ d av t − C (cid:18) t κ λ γ − κ + t κ λ γ − κ (cid:19) . (3.3)Since κ j < j = 1 ,
2, this immediately implies (3.1).The lower bound (3.2) also shows that E d av λ = inf (cid:8) H ( f ) : k f k = λ (cid:9) ≥ inf t> (cid:18) d av t − C (cid:18) t κ λ γ − κ + t κ λ γ − κ (cid:19)(cid:19) > −∞ . The coercivity expressed in (3.3) also makes it easy to see that any minimizing sequence( f n ) n ⊂ H ( R ) for E d av λ is bounded in the H norm. Indeed, if f n is such that k f n k = λ > H ( f n ) → E d av λ > −∞ as n → ∞ , then the lower bound (3.3) shows that k f ′ n k staysbounded and hence also k f n k H = k f n k + k f ′ n k stays bounded.To finish the proof of Lemma 3.1, we have to show that for λ > d av ≥ E d av λ ≤ E d av λ <
0, if, in addition, assumption A4 on V holds. We will do thisby computing the energy of suitable Gaussians.For this, we let g σ be the centered Gaussian from (B.11) with σ >
0. Then k g σ k = λ > k g ′ σ k = λ/σ , and its time evolution is given in Lemma B.3 by T r g σ ( x ) = (cid:18) λ πσ (cid:19) / (cid:18) σ σ ( r ) (cid:19) / e − x σ ( r ) with σ ( r ) = σ + 4 ir . The first bound from Lemma 2.14 shows | N ( g σ ) | ≤ Z Z R | V ( | T r g σ ( x ) | ) | dxψdr . k ψ k L (cid:0) k T r g σ k γ L γ + k T r g σ k γ L γ (cid:1) and Lemma B.3 gives k T r g σ k γL γ = (cid:18) πγ (cid:19) / (cid:18) λ π (cid:19) γ/ σ − γ − (cid:18) | σ || σ ( r ) | (cid:19) γ − ≤ (cid:18) πγ (cid:19) / (cid:18) λ π (cid:19) γ/ σ − γ − . Thus, H ( g σ ) = d av k g ′ σ k − N ( g σ ) ≤ d av · λσ + C k ψ k L (cid:18) πγ (cid:19) / "(cid:18) λ π (cid:19) γ / σ − γ − + (cid:18) λ π (cid:19) γ / σ − γ − for some constant C >
0. Since 2 < γ ≤ γ we can let σ → ∞ to seelim σ →∞ H ( g σ ) = 0which clearly implies E d av λ ≤ A4 when d av >
0, we consider σ large enough so that | T r g σ ( x ) | ≤ (cid:18) λ πσ (cid:19) / (cid:18) | σ || σ ( r ) | (cid:19) / ≤ (cid:18) λ πσ (cid:19) / < ǫ. Then A4 implies the lower bound N ( g σ ) = Z Z R V ( | T r g σ ( x ) | ) dxψdr & Z Z R | T r g σ ( x ) | κ dxψdr = (cid:18) πκ (cid:19) / (cid:18) λ π (cid:19) κ σ − κ Z R (cid:18) σ | σ ( r ) | (cid:19) κ − ψ ( r ) dr = (cid:18) πκ (cid:19) / (cid:18) λ π (cid:19) κ σ − κ Z R ψ ( r )[1 + (4 r/σ ) ] κ − dr, where in the second line we used (B.13). Thus the energy of this Gaussian test function isbounded above by H ( g σ ) ≤ d av λ σ " − Cd av λ (cid:18) πκ (cid:19) / (cid:18) λ π (cid:19) κ σ − κ Z R ψ ( r )(1 + (4 r/σ ) ) κ − dr for some constant C . So, using a large enough σ , we get H ( g σ ) < < κ < Z R ψ ( r )[1 + (4 r/σ ) ] κ − dr → k ψ k L as σ → ∞ by Lebesgue’s dominated convergence theorem.If d av = 0, we again use the Gaussian g σ with σ so large that 0 < | T r g σ | ≤ ε . Then A4 implies H ( g σ ) = − N ( g σ ) < , so E λ < Lemma 3.2. V obeys A2 if and only if for all t ≥ we have V ( ta ) ≥ t γ V ( a ) for all a > . (3.4) Proof.
Assume that V obeys A2 . Then ddt V ( ta ) = V ′ ( ta ) a ≥ γ t V ( ta )for all a > t >
1. Thus ddt ( t − γ V ( ta )) ≥ t = 1, we candifferentiate it at t = 1 to get A2 .The lower bound from Lemma 3.2 will be the main input for the following quantita-tive version of strict subadditivity of E d av λ , which in turn will be crucial in the proof ofPropositions 4.3 and 4.4. Proposition 3.3 (Strict Subadditivity) . Under assumptions A1 and A2 and for any λ > , < δ < λ/ , and λ , λ ≥ δ with λ + λ ≤ λ , we have E d av λ + E d av λ ≥ " − (2 γ − (cid:18) δλ (cid:19) γ E d av λ , for γ > as in assumption A2 . Remark 3.4.
Since 1 − (2 γ − (cid:0) δλ (cid:1) γ < δ > γ >
2, the energy is strictlysubadditive whenever E d av λ <
0, since then E d av λ + E d av λ > E λ for all λ , λ > λ + λ = λ >
0, by Proposition 3.3.
HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 21
Proof of Proposition 3.3:
First we show that for all λ > < µ ≤ E d av µλ ≥ µ γ E d av λ . (3.5)Setting e λ = µλ and µ = ρ − , one sees that the inequality (3.5) is equivalent to E d av ρ ˜ λ ≤ ρ γ E d av ˜ λ for all ρ ≥ , ˜ λ > f ∈ H ( R ), or f ∈ L ( R ) if d av = 0, with k f k = λ and ρ ≥
1, we get from Lemma 3.2 N ( ρ / f ) = Z Z R V ( ρ / | T r f ( x ) | ) dxψdr ≥ ρ γ N ( f ) . Since k ρ / f k = ρλ , ρ ≥ γ > H ( ρ / f ) ≤ ρ d av k f ′ k − ρ γ N ( f ) ≤ ρ γ H ( f ) , which proves (3.6).Now let λ = µ λ and λ = µ λ with µ + µ ≤ µ , µ ≥ δ/λ . Using (3.5), we get E d av λ + E d av λ = E d av µ λ + E d av µ λ ≥ ( µ γ + µ γ ) E d av λ . (3.7)Without loss of generality, we may assume that δ ≤ µ ≤ µ . Using this and µ + µ ≤ µ γ + µ γ = ( µ + µ ) γ − (cid:18) ( µ + µ ) γ − µ γ − µ γ (cid:19) = ( µ + µ ) γ − µ γ (cid:18) µ µ (cid:19) γ − − (cid:18) µ µ (cid:19) γ ! ≤ − µ γ (cid:16) γ − (cid:17) ≤ − (cid:18) δλ (cid:19) γ (cid:16) γ − (cid:17) (3.8)where we have also used that the function t (1 + t ) γ − − t γ is increasing on [1 , ∞ ).Since by Lemma 3.1 we always have E d av λ ≤
0, we can use (3.8) in (3.7) to get E d av λ + E d av λ ≥ " − (cid:18) δλ (cid:19) γ (cid:16) γ − (cid:17) E d av λ which completes the proof. 4. The existence proof
In this section we will characterize when minimizing sequences are precompact modulotranlations and boosts. Recall the definition of the exponent α δ = α δ ( γ ) = − γ + δ . Theorem 4.1.
Let λ > and assume that V obeys A1 and A2 and that the density ψ hascompact support. (i) If d av > , < γ ≤ γ < , and ψ ∈ L α δ for some δ > , then every minimizingsequence for the variational problem (1.1) is precompact modulo translations if and onlyif E d av λ < . (ii) If d av = 0 , < γ ≤ γ < , and ψ ∈ L − γ + δ for some δ > , then every minimizingsequence for the variational problem (1.1) is precompact modulo translations and boostsif and only if E λ < . In both cases minimizers of (1.1) exist if E d av λ < , and these miniminzers are solutions ofthe dispersion management equation (1.12) for some Lagrange multiplier ω < E d av λ /λ < . Remark 4.2.
This theorem shows that compactness modulo translation, respectively mod-ulo translations and boost, for minimizing sequences is equivalent to strict negativity of theenergy.Key for our proof of Theorem 4.1 are the following propositions, which will help toeliminate splitting of minimizing sequences. First, we introduce notations. For s > < α ≤
1, define G α ( s ) := h ( s + 1) α α − i − / . (4.1)Note that G α is a decreasing function on (0 , ∞ ) which vanishes at infinity, which is impor-tant for us, and lim s → + G α ( s ) = ∞ (4.2)which is of less importance. Moreover, for x ∈ R , let x + := max { x, } . Proposition 4.3 (Fat-tail for positive average dispersion) . Assume V obeys A1 with <γ ≤ γ < and A2 , d av > and ψ ∈ L has compact support. Let λ > , f ∈ H with k f k = λ , and < δ < λ/ , and choose any a, b ∈ R with Z a −∞ | f ( x ) | dx ≥ δ and Z ∞ b | f ( x ) | dx ≥ δ (4.3) then H ( f ) ≥ " − (2 γ − (cid:18) δλ (cid:19) γ E d av λ − C k f k H (cid:0) k f k H (cid:1) G (( b − a − + ) , (4.4) where the constant C depends only on the support and the L norm of ψ . We have a similar bound in the case of vanishing average dispersion.
Proposition 4.4 (Fat-tail for zero average dispersion) . Assume V obeys A1 with < γ ≤ γ < and A2 , d av = 0 and ψ ∈ L β ( γ ,τ ) has compact support . Let λ > , f ∈ L with k f k = λ , and < δ < λ/ , and a, b ∈ R with either Z a −∞ | f ( x ) | dx ≥ δ and Z ∞ b | f ( x ) | dx ≥ δ (4.5) or Z a −∞ | b f ( η ) | dη ≥ δ and Z ∞ b | b f ( η ) | dη ≥ δ, (4.6) then H ( f ) ≥ " − (2 γ − (cid:18) δλ (cid:19) γ E λ − Cλ (1 + λ ) G min { α ( γ ,τ ) ,α ( γ ,τ ) } (( b − a − + ) (4.7) where the constant C depends only on the support and the L β ( γ ,τ ) norm of ψ . Recall the definition of α ( γ, τ ) and β ( γ, τ ) from Proposition 2.8. HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 23
Proof of Proposition 4.3. If b − a ≤
1, (4.4) holds immediately since its right hand sideis −∞ by (4.2). So now we assume that b − a >
1. Let a ′ and b ′ be arbitrary numberssatisfying a ≤ a ′ < b ′ ≤ b and b ′ − a ′ ≥
1, which we will suitably choose later. The estimateof k f ′ k is based on a one-dimensional version of the well-known IMS localization formula k f ′ k = X j h ( ξ j f ) ′ , ( ξ j f ) ′ i − X j h f, | ξ ′ j | f i (4.8)for any collection of functions { ξ j } which are smooth, 0 ≤ ξ j ≤
1, and P j ξ j = 1. Toconstruct such a partition which suits our needs, consider smooth functions { χ j } thatsatisfyi) 0 ≤ χ j ≤ j = − , , X j = − χ j = 1.iii) supp χ ⊂ [ − , ] , χ = 1 on [ − , ] , supp χ − ⊂ ( −∞ , − ] , χ − = 1 on ( −∞ , − ] , supp χ ⊂ [ , ∞ ) , χ = 1 on [ , ∞ ) . Let ξ j ( x ) = χ j x − ( a ′ + b ′ ) b ′ − a ′ ! for j = − , , . Since χ ′ j is bounded, we see that for some constant C > X j = − | ξ ′ j | ≤ C ( b ′ − a ′ ) . Plugging this into (4.8) yields k f ′ k ≥ k ( ξ − f ) ′ k + k ( ξ f ) ′ k + k ( ξ f ) ′ k − C k f k ( b ′ − a ′ ) ≥ k ( ξ − f ) ′ k + k ( ξ f ) ′ k − C k f k ( b ′ − a ′ ) . (4.9)Now we set f j := ξ j f for j = − , f := f − f − f − = (1 − ξ − − ξ ) f , where we notethat f is defined differently from f − and f !Obviously, k f j k ≤ k f k for j = − ,
1, and since the supports of ξ − and ξ are disjointalso | f | ≤ | f | , hence k f k ≤ k f k .Set h := f − + f . Then f = f + h and the bound (2.26) from Proposition 2.17 shows N ( f ) − N ( f ) − N ( h ) . k f kk h k (cid:0) k f k H + k h k H (cid:1) where we also used 1 + a γ − . a for all a ≥ γ <
10. Using Proposition 2.15,more precisely equation (2.23), which is one of its consequences, we have N ( f ) . k f k (cid:16) k f k γ − H + k f k γ − H (cid:17) . k f k (cid:0) k f k H (cid:1) , and combining the above two bounds we arrive at N ( f ) − N ( h ) . k f kk f k (1 + k f k H ) , (4.10)where used k f k , k h k ≤ k f k and also k f k H , k h k H . k f k H , the latter holds because ofour smoothness assumptions on the cut-off functions ξ j uniformly in b ′ − a ′ ≥ Since f − and f have supports separated by at least ( b ′ − a ′ ) /
2, (2.27) gives N ( h ) − N ( f − ) − N ( f ) . ( b ′ − a ′ ) − k f − k H k f k H (cid:0) k f − k H + k f k H (cid:1) . ( b ′ − a ′ ) − k f k H (cid:0) k f k H (cid:1) (4.11)where we again used that, because of our assumption that b ′ − a ′ ≥
1, the bound k f j k H . k f k H holds, where the implicit constant does not depend on a ′ and b ′ .Combining (4.10) and (4.11), we get N ( f ) − N ( f − ) − N ( f ) . (cid:18) k f kk f k + k f k H b ′ − a ′ (cid:19) (cid:0) k f k H (cid:1) so when combined with (4.9), this yields H ( f ) − H ( f − ) − H ( f ) & − (cid:20) k f k ( b ′ − a ′ ) + (cid:18) k f kk f k + k f k H b ′ − a ′ (cid:19) (cid:0) k f k H (cid:1)(cid:21) . (4.12)To choose a ′ and b ′ , we use a continuous version of the pigeon hole principle, as in ourprevious work [12]: Let 1 ≤ l ≤ b − a and note that Z b − la Z y + ly | f ( x ) | dxdy ≤ Z ba Z xx − l | f ( x ) | dydx ≤ l k f k . (4.13)Moreover, by the mean value theorem, there exists y ′ ∈ ( a, b − l ) such that( b − a − l ) Z y ′ + ly ′ | f ( x ) | dx = Z b − la Z y + ly | f ( x ) | dxdη. Thus, since f has support in [ a ′ , b ′ ] and | f | ≤ | f | , choosing a ′ = y ′ and b ′ = y ′ + l in theprevious identity together with (4.13) gives l = b ′ − a ′ and k f k ≤ k f [ a ′ ,b ′ ] k ≤ lb − a − l k f k . Plugging this into (4.12) yields H ( f ) − H ( f − ) − H ( f ) & − " k f k l + (cid:18) lb − a − l (cid:19) / k f k + k f k H l ! (cid:0) k f k H (cid:1) ≥ −k f k H (cid:0) k f k H (cid:1) " l + (cid:18) lb − a − l (cid:19) / + 1 l . Since k f k = λ , λ j = k f j k ≥ δ, j = − , λ + λ = k f − k + k f k ≤ λ , we certainlyhave H ( f − ) + H ( f ) ≥ E d av λ + E d av λ ≥ " − (2 γ − (cid:18) δλ (cid:19) γ E d av λ by Proposition 3.3. Thus we arrive at the bound H ( f ) − " − (2 γ − (cid:18) δλ (cid:19) γ E d av λ & −k f k H (cid:0) k f k H (cid:1) " l + (cid:18) lb − a − l (cid:19) / + 1 l for any 0 < δ < λ/ ≤ l ≤ b − a . Now we choose l = √ b − a . Then 1 ≤ l ≤ b − a since b − a ≥
1, andmax ( l , (cid:18) lb − a − l (cid:19) / , l ) = (cid:18) lb − a − l (cid:19) / = (cid:18) b − a ) / − (cid:19) / = G (( b − a − + ) HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 25 which completes the proof.
Proof of Proposition 4.4.
Since its proof is very analogous to that of Proposition 4.3, let usmention only the things which need to be changed: In the case of zero average dispersion,the energy contains no k f ′ k term, hence we do not need to use smooth cut-offs, that is,we can use f = f − + f + f where we set f − = f ( −∞ ,a ′ ) , f = f [ a ′ ,b ′ ] and f = f ( b ′ , ∞ ) ,and similarly for ˆ f .We can then simply repeat the argument in the proof of (4.12), again using (2.24) butnow combined with (2.25) instead of (2.27), to see that H ( f ) − H ( f − ) − H ( f ) & − (cid:18) k f kk f k + k f k ( b ′ − a ′ ) min { α ( γ ,τ ) ,α ( γ ,τ ) } (cid:19) (cid:0) k f k (cid:1) ≥ − λ (cid:0) λ (cid:1) "(cid:18) lb − a − l (cid:19) / + 1 l min { α ( γ ,τ ) ,α ( γ ,τ ) } (4.14)with the only restriction that l = b ′ − a ′ ≥ < b − a ≤
1, we note that (4.7) trivially holds since the right hand side equals −∞ .So let b − a >
1. We choose l := ( b − a ) { α ( γ ,τ ) ,α ( γ ,τ ) } . Then 1 < l < b − a and l { α ( γ ,τ ) ,α ( γ ,τ ) } = b − a > b − a − l > (cid:18) lb − a − l (cid:19) / ≥ l min { α ( γ ,τ ) ,α ( γ ,τ ) } . This together with (4.14) and our choice of G min { α ( γ ,τ ) ,α ( γ ,τ ) } (( b − a − + ), which satisfies0 < min { α ( γ , τ ) , α ( γ , τ ) } ≤
1, finishes the proof.Since the function G α is decreasing on R + and vanishes at infinity, similar results toProposition 2.4 in [12] follow from Propositions 4.3 and 4.4. Proposition 4.5 (Tightness for Positive Average Dispersion) . Under the conditions ofTheorem 4.1 on V , γ , γ , and ψ , let ( f n ) n ⊂ H ( R ) be a minimizing sequence for thevariational problem (1.1) for d av > with λ = k f n k > and assume E d av λ < . Then thereexists K < ∞ such that, for any L > , sup n ∈ N Z | η | >L | ˆ f n ( η ) | dη ≤ KL (4.15) i.e., the sequence is tight in Fourier space. Moreover, there exist shifts y n such that lim R →∞ sup n ∈ N Z | x | >R | f n ( x − y n ) | dx = 0 , (4.16) i.e., the shifted sequence is also tight. Proposition 4.6 (Tightness for Zero Average Dispersion) . Under the conditions of Theo-rem 4.1 on V , γ , γ , and ψ , let ( f n ) n ⊂ L ( R ) be a minimizing sequence for the variationalproblem (1.1) for d av = 0 with λ = k f n k > and assume E λ < . Then there exist shifts y n and boosts ξ n such that lim L →∞ sup n ∈ N Z | η − ξ n | >L | b f n ( η ) | dη = 0 (4.17) and lim R →∞ sup n ∈ N Z | x − y n | >R | f n ( x ) | dx = 0 . (4.18) Proof of Proposition 4.5.
Let ( f n ) n be a minimizing sequence. Lemma 3.1 shows that k f ′ n k is bounded, that is, K := sup n ∈ N k f ′ n k < ∞ . Thus, for every n ∈ N and L >
0, we obtain Z | η | >L | ˆ f n ( η ) | dη ≤ Z | η | >L | η | L | ˆ f n ( η ) | dη ≤ Z R | η | L | ˆ f n ( η ) | dη ≤ KL which is (4.15).To prove the second bound, we follow the argument of [12] closely. We give some detailsfor the readers’ convenience. Define a n,δ and b n,δ by a n,δ := inf (cid:26) a ∈ R : Z a −∞ | f n ( x ) | dx ≥ δ (cid:27) and b n,δ := sup (cid:26) b ∈ R : Z ∞ b | f n ( x ) | dx ≥ δ (cid:27) . Note that the measure | f n ( x ) | dx is absolutely continuous with respect to Lebesgue measureand hence Z a n,δ −∞ | f n ( x ) | dx = δ and Z ∞ b n,δ | f n ( x ) | dx = δ. Furthermore δ a n,δ and δ b n,δ are monotone, more precisely, for 0 < δ < δ < λ/ a n,δ ≤ a n,δ and b n,δ ≥ b n,δ . Let R n,δ := b n,δ − a n,δ and note that the abovemonotonicity yields R n,δ ≥ R n,δ for 0 < δ < δ < λ/
2. Lastly, for some fixed 0 < δ < λ/ y n := b n,δ + a n,δ ∈ [ a n,δ , b n,δ ] . In particular, a n,δ ≤ a n,δ ≤ y n ≤ b n,δ ≤ b n,δ for all 0 < δ ≤ δ . This implies b n,δ − y n ≤ b n,δ − a n,δ = R n,δ and y n − a n,δ ≤ b n,δ − a n,δ = R n,δ , hence we are guaranteed that[ a n,δ , b n,δ ] ⊂ [ y n − R nδ , y n + R n,δ ] . (4.19)Now assume that R δ := sup n ∈ N R n,δ < ∞ (4.20)for 0 < δ ≤ δ and put R δ := R δ for δ < δ < λ/
2. Then (4.19) yields Z | x − y n | >R δ | f n ( x ) | dx ≤ Z a n,δ −∞ | f n ( x ) | dx + Z ∞ b n,δ | f n ( x ) | dx = 2 δ for all 0 < δ ≤ δ but the same bound also holds when δ < δ < λ/ Z | x − y n | >R δ | f n ( x ) | dx = Z | x − y n | >R δ | f n ( x ) | dx ≤ δ < δ. It remains to show (4.20): Recall that K := sup n ∈ N k f ′ n k < ∞ , set e K = √ λ + K , andnote that the minimizing sequence f n obeys k f n k H ≤ e K for all n ∈ N . Using b = b n,δ and a = a n,δ , rearranging (4.4) from Proposition 4.3 yields E d av λ − (2 γ − (cid:18) δλ (cid:19) γ E d av λ − H ( f n ) ≤ C e K (1 + e K ) G (( R n,δ − + ) . HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 27
Thus, since H ( f n ) → E d av λ < < − (2 γ − (cid:18) δλ (cid:19) γ E d av λ ≤ C e K (1 + e K ) lim inf n →∞ G (( R n,δ − + ) . Since G is monotone decreasing, we get G ((lim sup n →∞ R n,δ − + ) = lim inf n →∞ G (( R n,δ − + ) > n →∞ R n,δ < ∞ . Hence (4.20) holds.
Proof of Proposition 4.6.
Using the fact that the function G α is monotone decreasing, theproof is virtually identical to the proof of the bound (4.16) of Proposition 4.5.To prove Theorem 4.1, we need one more result for the continuity of the nonlinearfunctional N ( f ). Lemma 4.7. (i) If ≤ γ ≤ γ ≤ and ψ ∈ L − γ then the nonlinear nonlocal functional N : L ( R ) → R given by L ( R ) ∋ f N ( f ) = Z Z R V ( | T r f | ) dxψdr is locally Lipshitz continuous on L in the sense that | N ( f ) − N ( f ) | . (cid:0) k f k γ − + k f k γ − (cid:1) k f − f k where the implicit constant depends only on the L − γ norm of ψ . (ii) If ≤ γ ≤ γ < ∞ and ψ ∈ L , then the nonlinear nonlocal functional N : H ( R ) → R given by H ( R ) ∋ f N ( f ) = Z Z R V ( | T r f | ) dxψdr is locally Lipschitz continuous in the sense that | N ( f ) − N ( f ) | . (cid:16) k f k γ − H + k f k γ − H (cid:17) ( k f k + k f k ) k f − f k . Remark 4.8.
Note that the second part of Lemma 4.7 shows that the Lipschitz constantof N on H depends on the H norm, however, if f and f are bounded in H , then thedifference N ( f ) − N ( f ) is small whenever f is close to f in the much weaker L norm! Proof.
We always have | N ( f ) − N ( f ) | ≤ Z Z R | V ( | T r f | ) − V ( | T r f | ) | dxψdr and from (2.17) we see that | V ( | T r f | ) − V ( | T r f | ) | . | T r ( f − f ) | (cid:0) ( | T r f | + | T r f | ) γ − + ( | T r f | + | T r f | ) γ − (cid:1) so | N ( f ) − N ( f ) | . k T r ( f − f )( | T r f | + | T r f | ) γ − k L ( R , dxψdr ) + k T r ( f − f )( | T r f | + | T r f | ) γ − k L ( R , dxψdr ) . (4.21) If 2 ≤ γ ≤ γ , and γ/ ( γ − k T r ( f − f )( | T r f | + | T r f | ) γ − k L ( R , dxψdr ) ≤k T r ( f − f ) k L γ ( R ,dxψdr ) k| T r f | + | T r f |k γ − L γ ( R ,dxψdr ) . Applying the triangle inequality and Lemma 2.1 then yields k T r ( f − f )( | T r f | + | T r f | ) γ − k L ( R , dxψdr ) . k f − f k (cid:0) k f k γ − + k f k γ − (cid:1) where the implicit constant depends only on the L − γ norm of ψ . Using these bounds in(4.21) shows that | N ( f ) − N ( f ) | . (cid:0) k f k γ − + k f k γ − + k f k γ − + k f k γ − (cid:1) k f − f k . (cid:0) k f k γ − + k f k γ − (cid:1) k f − f k where we also used that a γ − + a γ − . a γ − for all a ≥ ≤ γ ≤ γ . Thisproves the first part of Lemma 4.7 when 2 ≤ γ ≤ γ ≤ γ ≥ k T r ( f − f )( | T r f | + | T r f | ) γ − k L ( R , dxψdr ) ≤ k T r ( f − f ) k L ( R , dxψdr ) k ( | T r f | + | T r f | ) γ − k / L ( R , dxψdr ) ≤ k ψ k L k f − f kk ( | T r f | + | T r f | ) γ − k / L ( R , dxψdr ) using Lemma 2.1. Since γ ≥
2, we can further split k ( | T r f | + | T r f | ) γ − k / L ( R , dxψdr ) ≤ sup r ∈ R ( k T r f k L ∞ + k T r f k L ∞ ) γ − k| T r f | + | T r f |k L ( R , dxψdr ) . Because of (2.10) the first factor is bounded by ( k f k H + k f k H ) γ − and using Lemma2.1 and the triangle inequality, the second factor is bounded by k ψ k L ( k f k + k f k ). Usingthis in (4.21) proves the second part of the lemma for 2 ≤ γ ≤ γ < ∞ . Lemma 4.9. If ≤ γ ≤ γ ≤ and ψ ∈ L ∩ L − γ , respectively if ≤ γ ≤ γ < ∞ and ψ ∈ L , then for any f, h ∈ L ( R ) , respectively f, h ∈ H ( R ) , the functional N as abovehas directional derivative given by D h N ( f ) = Z R Re (cid:10) T r h, V ′ ( | T r f | )sgn( T r f ) (cid:11) ψdr. Proof.
Let f ∈ L ( R ) and ǫ = 0. Fix any h ∈ L ( R ) and the quotient of N is N ( f + ǫh ) − N ( f ) ǫ = 1 ǫ (cid:20)Z Z R V ( | T r ( f + ǫh ) | ) − V ( | T r f | ) dxψdr (cid:21) = 1 ǫ Z Z R Z dds V ( | T r ( f + sǫh ) | ) dsdxψdr. (4.22)By straightforward calculations, we obtain dds V ( | T r ( f + sǫh ) | ) = V ′ ( | T r ( f + sǫh ) | ) ǫ ( T r f T r h + T r hT r f + 2 sǫ | T r h | )2 | T r ( f + sǫh ) | HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 29 and thus(4.22) =
Z Z R Z V ′ ( | T r ( f + sǫh ) | ) T r f T r h + T r hT r f + 2 sǫ | T r h | | T r ( f + sǫh ) | dsdxψdr. By Lebesgue’s dominated convergence theorem, letting ǫ →
0, we get D h N ( f ) = Z Z R Z V ′ ( | T r f | ) Re( T r f T r h ) | T r f | dsdxψdr = Z Z R V ′ ( | T r f | ) Re( T r f T r h ) | T r f | dxψdr which completes the proof when 2 ≤ γ ≤ γ ≤
6. The case 2 ≤ γ ≤ γ < ∞ is similar.Now we are ready to give the Proof of Theorem 4.1:
It is easy to see that if E d av λ = 0 for some λ >
0, then there areminimizing sequences which are not precompact modulo translations. Indeed, assume that E d av λ = 0 and let g n be the centered Gaussian from (B.11) with the choice σ = n ∈ N . Thisgives a sequence of Gaussians which weakly converges to zero and no translates or boostsof g n converges strongly and, as the proof of Lemma 3.1 shows, we have H ( g n ) → E d av λ as n → ∞ . By contrapositive, this is equivalent to that if every sequence is modulotranslations, respectively modulo translations and boosts, then necessarily E d av λ < E d av λ < d av >
0. Let ( f n ) n ⊂ H ( R ) be a minimizing sequence of the variational problem (1.1). Since k f n k = λ > f n is bounded in the H norm, K := sup n ∈ N k f n k H < ∞ . In addition, applying Proposition 4.5, there exist shifts y n such that for the shiftedsequence h n , h n ( x ) := f n ( x − y n ) for x ∈ R , we havelim R →∞ sup n ∈ N Z | x | >R | h n ( x ) | dx = 0 . (4.23)Clearly, by translation invariance of Lebesgue measure, we still have k h n k = λ . On theFourier side, shifts correspond to modulations with e iy n η , so for the shifted sequence h n Proposition 4.5 also yields that there exists K < ∞ such that for any L > n ∈ N Z | η | >L | ˆ h n ( η ) | dη ≤ K L . (4.24)Thus, by translation invariance of the minimization problem, the shifted sequence is aminimizing sequence for E d av λ which is tight in the sense of Lemma A.1. The shifted sequence h n is certainly also bounded in H , hence also bounded in L . By the weak sequentialcompactness of bounded sets in L and H , we can extract a subsequence, which by someslight abuse of notation, we still denote by h n , which converges weakly both to some f in L and some e f in H . By uniqueness of weak limits, we must have f = e f and by thecharacterization of strong convergence in L from Lemma A.1, we know that h n converges even strongly in L to f . In particular, k f k = lim n →∞ k h n k = λ > f = 0. Since H is a Hilbert space, we also have the weak sequential lower semi–continuityof the H norm, that is, k f k H ≤ lim inf n →∞ k h n k H (4.25) Since k f k = λ and k h n k = λ this implies k f ′ k ≤ lim inf n →∞ k h ′ n k , that is, the kinetic energy is lower semi–continuous.Since h n is bounded in H and h n converges strongly to f in L , Lemma 4.7 shows thatlim n →∞ N ( h n ) = N ( f ) . Together with the lower semi–continuity of the kinetic energy this implies E d av λ ≤ H ( f ) ≤ lim inf n →∞ H ( f n ) = E d av λ and since k f k = λ , this shows that f is a minimizer for the variational problem (1.1).It remains to show that the existence of a minimizer of (1.1) for d av = 0. Again, themain task is to use translations and boosts to massage an arbitrary minimizing sequenceinto one having a strongly convergent subsequence.Let ( f n ) n ⊂ L ( R ) be an arbitrary minimizing sequence of the variational problem (1.1)with k f n k = λ >
0. Proposition 4.6 guarantees the existence of shifts y n ∈ R andboosts ξ n ∈ R such that (4.17) and (4.18) hold. Define the shifted and boosted sequence( h n ) n = ( f ξ n ,y n ,n ) n by h n ( x ) = f ξ n ,y n ,n ( x ) := e iξ n x f n ( x − y n ) for x ∈ R . Note that k h n k = k f n k = λ since shifts and boost are unitary operations on L ( R ) and N ( f n ) = N ( h n ), see Appendix B. Hence ( h n ) n is also a minimizing sequence. Certainly | h n ( x ) | = | f n ( x − y n ) | for all n ∈ N . The Fourier transform of h n is given by b h n ( η ) = 1 √ π Z e − ixη e ixξ n f n ( x − y n ) dx = e − iy n η b f n ( η − ξ n ) . (4.26)Thus also | b h n ( η ) | = | b f n ( η − ξ n ) | . In particular, (4.17) and (4.18) show that the minimizingsequence ( h n ) n is tight in the sense of Lemma A.1.Since ( h n ) n is bounded in L ( R ), the weak compactness of the unit ball, guarantees theexistence of a weakly converging subsequence of ( h n ) n , denoted again by ( h n ) n . Obviously,this subsequence is also tight in the sense of Lemma A.1 and thus hence converges evenstrongly in L ( R ). We set f = lim n →∞ h n . By strong convergence k f k = lim n →∞ k h n k = λ . To conclude that f is the soughtafter minimizer we note that by Lemma 4.7 the map f N ( f ) = RR R V ( | T r f | ) dxψdr iscontinuous on L ( R ). Hence E λ ≤ H ( f ) = − N ( f ) = lim n →∞ − N ( h n ) = E λ where the last equality follows since ( h n ) n is a minimizing sequence for (1.1). Thus f is aminimizer for the variational problem (1.1).To prove that the above minimizer is a weak solution of the associated Euler-Lagrangeequation (1.12) is standard in the calculus of variations. One has to be a bit careful here,since we only have the directional derivative of N and hence of the energy functional H atour disposal. Let f be a minimizer of (1.1).Recall H ( f ) = d av k f ′ k − N ( f ). By Lemma 4.9 the directional derivative of the functionalof H at f ∈ H in direction h ∈ H is given by D h H ( f ) = d av Re h h ′ , f ′ i − D h N ( f ) . HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 31
Similarly, one can check that the derivative of ϕ ( f ) := k f k = h f, f i is given by D h ϕ ( f ) =2Re h h, f i .Now let f be any minimizer of the constraint variational problem (1.1) and h ∈ H arbitrary. Define, for any ( s, t ) ∈ R , F ( s, t ) := H ( f + sf + th ) ,G ( s, t ) := ϕ ( f + sf + th ) . Note that ∇ F ( s, t ) = (cid:18) D f H ( f + sf + th ) D h H ( f + sf + th ) (cid:19) and ∇ G ( s, t ) = (cid:18) D f ϕ ( f + sf + th ) D h ϕ ( f + sf + th ) (cid:19) = 2 (cid:18) Re h f, f + sf + th i Re h h, f + sf + th i (cid:19) . Since h f, f i = k f k = λ = 0, ∇ G (0 ,
0) = 2 (cid:18) h f, f i Re h h, f i (cid:19) = 2 (cid:18) λ Re h h, f i (cid:19) is not the zero vector in R and since ∇ G ( s, t ) depends multi-linearly, in particular contin-uously, on ( s, t ), the implicit function theorem [24] shows that there exists an open interval I ⊂ R containing 0 and a differentiable function φ on I with φ (0) = 0 such that λ = k f k = G (0 ,
0) = G ( φ ( t ) , t )for all t ∈ I . Consider the function I ∋ t F ( φ ( t ) , t ). Since f is a minimizer for theconstraint variational problem (1.1), F ( φ ( t ) , t ) has a local minimum at t = 0. Hence, usingthe chain rule,0 = dF ( φ ( t ) , t ) dt (cid:12)(cid:12)(cid:12) t =0 = ∇ F (0 , · (cid:18) φ ′ (0)1 (cid:19) = D f H ( f ) φ ′ (0) + D h H ( f ) . Since λ = G ( φ ( t ) , t ), the chain rule also yields0 = dG ( φ ( t ) , t ) dt (cid:12)(cid:12)(cid:12) t =0 = ∇ G (0 , · (cid:18) φ ′ (0)1 (cid:19) = 2 h f, f i φ ′ (0) + 2Re h h, f i . Solving this for φ ′ (0) and plugging it back into the expression for the derivative of F , wesee that D f H ( f ) h f, f i Re h h, f i = D h H ( f ) . In other words, with ω := D f H ( f ) λ (4.27)and f any minimizer of (1.1) we haveRe( ω h h, f i ) = D h H ( f ) = Re (cid:18) d av h h ′ , f ′ i − Z R (cid:10) T r h, V ′ ( | T r f | )sgn( T r f ) (cid:11) ψdr (cid:19) (4.28)for any h ∈ H , using the formula for D h N ( f ) from Lemma 4.9. Replacing h by ih in(4.28), one getsIm( ω h h, f i ) = Im (cid:18) d av h h ′ , f ′ i − Z R (cid:10) T r h, V ′ ( | T r f | )sgn( T r f ) (cid:11) ψdr (cid:19) (4.29) for all h ∈ H . (4.28) and (4.29) together show ω h h, f i = d av h h ′ , f ′ i − Z R (cid:10) T r h, V ′ ( | T r f | )sgn( T r f ) (cid:11) ψdr for any h ∈ L ( R ), that is, f is a weak solution of the dispersion management equation(1.12).It remains to prove ω < E d av λ /λ . For this, recall that assumption A2 states that V ′ ( a ) a ≥ γ V ( a ) for all a > . Thus D f N ( f ) = Z Z R V ′ ( | T r f | ) | T r f | dxψdr ≥ γ Z Z R V ( | T r f | ) dxψdr = γ N ( f )and since E d av λ <
0, we must have N ( f ) > f , so (4.27) gives ω ( f ) λ = D f H ( f ) = d av h f ′ , f ′ i − D f N ( f ) ≤ d av h f ′ , f ′ i − γ N ( f )= 2 H ( f ) − ( γ − N ( f ) < H ( f ) = 2 E d av λ < Threshold phenomena
As we showed in the previous section, assumptions A1 and A2 guarantee the existenceof minimizers for arbitrary λ > d av ≥ E d av λ is strictly negative . In this section we will prove there exists a threshold 0 ≤ λ d av cr ≤ ∞ suchthat solutions exist if the power λ = k f k > λ d av cr . Furthermore λ d av cr < ∞ if assumption A3 holds.For pure power law nonlinearities and the model case d = [0 , − [1 , for the diffractionprofile, this had been partly investigated earlier in [18] for the diffraction managementequation and for pure power nonlinearities in [28] for the discrete nonlinear Schr¨odingerequation. We are not aware of any work which investigates threshold phenomena for generalnonlinearities for dispersion management solitons in the continuum.In the following we will always assume, without explicitly mentioning it every time, that ψ is a probability density on R with compact support, that is, there exists 0 < R < ∞ such that supp ( ψ ) ⊂ [ − R, R ] together with further L p properties, depending on the rangeof γ ≤ γ . Our main result in this section is Theorem 5.1 (Threshold phenomenon) . Assume that V obeys A1 for some < γ ≤ γ < if d av > and some < γ ≤ γ < if d av = 0 and also A2 . Then (i) The map λ E d av λ is decreasing on (0 , ∞ ) , strictly decreasing where E d av λ < , andthere exists a critical threshold ≤ λ d av cr ≤ ∞ such that for < λ < λ d av cr we have E d av λ = 0 and for λ > λ d av cr we have −∞ < E d av λ < . (ii) If λ > λ d av cr , then minimizers of (1.1) exist and any minimizing sequence is, up totranslations, precompact in L ( R ) if d av > , respectively, precompact up to translationsand boosts in L if d av = 0 . In both cases the suitably translated, respectively translatedand boosted, minimizing sequence has a subsequence which converges to a minimizer. (iii) If < λ < λ d av cr and d av > , no minimizers of the variational problem (1.1) exist. (iv) If ≤ γ ≤ γ < , then λ d av cr > for all d av > . (v) If there exists f ∈ H ( R ) such that N ( f ) > , then one has λ d av cr < ∞ for all d av ≥ . HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 33
Remarks 5.2.
The precise definition of λ d av cr is given below in Definition 5.4. When λ > λ d av cr we have E d av λ < d av >
0, respectively, precompact modulo translations and boosts if d av = 0, and thus minimizers exist yielding solutions of (1.12) for some Lagrange multiplier ω < E d av λ /λ < E d av λ = 0 when 0 < λ < λ d av cr , Theorem 4.1 also shows that there are minimizingsequences which are not precompact modulo translations and boosts in this case. Never-theless, it could be that minimizers still exist. At least when d av >
0, Theorem 5.1 showsthat this cannot be the case. At the moment, we need d av > < λ < λ cr .We give the proof of Theorem 5.1 at the end of this section after some preparations.Recall the definition of the energy functional H ( f ) = d av k f ′ k − N ( f )and E d av λ = inf { H ( f ) : f ∈ H ( R ) , k f k = λ } . For strictly positive average dispersion, we write f ∈ H ( R ) with λ = k f k > f = √ λh .Then h ∈ H ( R ) with k h k = 1 and H ( f ) = d av k f ′ k − N ( f ) = k f ′ k d av − N ( √ λh ) λ k h ′ k ! . (5.1)In the case of vanishing average dispersion, we simply write, again for f ∈ H , H ( f ) = − N ( f ) = −k f ′ k N ( √ λh ) λ k h ′ k ! , so defining for d av ≥ R ( λ, h ) := N ( √ λh ) λ k h ′ k and R ( λ ) := sup (cid:8) R ( λ, h ) : h ∈ H , k h k = 1 (cid:9) we see that the following holds Lemma 5.3.
For any d av ≥ and λ > one has E d av λ < if and only if R ( λ, h ) > d av forsome h ∈ H ( R ) with k h k = 1 and this is the case if and only if R ( λ ) > d av .Proof. If d av > R ( λ ). Now let d av = 0. We note that E λ < f ∈ L with k f k = λ and N ( f ) >
0. For d av = 0, we consider only the range2 < γ ≤ γ ≤ L p properties of ψ which guarantee that L ∋ f N ( f ) is continuous, see Lemma 4.7. Since H is dense in L , we can find e f ∈ H with k e f k = k f k such that N ( e f ) > N ( f ) >
0. Thus we also have E λ < R ( λ ) > Note that the null space of ∂ x on H ( R ) is trivial, so R ( λ, h ) is well defined for any h ∈ H \ { } . We combine the cases d av > d av = 0 only so that we do not have to distinguish the two cases inLemma 5.6 and Corollary 5.7. Definition 5.4 (Threshold) . For d av ≥ λ d av cr := inf { λ > R ( λ ) > d av } . Remark 5.5.
It is clear from Lemma 5.3 that E d av λ < λ > d av ≥ R ( λ ) = ∞ , in which case λ d av cr = 0 for all d av ≥
0. Thus it is important toknow when R is finite. Using the bound from Proposition 2.15 with κ = κ = 4, which isallowed if 6 ≤ γ ≤ γ <
10, one sees that R ( λ ) . λ γ − + λ γ − < ∞ . Thus R ( λ ) < d av for small enough λ >
0, hence λ d av cr > d av > R in the study ofthreshold phenomena.For a pure power law nonlinearity, given by V ( a ) = a γ for some γ >
2, one can explicitlycalculate R ( λ ) = sup k h k =1 N ( √ λh ) λ k h ′ k = λ γ − R with R = sup k h k =1 R R k T r h k γL γ ψdr k h ′ k ∈ (0 , ∞ ]and Remark 5.5 shows that R < ∞ if γ ≥
6. This scaling property for R for pure powernonlinearities shows that R ( λ ) > d av for all λ > λ d av cr and R ( λ ) < d av for all 0 < λ < λ d av cr .Thus for pure power nonlinearities one immediately sees that the first claim of Theorem1.2 and 1.4 holds and λ d av cr > γ ≥
6. For the general class of nonlinearities one cannotexpect a simple scaling property of R to hold. However, condition A2 ensures a lowerbound of the same type which is enough to conlcude all necessary properties of R and thethreshold λ d av cr . This is made precise in the following Lemma 5.6.
Assume that V obeys assumption A2 . Then R ( λ ) ≥ (cid:18) λ λ (cid:19) γ − R ( λ ) (5.2) for all < λ ≤ λ . Before we give the proof we state and prove an important consequence.
Corollary 5.7.
Assume that V obeys assumption A2 . Then (i) If λ d av cr < ∞ , then R ( λ ) > d av for all λ > λ d av cr . (ii) If λ d av cr > , then R ( λ ) < d av for all < λ < λ d av cr . (iii) R ( λ ) < ∞ for some λ > if and only if lim sup λ → R ( λ ) ≤ . (iv) R ( λ ) > for some λ > if and only if lim inf λ →∞ R ( λ ) = ∞ .Proof. Take any λ > λ d av cr . The definition of λ d av cr shows that there exists λ with λ > λ >λ d av cr and R ( λ ) > d av . Then Lemma 5.6 shows R ( λ ) ≥ (cid:18) λλ (cid:19) γ − R ( λ ) ≥ R ( λ ) > d av , which proves the first claim.For the second claim, assume that R ( λ ) ≥ d av for some 0 < λ < λ d av cr . Then the boundfrom Lemma 5.6 shows that for every λ with λ < λ < λ d av cr we have R ( λ ) ≥ (cid:18) λ λ (cid:19) γ − R ( λ ) > R ( λ ) ≥ d av HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 35 which is in conflict with the definition of λ d av cr .For the third claim assume that R ( λ ) < ∞ . Setting λ = λ and λ = λ in Lemma 5.6,we get lim sup λ → R ( λ ) ≤ lim sup λ → (cid:18) λλ (cid:19) γ − R ( λ ) ≤ . The converse is easy.For the last claim assume that R ( λ ) > λ >
0. Arguing similarly as abovewe see that Lemma 5.6 implieslim inf λ →∞ R ( λ ) ≥ lim inf λ →∞ (cid:18) λλ (cid:19) γ − R ( λ ) = ∞ . Again, the converse is easy.It remains to give the
Proof of Lemma 5.6.
Fix h ∈ H ( R ) \ { } and define A ( s ) := s − N ( sh )for s >
0. Because of Lemma 4.9 , A is differentiable with derivative A ′ ( s ) = s − (cid:16) sD h N ( sh ) − N ( sh ) (cid:17) where sD h N ( sh ) − N ( sh ) = Z Z R (cid:2) V ′ ( | T r ( sh )( x ) | ) | T r ( sh )( x ) | − V ( | T r ( sh )( x ) | ) (cid:3) dxψdr ≥ ( γ − N ( sh )and the lower bound follows from assumption A2 . Thus we arrive at the first order differ-ential inequality A ′ ( s ) ≥ γ − s A ( s ) (5.3)for all s >
0. Using the integrating factor s − γ , one sees that this implies dds ( s − γ A ( s )) ≥ s − γ A ( s ) ≥ s − γ A ( s )for all 0 < s ≤ s . Since R ( λ, h ) = A ( √ λ ) / k h ′ k , this proves R ( λ , h ) ≥ (cid:18) λ λ (cid:19) γ − R ( λ , h )for all 0 < λ ≤ λ and taking the supremum over all h ∈ H ( R ) with k h k = 1 gives(5.2).Now we can give the proof of Proof of Theorem 5.1:
By Lemma 3.1 we know that E d av λ ≤ λ > d av ≥ E d av λ ≥ E d av λ + E d av λ ≥ E d av λ + λ where the last inequality is strict, when E d av λ + λ <
0. Thus 0 < λ E d av λ is decreasing andstrictly decreasing where E d av λ <
0. Corollary 5.7 and Lemma 5.3 show that E d av λ < λ > λ d av cr and if 0 < λ < λ d av cr , Corollary 5.7 and Lemma 5.3 yields E d av λ ≥
0, which togetherwith Lemma 3.1 shows E d av λ = 0 in this case. This proves the first claim.If λ > λ d av cr , we know by the first part that E d av λ <
0. So Theorem 4.1 applies. Thisproves the second part.To prove the third claim, assume that d av >
0, 0 < λ < λ d av cr , and f ∈ H with k f k = λ > E d av λ . Using (5.1) we must have0 = E d av λ = H ( f ) ≥ k f ′ k (cid:18) d av − R ( λ ) (cid:19) . From Corollary 5.7.ii we know that R ( λ ) < d av . So the above inequality implies k f ′ k = 0.On H the null–space of ∂ x on H is trivial, hence f = 0, which is a contradiction to k f k > d av >
0. Since the proof of the fourth claim was already given in Remark5.5, we finish with the proof of the last claim.Assume that there exists f ∈ H with N ( f ) >
0. Then R ( λ ) > λ = k f k .Corollary 5.7.iv then shows that lim inf λ →∞ R ( λ ) = ∞ , which implies that for every d av ≥ λ > R ( λ ) > d av . By the definition of the threshold, this shows λ d av cr < ∞ for all d av ≥ Proof of Theorems 1.2 and 1.4:
The first three claims of Theorem 1.2, respectively thefirst two claims of Theorem 1.4, follow from Theorem 4.1 in tandem with Theorem 5.1.It remains to prove that assumption A3 guarantees that λ d av cr < ∞ and λ d av cr = 0, if,additionally, A4 holds.Under assumption A4 Lemma 3.1 shows that E d av λ < d av ≥ λ d av cr then yields R ( λ ) = ∞ for λ >
0, so λ d av cr = 0.Now assume that A1 , A2 , and A3 hold. First, note that assumptions A2 and A3 ,together with Lemma 3.2 show that there exists a > V ( a ) & a γ for all a ≥ a and using assumption A1 , we have V ( a ) & − a γ for 0 ≤ a < a . Thus, with γ := min( γ , γ ), we see that the lower bound V ( a ) & − a γ [0 ,a ) ( a ) + a γ [ a , ∞ ) ( a ) (5.4)holds. In particular, V is bounded from below.Let σ > g σ from (B.11). Clearly g σ ∈ H for all σ >
0. Then | T r g σ ( x ) | = A (cid:18) σ | σ ( r ) | (cid:19) / e − σ x | σ ( r ) | with A = ( λ πσ ) / . Then k g σ k = λ and, moreover, since | σ ( r ) | = σ + (4 r ) , we have,for any | x | ≤ √ σ , | T r g σ ( x ) | ≥ A (cid:18) σ σ + (4 r ) (cid:19) / e − σ σ r )2 ≥ A σ and all | r | ≤ R , where R > ψ ) ⊂ [ − R, R ].On the other hand, for a large enough σ and any | x | ≥ σ we also have | T r g σ ( x ) | ≤ A e − | x | . (5.6) HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 37
By (5.5), we can choose σ and A large enough such that | T r g σ ( x ) | ≥ A ≥ a for all | x | ≤ √ σ and | r | ≤ R . Then (5.4) yields I := Z | x |≤√ σ V ( | T r g σ ( x ) | ) dx & √ σ A γ . Since V is bounded from below, we also have II := Z √ σ ≤| x |≤ σ V ( | T r g σ ( x ) | ) dx & − σ , and (5.4) together with (5.6) gives III := Z | x |≥ σ V ( | T r g σ ( x ) | ) dx & − A γ Z | x |≥ σ e − γ | x | / dx & − A γ γ e − γσ / for all | r | ≤ R . Thus, since ψ is integrable, this gives the lower bound N ( g σ ) = Z R ( I + II + III ) ψ dr & √ σ A γ − σ − A γ γ e − γσ / for all large enough A and σ . Setting λ = σ , that is, A = (2 σ /π ) / shows N ( g σ ) > σ >
0. 6.
Nonexistence
In this section, we will make the standing assumption that V is a power–law nonlinearitygiven by V ( a ) = ca γ for a ≥ c > , γ ≥ Proof of Theorem 1.6:
For the proof of the first part of Theorem 1.6 assume first that γ > c >
0, and fix λ >
0. Let g σ be the Gaussian from (B.11) with σ >
0. Since V isa power–law, Lemma B.3 shows that the nonlinearity N ( g σ ) is given by N ( g σ ) = c (cid:18) πγ (cid:19) / (cid:18) λ π (cid:19) γ/ σ − γ Z R (cid:18) σ | σ ( r ) | (cid:19) γ − ψ ( r ) dr = c (cid:18) πγ (cid:19) / (cid:18) λ π (cid:19) γ/ σ − γ Z R (cid:18)
11 + (4 r/σ ) (cid:19) γ − ψ ( r ) dr = c (cid:18) πγ (cid:19) / (cid:18) λ π (cid:19) γ/ σ − γ Z R (cid:18)
11 + (4 s ) (cid:19) γ − ψ ( σ s ) ds (6.2)where we also did a change of variables r = σ s . Since ψ is bounded below by m , say, in apossibly one-sided neighborhood of zero and ψ has compact support, we havelim inf σ → Z R (cid:18)
11 + (4 s ) (cid:19) γ − ψ ( σ s ) ds ≥ m Z ∞ (cid:18)
11 + (4 s ) (cid:19) γ − ds > C γ,λ = c (cid:16) πγ (cid:17) / (cid:16) λ π (cid:17) γ/ m R ∞ (cid:16) s ) (cid:17) γ − ds > N ( g σ ) ≥ C γ,λ σ − γ (6.3) for all small enough σ >
0. Lemma B.3 also yields k g σ k = λ and k g ′ σ k = λ/σ , so H ( g σ ) = d av k g ′ σ k − N ( g σ ) ≤ σ (cid:18) d av λ − C γ,λ σ − γ (cid:19) → −∞ as σ ↓ γ >
10. If γ = 10, we can still conclude that H ( g σ ) → −∞ as σ ↓
0, as long as C γ,λ > d av λ , which is the case if c > f H ( f ) isunbounded from below on H even for fixed L norm of f .If d av = 0 and γ >
6, then (6.3) shows H ( g σ ) = − N ( g σ ) ≤ − C γ,λ σ − γ → −∞ as σ ↓ , so in this case the energy functional f H ( f ) is again unbounded from below on L evenfor fixed L norm of f .If γ = 6 and ψ = [0 , , we modify an argument of [22]. Set C s ( λ ) := sup (cid:26)Z s Z R | T r f ( x ) | dxdr : k f k = λ (cid:27) (6.4)and note that E λ has a minimizer for ψ = [0 , if there is a maximizer for C ( λ ). Themain point for the argument is that C s ( λ ) is independent of s >
0: To see this, note thatif u : R → C solves the free Schr¨odinger equation, i∂ r u = − ∂ x u , u (0 , · ) = f ∈ L , then u δ defined by u δ ( r, x ) := δ / u ( δ r, δx ) solves again the free Schr¨odinger equation with initialcondition u δ (0 , x ) = f δ ( x ) := δ / f ( δx ), x ∈ R . Then Z s Z R | T r f δ ( x ) | dxdr = Z s Z R | u δ ( r, x ) | dxdr = Z s Z R δ | u ( δ r, δx ) | dxdr = Z δ s Z R | u ( r, x ) | dxdr = Z δ s Z R | T r f ( x )) | dxdr and noting k f δ k = k f k = λ we get C s ( λ ) = C δ s ( λ )for all s, δ, λ >
0, in particular, C s ( λ ) = C ( λ ) for all s > λ >
0. Assume that f is aminimizer for E λ for ψ = [0 , , that is, f is a maximizer for C ( λ ): k f k = λ > C ( λ ) = Z Z R | T r f ( x ) | dxdr. Then 0 = C ( λ ) − C ( λ ) ≥ Z Z R | T r f ( x ) | dxdr − Z Z R | T r f ( x ) | dxdr = Z Z R | T r f ( x ) | dxdr ≥ . So | T r f ( x ) | = 0 for Lebesque almost all pairs 1 ≤ r ≤ x ∈ R and hence, since T r isunitary on L , 0 = Z Z R | T r f ( x ) | dxdr = k f k , which contradicts k f k = λ >
0. So no minimizer of (1.1) exists if γ = 6 in the model casewhere ψ = [0 , . HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 39
Appendix A. Tightness and strong convergence in L A key step in our existence proof of minimizers of the variational problems (1.1) is thefollowing characterization of strong convergence in L ( R ) which is given in [12]. Lemma A.1.
A sequence ( f n ) n ⊂ L ( R ) is strongly converging to f in L ( R ) if and onlyif it is weakly convergent to f and lim L →∞ lim sup n →∞ Z | η | >L | b f n ( η ) | dη = 0 , (A.1)lim R →∞ lim sup n →∞ Z | x | >R | f n ( x ) | dx = 0 , (A.2) where b f is the Fourier transform of f . Appendix B. Galilei transformations and space-time localization propertiesof Gaussian coherent states
We will only discuss the one-dimensional case which is somewhat easier since we do nothave to deal with rotations in one dimension. The unitary operator implementing the shift S y : L ( R ) → L ( R ), ( S y f )( x ) = f ( x − y ) is given by S y = e − iyP (B.1)where P = − i∂ x is the momentum operator. Indeed, since e − iyP corresponds to multipli-cation by e − iyk in Fourier space, we have( e − iyP f )( x ) = 1 √ π Z R e i ( x − y ) k b f ( k ) dk = f ( x − y ) . Boosts, i.e., shifts in momentum space are given by e iv · : L ( R ) → L ( R ), i.e., multiplicationby e ivx , since d e iv · f ( k ) = 1 √ π Z R e − ix ( k − v ) f ( x ) dx = b f ( k − v ) . (B.2)Finally, if G is a bounded (measurable) function then G ( P ) is defined by \ G ( P ) f ( k ) = G ( k ) b f ( k ) . Of course, for any y ∈ R , the operators G ( P ) and e − iyP commute, G ( P ) e − iyP = e − iyP G ( P ). Moreover, for any v ∈ R the commutation relation G ( P ) e iv · = e iv · G ( P + v ) (B.3)holds. Indeed, Computing the Fourier transform F yields F (cid:0) G ( P ) e iv · f (cid:1) ( k ) = G ( k ) d e iv · f ( k ) = G ( k ) b f ( k − v )= ( G ( · + v ) b f )( k − v ) = F (cid:0) G ( P + v ) f (cid:1) ( k − v )= F (cid:0) e iv · G ( P + v ) f (cid:1) ( k ) . In particular, choosing G ( P ) = e − irP , we arrive at the commutation relation e − irP e iv · e − iyP = e iv · e − iyP e − ir ( P + v ) = e iv · e − iyP e − ir ( P +2 vP + v ) = e − irv e iv · e − i ( y +2 rv ) P e − irP . (B.4)Now let f ∈ L ( R ). Then u ( r ) = T r f = e − irP f is the solution of the (one-dimensional)Schr¨odinger equation − i∂ r u = P u = − ∂ x u with initial condition u (0) = f . Using (B.4), the solution of the free Schr¨odinger equation for the translated and boosted initial condition f y,v = e iv · e − iyP f is given by u y,v ( r, x ) := T r f y,v ( x ) = (cid:0) e − irP e iv · e − iyP f (cid:1) ( x )= (cid:0) e − irv e iv · e − i ( y +2 rv ) P e − irP f (cid:1) ( x )= e − irv e ivx (cid:0) e − i ( y +2 rv ) P e − irP f (cid:1) ( x )= e − irv e ivx (cid:0) e − irP f (cid:1) ( x − y − rv )= e − irv e ivx ( T r f )( x − y − rv ) , (B.5)that is, on the level of the solutions of the free time-dependent Schr¨odinger equation, trans-lations and boosts of the initial condition are implemented by the Galilei transformations G y,v given by ( G y,v u )( r, x ) := u y,v ( r, x ) = e − irv e ivx u ( r, x − y − rv ). Except for the time-dependent phase factor e − irv , formula (B.5) is exactly what one would have guessed fromclassical mechanicsA simple calculation now shows that any functional of the form f N ( f ) = Z Z R V ( | T r f ( x ) | ) dxψdr is invariant under translations and boosts of f in L ( R ).Now, we come to one of the major tools for our analysis, the so-called coherent states. Definition B.1 (Coherent states) . Let h ∈ L , k h k = 1, y, v ∈ R and h y,v := e iv · e − iyP h ,i.e., h y,v ( x ) = e ivx h ( x − y ) (B.6)for x ∈ R and define the coherent rank-one projection P y,v := | h y,v ih h y,v | in Dirac’s notation,i.e., given by f P y,v f := h y,v h h y,v , f i . (B.7)A well-known property of coherent states is their completeness expressed in Lemma B.2 (Completeness of coherent states) . Let h ∈ L ( R ) with k h k = 1 and h y,v theshifted and boosted h as above. Then, in a weak sense, π Z Z R dydvP y,v = 12 π Z Z R dydv | h y,v ih h y,v | = (B.8) on L . Moreover, π Z R dv h f, P y,v f i = Z | h ( x − y ) | | f ( x ) | dx (B.9) and π Z R dy h f, P y,v f i = Z | ˆ h ( η − v ) | | ˆ f ( η ) | dη. (B.10) Proof.
The completeness expressed in (B.8) is well-known, see [19, 21], the other two areless known. We give a short proof for the convenience of the reader: In order to see thatthe operator A given by its matrix elements h f , Af i := 12 π Z R Z R dydv h f , h y,v ih h y,v , f i HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 41 is the identity on L it is enough, by polarization, to take f = f = f and to check h f, Af i = h f, f i for all f ∈ L . Note h h y,v , f i = Z R e − ivx h ( x − y ) f ( x ) dx = (2 π ) / \ ( h y, f )( v ) . and thus by Plancherel,12 π Z R dv h f, P y,v f i = 12 π Z R dv |h h y,v , f i| = Z R dx | h y, ( x ) f ( x ) | = Z R dx | h ( x − y ) f ( x ) | , so (B.9) follows and we also see h f, Af i = 12 π Z R dy Z R dv |h h y,v , f i| = Z R dy Z R dx | h ( x − y ) f ( x ) | = Z R | f ( x ) | dx thus, in addition, (B.8) follows. For (B.10) we note that a short calculation reveals d h y,v ( η ) = e − iy ( η − v ) ˆ h ( η − v ) = e iyv ˆ h v, − y ( η ) . By Plancherel h h y,v , f i = h d h y,v , b f i = Z R e iy ( η − v ) ˆ h ( η − v ) b f ( η ) dη = (2 π ) / e − iyv F − hb h v, b f i ( y )where F − denotes the inverse Fourier transform. Again by Plancherel, we thus have12 π Z R dy h f, P y,v f i = 12 π Z R dy |h d h y,v , b f i| = Z R dη (cid:12)(cid:12)(cid:12)b h v, ( η ) b f ( η ) (cid:12)(cid:12)(cid:12) = Z R dη (cid:12)(cid:12)(cid:12)b h ( η − v ) b f ( η ) (cid:12)(cid:12)(cid:12) and (B.10) follows.We use coherent states in order to localize a wave function simultaneously in real andFourier spaces and since Gaussians have nice localization properties simultaneously in realand Fourier spaces, it is natural to use Gaussian coherent states for this.First we note some important properties of Gaussians, which are needed in several placesof this work. Lemma B.3 (Properties of Gaussians) . Let λ > , σ ∈ C with Re( σ ) > , and g σ ( x ) = (cid:18) σ ) λ π | σ | (cid:19) / e − x σ . (B.11) Then k g σ k = λ , k g ′ σ k = λ Re σ , and its time evolution is given by T r g σ ( x ) = (cid:18) σ ) λ π | σ | (cid:19) / (cid:18) σ σ ( r ) (cid:19) / e − x σ ( r ) (B.12) with σ ( r ) = σ + 4 ir . In particular, for all γ ≥ , k T r g σ k γL γ ( R ,dx ) = (cid:18) πγ (cid:19) / (cid:18) λ π (cid:19) γ/ (cid:18) Re( σ ) | σ | (cid:19) γ − (cid:18) | σ || σ ( r ) | (cid:19) γ − (B.13) Proof.
Write g σ as g ( x ) = A e − x /σ with A , σ ∈ C with Re( σ ) >
0. Then | g ( x ) | = | A | e − Re( σ x | σ | and thus k g k = | A | Z R e − σ x | σ | dx = | A | (cid:18) π | σ | σ ) (cid:19) /
22 M.-R. CHOI, D. HUNDERTMARK, Y.-R. LEE using the Gaussian integral R R e − βx dx = ( πβ ) / . Thus with the choice A = ( σ ) λ π | σ | ) / (B.14)we have k g k = λ . In addition, k g ′ k = | A | | σ | − Z R x e − σ x | σ | dx = | A | | σ | − (cid:18) − ∂∂β Z R e − βx dx (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) β = σ | σ | = 2 | A | | σ | − π / (cid:18) | σ | σ ) (cid:19) / = λ Re( σ ) . To prove formula (B.12) note that for a centered Gaussian the time evolution T r g canbe found by making the ansatz( T r g )( x ) = A ( r ) e − x /σ ( r ) =: u ( r, x ) . (B.15)A short calculation, using that u ( r, x ) solves i∂ r u = − ∂ x u , reveals that A and σ solve iA ′ = 2 Aσ and σ ′ = 4 i, thus A ( r ) and σ ( r ) are given by A ( r ) = A (cid:18) σ σ ( r ) (cid:19) / and σ ( r ) = σ + 4 ir (B.16)which proves (B.12).Using (B.15), (B.16), and Re( σ ( r )) = Re( σ ) we get k T r g σ k γL γ ( R ,dx ) = | A | γ (cid:12)(cid:12)(cid:12)(cid:12) σ σ ( r ) (cid:12)(cid:12)(cid:12)(cid:12) γ/ Z R e − γ Re( σ x | σ ( r ) | dx = | A | γ (cid:12)(cid:12)(cid:12)(cid:12) σ σ ( r ) (cid:12)(cid:12)(cid:12)(cid:12) γ/ (cid:18) π | σ ( r ) | γ Re( σ ) (cid:19) / and with the choice (B.14) for A and rearranging the terms this shows (B.13).The localization properties of Gaussian coherent states are the content of Lemma B.4 (Space-time localization properties of Gaussian coherent states) . Let g ( x ) = π − / e − x / be the standard L normalized Gaussian and g y,v ( x ) := e ivx g ( x − y ) (B.17) its shifted and boosted version. Let P ≤ L := 12 π Z R dy Z | v |≤ L dv | g y,v ih g y,v | (B.18) and P > L := 12 π Z R dy Z | v | >L dv | g y,v ih g y,v | . (B.19) Then P ≤ L + P > L = , ≤ P ≤ L ≤ , and ≤ P > L ≤ as operators. Moreover P > L localizes awave function in the region of large frequencies | η | & L in the sense that for any f ∈ H α we have k P > L f k . L − α k f k H α (B.20) where the implicit constant does not depend on f nor L . HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 43
Moreover, the time-evolution of the shifted and boosted Gaussian g y,v is given by ( T r g y,v )( x ) = 1 π / √ ir e − irv e ivx e − ( x − y − rv )22(1+2 ir ) (B.21) and for any f , f ∈ L which have separated supports we have the bilinear estimate sup | r |≤ R k T r P ≤ L f T r P ≤ L f k L px . A R L e L /p − B p,R s k f kk f k , ≤ p < ∞ , (B.22) where A R := √ R , B p,R := 2 − ( p p (1 + 4 R ) + 1) − , and s := dist(supp f , supp f ) .Proof. The first assertions are clear, since by Lemma B.2 we have P ≤ L + P > L = and certainly P ≤ L and P > L ≥ P ≤ L = − P > L ≤ and similarly P > L ≤ .To prove (B.20), we first note that because of 0 ≤ P > L ≤ , one has k P > L f k = h P > L / f, P > L P > L / f i ≤ h f, P > L f i . Let P y,v := | g y,v ih g y,v | , then h f, P > L f i = 12 π Z R dy Z | v | >L dv h f, P y,v f i = Z | v | >L Z R | ˆ g ( η − v ) | | ˆ f ( η ) | dηdv = 1 √ π Z | v | >L Z R e − ( η − v ) | ˆ f ( η ) | dηdv = Z R H L ( η ) | ˆ f ( η ) | dη (B.23)due to (B.10) and ˆ g = g where we set H L ( η ) := 1 √ π Z | v | >L e − ( η − v ) dv. Note that H L is even, 0 < H L ≤
1, increasing on [0 , ∞ ), and lim η →∞ H L ( η ) = 1. A shortcalculation reveals H L ( L ) = 12 + 1 √ π Z ∞ L e − v dv so H L ( L ) is extremely close to 1 / L . For | η | ≤ L/ | v | ≥ L , one has | v − η | ≥ | v | − | η | ≥ | v | − L/ ≥ L/
2, hence H L ( η ) ≤ √ π Z ∞ L e − L ( v − L ) dv = 4 √ πL e − L for all | η | ≤ L . So Z R H L ( η ) | ˆ f ( η ) | dη = Z | η |≤ L/ H L ( η ) | ˆ f ( η ) | dη + Z | η | >L/ H L ( η ) | ˆ f ( η ) | dη ≤ √ πL e − L k f k + Z | η | >L/ | ˆ f ( η ) | dη. Using Z | η | >L/ | ˆ f ( η ) | dη ≤ ( L/ − α Z | η | >L/ | η | α | ˆ f ( η ) | dη ≤ ( L/ − α k f k H α completes the proof of (B.20).To prove formula (B.21) first note that for the centered Gaussian from (B.11) with σ = 2and λ = 1 Lemma B.3 gives the time evolution as( T r g , )( x ) = π − / √ ir e − x ir ) . Now we use the Galilei transformation formula (B.5) to arrive at( T r g y,v )( x ) = π − / e − irv e ivx √ ir e − ( x − y − rv )22(1+2 ir ) which is (B.21).To prove (B.22), fix | r | ≤ R and note that( T r P ≤ L f )( x ) = 12 π Z R dy Z | v |≤ L dv ( T r g y,v )( x ) h g y,v , f i . Thus using (B.21) and the triangle inequality | ( T r P ≤ L f )( x ) | ≤ π ( π (1 + 4 r )) / Z R dy Z | v |≤ L dv e − ( x − y − rv )22(1+4 r |h g y,v , f i| together with A ( r, L ) := Z R dy Z | v |≤ L dv e − ( x − y − rv )22(1+4 r = 2 L (2 π (1 + 4 r )) / , which is independent of x , by translation invariance of Lebesgue measure we can thus bound | ( T r P ≤ L f )( x ) | ≤ A ( r, L )2 π ( π (1 + 4 r )) / Z R Z R ν x ( dy, dv ) |h g y,v , f i| with the probability measure ν x ( dy, dv ) := A ( r,L ) e − ( x − y − rv )22(1+4 r | v |≤ L dydv . Hence Jensen’sinequality [16] for the convex function r → | r | p , ≤ p < ∞ , shows (cid:12)(cid:12) ( T r P ≤ L f )( x ) (cid:12)(cid:12) p ≤ A ( r, L ) p (2 π ) p ( π (1 + 4 r )) p/ Z R Z R ν x ( dy, dv ) |h g y,v , f i| p . L p − (1 + 4 r ) p − Z R dy Z | v |≤ L dv e − ( x − y − rv )22(1+4 r |h g y,v , f i| p . Therefore, k ( T r P ≤ L f )( T r P ≤ L f ) k pL px . L p − (1 + 4 r ) p − Z R dy Z | v |≤ L dv Z R dy Z | v |≤ L dv |h g y ,v , f i| p |h g y ,v , f i| p Z R dx e − ( x − y − rv x − y − rv r . L p − (1 + 4 r ) p − Z R dy Z | v |≤ L dv Z R dy Z | v |≤ L dv |h g y ,v , f i| p |h g y ,v , f i| p e − [( y − y r ( v − v r (B.24)where we used Z R dx e − ( x − y − rv x − y − rv r = ( π (1 + 4 r )) / e − (( y − y r ( v − v r by a simple convolution of Gaussians. Since ( a + b ) ≥ a − b for any a, b ∈ R , the lowerbound [( y − y ) + 2 r ( v − v )] ≥
12 ( y − y ) − r L holds for all y , y , and | v | , | v | ≤ L . Moreover, |h g y,v , f i| ≤ Z R | g y,v ( x ) || f ( x ) | dx = π − / Z R e − ( x − y ) | f ( x ) | dx = ( g , ∗ | f | )( y ) , HRESHOLDS FOR EXISTENCE OF DMS FOR GENERAL NONLINEARITIES 45 and thus (B.24) gives the upper bound k T r P ≤ L f T r P ≤ L f k pL px . L p e L (1 + 4 r ) p − Z R dy Z R dy e − ( y − y r [ g , ∗ | f | ( y )] p [ g , ∗ | f | ( y )] p . (B.25)Let K j := supp f j , j = 1 , f j . Recall that we assume s := dist( K , K ) >
0. Given 0 < ˜ s < s/
2, we will enlarge K j a little bit, e K j := { y ∈ R | dist( y, K j ) ≤ ˜ s } . Note that dist( e K , e K ) = s − e s > R × R = ( e K c × R ) ∪ ( e K × R ) = ( e K c × R ) ∪ ( e K × e K c ) ∪ ( e K × e K ). As afurther preparation, note that the Cauchy-Schwartz inequality implies (cid:12)(cid:12)(cid:12)(cid:12)Z Z R e − c ( y − y ) h ( y ) h ( y ) dy dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:20)Z Z R e − c ( y − y ) | h ( y ) | dy dy (cid:21) / (cid:20)Z Z R e − c ( y − y ) | h ( y ) | dy dy (cid:21) / = √ cπ k h kk h k . (B.26)for any h , h ∈ L ( R ) and c >
0. Using this, we can bound I := Z f K c dy Z R dy e − ( y − y r (cid:2) ( g , ∗ | f | )( y ) (cid:3) p (cid:2) ( g , ∗ | f | )( y ) (cid:3) p . (1 + 4 r ) / (cid:20)Z f K c h ( g , ∗ | f | )( y ) i p dy (cid:21) / (cid:20)Z R h ( g , ∗ | f | )( y ) i p dy (cid:21) / . (B.27)Moreover, by Young’s inequality, Z R h ( g , ∗ | f | )( y ) i p dy . k f k p (B.28)and, on the other hand, Z f K c h ( g , ∗ | f | )( y ) i p dy = 1(2 π ) p Z f K c dy (cid:20)Z K e − ( y − z ) | f ( z ) | dz (cid:21) p . e − p [dist( K , f K c )] k e − |·| ∗ | f |k pL p . e − p [dist( K , f K c )] k f k p , (B.29)where again Young’s inequality, similar as for (B.28), has been used in the last inequality.Plugging (B.28) and (B.29) into (B.27), we obtain I . (1 + 4 r ) / e − p [dist( K , f K c )] k f k p k f k p . (B.30)Furthermore, the bound I := Z f K dy Z f K c dy e − ( y − y r h ( g , ∗ | f | )( y ) i p h ( g , ∗ | f | )( y ) i p . (1 + 4 r ) / e − p [dist( K , f K c )] k f k p k f k p (B.31)follows as the one for I , by symmetry. It remains to get a bound on I := Z f K dy Z f K dy e − ( y − y r h ( g , ∗ | f | )( y ) i p h ( g , ∗ | f | )( y ) i p . (B.32)Since ( y − y ) ≥ ( y − y ) / e K , e K )] / I ≤ e − r [dist( f K , f K )] Z f K dy Z f K dy e − ( y − y r h ( g , ∗ | f | )( y ) i p h ( g , ∗ | f | )( y ) i p . (1 + 4 r ) / e − r [dist( f K , f K )] k g , ∗ | f | k pL p k g , ∗ | f | k pL p . (1 + 4 r ) / e − r [dist( f K , f K )] k f k p k f k p (B.33)using again (B.28). Combining k T r P ≤ L f T r P ≤ L f k pL px . L p e L (1 + 4 r ) p − (cid:16) I + I + I (cid:17) with (B.30), (B.31), (B.33), dist( K j , e K cj ) = ˜ s for j = 1 ,
2, and dist( e K , e K ) = s − s , weobtain k T r P ≤ L f T r P ≤ L f k pL px . L p e L (1 + 4 r ) p h e − p e s + e − ( s − e s )216(1+4 r i k f k p k f k p choosing ˜ s = s/ (2 p p (1 + 4 r ) + 2), which makes p ˜ s / s − s ) / (16(1 + 4 r )), gives theupper bound k T r P ≤ L f T r P ≤ L f k L px . (1 + 4 r ) / L e L /p e − s √ p (1+4 r k f kk f k≤ (1 + 4 R ) / L e L /p e − s √ p (1+4 R k f kk f k for all | r | ≤ R , which proves (B.22). Acknowledgements:
Mi-Ran Choi and Young-Ran Lee thank the Department of Math-ematics at KIT and Dirk Hundertmark thanks the Department of Mathematics at SogangUniversity for their warm hospitality. We would also like to thank the anonymous refereesfor constructive criticism on an earlier version of this work, making us rethink some of ourresults, leading to, in part, strong improvements of our previous results. Dirk Hundertmarkgratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG)through CRC 1173. He also thanks the Alfried Krupp von Bohlen und Halbach Founda-tion for financial support. Young-Ran Lee thanks the National Research Foundation ofKorea(NRF) for financial support funded by the Korea government under grants (MSIP)No. 2011-0013073 and (MOE) No. 2014R1A1A2058848.
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Department of Mathematics, Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul 04107,Korea.
E-mail address : [email protected] Department of Mathematics, Institute for Analysis, Karlsruhe Institute of Technology,76128 Karlsruhe, Germany.
E-mail address : [email protected] Department of Mathematics, Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul 04107,South Korea.
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