Periodic solutions for a class of nonlinear partial differential equations in higher dimension
aa r X i v : . [ m a t h . A P ] A p r Periodic solutions for a class of nonlinear partialdifferential equations in higher dimension
Guido Gentile and Michela Procesi Dipartimento di Matematica, Universit`a di Roma Tre, Roma, I-00146, Italy Dipartimento di Matematica, Universit`a di Napoli “Federico II”, Napoli, I-80126, ItalyE-mail: [email protected], [email protected]
Abstract
We prove the existence of periodic solutions in a class of nonlinear partial differential equations,including the nonlinear Schr¨odinger equation, the nonlinear wave equation, and the nonlinear beamequation, in higher dimension. Our result covers cases where the bifurcation equation is infinite-dimensional, such as the nonlinear Schr¨odinger equation with zero mass, for which solutions whichat leading order are wave packets are shown to exist.
The problem of the existence of finite-dimensional tori for infinite-dimensional systems, such as nonlinearPDE equations, has been extensively studied in the literature. Up to very recent times, the only availableresults were confined to the case of one space dimension ( D = 1). In this context the first results wereobtained by Wayne, Kuksin, and P¨oschel [24, 20, 21, 22], for the nonlinear Schr¨odinger equation (NLS)and the nonlinear wave equation (NLW) with Dirichlet boundary conditions, by using KAM techniques.Later on, Craig and Wayne proved similar results, for both Dirichlet and periodic boundary conditions[11], with a rather different method based on the Lyapunov-Schmidt decomposition. The case of periodicboundary condition within the framework of KAM theory was then obtained by Chierchia and You[10]. The case of completely resonant systems, i.e. systems where all eigenvalues of the linear operatorare commensurate with each other, was discussed by several authors, and theorems on the existence ofperiodic solutions for a large measure set of frequencies were obtained by Bourgain [8] for the NLW withperiodic boundary conditions, by Gentile, Mastropietro and Procesi [16], and Berti and Bolle [3] for theNLW with Dirichlet boundary conditions, and by Gentile and Procesi [17] for the NLS with Dirichletboundary conditions. The existence of quasi-periodic solutions for the completely resonant NLW withperiodic boundary conditions has been proved by Procesi [23] for a zero-measure set of two-dimensionalrotation vectors, by Baldi and Berti [2] for a large measure set of two-dimensional rotation vectors, andby Yuan [25] for a large measure set of – at least three-dimensional – rotation vectors.Extending the results to higher space dimensions ( D >
1) introduces a lot of difficulties, mainly dueto the high degeneracy of the eigenvalues of the linear operator. The first achievements in this directionwere due to Bourgain, and concerned the existence of periodic solutions for NLW [6] and of periodicsolutions (also quasi-periodic in D = 2) for the NLS [7]. The case of quasi-periodic solutions in arbitrarydimension was solved by Bourgain [9] for the NLS and the NLW. Bourgain’s method is based on aNash-Moser algorithm, which does not imply the linear stability.A proof of existence and stability of quasi-periodic solutions in high dimension was given by Gengand You using KAM theory. Their result holds for a class of PDE’s, which includes the nonlinear beam1quation (NLB) [13] and the NLS with a smoothing nonlinearity [14], with periodic boundary conditionsand with nonlinearities which do not depend on the space variable. Both conditions are required in orderto ensure a symmetry for the Hamiltonian which simplifies the problem in a remarkable way. Theirapproach does not extend to the NLS with local nonlinearities – mainly because it requires a “secondMelnikov condition” at each iterative KAM step, and such a condition does not appear to be satisfied bythe local NLS.Successively, Eliasson and Kuksin [12], by using KAM techniques, proved the existence and stabilityof quasi-periodic solutions for the NLS with local nonlinearities. In their paper the main point is indeedto prove that one may impose a second Melnikov condition at each iterative KAM step. However, givena PDE equation, in general (see for instance the case of the NLW in D > D = 2, for the NLS with periodic boundary conditions. In the case of Dirichletboundary conditions, proving the non-degeneracy of the solutions becomes rather involved. We use acombinatorial lemma, proved in [18], and some results in algebraic number theory. With respect to thenonlocal NLS considered in [18], the proof we give here is much simpler, however it has the drawbackthat a stronger assumption on the nonlinearity is required.In the remaining part of this section, we give a rigorous description of the PDE systems we shallconsider, and a formal statement of the results that we shall prove in the paper. Throughout the paperwe shall call a function F ( x, t ), with x = ( x , . . . , x D ) ∈ R D and t ∈ R , even [resp. odd] in x – or even[resp. odd] tout court – if it even [resp. odd] in each of its arguments x i .Let S be the D dimensional square [0 , π ] D , and let ∂ S be its boundary. We consider for instance thefollowing class of equations ( (i ∂ t + P ( − ∆) + µ ) v = f ( x, v, ¯ v ) , ( x, t ) ∈ S × R ,v ( x, t ) = 0 ( x, t ) ∈ ∂ S × R , (1.1)where ∆ is the Laplacian operator, P ( x ) is a strictly increasing convex C ∞ function with P (0) = 0, µ is a real parameter which – we can assume – belongs to some finite interval (0 , µ ), with µ >
0, and x → f ( x, v ( x, t ) , ¯ v ( x, t )) is an analytic function which is super-linear in v, ¯ v and odd (in x ) for odd v ( x, t ): f ( x, v, ¯ v ) = X r,s ∈ N : r + s ≥ N +1 a r,s ( x ) v r ¯ v s , N ≥ , (1.2)with a r,s ( x ) even for odd r + s and odd otherwise. We shall look for odd 2 π -periodic solutions withperiodic boundary conditions in [ − π, π ] D .We require for f in (1.2) to be of the form f ( x, v, ¯ v ) = ∂∂ ¯ v H ( x, v, ¯ v ) + g ( x, ¯ v ) , H ( x, v, ¯ v ) = H ( x, v, ¯ v ) . (1.3)We also consider the class of equations ((cid:0) ∂ tt + ( P ( − ∆) + µ ) (cid:1) v = f ( x, v ) , ( x, t ) ∈ S × R ,v ( x, t ) = 0 ( x, t ) ∈ ∂ S × R , (1.4)and finally the wave equation ( ( ∂ tt − ∆ + µ ) v = f ( x, v ) , ( x, t ) ∈ S × R ,v ( x, t ) = 0 , ∀ ( x, t ) ∈ ∂ S × R , (1.5)where f ( x, v ) is of the form (1.2) with s identically zero and a r ( x ) := a r, ( x ) real (by parity a r ( x ) is evenfor odd r and odd for even r ).We shall consider also (1.1), (1.4) and (1.5) with periodic boundary conditions: in that case, we shalldrop the condition for f to be odd.For all these classes of equations we prove the existence of small periodic solutions with frequency ω close to the linear frequency ω = P ( D ) + µ for (1.1) and (1.4) and ω = p P ( D ) + µ for (1.5), with ω in an appropriate Cantor set of positive measure. We introduce a smallness parameter by rescaling v ( x, t ) = ε /N u ( x, ωt ) , ε > , (1.6)3ith ω = P ( D ) + µ − ε for (1.1) and (1.4) and ω = P ( D ) + µ − ε for (1.5).We shall formulate our results in a more abstract context, by considering the following classes ofequations with Dirichlet boundary conditions:(I) ( D ( ε ) u = εf ( x, u, ¯ u, ε /N ) , ( x, t ) ∈ S × T ,u ( x, t ) = 0 , ( x, t ) ∈ ∂ S × T , (1.7a)(II) ( D ( ε ) u = εf ( x, u, ε /N ) , ( x, t ) ∈ S × T ,u ( x, t ) = 0 , ( x, t ) ∈ ∂ S × T , (1.7b)where T := R / π Z and D ( ε ) is a linear (possibly integro-)differential wave-like operator with constantcoefficients depending on a (fixed once and for all) real parameter ω and on the parameter ε .We can treat the case of periodic boundary conditions in the same way:(I) D ( ε ) u = εf ( x, u, ¯ u, ε /N ) , ( x, t ) ∈ T D × T , (1.8a)(II) D ( ε ) u = εf ( x, u, ε /N ) , ( x, t ) ∈ T D × T , (1.8b)with the same meaning of the symbols as in (1.7).In Case (I) we assume that f ( x, u, ¯ u, ε /N ) is a rescaling of a function f ( x, u, ¯ u ) defined as in (1.2)and satisfying (1.3). In Case (II) we suppose D ( ε ) real and f real for real u , so that it is natural to lookfor real solutions u = ¯ u .For ν ∈ Z D +1 set ν = ( ν , m ), with ν ∈ Z and m = ( ν , . . . , ν D ) ∈ Z D and | ν | = | ν | + | m | = | ν | + | ν | + . . . + | ν D | . For x = ( t, x ) = ( t, x , . . . , x D ) ∈ R D +1 set ν · x = ν t + m · x = ν t + ν x + . . . + ν D x D .Set also Z + = { } ∪ N and Z D +1 ∗ = Z D +1 \ { } . Finally denote by δ ( i, j ) the Kronecker delta, i.e. δ ( i, j ) = 1 if i = j and δ ( i, j ) = 0 otherwise. Given a finite set A we denote by | A | the cardinality of theset. Throughout the paper, for z ∈ C we denote by z the complex conjugate of z .Since all the results of the paper are local (that is, they concern small amplitude solutions), we shallalways assume that the hypotheses below are satisfied for all ε sufficiently small. Hypothesis 1. (Conditions on the linear part). D ( ε ) is diagonal in the Fourier basis { e i ν · x } ν ∈ Z D +1 with real eigenvalues δ ν ( ε ) which are C ∞ inboth ν and ε .2. For all ν ∈ Z D +1 ∗ one has either δ ν (0) = 0 or | δ ν (0) | ≥ γ | ν | − τ , for suitable constants γ , τ > .3. For all ν ∈ Z D +1 ∗ one has | ∂ ε δ ν ( ε ) | < c | ν | c and, if | δ ν ( ε ) | < / , one has | ∂ ε δ ν ( ε ) | > c | ν | c aswell, for suitable ε -independent constants c , c , c > .4. For all ν ∈ Z D +1 ∗ such that | δ ν ( ε ) | < / one has | ∂ ε ∂ ν δ ν ( ε ) | ≤ c | ν | c − , for a suitable ε -indepen-dent constant c > .5. In case (I) we require that if for some ε and for some ν , ν ∈ Z D +1 one has | δ ν ( ε ) | , | δ ν ( ε ) | < / then | ν − ν | ≤ | ν + ν | . We now pass to the equation for the Fourier coefficients. We write u ( x, t ) = X ν ∈ Z D +1 u ν e i ν · x , (1.9)and introduce the coefficients u ± ν by setting u + ν := u ν and u − ν := u ν . Analogously we define f ν ( { u } , η ) := [ f ( x, u, ¯ u, η )] ν = X r,s ∈ N : r + s = N +1 [ a r,s ( x ) u r ¯ u s ] ν + X r,s ∈ N : r + s>N +1 η r + s − N − [ a r,s ( x ) u r ¯ u s ] ν { u } = { u σ ν } σ = ± ν ∈ Z D +1 , [ · ] ν denotes the Fourier coefficient with label ν , and we set f + ν := f ν and f − ν := f ν . Naturally f ν depends also on the Fourier coefficients of the functions a r,s ( x ), which we denoteby a r,s,m , with m ∈ Z D ; we set a + r,s,m := a r,s,m and a − r,s,m := a r,s,m .Then in Fourier space the equations (1.7) and (1.8) give δ ν ( ε ) u σ ν = εf σ ν ( { u } , ε /N ) , ν ∈ Z D +1 , σ = ± , (1.10)and in the case of Dirichlet boundary conditions we shall require u ν = − u S i ( ν ) for all i = 1 , . . . , D , where S i ( ν ) is the linear operator that changes the sign of the i -th component of ν . Remark 1.
The reality condition on H in (1.3) spells ( s + 1) a − s +1 ,r − ,m = r a + r,s, − m . (1.11) Moreover, by the analyticity assumption on the nonlinearity, one has | a r,s,m | ≤ A r + s e − A | m | for suitablepositive constants A and A independent of r and s . Remark 2.
We have doubled our equations by considering separately the equations for u + ν and u − ν – whichclearly must satisfy a compatibility condition. In Case (II) one can work only on u + ν , since u − ν = u + − ν .In other examples it may be possible to reduce to solutions with u ν real for all ν ∈ Z D +1 , but we foundmore convenient to introduce the doubled equations in order to deal with the general case. Following the standard Lyapunov-Schmidt decomposition scheme we split Z D +1 into two subsetscalled P and Q and treat the equations separately. By definition we call Q the set of those ν ∈ Z D +1 such that δ ν (0) = 0; then we define P = Z D +1 \ Q . The equations (1.10) restricted to the P and Q subset are called respectively the P and Q equations. Hypothesis 2. (Conditions on the Q equation).
1. For all ν ∈ Q one has λ ν ( ε ) := ε − δ ν ( ε ) ≥ c > , where c is ε -independent.2. The Q equation at ε = 0 , λ ν (0) u σ ν = f σ ν ( { u } , , ν ∈ Q , σ = ± , has a non-trivial non-degenerate solution q (0) ( x, t ) = X ν ∈ Q u (0) ν e i ν · x , where non-degenerate means that the matrix J σ,σ ′ ν , ν ′ = λ ν (0) δ ( ν , ν ′ ) δ ( σ, σ ′ ) − ∂f σ ν ∂u σ ′ ν ′ ( { q (0) } , is invertible. Moreover one has | u (0) ν | ≤ Λ e − λ | ν | and (cid:12)(cid:12)(cid:12) ( J − ) σ,σ ′ ν , ν ′ (cid:12)(cid:12)(cid:12) ≤ Λ e − λ | ν − ν ′ | , for suitableconstants Λ and λ . Remark 3.
The solution of the bifurcation equation , i.e. of the Q equation at ε = 0 , could be assumedto be only Gevrey-smooth. Note also that, even when Q is infinite-dimensional, the number of non-zeroFourier components of q (0) ( x, t ) can be finite. Definition 1. (The sets E , O ( ε ) and O ). Given ε ∈ E := [0 , ε ] we set O ( ε ) := { ν ∈ P : | δ ν ( ε ) | < / } and O = ∪ ε ∈ E O ( ε ) . Finally we call R the subset P \ O . emark 4. Note that ν ∈ R means that | δ ν ( ε ) | ≥ / for all ε ∈ E . The following definitions appear (in a slightly different form) in the papers by Bourgain. The notationswhich we use are those proposed by Berti and Bolle in [5].
Definition 2. (The equivalence relation ∼ ). We say that two vectors ν , ν ′ ∈ O ( ε ) are equivalent,and we write ν ∼ ν ′ , if for β small enough the following happens: one has | δ ν ( ε ) | , | δ ν ′ ( ε ) | < / andthere exists a sequence { ν , . . . , ν N } in O ( ε ) , with ν = ν and ν N = ν ′ , such that | δ ν k ( ε ) | < , | ν k − ν k +1 | ≤ C | ν k | + | ν k +1 | ) β , k = 1 , . . . , N − , ] where C is a universal constant. Denote by ∆ j ( ε ) , j ∈ N , the equivalence classes with respect to ∼ . Remark 5.
The equivalence relation ∼ induces a partition of O ( ε ) into disjoint sets { ∆ j ( ε ) } j ∈ N . Notealso that, if ν , ν ′ ∈ ∆ j ( ε ) , then it is not possible that for some ε ′ one has ν ∈ ∆ j ( ε ′ ) and ν ′ ∈ ∆ j ( ε ′ ) with j = j . Hypothesis 3. (Conditions on the set O ( ε ) : separation properties). There exist three ε -independent positive constants α, β, C , with α small enough and β < α , such that | ∆ j ( ε ) | ≤ C p αj ( ε ) ,where p j ( ε ) = min ν ∈ ∆ j ( ε ) | ν | , for all j ∈ N . Remark 6.
Hypothesis 3 implies the following properties: dist(∆ j ( ε ) , ∆ j ′ ( ε )) ≥ C p j ( ε ) + p j ′ ( ε )) β ∀ j, j ′ ∈ N such that j = j ′ , diam(∆ j ( ε )) ≤ C C p α + βj ( ε ) , max ν ∈ ∆ j ( ε ) | ν | ≤ p j ( ε ) ∀ j ∈ N , and, furthermore, we can always assume that c − C C p α + βj ≤ ζp j , with ζc < c / , where the constants c and c are defined in Hypothesis 1. Remark 7.
Given
N > and for all ε outside a finite set (depending on N ) the sets ∆ j ( ε ) ∩{ ν : | ν | ≤ N } are locally constant, namely for all ¯ ε outside a finite set there exists an interval I such that ¯ ε ∈ I with thefollowing property: There exists an ε -independent numbering of the sets ∆ j ( ε ) contained in { ν : | ν | ≤ N } so that ∆ j ( ε ) = ∆ j (¯ ε ) for all ε ∈ I . We can now state our main result.
Theorem 1.
Consider an equation in the class described by (1.7) and (1.8), such that the Hypotheses1, 2 and 3 hold. There exist a positive constant ε and a Cantor set E ⊂ [0 , ε ] , such that for all ε ∈ E the equation admits a solution u ( x, t ) , which is π -periodic in time and Gevrey-smooth both in time andin space, and such that (cid:12)(cid:12)(cid:12) u ( x, t ) − q (0) ( x, t ) (cid:12)(cid:12)(cid:12) ≤ Cε /N , uniformly in ( x, t ) . The set E has positive Lebesgue measure and lim ε → + meas( E ∩ [0 , ε ]) ε = 1 , (1.12) where meas denotes the Lebesgue measure. Let us prove that the equations (1.1), (1.4), and (1.5) – in particular the NLS, the NLB and the NLW –comply with all the Hypotheses and therefore admit a periodic solution by Theorem 1.6 .1.1 The NLS equationTheorem 2.
Consider the nonlinear Schr¨odinger equation in dimension D i ∂ t v − ∆ v + µ v = f ( x, v, ¯ v ) , with Dirichlet boundary conditions on the square [0 , π ] D , where µ ∈ (0 , µ ) ⊂ R and f is given accordingto (1.2) and (1.3), with N = 2 , a , = 1 and a r,s = 0 for r, s such that r + s = 3 and ( r, s ) = (2 , , thatis f ( x, v, ¯ v ) = | v | v + O ( | v | ) . There exist a full measure set M ⊂ (0 , µ ) and a positive constant ε suchthat the following holds. For all µ ∈ M there exists a Cantor set E ( µ ) ⊂ [0 , ε ] , such that for all ε ∈ E ( µ ) the equation admits a solution v ( x, t ) , which is π/ω -periodic in time and Gevrey-smooth both in timeand in space, and such that (cid:12)(cid:12) v ( x, t ) − √ εq e i ωt sin x . . . sin x D (cid:12)(cid:12) ≤ Cε, ω = D + µ − ε, | q | = (cid:16) (cid:17) D/ , uniformly in ( x, t ) . The set E = E ( µ ) has positive Lebesgue measure and satisfies (1.12). With the notations of Section 1 one has δ ν ( ε ) = − ωn + | m | + µ , with ω = ω − ε and ω = D + µ .Then it is easy to check that all items of Hypothesis 1 are satisfied provided µ is chosen in such a waythat | − ω n + | m | | ≥ γ | n | − τ . This is possible for µ in a full measure set; cf. equation (2.1) in [18].Then Hypothesis 1 holds with c = c = c = 1 and c = 1 / √ ω .The subset Q is defined as Q := { ( n, m ) ∈ Z D : n = 1 , | m i | = 1 ∀ i = 1 , . . . D } , and one canassume take q to be real, so that, by the Dirichlet boundary conditions, Q is in fact one-dimensional,and u n,m = ± q for all ( n, m ) ∈ Q . The leading order of the Q equation is explicitly studied in [18],where it is proved that Hypothesis 2 is satisfied.Finally, Hypothesis 3 has been proven by Bourgain [7] (see also Appendix A6 in [18]).Of course, Theorem 2 refers to solutions with m = (1 , , . . . , N can be any integer N >
1, andno other conditions must be assumed on the functions a r,s ( x ) beyond those mentioned after (1.2). In thatcase (for simplicity we consider the same solution of the linear equation as in Theorem 2), the leadingorder of the Q equation becomes q = sign( ε ) A q N (again by taking for simplicity’s sake q to be real),where A is a constant depending on the nonlinearity. If A is non-zero, this surely has a non-trivialnon-degenerate solution q either for positive or negative values of ε . In general the non-degeneracycondition in item 2 of Hypothesis 2 has to be verified case by case by computing A . Consider the nonlinear wave equation in dimension D∂ tt v − ∆ v + µ v = f ( x, v ) , with Dirichlet boundary conditions on the square [0 , π ] D , where µ ∈ (0 , µ ) ⊂ R and f is given accordingto (1.2), with s = 0 , N = 2 , a , = 1 , that is f ( x, v ) = v + O ( v ) . There exist a full measure set M ⊂ (0 , µ ) and a positive constant ε such that the following holds. For all µ ∈ M there exists a Cantorset E ( µ ) ⊂ [0 , ε ] , such that for all ε ∈ E ( µ ) the equation admits a solution v ( x, t ) , which is π/ω -periodicin time and Gevrey-smooth both in time and in space, and such that (cid:12)(cid:12) v ( x, t ) − q √ ε cos ωt sin x . . . sin x D (cid:12)(cid:12) ≤ Cε, ω = p D + µ − ε, q = (cid:18) (cid:19) ( D +1) / , uniformly in ( x, t ) . The set E = E ( µ ) has positive Lebesgue measure and satisfies (1.12).
7n that case one has δ ν ( ε ) = − ω n + | m | + µ , with ω = ω − ε and ω = D + µ . Once more, it is easyto check that Hypothesis 1 is satisfied provided µ is chosen in a full measure set, with c = c = c = 1and c = 1 / (1 + 4 ω ).The subset Q is given by Q := { ( n, m ) ∈ Z D : n = ± , | m i | = 1 ∀ i = 1 , . . . D } , and, if one choosesto look for solutions that are even in time, then Q is one-dimensional. The Q equation at ε = 0 can bediscussed as in the case of the nonlinear Schr¨odinger equation. For instance for f as in the statement ofTheorem 3 the non-degeneracy in item 2 of Hypothesis 2 can be explicitly verified. Again, the analysiseasily extends to more general situations, under the assumption that the Q equation at ε = 0 admits anon-degenerate solution. For a fixed nonlinearity, this can be easily checked with a simple computation.Hypothesis 3 has been verified by Bourgain [6], under some strong conditions on ω . Recently the sameseparation estimates have been proved by Berti and Bolle [5], by only requiring that ω be Diophantine. Of course, the separation properties for the NLS equation imply similar separation also for the nonlinearbeam (NLB) equation ∂ tt v + (∆ + µ ) v = f ( x, v ) , and in that case we can also consider nonlinearities with one or two space derivatives.As in the previous cases one restricts µ to some full measure set, and Hypothesis 1 holds with c = c = 2, c = 1 and c = 1 / √ ω . This implies that the subset Q is one-dimensional, providedwe look for real solutions which are even in time.The same kind of arguments holds for all equations of the form (1.1) and (1.4). The separation of thepoints ( m, | m | ) in Z D +1 implies, by convexity, also the separation of ( m, P ( | m | )), with P ( x ) definedafter (1.1). Here we describe an application to completely resonant NLS and NLB equations, namely equations (1.1)and (1.4) with P ( x ) = x and µ = 0, and with Dirichlet boundary conditions (the case of periodic boundaryconditions is easier for fully resonant equations). Since the equation is completely resonant we need someassumption on the nonlinearity in order to comply with Hypothesis 2. We set f ( x, v, ¯ v ) = | v | v for theNLS and f ( x, v ) = v for the NLB (the NLB falls in case (II) and we look for real solutions), but ourproofs extend easily to deal with higher order corrections which are odd and do not depend explicitly onthe space variables. In the case of the NLS we say that the leading term of the nonlinearity is cubic andgauge-invariant. The validity of Hypothesis 1 can be discussed as in the non-resonant equations of Subsections 2.1.The separation properties (Hypothesis 3) do not change in the presence of a mass term, and they havebeen already discussed in the non-resonant examples of Subsection 2.1. Thus, we only need to prove thenon-degeneracy of the solution of the Q equation. Since the nonlinearity does not depend explicitely on x we look for solutions such that u ν ∈ R . We follow closely [18], but we set ω = 1. This is done forpurely notational reasons, and is due to the fact that a trivial rescaling of time allows us to put ω = 1. The subset Q is infinite-dimensional, i.e. Q := { ( n, m ) ∈ N × Z D : n = | m | } . We set u ( n,m ) = q m = a m + O ( ε / ) for ( n, m ) ∈ Q and restrict our attention to the case q m ∈ R . At leading order, the Q i.e. the equation up to the third order is invariant under the transformation v → v iα for any α ∈ R . | m | a m = X m ,m ,m m + m − m = m h m − m ,m − m i =0 a m a m a m . (2.1)Note that in the case of [18], the left hand side of (2.1) was | m | s D − a m , with s a free parameter; then(2.1) is recovered by setting s = 0 and rescaling by 1 / √ D the coefficients q m .By Lemma 17 of [18] – which holds for all values of s –, for each N ≥ M + ⊂ Z D + with N elements such that equation (2.1) admits the solution (due to the Dirichletboundary conditions we describe the solution in Z D + ) a m = , m ∈ Z D + \ M + s D +1 − D (cid:16) | m | − c X m ′ ∈M + | m ′ | (cid:17) , m ∈ M + , with c = 2 D +1 / (2 D +1 ( N −
1) + 3 D ). The set M + defines a matrix J on Z D such that( JQ ) m = | m | − X m ,m ,m m + m − m = m h m − m ,m − m i =0 Q m a m a m − X m >m ,m m + m − m = m h m − m ,m − m i =0 a m a m Q m , (2.2)where m > m refers, say, to lexicographic ordering of Z D ; see in particular equations (8.5) and (8.7) of[18].Moreover we know (Lemma 18 of [18]) that the matrix J is block-diagonal with blocks of size dependingonly on N , D : we denote by K ( N , D ) the bound on such a size. Whatever the block structure, thematrix J has the form diag( | m | ) + 2 T where all the entries of T are linear combinations of terms q m i q m j with integer coefficients. If we multiply J by z := (2 D +1 − D )(2 D +1 ( N −
1) + 3 D ) – which is odd –, weobtain a matrix J ′ := diag( z | m | ) + 2 T ′ , where all the entries of T ′ are integral linear combinations of thesquare roots of a finite number of integers. Let us call the prime factors of such integers p = 1 , p , p , . . . . Definition 3. (The lattice Z D ). Let Z D := (1 , , . . . ,
0) + 2 Z D be the affine lattice of integer vectorssuch that the first component is odd and the others even. Let Z D , + be its intersection with Z D + . Of course,for all m ∈ Z D one has | m | odd. Since we are working with odd nonlinearities which do not depend explicitly on the space variableswe look for solutions such that u n,m = 0 if m / ∈ Z D .Let 1 , p , . . . , p k be prime numbers (as above), and let a , . . . , a K be the set of all products of squareroots of different numbers p i , i.e. a = 1, a = √ p , a = √ p p , etc. It is clear that the set of integrallinear combinations of a i is a ring (of algebraic integers). We denote it by a . The following Lemma is asimple consequence of Galois theory [1]. For completeness, the proof is given in Appendix A. Lemma 1.
The numbers a i are linearly independent over the rationals. Immediately we have the following corollary ( I denotes the identity). Corollary 1. In a consider a , i.e. the set of linear combinations with even coefficients. • a is a proper ideal, and the quotient ring a / a is thus a non-zero ring. • if a matrix M with entries in a is such that M − I has all entries in a , then M is invertible. M = I + 2 M ′ , with the entries of M ′ in a , is1 + 2 α , with α ∈ a . Hence, by Lemma 1, 2 α = ± Lemma 2.
For all N and for all M + ⊂ Z D , + the matrix J defined by M + is invertible. Its inverse isa block matrix with blocks of dimension depending only on N , D so that for some appropriate C one has ( J − ) m,m ′ ≤ C if | m − m ′ | ≤ K ( N , D ) , while ( J − ) m,m ′ = 0 otherwise.Proof. We use Corollary 1, the fact that the matrix J ′ has entries in a and the fact that z | m | is odd forall m ∈ Z D , + .Now, we can state our result on the completely resonant NLS. Theorem 4.
Consider the nonlinear Schr¨odinger equation in dimension D i ∂ t v − ∆ v = f ( v, ¯ v ) , with Dirichlet boundary conditions on the square [0 , π ] D , where f is given according to (1.2) and (1.3),with N = 2 , a , = 1 , a r,s = 0 for r, s such that r + s = 3 and ( r, s ) = (2 , , and a r,s ( x ) independent of x for r + s > (so that in particular a r,s = 0 for even r + s ). For any N ≥ there exist sets M + of N vectors in Z D + and real amplitudes { a m } m ∈M + such that the following holds. There exist a positiveconstant ε and a Cantor set E ⊂ [0 , ε ] , such that for all ε ∈ E the equation admits a solution v ( x, t ) ,which is π/ω -periodic in time and Gevrey-smooth both in time and in space, and such that, setting q ( x, t ) = (2i) D X m ∈M + a m e i | m | t sin m x . . . sin m D x D , ω = 1 − ε, (2.3) one has (cid:12)(cid:12) v ( x, t ) − √ εq ( x, ωt ) (cid:12)(cid:12) ≤ Cε, uniformly in ( x, t ) . The set E has positive Lebesgue measure and satisfies (1.12). We set ω = ω − ε = 1 − ε (recall that we are assuming ω = 1 by a suitable time rescaling). The subset Q is given by Q := { ( n, m ) ∈ N × Z D : | n | = | m | } . We set u n,m = q + m for n = | m | and u n,m = q − m for n = −| m | . We can require that q + m = q − m ≡ q m for all m (we obtain a solution which is even in time).Since we look for real solutions, this implies that q m ∈ R if D is even and q m ∈ i R if D is odd. Sincethe nonlinearity does not depend explicitly on x , we can look for solutions u n,m such that m ∈ Z D (seeDefinition 3).Finally the separation properties of the small divisors do not depend on the presence of the massterm, so that we only need to prove the existence and non-degeneracy of the solutions of the bifurcationequation.The Q equation at leading order is | m | a m = ( − D X m + m + m = m ±| m | ±| m | ±| m | = ±| m | a m a m a m , where we have set | q m | = a m + O ( ε / ). Lemma 3.
The condition ±| m | ± | m | ± | m | = ±| m | , for m i , m ∈ Z D , is equivalent to h m + m , m + m i = 0 . roof. The condition | m | + | m | + | m | = ( m + m + m ) is equivalent to h m , m + m i + h m , m i =0, which is impossible since the left hand side is an odd integer. The same happens with the condition | m | − | m | − | m | = ( m + m + m ) . Thus, we are left with | m | + | m | − | m | = ( m + m + m ) ,which implies h m + m , m + m i = 0.Lemma 3 implies that the bifurcation equation, restricted to Z D , is identical to that of a smoothingNLS with s = 2; cf. [18]. Indeed by recalling that q m = ( − D q − m one has | m | a m = X m + m − m = m h m − m ,m − m i =0 a m a m a m . (2.4)Then we can repeat the arguments of the previous subsection. By Lemma 17 of [18] – which holds for allvalues of s – for each N ≥ M + ⊂ Z D , + with N elements suchthat the equation (2.4) has the solution a m = , m ∈ Z D + \ M + s D +1 − D (cid:16) | m | − c X m ′ ∈M + | m ′ | (cid:17) , m ∈ M + , with c = 2 D +1 / (2 D +1 ( N −
1) + 3 D ).The matrix J is defined as in (2.2), only with | m | on the diagonal. We know (Lemma 18 of [18]does not depend on the values of s ) that the matrix J is block-diagonal with blocks of size boundedby K ( N , D ) (defined as in subsection 2.2.1). Whatever the block structure, the matrix J has theform diag( | m | ) + 2 T , where all the entries of T are linear combinations of terms a m i a m j with integercoefficients. If we multiply J by z := (2 D +1 − D )(2 D +1 ( N −
1) + 3 D ) – which is odd –, we obtain amatrix J ′ := diag( z | m | ) + 2 T ′ , where all the entries of T ′ are linear combinations of the square roots of afinite number of integers; finally z | m | is clearly odd and we can apply Lemma 1 to obtain the analogousof Lemma 2. Thus, a theorem analogous to Theorem 4 is obtained, with q ( x, t ) in (2.3) replaced with q ( x, t ) = 2 D +1 X m ∈M + a m cos | m | t sin m x . . . sin m D x D , ω = 1 − ε. We leave the formulation to the reader. P - Q equations Group the equations (1.10) for ν ∈ O as a matrix equation. Setting U = { u σ ν } σ = ± ν ∈ O , V = { u σ ν } σ = ± ν ∈ R , Q = { u σ ν } σ = ± ν ∈ Q , F = { f σ ν } σ = ± ν ∈ O , D ( ε ) = diag { δ ν ( ε ) } σ = ± ν ∈ O , (3.1)the P equations spell ( D ( ε ) U = εF ( U, V, Q, ε /N ) ,u σ ν = εδ − ν ( ε ) f σ ν ( U, V, Q, ε /N ) , ν ∈ R , (3.2)with a reordering of the arguments of the coefficients f σ ν .We want to introduce an appropriate “correction” to the left hand side of (3.2). We shall considerself-adjoint matrices c M ( ε ) := { c M σ,σ ′ ν , ν ′ ( ε ) } σ,σ ′ = ± ν , ν ′ ∈ O , which for each fixed ε are block-diagonal on the sets11 j ( ε ) (cf. Definition 2), namely c M σ,σ ′ ν , ν ′ ( ε ) = 0 can hold only if ν , ν ′ ∈ ∆ j ( ε ) for some j . Moreover werequire for c M σ,σ ′ ν , ν ′ ( ε ) to depend smoothly on ε , at least in a large measure set.We shall first introduce the self-adjoint matrices c M as independent parameters, and eventually weshall manage to fix them as functions of the parameter ε . Note that in order to have u + ν = u − ν we mustrequire that c M σ,σ ′ ν , ν ′ = c M − σ ′ , − σ ν ′ , ν . Definition 4. (The set G and the matrix b χ ). Call G = { / > ¯ γ > || δ ν (0) | − ¯ γ | ≥ ¯ γ / | ν | ¯ τ for all ν ∈ Z D +1 ∗ } , for suitable constants ¯ γ , ¯ τ > . For ¯ γ ∈ G , we introduce the step function ¯ χ ( x ) such that ¯ χ ( x ) = 0 if | x | ≥ ¯ γ and ¯ χ ( x ) = 1 if | x | < ¯ γ , and set ¯ χ ( x ) = 1 − ¯ χ ( x ) . We thenintroduce the ( ε -dependent) diagonal matrices b χ = diag { ¯ χ ( δ ν ( ε )) } σ = ± ν ∈ O and b χ = diag { ¯ χ ( δ ν ( ε )) } σ = ± ν ∈ O . Remark 8.
One has G = ∅ . Moreover, for any interval U ⊂ (0 , / , the relative measure of the set U ∩ G tends to 1 as ¯ γ tends to 0, provided ¯ τ is large enough Remark 9.
Note that b χ = b χ and b χ b χ = 0 , with the null matrix. Definition 5. (Resonant sets).
A set N = { ν , . . . , ν m } ⊂ O is resonant if there exists ε ∈ E and j ∈ N such that ν , . . . , ν m ∈ ∆ j ( ε ) . A resonant set { ν , ν } with m = 2 will be called a resonant pair .Given a resonant set N = { ν , . . . , ν m } we call C N the set of all ν ∈ O such that N ∪ { ν } is still aresonant set. Finally set C N ( ε ) := { ν ′ ∈ C N : | δ ν ′ ( ε ) | < ¯ γ } . Define the renormalised P equation as ((cid:16) D ( ε ) + c M (cid:17) U = η N F ( U, V, Q, η ) +
L U,u σ ν = η N δ − ν ( ε ) f σ ν ( U, V, Q, η ) , ν ∈ R , (3.3)with c M = b χ M b χ , where η is a real parameter, while M = { M σ,σ ′ ν , ν ′ } σ,σ ′ = ± ν , ν ′ ∈ O and L = { L σ,σ ′ ν , ν ′ } σ,σ ′ = ± ν , ν ′ ∈ O areself-adjoint matrices of free parameters with the properties:1. M σ,σ ′ ν , ν ′ = L σ,σ ′ ν , ν ′ = 0 if { ν , ν ′ } is not a resonant pair.2. M σ,σ ′ ν , ν ′ = M − σ ′ , − σ ν ′ , ν and L σ,σ ′ ν , ν ′ = L − σ ′ , − σ ν ′ , ν .The renormalised Q equation is defined as u σ ν = X ν ∈ Q X σ ′ = ± ( J − ) σ,σ ′ ν , ν ′ f σ ′ ν ′ ( U, V, Q, η ) , ν ∈ Q . (3.4)The parameter η and the counterterms L will have to satisfy eventually the identities ( compatibilityequation ) η = ε /N , c M = L. (3.5)We proceed in the following way: first we solve the renormalised P and Q equations (3.3) and (3.4),then we impose the compatibility equation (3.5). Here we introduce some notations and properties that we shall need in the following.12 efinition 6. (The Banach space B κ ). We consider the space of infinite-dimensional self-adjointmatrices { M σ,σ ′ ν , ν ′ } σ,σ ′ = ± ν , ν ′ ∈ O such that M σ,σ ′ ν , ν ′ = 0 if { ν , ν ′ } is not resonant. For ρ, κ > we equip such aspace with the norm | M | κ := sup ν , ν ′ ∈ O sup σ,σ ′ = ± (cid:12)(cid:12)(cid:12) M σ,σ ′ ν , ν ′ (cid:12)(cid:12)(cid:12) e κ | ν − ν ′ | ρ , so obtaining a Banach space that we call B κ . For L a linear operator on B κ define the operator norm | L | op = sup M ∈B κ | LM | κ | M | κ . Definition 7. (Matrix norms).
Let A be a d × d self-adjoint matrix, and denote with A ( a, b ) and λ ( a ) ( A ) its entries and its eigenvalues, respectively. We define the norms | A | ∞ := max ≤ a,b ≤ d | A ( a, b ) | , k A k := 1 √ d p tr( A ) , k A k := max | x | ≤ | Ax | , where, given a vector x ∈ R d , we denote by | x | its Euclidean norm. Lemma 4.
Given d × d self-adjoint matrix A , the following properties hold.1. The norm k A k depends smoothly on the coefficients A ( a, b ) .2. One has k A k / √ d ≤ | A | ∞ ≤ √ d k A k .3. One has max ≤ a ≤ d | λ ( a ) ( A ) | / √ d ≤ k A k ≤ max ≤ a ≤ d | λ ( a ) ( A ) | .4. For invertible A one has ∂ A ( a,b ) A − ( a ′ , b ′ ) = − A − ( a ′ , a ) A − ( b, b ′ ) and ∂ A ( a,b ) k A k = A ( a, b ) /d k A k . Here and henceforth we shall write A = D ( ε ) + c M in (3.3). Definition 8. (Small divisors).
For ν ∈ O define A ν ( ε ) as the matrix with entries ¯ χ ( δ ν ( ε )) A σ ,σ ν , ν such that ν , ν ∈ C ν ( ε ) and σ , σ = ± . If | δ ν ( ε ) | < ¯ γ , define also d ν ( ε ) := 2 |C ν ( ε ) | and p ν ( ε ) =min {| ν ′ | : ν ′ ∈ C ν ( ε ) } . For real positive ξ , define the small divisor x ν ( ε ) := 1 p ξ ν ( ε ) (cid:13)(cid:13) ( A ν ( ε )) − (cid:13)(cid:13) − , if A is invertible, and set x ν ( ε ) = 0 if A is not invertible. Remark 10.
Note that for ν ∈ ∆ j ( ε ) one has p ν ( ε ) = p j ( ε ) , d ν ( ε ) ≤ | ∆ j ( ε ) | , and A ν ( ε ) = A ν ′ ( ε ) forall ν ′ ∈ C ν ( ε ) . This shows that d ν ( ε ) , x ν ( ε ) and p ν ( ε ) are the same for all ν ′ ∈ C ν ( ε ) . Note also that,if ν ∈ ∆ j ( ε ) for some j ∈ N , then one has C ν ( ε ) = { ν ′ ∈ ∆ j ( ε ) : | δ ν ′ ( ε ) | < ¯ γ } . Hypothesis 3 implies d ν ( ε ) ≤ C p α ν ( ε ) . Definition 9. (The sets D , D ( γ ) , D ( γ ) , and D ( γ ) ). We define D = { ( ε, M ) : ε ∈ E , | M | κ ≤ C ε } , for a suitable positive constant C , and, for fixed τ, τ > and γ < ¯ γ , we set D ( γ ) = { ( ε, M ) ∈ D : x ν ≥ γ/p τ ν ( ε ) for all j ∈ N } , D ( γ ) = { ( ε, M ) ∈ D : || δ ν ( ε ) | − ¯ γ | ≥ γ/ | ν | τ for all ν ∈ O } , and D ( γ ) = D ( γ ) ∩ D ( γ ) . Definition 10. (The sets I N ( γ ) and I N ( γ ) ). Given a resonant set N we define I N ( γ ) := { ε ∈ E : ∃ ν ∈ C N such that || δ ν ( ε ) | − ¯ γ | < γ | ν | − τ } , and set I N ( γ ) := { ( ε, M ) ∈ D : ε ∈ I N ( γ ) } . .3 Main propositions We state the propositions which represent our main technical results. Theorem 1 is an immediate conse-quence of Propositions 1 and 2 below.
Proposition 1.
There exist positive constants K , K , κ, ρ, η such that the following holds true. For ( ε, M ) ∈ D ( γ ) , there exists a matrix L ( η, ε, M ) ∈ B κ , such that the following holds.1. For each ε the matrix L ( η, ε, M ) is block-diagonal so as to satisfy L ( η, ε, M ) = b χ L ( η, ε, M ) b χ .2. There exists a unique solution u σ ν ( η, M, ε ) , with ν ∈ Z D +1 , of equations (3.3) and (3.4), which isanalytic in η for | η | ≤ η , and such that for all ν ∈ Z D +1 and σ = ±| u σ ν ( η, M, ε ) | ≤ | η | K e − κ | ν | / .
3. The matrix elements L σ,σ ′ ν , ν ′ ( η, ε, M ) are analytic in η for | η | ≤ η , and uniformly bounded for ( ε, M ) ∈ D ( γ ) as | L ( η, ε, M ) | κ ≤ | η | N K .
4. The functions u σ ν ( η, ε, M ) can be extended on the set D to C functions u E σ ν ( η, ε, M ) , andthe matrix elements L σ,σ ′ ν , ν ′ ( η, ε, M ) can be extended on the set D \ I { ν , ν ′ } ( γ ) to C functions L E σ,σ ′ ν , ν ′ ( η, ε, M ) , such that L E σ,σ ′ ν , ν ′ ( η, ε, M ) = L σ,σ ′ ν , ν ′ ( η, ε, M ) and u E σ ν ( η, ε, M ) = u σ ν ( η, ε, M ) forall ( ε, M ) ∈ D (2 γ ) .5. The matrix elements L E σ,σ ′ ν , ν ′ ( η, ε, M ) satisfy for all ( ε, M ) ∈ D \ I { ν , ν ′ } ( γ ) the bounds (cid:12)(cid:12)(cid:12) L E σ,σ ′ ν , ν ′ ( η, ε, M ) (cid:12)(cid:12)(cid:12) ≤ e − κ | ν − ν ′ | ρ | η | N K , | ∂ ε L E σ,σ ′ ν , ν ′ ( η, ε, M ) | ≤ e − κ | ν − ν ′ | ρ | η | N K | p ν | c , | ∂ η L E σ,σ ′ ν , ν ′ ( η, ε, M ) | ≤ e − κ | ν − ν ′ | ρ N | η | N − K , for all ( ε, M ) ∈ D \ ∪I { ν , ν ′ } ( γ ) , where the union is taken over all the resonant pairs { ν , ν ′ } , onehas (cid:12)(cid:12) ∂ M L E ( η, ε, M ) (cid:12)(cid:12) op ≤ X ν ∈ O X ν ′ ∈C ν X σ,σ ′ = ± (cid:12)(cid:12)(cid:12)(cid:12) ∂ M σ,σ ′ ν , ν ′ L E ( η, ε, M ) (cid:12)(cid:12)(cid:12)(cid:12) κ ≤ | η | N K , and, finally, one has (cid:12)(cid:12) u E σ ν ( η, ε, M ) (cid:12)(cid:12) ≤ | η | N K e − κ | ν | / , uniformly for ( ε, M ) ∈ D . Remark 11.
In our analysis we choose M ∈ B κ because eventually we obtain L ∈ B κ , but – as thebound on the M -derivative in item 5 of Proposition 1 suggests – we could also take M in a larger space,say B ∞ with norm | M | ∞ = sup ν , ν ′ ∈ O sup σ,σ ′ = ± | M σ,σ ′ ν , ν ′ | . Once we have proved Proposition 1, we solve the compatibility equation (3.5) for the extended coun-terterms L E ( ε /N , ε, M ), which are well defined provided we choose ε < ε , with ε = η N . Proposition 2.
There exist C functions ε → ( ε, M σ,σ ′ ν , ν ′ ( ε )) from E \ I { ν , ν ′ } ( γ ) → D , with an appro-priate choice of C in Definition 9, such that the following holds.1. M ( ε ) verifies the equation M σ,σ ′ ν , ν ′ ( ε ) = L E σ,σ ′ ν , ν ′ ( ε /N , ε, M ( ε )) , (3.6)14 nd the bounds (cid:12)(cid:12)(cid:12) M σ,σ ′ ν , ν ′ ( ε ) (cid:12)(cid:12)(cid:12) ≤ K ε e − κ | ν − ν ′ | ρ , (cid:12)(cid:12)(cid:12) ∂ ε M σ,σ ′ ν , ν ′ ( ε ) (cid:12)(cid:12)(cid:12) ≤ K (1 + εp c ν ( ε )) e − κ | ν − ν ′ | ρ , for a suitable constant K .2. The set E (2 γ ) := { ε ∈ E : ( ε, M ( ε )) ∈ D (2 γ ) } has large relative Lebesgue measure, namely lim ε → + ε − meas( E (2 γ ) ∩ (0 , ε )) = 1 . By item 1 in Proposition 1 for all ( ε, M ) ∈ D ( γ ) we can find a matrix L ( η, ε, M ) so that there exists aunique solution u σ ν ( η, ε, M ) of (3.3) and (3.4) for all | η | ≤ η , for a suitable η , and for ε small enough. Byitem 3 in Proposition 1 the matrix elements L σ,σ ′ ν , ν ′ ( η, ε, M ) and the solution u σ ν ( η, ε, M ) can be extendedto C functions – denoted by L E σ,σ ′ ν , ν ′ ( η, ε, M ) and u E σ ν ( η, ε, M ) – for all ( ε, M ) ∈ D \ I { ν , ν ′ } ( γ ) and forall ( ε, M ) ∈ D , respectively. Moreover L E σ,σ ′ ν , ν ′ ( η, ε, M ) = L σ,σ ′ ν , ν ′ ( η, ε, M ) and u E σ ν ( η, ε, M ) = u σ ν ( η, ε, M )for all ( ε, M ) ∈ D (2 γ ).Equation (3.3) coincides with our original (3.2) provided the compatibility equation (3.5) is satisfied.Now we fix ε < η N so that L E ( ε /N , ε, M ) and u E σ ν ( ε /N , ε, M ) are well defined for | ε | < ε . By item1 in Proposition 2, there exists a matrix M ( ε ) which satisfies the extended compatibility equation (3.6).Finally by item 2 in Proposition 2 the Cantor set E (2 γ ) is well defined and of large relative measure.For all ε ∈ E (2 γ ) the pair ( ε, M ( ε )) is by definition in D (2 γ ), so that by item 3 in Proposition 1 one has L σ,σ ′ ν , ν ′ ( ε /N , ε, M ( ε )) = L E σ,σ ′ ν , ν ′ ( ε /N , ε, M ( ε )) and u σ ( ε /N , ε, M ( ε ); x, t ) = u E σ ( ε /N , ε, M ( ε ); x, t ), andhence u σ ν ( ε /N , ε, M ( ε )) solves (3.3) for η = ε /N . So, by item 1 in Proposition 2, M ( ε ) solves the truecompatibility equation (3.5) for all ε ∈ E (2 γ ). Then u σ ( ε /N , ε, M ( ε ); x, t ) is a true nontrivial solution of(3.3) and (3.4) in E (2 γ ). Then by setting E = E (2 γ ) the result follows. In this section we find a formal solution u σ ν , L of (3.3) and (3.4) as a power series on η ; the solution u σ ν , L depends on the matrix M and it will be written in the form of a tree expansion.We assume for u σ ν ( η, ε, M ) for all ν ∈ P and for the matrix L ( η, ε, M ) a formal series expansion in η : u σ ν ( η, ε, M ) = ∞ X k = N η k u ( k ) σ ν , L ( η, ε, M ) = ∞ X k = N η k L ( k ) , (4.1)with the Ansatz that L ( k ) σ,σ ′ ν , ν ′ = 0 if either ¯ χ ( δ ν ( ε )) ¯ χ ( δ ν ′ ( ε )) = 0 or the pair { ν , ν ′ } is not resonant, sothat L = b χ L b χ . We set also u ( k ) σ ν = 0 for all k ≤ N and ν , ν ′ ∈ P same for L ( k ) σ,σ ′ ν , ν ′ for ν , ν ′ ∈ O .For ν ∈ Q we set u σ ν ( η, ε, M ) = u (0) σ ν + ∞ X k = N η k u ( k ) σ ν . (4.2)with u (0)+ ν = u (0) ν and u (0) − ν = u (0) ν (cf. item 2 in Hypothesis 2 for notations). Again we set u ( k ) σ ν = 0 for0 < k < N and ν ∈ Q . 15nserting the series expansions (4.1) and (4.2) into (3.3) we obtain u ( k ) σ ν = f ( k − N ) σ ν δ ν ( ε ) , ν ∈ R ,u ( k ) σ ν = X ν ′ ∈ Q , σ ′ = ± ( J − ) σ,σ ′ ν , ν ′ f ( k ) σ ′ ν ′ , ν ∈ Q , (cid:16) D ( ε ) + c M (cid:17) U ( k ) = F ( k − N ) + k − N X r = N L ( r ) U ( k − r ) . (4.3) Definition 11. (The scale functions).
Let χ be a non-increasing function C ∞ ( R + , [0 , , such that χ ( x ) = 0 if x ≥ γ and χ ( x ) = 1 if x ≤ γ ; moreover one has | ∂ x χ ( x ) | ≤ Γ γ − for some positive constant Γ . Let χ h ( x ) = χ (2 h x ) − χ (2 h +1 x ) for h ≥ , and χ − ( x ) = 1 − χ ( x ) . Recall that for each ε the matrix A = D ( ε ) + c M is block diagonal with a diagonal part whoseeigenvalues are larger than ¯ γ > γ and a list of C p α ν ( ε ) × C p α ν ( ε ) blocks A ν containing small entries. Inthe following if A ν is invertible – i.e. if x ν = 0 – we will denote the entries of ( A ν ) − by ( A − ) σ,σ ′ ν , ν ′ eventhough it may be possible that the whole matrix A is not invertible. Definition 12. (Propagators).
For ν , ν ′ ∈ O , we define the propagators( G i,h ) σ,σ ′ ν , ν ′ = χ h ( x ν ( ε )) ¯ χ ( δ ν ( ε )) ¯ χ ( δ ν ′ ( ε ))( A − ) σ,σ ′ ν , ν ′ , if i = 1 and χ h ( x ν ( ε )) = 0 , ¯ χ ( δ ν ( ε )) δ − ν ( ε ) , if i = 0 , ν = ν ′ , σ = σ ′ and h = − , , otherwise. In terms of the propagators we obtain A − = X i =0 , ∞ X h = − G i,h , (4.4)which provides the multiscale decomposition. Notice that if ( A − ) σ,σ ′ ν , ν ′ = 0 then x ν ( ε ) = x ν ′ ( ε ) (seeRemark 10), so that the matrices G i,h are indeed self-adjoint. Remark 12.
Only the propagator G ,h can produce small divisors while the propagator G , − is diagonaland of order one. Hence, there exists a positive constant C such that we can bound the propagators as | G , − | ∞ ≤ Cγ − , (cid:12)(cid:12)(cid:12) ( G ,h ) σ,σ ′ ν , ν ′ (cid:12)(cid:12)(cid:12) ≤ h Cγ − p − ξ ν ( ε ) p p α ν ( ε ) , (4.5) where the condition d ν ( ε ) ≤ C p α ν ( ε ) – cf. Remark 10 – and item 2 of Lemma 4 have been used. We write L ( k ) in (4.1) as L ( k ) σ ,σ ν , ν = ∞ X h = − χ h ( x ν ( ε )) L ( k ) σ ,σ h, ν , ν , (4.6)for all resonant pairs { ν , ν } ; we denote by L ( k ) h the matrix with entries L ( k ) σ ,σ h, ν , ν . Finally we set U ( k ) = X i =0 , ∞ X h = − U ( k ) i,h , (4.7)16o that (4.3) gives u ( k ) σ ν = X ν ′ ∈ Q ( J − ) σ,σ ′ ν , ν ′ f ( k ) σ ′ ν ′ , ν ∈ Q ,u ( k ) σ ν = f ( k − N ) σ ν δ ν ( ε ) , ν ∈ R ,U ( k ) i,h = G i,h F ( k − N ) + δ ( i, G ,h ∞ X h = − k − N X r = N L ( r ) h U ( k − r )1 ,h , i = 0 , , h ≥ − , (4.8)which are the recursive equations we want to study. A connected graph G is a collection of points (vertices) and lines connecting all of them. We denotewith V ( G ) and L ( G ) the set of nodes and the set of lines, respectively. A path between two nodes is theminimal subset of L ( G ) connecting the two nodes. A graph is planar if it can be drawn in a plane withoutgraph lines crossing. Definition 13. (Trees).
A tree is a planar graph G containing no closed loops. One can considera tree G with a single special node v : this introduces a natural partial ordering on the set of lines andnodes, and one can imagine that each line carries an arrow pointing toward the node v . We can add anextra (oriented) line ℓ exiting the special node v ; the added line ℓ will be called the root line and thepoint it enters (which is not a node) will be called the root of the tree. In this way we obtain a rooted tree θ defined by V ( θ ) = V ( G ) and L ( θ ) = L ( G ) ∪ ℓ . A labelled tree is a rooted tree θ together with a labelfunction defined on the sets L ( θ ) and V ( θ ) . We shall call equivalent two rooted trees which can be transformed into each other by continuouslydeforming the lines in the plane in such a way that the latter do not cross each other (i.e. withoutdestroying the graph structure). We can extend the notion of equivalence also to labelled trees, simplyby considering equivalent two labelled trees if they can be transformed into each other in such a way thatalso the labels match.Given two nodes v, w ∈ V ( θ ), we say that v ≺ w if w is on the path connecting v to the root line.We can identify a line with the nodes it connects; given a line ℓ = ( w, v ) we say that ℓ enters w and exits(or comes out of) v , and we write ℓ = ℓ v . Given two comparable lines ℓ and ℓ , with ℓ ≺ ℓ , we denotewith P ( ℓ , ℓ ) the path of lines connecting ℓ to ℓ ; by definition the two lines ℓ and ℓ do not belong to P ( ℓ , ℓ ). We say that a node v is along the path P ( ℓ , ℓ ) if at least one line entering or exiting v belongsto the path. If P ( ℓ , ℓ ) = ∅ there is only one node v along the path (such that ℓ enters v and ℓ exits v ). Definition 14. (Lines and nodes).
We call internal nodes the nodes such that there is at least oneline entering them; we call internal lines the lines exiting the internal nodes. We call end-nodes the nodeswhich have no entering line. We denote with L ( θ ) , V ( θ ) and E ( θ ) the set of lines, internal nodes andend-nodes, respectively. Of course V ( θ ) = V ( θ ) ∪ E ( θ ) . We associate with the nodes (internal nodes and end-nodes) and lines of any tree θ some labels,according to the following rules. Definition 15. (Diagrammatic rules).
1. For each node v there are p v ≥ entering lines. If p v = 0 then v ∈ E ( θ ) , if p v > then either p v = 1 or p v ≥ N + 1 and v ∈ V ( θ ) . If L ( v ) is the set of lines entering v one has p v = | L ( v ) | . . With each internal line ℓ ∈ L ( θ ) one associates a label q , p or r . We say that ℓ is a p -line, a q -lineor an r -line, respectively, and we call L q ( θ ) , L p ( θ ) and L r ( θ ) the set of internal lines ℓ ∈ L ( θ ) which are q -lines, p -lines and r -lines, respectively. If p v = 1 then the line ℓ exiting v and the line ℓ entering v are both p -lines.3. With each line ℓ ∈ L ( θ ) one associates the type label i ℓ = 0 , .4. With each line ℓ ∈ L ( θ ) except the root line ℓ one associates a sign label σ ℓ = ± .5. With each internal line ℓ ∈ L ( θ ) one associates the momenta ( ν ℓ , ν ′ ℓ ) ∈ Z D +1 × Z D +1 .6. With each line ℓ ∈ L ( θ ) exiting an end-node one associates the momentum ν ℓ .7. With each line ℓ ∈ L ( θ ) one associates the scale label h ℓ ∈ N ∪ {− , } .8. With each end-node v ∈ E ( θ ) one associates the mode label ν v ∈ Q , the order label k v = 0 , and the sign label σ v = ± .9. With each internal node v ∈ V ( θ ) one associates the mode label m v ∈ Z D , the order label k v ∈ N ,and the sign label σ v = ± .10. For each internal node v ∈ V ( θ ) one defines r v as the number of lines ℓ ∈ L ( v ) with σ ℓ = σ v , andone sets s v = p v − r v .11. If a line ℓ ∈ L ( θ ) is not a p -line one sets i ℓ = 0
12. If a line ℓ ∈ L ( θ ) has i ℓ = 0 , then h ℓ = − .13. Let ℓ ∈ L ( θ ) be an internal line. If ℓ is a p -line with i ℓ = 0 , then ν ℓ = ν ′ ℓ . If ℓ is a p -line with i ℓ = 1 , then { ν ℓ , ν ′ ℓ } is a resonant pair. If ℓ is a q -line, then ν ℓ , ν ′ ℓ ∈ Q . If ℓ is an r -line, then ν ℓ = ν ′ ℓ ∈ R .14. If ℓ exits an end-node v , then ν ℓ = ν v .15. If two p -lines ℓ and ℓ ′ have i ℓ = i ℓ ′ = 1 and are such that { ν ℓ , ν ′ ℓ , ν ℓ ′ , ν ′ ℓ ′ } is a resonant set, then | h ℓ − h ℓ ′ | ≤ .16. If ℓ ∈ L ( θ ) exits an end-node v ∈ E ( θ ) , then one sets σ ℓ = σ v .17. If ℓ is the line exiting v and ℓ , . . . , ℓ p v are the lines entering v one has ν ′ ℓ = (0 , m v ) + σ v ( σ ℓ ν ℓ + . . . + σ ℓ pv ν ℓ pv ) = (0 , m v ) + σ v X ℓ ′ ∈ L ( v ) σ ℓ ′ ν ℓ ′ , which represents a conservation rule for the momenta.18. Given an internal node v ∈ V ( θ ) , if p v = 1 one has k v ≥ N , while if p v ≥ N one has k v = p v − .19. Given an internal node v ∈ V ( θ ) , if p v = 1 , let ℓ be the line entering v and ℓ be the line exiting v . One has i ℓ = i ℓ = 1 and { ν ′ ℓ , ν ℓ } is a resonant pair.20. With each end-node v ∈ E ( θ ) one associates the node factor η v = u (0) σ v ν v ; cf. item 2 in Hypothesis2 and (4.2) for notations.21. With each internal node v ∈ V ( θ ) with p v > one associates the node factor η v = a σ v r v ,s v ,m v , where a σr,s,m satisfies equation (1.11).
2. With each internal node v ∈ V ( θ ) with p v = 1 one associates the node factor η v = L ( k v ) σ v ,σ ℓ h ℓ , ν ′ ℓ , ν ℓ ,still to be defined (see Definition 25 below), where ℓ and ℓ are the lines exiting and entering v ,respectively.23. One associates with each line ℓ ∈ L ( θ ) a line propagator g ℓ ∈ C with the following rules. If ℓ is a p -line exiting the internal node v one sets g ℓ := ( G i ℓ ,h ℓ ) σ ℓ ,σ v ν ℓ , ν ′ ℓ , if ℓ is an r -line one sets g ℓ := 1 /δ ν ℓ ( ε ) ,if ℓ is a q -line exiting the internal node v one sets g ℓ := ( J − ) σ ℓ ,σ v ν ℓ , ν ′ ℓ , if ℓ exits an end-node one sets g ℓ = 1 .24. One defines the order of the tree θ as k ( θ ) := X v ∈ V ( θ ) k v , the momentum of θ as the momentum ν ℓ of the root line ℓ , and the sign of θ as the sign σ v of thenode v which the root line exits. Definition 16. (The sets of trees Θ ( k ) σν and Θ ). We call Θ ( k ) σ ν the set of all the nonequivalenttrees of order k , momentum ν and sign σ , defined according to the diagrammatic rules of Definition 15.We call Θ the sets of trees belonging to Θ ( k ) σ ν for some k ≥ , σ = ± and ν ∈ Z D +1 . Definition 17. (Clusters).
Given a tree θ ∈ Θ ( k ) σ ν a cluster T on scale h is a connected maximal setof nodes and lines such that all the lines ℓ have a scale label ≤ h and at least one of them has scale h ;we shall call h T = h the scale of the cluster. We shall denote by V ( T ) , V ( T ) and E ( T ) the set of nodes,internal nodes and the set of end-nodes, respectively, which are contained inside the cluster T , and with L ( T ) the set of lines connecting them. Finally k ( T ) = P v ∈ V ( T ) k v will be called the order of T . An inclusion relation is established between clusters, in such a way that the innermost clusters arethe clusters with lowest scale, and so on. A cluster T can have an arbitrary number of lines entering it( entering lines ), but only one or zero line coming out from it ( exiting line or root line of the cluster);we shall denote the latter (when it exists) with ℓ T . Notice that, by definition, | V ( T ) | > i ℓ = 1. Definition 18. (Resonances).
We call resonance on scale h a cluster T on scale h T = h such that1. the cluster has only one entering line ℓ T and one exiting line ℓ T of scale h ℓ T ≥ h + 2 ,2. one has that { ν ′ ℓ T , ν ℓ T } is a resonant pair and min {| ν ℓ T | , | ν ′ ℓ T |} ≥ ( h − /τ ,3. for all ℓ ∈ P ( ℓ T , ℓ T ) with i ℓ = 1 the pair { ν ′ ℓ , ν ℓ T } is not resonant,4. for all ℓ ∈ L ( T ) \ P ( ℓ T , ℓ T ) the pair { ν ′ ℓ , ν ℓ T } is not resonant.The line ℓ T of a resonance will be called the root line of the resonance. Definition 19. (The sets of trees R ( k ) σ , σ ′ h , ν , ν ′ and R ). For k ≥ N , h ≥ and a resonant pair { ν , ν ′ } such that min {| ν | , | ν ′ |} ≥ ( h − /τ , we define R ( k ) σσ ′ h, ν , ν ′ as the set of trees with the following differenceswith respect to Θ ( k ) σ ν .1. There is a single end-node, called e , with node factor η e = 1 (but no label no labels ν e nor σ e ). . The line ℓ e exiting e is a p -line. We associate with ℓ e the labels ν ℓ e = ν ′ , σ ℓ = σ ′ , and i ℓ e = 1 (butno labels ν ′ ℓ nor h ℓ ), and the corresponding line propagator is g ℓ e = ¯ χ ( δ ν ′ ( ε )) .3. The root line ℓ is a p -line. We associate with ℓ the labels i ℓ = 1 and ν ′ ℓ = ν (but no labels ν ℓ nor h ℓ ), and the corresponding line propagator is g ℓ = ¯ χ ( δ ν ( ε )) . Let v be the node which theline ℓ exits: we set σ v = σ .4. One has max ℓ ∈ L ( θ ) \{ ℓ ,ℓ e } h ℓ = h .5. If ℓ ∈ P ( ℓ e , ℓ ) is such that { ν ′ ℓ , ν ′ } is resonant, then i ℓ = 0 .6. For ℓ / ∈ P ( ℓ e , ℓ ) one has that { ν ′ ℓ , ν ′ } is not a resonant pair.We call R the sets of trees belonging to R ( k ) σσ ′ h, ν , ν ′ for some k ≥ , h ≥ , σ, σ ′ = ± , and ν , ν ∈ O such that { ν , ν ′ } is resonant and min {| ν | , | ν ′ |} ≥ ( h − /τ . Definition 20. (Clusters for trees in R ). Given a tree θ ∈ R , a cluster T on scale h T ≤ h is aconnected maximal set of nodes v ∈ V ( θ ) and lines ℓ ∈ L ( θ ) \ { ℓ , ℓ e } such that all the lines ℓ have a scalelabel ≤ h T and at least one of them has scale h T . Note that if θ ∈ R ( k ) σ,σ ′ h, ν , ν ′ , then for any cluster T in θ one necessarily has h T ≤ h . Definition 21. (Resonances for trees in R ). Given a tree θ ∈ R , a cluster T is a resonance if thefour items of Definition 18 are satisfied. Remark 13.
There is a one-to-one correspondence between resonances T of order k and scale h with ν ℓ T = ν ′ , ν ′ ℓ T = ν , σ v = σ , σ ℓ T = σ ′ (here v is the node which ℓ T exits) and trees θ ∈ R ( k ) σ,σ ′ h, ν , ν ′ ; cf.[18], Section 3.4 and Figure 7. Definition 22. (The sets of renormalised trees Θ ( k ) σR , ν , R ( k ) σ , σ ′ R , h , ν , ν ′ , Θ R and R R ). We define theset of renormalised trees Θ ( k ) σR, ν and R ( k ) σ,σ ′ R,h, ν , ν ′ as the set of trees defined as Θ ( k ) σ ν and R ( k ) σ,σ ′ h, ν , ν ′ , respectively,but with no resonances and no nodes v with p v = 1 . Analogously we define the sets Θ R and R R . In the following it will turn out to be convenient to introduce also the following set of trees.
Definition 23. (The set of renormalised trees S ( k ) σ , σ ′ R , h , ν , ν ′ and S R ). For k ≥ N , h ≥ and ν , ν ′ ∈ O such that | ν ′ | ≥ ( h − /τ we define the set of renormalised trees S ( k ) σ,σ ′ R,h, ν , ν ′ as the set of treeswith the following differences with respect to R ( k ) σ,σ ′ R,h, ν , ν ′ (see Definition 19).Items and are unchanged.3 ′ One assigns to the line ℓ the further label h ℓ ≤ h , and requires | ν | ≥ ( h ℓ − /τ .4 ′ One has max ℓ ∈ L ( θ ) \{ ℓ e } h ℓ = h Items and are unchanged.The set S R is defined analogously as R R . Remark 14.
Note that if θ ∈ R ( k ) σ,σ ′ R,h, ν , ν ′ then Val( θ ) = Val( θ ′ ) with θ ′ ∈ S ( k ) σ,σ ′ R,h, ν , ν ′ such that h ℓ = h − .Thus, it is enough to study the set S R in order to obtain bounds for trees in R R . Definition 24. (Tree values).
For any tree or renormalised tree θ call Val( θ ) = (cid:16) Y ℓ ∈ L ( θ ) g ℓ (cid:17)(cid:16) Y v ∈ V ( θ ) η v (cid:17) the value of the tree θ . To make explicit the dependence of the tree value on ε and M , sometimes we shallwrite Val( θ ) = Val( θ ; ε, M ) . efinition 25. (Counterterms). We define the node factors L ( k ) σ,σ ′ h, ν , ν ′ (cf. item 21 in Definition 15)by setting L ( k ) σ,σ ′ h, ν , ν ′ = X h ′ For any tree θ ∈ R ( k ) σ,σ ′ R,h, ν , ν ′ there exists a tree θ ′ ∈ R ( k ) − σ ′ , − σR,h, ν ′ , ν such that Val( θ ) = Val( θ ′ ) .Proof. Given a tree θ ∈ R ( k ) σ,σ ′ R,h, ν , ν ′ , consider the path P = P ( ℓ e , ℓ ), and set P = { ℓ , . . . , ℓ N } , with ℓ ≻ ℓ ≻ . . . ≻ ℓ N ≻ ℓ N +1 = ℓ e (if P = ∅ , set N = 0 in the forthcoming discussion). For k = 0 , . . . , N ,denote by v k the node which the line ℓ k exits and by L ( v k ) the set L ( v k ) \{ ℓ k +1 } (cf. item 1 in Definition15).We construct a tree θ ′ ∈ R ( k ) − σ ′ , − σR,h, ν ′ , ν in the following way.1. We shift the sign labels down the path P and change their sign, so that σ ℓ k → − σ v k and σ v k →− σ ℓ k +1 for k = 0 , . . . , N . In particular ℓ acquires the label − σ v , while ℓ e loses its label σ ℓ e (whichwith the opposite sign becomes associated with the node v N ).2. The end-node e becomes the root, and the root line becomes the end-node e . In particular the line ℓ e becomes the root line, and the line ℓ becomes the entering line, so that the arrows of all thelines ℓ ∈ P are reverted, while the ordering of all the lines and nodes outside P is not changed.3. For all the lines ℓ ∈ P we exchange the labels ν ℓ , ν ′ ℓ , so that ν ℓ k → ν ′ ℓ k and ν ′ ℓ k → ν ℓ k for k = 1 , . . . , N , and we set ν ′ ℓ e = ν ′ and ν ℓ = ν .4. For all k = 0 , . . . , N we replace m v k → − σ v k σ ℓ k +1 m v k .By construction, the tree θ ′ belongs to R ( k ) − σ ′ , − σR,h, ν ′ , ν , and all line propagators and node factors of the linesand nodes, respectively, which do not belong to P remain the same.Moreover, the line propagator of each ℓ k ∈ P in θ ′ is ( G i ℓk ,h ℓk ) − σ vk , − σ ℓk ν ℓ , ν ℓ ′ k = ( G i ℓk ,h ℓk ) σ ℓk ,σ vk ν ℓ ′ k , ν ℓ , hence itdoes not change with respect with the line propagator of the corresponding line in θ . For each node v k ,the conservation law ν ℓ k +1 = (0 , − σ v k σ ℓ k +1 m v k ) − σ ℓ k +1 (cid:16) − σ v k ν ′ ℓ k + X ℓ ′ ∈ L ( v k ) σ ℓ ′ ν ℓ ′ (cid:17) (4.10)is assured by the conservation law (cf. item 17 in Definition 15) ν ′ ℓ k = (0 , m v k ) + σ v k (cid:16) σ ℓ k +1 ν ℓ k +1 + X ℓ ′ ∈ L ( v k ) σ ℓ ′ ν ℓ ′ (cid:17) (4.11)for the corresponding node v k in θ : simply multiply (4.11) times σ v k σ ℓ k +1 in order to obtain (4.10).Finally we want to show that the product of the combinatorial factors times the node factors of thenodes v , . . . , v N do not change. Take a node v = v k , for k = 0 , . . . , N , and call r ′ v and s ′ v the number oflines ℓ ′ ∈ L ( v ) with σ ℓ ′ = σ v and σ ℓ ′ = − σ v , respectively. Set σ v = σ and σ ℓ k +1 = σ ′ .Consider first the case σ ′ = σ . In that case in θ one has r v = r ′ v + 1 and s v = s ′ v , and thecombinatorial factor contains a factor r v because there are r v lines ℓ entering v with σ ℓ = σ . In θ ′ onehas σ v → − σ , r v → s ′ v + 1, s v → r ′ v and m v → − m v (because σσ ′ = 1). Moreover the correspondingcombinatorial factor contains a factor ( s v + 1) because there are s v + 1 lines ℓ entering v with σ ℓ = − σ .21herefore, taking into account also the combinatorics, the node factor associated with the node v in θ is( s v + 1) a − σs v +1 ,r v − , − m v = r v a σr v ,s v ,m v , i.e. the same as in θ , by the condition (1.11).Now, we pass to the case σ = − σ ′ . In that case in θ one has r v = r ′ v , s v = s ′ v + 1. In θ ′ one has thesame values for r v , s v and σ v , so that, by using also that − σσ ′ m v = m v in such a case, the node factors a σ v r v ,s v ,m v do not change. Of course the combinatorial factors do not change either.In conclusion, one has Val( θ ) = Val( θ ′ ), which yields the assertion. Remark 15. By Lemma 5 we have that the matrix L ( k ) h is self-adjoint, and the Definition 25 togetherwith (4.6) implies that we can write L ( k ) σ,σ ′ ν , ν ′ = ∞ X h = − C h ( x ν ( ε )) X θ ∈R ( k ) σ,σ ′ R,h, ν , ν ′ Val( θ ) , C h ( x ) = ∞ X h ′ = h +2 χ h ( x ) , σ = ± , for all k ≥ N , all h ≥ , and all resonant pairs { ν , ν ′ } . By construction x ν ( ε ) = x ν ′ ( ε ) whenever L ( k ) σ,σ ′ h, ν , ν ′ = 0 , so that also L ( k ) is self-adjoint. Finally we have that L ( k ) = b χ L ( k ) b χ (cf. the definition ofthe line propagators g ℓ and g ℓ e for trees θ ∈ R ( k ) σ,σ ′ R,h, ν , ν ′ in Definition 19). Lemma 6. One has u ( k ) σ ν = X θ ∈ Θ ( k ) σR, ν Val( θ ) , σ = ± , (4.12) for all k ≥ and all ν ∈ Z D +1 .Proof. For any given counterterm L , the coefficients u ( k ) σ ν can be written as sums over tree values u ( k ) σ ν = X θ ∈ Θ ( k ) σ ν Val( θ ) . This can be easily proved by induction, using the diagrammatic rules and definitions given in this section;we refer to Lemma 3.6 of [18] for details. Then, defining the counterterms according to Definition 25, allcontributions arising from trees belonging to the set Θ ( k ) σ ν but not to the set Θ ( k ) σR, ν cancel out exactly –see Lemma 3.13 of [18] for further details – and hence the assertion follows. Given a tree θ ∈ Θ R , call S ( θ, γ ) the set of ( ε, M ) ∈ D such that for all ℓ ∈ L p ( θ ) with i ℓ = 1 one has ( − h ℓ − γ ≤ | x ν ℓ ( ε ) | ≤ − h ℓ +1 γ, h ℓ = − , | x ν ℓ ( ε ) | ≥ γ, h ℓ = − , (5.1)and for all ℓ ∈ L p ( θ ) one has ( | δ ν ℓ ( ε ) | ≤ ¯ γ, | δ ν ′ ℓ ( ε ) | ≤ ¯ γ, i ℓ = 1 , ¯ γ ≤ | δ ν ℓ ( ε ) | , i ℓ = 0 . (5.2)Define also D ( θ, γ ) ⊂ D as the set of ( ε, M ) ∈ D such that for all ℓ ∈ L p ( θ ) with i ℓ = 0 one has | δ ν ℓ ( ε ) ± ¯ γ | ≥ γ/ | ν ℓ | τ , while for all ℓ ∈ L p ( θ ) with i ℓ = 1 one has x ν ℓ ( ε ) ≥ γp τ ν ℓ ( ε ) , | δ ν ( ε ) ± ¯ γ | ≥ γ | ν | τ ∀ ν ∈ C ν ℓ ∪ C ν ′ ℓ , (5.3)22or some τ, τ > 0. Note that the second condition in (5.3) does not depend on M .Analogously, given a tree θ ∈ S R , we call e S ( θ, γ ) the set of ( ε, M ) ∈ D such that (5.1) holds forall ℓ ∈ L p ( θ ) \ { ℓ e , ℓ } with i ℓ = 1 and (5.2) holds for all ℓ ∈ L p ( θ ), and we call e D ( θ, γ ) as the set of( ε, M ) ∈ D such that (5.3) holds for all ℓ ∈ L p ( θ ) \ { ℓ e , ℓ } with i ℓ = 1, while for all ℓ ∈ L p ( θ ) with i ℓ = 0 one has | δ ν ℓ ( ε ) ± ¯ γ | ≥ γ/ | ν ℓ | τ . Remark 16. If ( ε, M ) ∈ S ( θ, γ ) then Val( θ ; ε, M ) = 0 , while ( ε, M ) ∈ D ( θ, γ ) means that we can usethe bounds (5.3) to estimate Val( θ ; ε, M ) . Analogous considerations hold for trees θ ∈ S R . Remark 17. If for some ε one has Val( θ ; ε, M ) = 0 and for two comparable lines ℓ, ℓ ′ ∈ L ( θ ) the pair { ν ′ ℓ , ν ℓ ′ } is resonant, then all the set { ν ℓ , ν ′ ℓ , ν ℓ ′ , ν ′ ℓ ′ } is resonant. This motivates the condition in item15 in Definition 15. Remark 18. If θ ∈ R ( k ) σ,σ ′ R,h, ν , ν ′ is such that Val( θ ; ε, M ) = 0 , then ν , ν ′ ∈ ∆ j ( ε ) for some j , so that p ν ( ε ) = p ν ′ ( ε ) and | ν − ν ′ | ≤ C C p α + β ν ( ε ) ≤ C C p α ν ( ε ) . Moreover p ν ( ε ) ≤ | ν | , | ν ′ | ≤ p ν ( ε ) . Suchproperties follow from Hypothesis 3 – cf. also Remark 6. Definition 26. (The quantity N h ( θ ) ). Define N h ( θ ) as the set of lines ℓ ∈ L ( θ ) with i ℓ = 1 andscale h ℓ ≥ h . Definition 27. (The quantity K ( θ ) ). Define K ( θ ) = k ( θ ) + X v ∈ V ( θ ) | m v | + X ℓ ∈ L q ( θ ) | ν ℓ − ν ′ ℓ | + X v ∈ E ( θ ) | ν v | , where k ( θ ) is the order of θ . Lemma 7. There exists a constant B such the following holds.1. For all θ ∈ Θ R and all lines ℓ ∈ L ( θ ) one has | ν ℓ | ≤ B ( K ( θ )) α .2. If θ ∈ S R , for all lines ℓ ∈ L ( θ ) \ ( P ( ℓ e , ℓ ) ∪ { ℓ , ℓ e } ) one has | ν ℓ | ≤ B ( K ( θ )) α , while for alllines ℓ ∈ P ( ℓ e , ℓ ) ∪ { ℓ } one has | ν ′ ℓ | ≤ B ( | ν ℓ e | + K ( θ )) α .3. Given a tree θ let ℓ, ℓ ′ ∈ L ( θ ) be two comparable lines, with ℓ ≺ ℓ ′ , such that i ℓ = i ℓ ′ = 1 and i ℓ ′′ = 0 for all the lines ℓ ′′ ∈ P ( ℓ, ℓ ′ ) . If | ν ′ ℓ − ν ℓ ′ | ≥ BK ( θ ) α , then one has Val( θ ) = 0 for all ε .4. If θ ∈ S R , ℓ ∈ P ( ℓ e , ℓ ) ∪ { ℓ } and, moreover, i ℓ ′ = 0 for all lines ℓ ′ ∈ P ( ℓ e , ℓ ) , then | ν ′ ℓ | ≤| ν ℓ e | + B ( K ( θ )) α .Proof. Let us consider first trees θ ∈ Θ R . The proof is by induction on the order of the tree k = k ( θ ). For k = 1 the bound is trivial. If the root line ℓ is either a q -line or an r -line or a p -line with i ℓ = 0, again thebound follows trivially from the inductive bound. If ℓ is a p -line with i ℓ = 1, call v the node such that ℓ = ℓ v and θ , . . . , θ s the subtrees with root in v . By the inductive hypothesis and Hypothesis 3 oneobtains, for a suitable constant C and taking B large enough, | ν ℓ | ≤ | m v | + B ( K ( θ ) − − | m v | ) α + C ( | m v | + B ( K ( θ ) − − | m v | )) α (1+4 α ) ≤ B ( K ( θ )) α , which proves the assertion for Θ R in item 1.As a byproduct also the bound for S R is obtained, as far as lines ℓ / ∈ P ( ℓ e , ℓ ) ∪ { ℓ , ℓ e } are concerned.The bound | ν ′ ℓ | ≤ B ( | ν ℓ e | + K ( θ )) α for the lines ℓ ∈ P ( ℓ e , ℓ ) ∪ { ℓ } can be proved similarly byinduction. Thus, also item 2 is proved.Given two comparable lines ℓ, ℓ ′ such that i ℓ ′′ = 0 for all lines ℓ ′′ ∈ P ( ℓ, ℓ ′ ), then by momentumconservation one has min {| ν ′ ℓ − ν ℓ ′ | , | ν ′ ℓ + ν ℓ ′ |} ≤ B ( K ( θ )) α in case (I) and | ν ′ ℓ − ν ℓ ′ | ≤ B ( K ( θ )) α in case (II). This proves the bounds in item 3 in case (II) and in item 4 for both cases (I) and (II).23n case (I), if i ℓ = i ℓ ′ = 1 and max {| δ ν ′ ℓ ( ε ) | , | δ ν ℓ ′ ( ε ) |} < / 2, then | ν ′ ℓ − ν ℓ ′ | ≤ | ν ′ ℓ + ν ℓ ′ | by item5 in Hypothesis 1. On the other hand if i ℓ = i ℓ ′ = 1 and max {| δ ν ′ ℓ ( ε ) | , | δ ν ℓ ′ ( ε ) |} ≥ / 2, one hasVal( θ ; ε, M ) = 0. Hence item 3 follows also in case (I). Lemma 8. Given a tree θ ∈ Θ R such that D ( θ, γ ) ∩ S ( θ, γ ) = ∅ , for all h ≥ one has N h ( θ ) ≤ max { , c K ( θ )2 (2 − h ) β/ τ − } , where c is a suitable constant.Proof. Define E h := c − ( h − β/ τ . So, we have to prove that N h ( θ ) ≤ max { , K ( θ ) E − h − } .If a line ℓ is on scale h ≥ γ/p τ ν ℓ ( ε ) < x ν ℓ ( ε ) ≤ − h +1 γ by (5.1) and (5.3). Hence B ( K ( θ )) ≥ B ( K ( θ )) α ≥ | ν ℓ | ≥ p ν ℓ ( ε ) > ( h − /τ , by Lemma 7, so that K ( θ ) E − h ≥ cB − / ( h − / τ (2 − h ) β/ τ ≥ c suitably large. Therefore if a tree θ contains a line ℓ on scale h one has max { , K ( θ ) E − h − } = K ( θ ) E − h − ≥ N h ( θ ) ≤ max { , K ( θ ) E − h − } will be proved by induction on the order of the tree. Let ℓ be the root line of θ and call θ , . . . , θ m the subtrees of θ whose root lines ℓ , . . . , ℓ m are the lines onscale h ℓ i ≥ h − i ℓ i = 1 which are the closest to ℓ .If h ℓ < h we can write N h ( θ ) = N h ( θ ) + . . . + N h ( θ m ), and the bound follows by induction. If h ℓ ≥ h then ℓ , . . . , ℓ m are the entering lines of a cluster T with exiting line ℓ ; in that case we have N h ( θ ) = 1 + N h ( θ ) + . . . + N h ( θ m ). Again the bound follows by induction for m = 0 and m ≥ 2. Thecase m = 1 can be dealt with as follows.If { ν ′ ℓ , ν ℓ } is a resonant pair, then either there exists a line ℓ ∈ P ( ℓ , ℓ ) with i ℓ = 1 such that { ν ′ ℓ , ν ℓ } is a resonant pair or there must be a line ℓ ∈ L ( T ) \ P ( ℓ , ℓ ) with { ν ′ ℓ , ν ℓ } a resonant pair.In fact, the first case is not possible: indeed, also { ν ′ ℓ , ν ′ ℓ } would be resonant (cf. Remark 17), so that | h ℓ − h ℓ | ≤ h − ≥ h ℓ ≥ h ℓ − ≥ h − | ν ′ ℓ | ≥ p ν ℓ ( ε ) > ( h − /τ , hence if θ ′ is the subtree with rootline ℓ , then one has K ( θ ) − K ( θ ) > K ( θ ′ ) > E h , and the bound follows once more by the inductivehypothesis.If { ν ′ ℓ , ν ℓ } is not a resonant pair, call ¯ ℓ the line along the path P ( ℓ , ℓ ) ∪ { ℓ } with i ¯ ℓ = 1 closest to ℓ . Since i ¯ ℓ = 1 and by hypothesis h ¯ ℓ < h − { ν ¯ ℓ , ν ℓ } is not a resonant pair (see item 15 in Definition15). Call ˜ T the set of nodes and lines preceding ℓ and following ¯ ℓ , and define K ( T ) = K ( θ ) − K ( θ )and K ( ˜ T ) = K ( θ ) − K (¯ θ ), where ¯ θ is the tree with root line ¯ ℓ . Set also ¯ ν = ν ¯ ℓ and ν = ν ′ ℓ .One has 2 | ¯ ν − ν | ≥ C ( p ¯ ν ( ε ) + p ν ( ε )) β ≥ C p β ν ( ε ) (see Remark 6), so that by Lemma 7 one finds B ( K ( θ ) − K ( θ ) ≥ B ( K ( ˜ T )) ≥ | ¯ ν − ν | ≥ C p β ν ( ε ) / ≥ C ( h − β/τ / 2. Hence ( K ( θ ) − K ( θ )) E − h ≥ K ( T ) E − h ≥ K ( ˜ T ) E − h ≥ 2, provided c is large enough. This proves the bound. Lemma 9. There exists positive constants ξ and D such that, if ξ > ξ in Definition 8, then for alltrees θ ∈ Θ R and for all ( ε, M ) ∈ D ( θ, γ ) ∩ S ( θ, γ ) one has | Val( θ ) | ≤ D k e − κK ( θ ) Y ℓ ∈ L ( θ ) i ℓ =1 p − ( ξ − ξ ) ν ℓ ( ε ) , (5.4a) | ∂ ε Val( θ ) | ≤ D k e − κK ( θ ) Y ℓ ∈ L ( θ ) i ℓ =1 p − ( ξ − ξ ) ν ℓ ( ε ) , (5.4b) X ν ∈ O X ν ′ ∈C ν X σ,σ ′ = ± (cid:12)(cid:12)(cid:12)(cid:12) ∂ M σ,σ ′ ν , ν ′ Val( θ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ D k e − κK ( θ ) Y ℓ ∈ L ( θ ) i ℓ =1 p − ( ξ − ξ ) ν ℓ ( ε ) . (5.4c)24 roof. The propagators are bounded according to (4.5), so that for all trees θ ∈ Θ ( k ) R, ν one has | Val( θ ) | ≤ C k (cid:16) Y v ∈ V ( θ ) e − A | m v | (cid:17)(cid:16) Y ℓ ∈ L q ( θ ) e − λ | ν ℓ − ν ′ ℓ | (cid:17) ×× (cid:16) Y v ∈ E ( θ ) e − λ | ν v | (cid:17) kh (cid:16) ∞ Y h = h +1 hN h ( θ ) (cid:17) Y ℓ ∈ L ( θ ) i ℓ =1 p − ξ ν ℓ ( ε ) p a ν ℓ ( ε ) , for arbitrary h and for suitable constants C and a . For ( ε, M ) ∈ D ( θ, γ ) ∩ S ( θ, γ ) one can bound N h ( θ )through Lemma 8. Therefore, by choosing h large enough the bound (5.4a) follows, provided ξ − a > κ is suitably chosen.When bounding ∂ ε Val( θ ), one has to consider derivatives of the line propagators, i.e. ∂ ε g ℓ . If ℓ is an r -line then | ∂ ε g ℓ | is bounded proportionally to | ν ℓ | c , whereas if ℓ is a p -line, then the derivative producesfactors which admit bounds of the form Cp a ν ℓ ( ε ) 2 h ℓ p c ν ℓ ( ε ) p − ξ ν ℓ ( ε ) , (5.5)for suitable constants C and a ; see the proof of Lemma 4.2 in [18] for details (and use item 3 in Hypothesis1). The extra factor 2 h ℓ can be taken into account by bounding the product of line propagators with2 h k ∞ Y h = h +1 hN h ( θ ) . One can bound | ν ℓ | ≤ B ( K ( θ )) , and use part of the exponential decaying factors e − A | m v | , e − λ | ν ℓ − ν ′ ℓ | ,and e − λ | ν v | , to control the contribution P v ∈ V ( θ ) | m v | + P ℓ ∈ L q ( θ ) | ν ℓ − ν ′ ℓ | + P v ∈ E ( θ ) | ν v | to K ( θ ) (cf.Definition 27). Then, if ξ is large enough, so that ξ − a > a in (5.5), thebound (5.4b) follows.Also the bound (5.4c) can be discussed in the same way. We refer again to [18] for the details. Remark 19. Note that for ( ε, M ) ∈ D ( θ, γ ) the singularities of the functions ¯ χ are avoided, so that ∂ ε ¯ χ ( δ ν ℓ ( ε )) = 0 for all ℓ ∈ L ( θ ) . Note also that the bound (5.4c) is not really needed in the following. Lemma 10. There are two positive constants B and B such that the following holds.1. Given a tree θ ∈ S R such that Val( θ ; ε, M ) = 0 , if K ( θ ) ≤ B p β/ ν ℓe ( ε ) then for all lines ℓ ∈ P ( ℓ e , ℓ ) one has i ℓ = 0 . Moreover for all such lines ℓ , if { ν ′ ℓ , ν ℓ e } is not a resonant pair, then one has | δ ν ℓ ( ε ) | ≥ / .2. Given a tree θ ∈ R R such that Val( θ ; ε, M ) = 0 , one has (cid:12)(cid:12) ν ′ ℓ − ν ℓ e (cid:12)(cid:12) ≤ B ( K ( θ )) /ρ , with ρ dependingon α and β .Proof. Suppose that θ ∈ S ( k ) σ,σ ′ R,h, ν , ν ′ and P ( ℓ e , ℓ ) contains lines ℓ with i ℓ = 1 and consequently with { ν ′ ℓ , ν ′ } not resonant (cf. Definition 23). Let ¯ ℓ be the one closest to ℓ e ; thus, one has | ν ′ ¯ ℓ − ν ′ | ≥ C ( | ν ′ ¯ ℓ | + | ν ′ | ) β ≥ C p β ν ′ ( ε ) = C p β ν ( ε ), so that we can apply item 3 in Lemma 7 to obtain B ( K ( θ )) ≥ Cp β ν ( ε ), for somepositive constant C . This proves the first statement in item 1. The proof of the second statement isidentical, since | δ ν ℓ ( ε ) | < / ν ℓ ∈ ∆ j ( ε ) for some j , so that if { ν ′ ℓ , ν ′ } is not a resonantpair then ν ′ / ∈ ∆ j ( ε ), and therefore | ν ′ ℓ − ν ′ | ≥ C p β ν ′ ( ε ).To prove item 2, notice that | ν − ν ′ | ≤ C C p α + β ν ( ε ) (cf. Remark 18). If K ( θ ) > B p β/ ν ( ε ) then K ( θ ) ≥ C | ν − ν ′ | β/ α + β ) . If K ( θ ) ≤ B p β/ ν ( ε ) then P ( ℓ e , ℓ ) has only lines with i ℓ = 0, so that by item3 in Lemma 7 one finds | ν − ν ′ | ≤ BK ( θ ) . 25 emma 11. Given a tree θ ∈ S R such that e D ( θ, γ ) ∩ e S ( θ, γ ) = ∅ , if N h ( θ ) ≥ for some h ≥ , then cK ( θ )2 (2 − h ) β/ τ ≥ , with c the same constant as in Lemma 8.Proof. Consider a tree θ ∈ S ( k ) σ,σ ′ R, ¯ h, ν , ν ′ for some k ≥ 1, ¯ h ≥ σ, σ ′ = ± and ν , ν ′ ∈ O such that | ν ′ | ≥ (¯ h − /τ . Assume N h ( θ ) ≥ h ≥ h ≥ ℓ ∈ L ( θ ), which does not belong to P := P ( ℓ e , ℓ ), such that h ℓ ≥ h , then one canreason as at the beginning of the proof of Lemma 8 to obtain K ( θ ) E − h ≥ 2, with E h = c − ( h − β/ τ ≥ ℓ ∈ P on scale h ℓ ≥ h , and hence such that i ℓ = 1 and, consequently, { ν ′ ℓ , ν ′ } is not a resonant pair. Let ¯ ℓ be the one closest to ℓ e among such lines; thus, one has | ν ′ ¯ ℓ − ν ′ | ≥ C p β ν ′ ( ε ),so that one obtains B ( K ( θ )) ≥ Cp β ν ′ ( ε ) ≥ C (¯ h − β/τ , for some positive constant C . So, the desiredbound follows once more. Lemma 12. Given a tree θ ∈ S R such that e D ( θ, γ ) ∩ e S ( θ, γ ) = ∅ , for all h ≥ one has N h ( θ ) ≤ c K ( θ )2 (2 − h ) β/ τ , where c is the same constant as in Lemma 8.Proof. Consider a tree θ ∈ S ( k ) σ,σ ′ R, ¯ h, ν , ν ′ for some k ≥ 1, ¯ h ≥ σ, σ ′ = ± and ν , ν ′ ∈ O such that | ν ′ | ≥ (¯ h − /τ .For k ( θ ) = 1 one has N h ( θ ) ≤ 1, so that the bound follows from Lemma 11.For k ( θ ) > ℓ be the root line of θ and call θ , . . . , θ m the subtreesof θ whose root lines ℓ , . . . , ℓ m are the lines on scale h ℓ i ≥ h − i ℓ i = 1 which are the closest to ℓ . All the trees θ i such that ℓ i / ∈ P ( ℓ e , ℓ ) belong to some Θ ( k i ) ± R, ν i with k i < k . If K ( θ ) ≥ B p β/ ν ′ ( ε ) (cf.Lemma 10) it may be possible that a line, say ℓ , belongs to P ( ℓ e , ℓ ), so that Val( θ ) = g ℓ Val( θ ′ ), with θ ′ ∈ S ( k ) ,σ ,σ ′ R,h , ν , ν ′ with h ≤ ¯ h , σ = ± and k < k .If h ℓ < h one has N h ( θ ) = N h ( θ ) + . . . + N h ( θ m ), so that the bound N h ( θ ) ≤ K ( θ ) E − h follows bythe inductive hypothesis.If h ℓ ≥ h one has N h ( θ ) = 1 + N h ( θ ) + . . . + N h ( θ m ). For m = 0 the bound can be obtained oncemore from Lemma 11, while for m ≥ θ m , belongs to Θ ( k ′ ) ± R, ν ′ for some k ′ and ν ′ sothat we can apply Lemma 8 and the inductive hypothesis to obtain N h ( θ ) ≤ K ( θ ) + . . . + K ( θ m − )) E − h + (cid:0) K ( θ m ) E − h − (cid:1) ≤ ( K ( θ ) + . . . + K ( θ m − )) E − h + K ( θ m ) E − h ≤ K ( θ ) E − h , which yields the bound.Finally if m = 1 one has N h ( θ ) = 1 + N h ( θ ). Hence, if ℓ / ∈ P ( ℓ e , ℓ ), again the bound follows fromLemma 8. If on the contrary ℓ ∈ P ( ℓ e , ℓ ), one can adapt the discussion of the case m = 1 in the proofof Lemma 8. Lemma 13. There exists positive constants κ , ξ and D such that, if ξ > ξ in Definition 8, then for ll trees θ ∈ R R and for all ( ε, M ) ∈ e D ( θ, γ ) ∩ e S ( θ, γ ) , by setting ν = ν ′ ℓ and ν ′ = ν ℓ e , one has | Val( θ ) | ≤ D k − h e − κ | ν − ν ′ | ρ Y ℓ ∈ L ( θ ) i ℓ =1 p − ( ξ − ξ ) ν ℓ ( ε ) , (5.6a) | ∂ ε Val( θ ) | ≤ D k − h p c ν ( ε ) e − κ | ν − ν ′ | ρ Y ℓ ∈ L ( θ ) i ℓ =1 p − ( ξ − ξ ) ν ℓ ( ε ) , (5.6b) X ν ∈ O X ν ∈C ν X σ ,σ = ± (cid:12)(cid:12)(cid:12) ∂ M σ ,σ ν , ν Val( θ ) (cid:12)(cid:12)(cid:12) ≤ D k − h e − κ | ν − ν ′ | ρ Y ℓ ∈ L ( θ ) i ℓ =1 p − ( ξ − ξ ) ν ℓ ( ε ) , (5.6c) with ρ as in Lemma 10.Proof. Set for simplicity P = P ( ℓ e , ℓ ) andΣ( θ ) = X v ∈ V ( θ ) | m v | + X ℓ ∈ L q ( θ ) | ν ℓ − ν ′ ℓ | + X v ∈ E ( θ ) | ν v | , Π( θ ) = (cid:16) Y v ∈ V ( θ ) e A | m v | / (cid:17)(cid:16) Y ℓ ∈ L q ( θ ) e λ | ν ℓ − ν ′ ℓ | (cid:17)(cid:16) Y v ∈ E ( θ ) e λ | ν v | (cid:17) . If θ ∈ R ( k ) σ,σ ′ R,h, ν , ν ′ for some k ≥ h ≥ σ, σ ′ = ± and { ν , ν ′ } resonant, then N h ( θ ) ≥ 1, so that K ( θ ) = k + Σ( θ ) > C hβ/ τ , for some constant C , which imply 1 ≤ − h C k Π( θ ), for some constant C .This produces the extra factor 2 − h .By item 2 in Lemma 10 one has ( B − | ν − ν ′ | ) ρ ≤ K ( θ ), so that 1 ≤ e −| ν − ν ′ | ρ C k Π( θ ), for some constant C . The factor Π( θ ) can be bounded by using part of the factors e − A | m v | , e − λ | ν v | , and e − λ | ν ℓ − ν ′ ℓ | ,associated with the nodes and with the q -lines. This proves the bound (5.6a),To prove the bound (5.6b) one has to take into account the further ε -derivative acting on the linepropagator g ℓ , for some ℓ ∈ L ( θ ). If the line ℓ does not belong to P then one can reason as in the proofof (5.4b) in Lemma 9. If ℓ ∈ P one has to distinguish between two cases. If there exists a line ¯ ℓ ∈ P such that i ¯ ℓ = 1 then K ( θ ) > B p β/ ν ( ε ) by item 1 in Lemma 10, so that, by item 2 in Lemma 7, onehas p ν ℓ ( ε ) ≤ | ν ′ ℓ | ≤ B ( | ν ℓ e | + K ( θ )) α ≤ B (2 p ν ( ε ) + K ( θ )) α ≤ C ( K ( θ )) /β , for some constant C .If i ℓ = 0 for all lines ℓ ∈ P then, by item 3 in Lemma 7, one has p ν ℓ ( ε ) ≤ | ν ′ ℓ | ≤ | ν ℓ e | + B ( K ( θ )) . Thenitem 3 in Hypothesis 1 implies the bound (5.6b).To prove (5.6c) one has to study a sum of terms each containing a derivative ∂ M σ ,σ ν , ν g ℓ , for some ℓ ∈ L ( θ ). If ℓ ∈ P we distinguish between the two cases. If K ( θ ) > B p β/ ν ( ε ), the sum over ν , ν hasthe limitations | ν − ν | ≤ Cp α + β ν ( ε ), | ν − ν ℓ | ≤ Cp α + β ν ( ε ) and | ν ℓ | ≤ ( | ν ℓ e | + BK ( θ )) α ≤ C ( K ( θ )) /β ,for some constant C : hence the sum over ν , ν produces a factor C ( K ( θ )) C ′ for suitable constants C and C ′ , and one has ( K ( θ )) C ′ ≤ C k Π( θ ), for some constant C . If K ( θ ) ≤ B p β/ ν ( ε ), then i ℓ = 0 forall lines ℓ ∈ P , so that the line propagators g ℓ do not depend on M . Finally if ℓ 6∈ P then one has | ν ℓ | ≤ B ( K ( θ )) α , so that the sum over ν , ν is bounded once more proportionally to ( K ( θ )) C ′ , forsome constant C ′ , and again one can bound ( K ( θ )) C ′ ≤ C k Π( θ ), for some constant C . Remark 20. Both Lemma 12 and 16 deal with the first derivatives of Val( θ ) . One can easily extend theanalysis so to include derivatives of arbitrary order, at the price of allowing larger constants ξ and D – and a factor p c ν ( ε ) for any further ε -derivative. Therefore, one can prove that the function Val( θ ) is C r for any integer r , in particular for r = 1 , which we shall need in the following – cf. in particular theforthcoming Lemma 14. Proof of Proposition 1 Definition 28. (The extended tree values). Let the function χ − be as in Definition 11. Define Val E ( θ ) = (cid:16) Y ℓ ∈ L ( θ ) i ℓ =1 χ − ( | x ν ℓ ( ε ) | p τ ν ℓ ( ε )) (cid:17)(cid:16) Y ℓ ∈ L ( θ ) i ℓ =1 Y ν ∈C { ν ℓ, ν ′ ℓ } χ − ( || δ ν ( ε ) | − ¯ γ | | ν | τ ) (cid:17) ×× (cid:16) Y ℓ ∈ L p ( θ ) i ℓ =0 χ − ( || δ ν ℓ ( ε ) | − ¯ γ | | ν ℓ | τ ) (cid:17) Val( θ ) (6.1) for θ ∈ Θ ( k ) R, ν , and Val E ( θ ) = (cid:16) Y ℓ ∈ L ( θ ) \{ ℓ ,ℓ e } i ℓ =1 χ − ( | x ν ℓ ( ε ) | p τ ν ℓ ( ε )) (cid:17)(cid:16) Y ℓ ∈ L ( θ ) \{ ℓ ,ℓ e } i ℓ =1 Y ν ∈C { ν ℓ, ν ′ ℓ } χ − ( || δ ν ( ε ) | − ¯ γ | p τ ν ℓ ( ε )) (cid:17) ×× (cid:16) Y ℓ ∈ L p ( θ ) i ℓ =0 , ν ℓ / ∈C { ν , ν ′} χ − ( || δ ν ℓ ( ε ) | − ¯ γ | | ν ℓ | τ ) (cid:17) Val( θ ) (6.2) for θ ∈ R ( k ) R,h, ν , ν ′ . We call Val E ( θ ) the extended value of the tree θ . The following result proves Proposition 1. Lemma 14. Given θ ∈ R ( k ) σ,σ ′ R,h, ν , ν ′ , the function Val( θ ) can be extended to the function (6.1) defined and C in D \ I { ν , ν ′ } ( γ ) , such that, defining the “extended” counterterm L E σ,σ ′ ν , ν ′ according to Definition 25,with Val( θ ) replaced with Val E ( θ ) , the following holds.1. Possibly with different constants ξ and K , Val E ( θ ) satisfies for all ( ε, M ) ∈ D \ I { ν , ν ′ } ( γ ) thesame bounds in Lemma 13 as Val( θ ) in D ( γ ) .2. There exist constants ξ , K , κ , ρ and η , such that, if ξ > ξ in Definition 8, L E σ,σ ′ ν , ν ′ satisfies, forall ( ε, M ) ∈ D \ I { ν , ν ′ } ( γ ) and | η | ≤ η , the bounds (cid:12)(cid:12)(cid:12) L E σ,σ ′ ν , ν ′ (cid:12)(cid:12)(cid:12) ≤ | η | N K e − κ | ν − ν ′ | ρ , (cid:12)(cid:12)(cid:12) ∂ ε L E σ,σ ′ ν , ν ′ (cid:12)(cid:12)(cid:12) ≤ | η | N K p c ν e − κ | ν − ν ′ | ρ , (cid:12)(cid:12)(cid:12) ∂ η L E σ,σ ′ ν , ν ′ (cid:12)(cid:12)(cid:12) ≤ N | η | N − K e − κ | ν − ν ′ | ρ , X ν ∈ O ,σ = ± X ν ∈C ν ,σ = ± (cid:12)(cid:12)(cid:12) ∂ M σ ,σ ν , ν L E σ,σ ′ ν , ν ′ (cid:12)(cid:12)(cid:12) e κ | ν − ν ′ | ρ ≤ | η | N K . Val E ( θ ) = Val( θ ) for ( ε, M ) ∈ D (2 γ ) and Val E ( θ ) = 0 for ( ε, M ) ∈ D \ D ( γ ) .Analogously, given θ ∈ Θ ( k ) σR, ν , the function Val( θ ) can be extended to the function (6.2) defined and C in D , such that, defining u E ( k ) ν as in Lemma 6 with Val( θ ) replaced with Val E ( θ ) , the following holds.1. Possibly with different constants ξ and K , Val E ( θ ) satisfies for all ( ε, M ) ∈ D the same boundsin Lemma 9 as Val( θ ) in D ( γ ) .2. There exist constants ξ , K , κ and η such that, if ξ > ξ in Definition 8, u E σ ν satisfies, for all ( ε, M ) ∈ D and | η | ≤ η , the bounds (cid:12)(cid:12) u E σ ν (cid:12)(cid:12) ≤ | η | N K e − κ | ν | / for all ν ∈ Z D +1 . . Val E ( θ ) = Val( θ ) for ( ε, M ) ∈ D (2 γ ) and Val E ( θ ) = 0 for ( ε, M ) ∈ D \ D ( γ ) .Proof. We shall consider explicitly the case of trees θ ∈ R ( k ) σ,σ ′ R,h, ν , ν ′ . The case of trees θ ∈ Θ ( k ) σR, ν can bediscussed in the same way.Item 3 follows from the very definition. The bounds of item 1 can be proved by reasoning as inSection 5, by taking into account the further derivatives which arise because of the compact supportfunctions χ − in (6.2). On the other hand all such derivatives produce factors proportional to p a ν ℓ ( ε ) forsome constant a (again we refer to [18] for details); in particular we are using item 2 in Hypothesis 1to bound the derivatives of δ ν ℓ ( ε ) with respect to ε . Therefore by using Lemma 8 and possibly takinglarger constants ξ and K the bounds of Lemma 13 follow also for the extended function (6.2).Finally the bounds on L E in item 2 come directly from the definition. Indeed, the counterterms L E σ,σ ′ ν , ν ′ are expressed in terms of the values Val( θ ) according to Remark 15, and the factor 2 − h is usedto perform the summation over the scale labels. Hence we have to control the sum over the trees.Let us fix ε . For each v ∈ E ( θ ) the sum over | ν v | is controlled by using the exponential factorse − λ | ν v | . For each line ℓ ∈ L ( θ ) the labels ν ′ ℓ are fixed by the conservation rule of item 12 in Definition15, while the sum over ν ℓ gives a factor C p α ν ℓ ( ε ) for the p -lines (see item 2 in Hypothesis 3), and it iscontrolled by using the exponential factors e − λ | ν ℓ − ν ′ ℓ | for the q -lines. The sums over i ℓ and h ℓ can bebounded by a factor 4. Finally the sum over all the unlabelled trees of order k is bounded by C k forsome constant C . Thus, the bounds on L E ν , ν ′ are proved.Finally, the C smoothness follows from Remark 20. The following result proves item 1 in Proposition 2. Here and henceforth we write L = L ( η, ε, M ) and L E = L E ( η, ε, M ), and we fix η = ε /N . Lemma 15. There exists constants ε > such that there exist functions M σ,σ ′ ν , ν ′ ( ε ) = M σ,σ ′ ν ′ , ν ( ε ) welldefined and C for ε ∈ E \ I { ν , ν ′ } ( γ ) , such that the “extended” compatibility equation M σ,σ ′ ν , ν ′ ( ε ) = L E σ,σ ′ ν , ν ′ ( ε /N , ε, M ( ε )) holds for all ε ∈ (0 , ε ) \ I { ν , ν ′ } ( γ ) .Proof. By definition we set M σ,σ ′ ν , ν ′ ( ε ) = 0 for all ε such that ¯ χ ( δ ν ( ε )) ¯ χ ( δ ν ′ ( ε )) = 0. Consider the Banachspace B of lists { M σ,σ ′ ν , ν ′ ( ε ) } , with { ν , ν ′ } a resonant pair, such that each M σ,σ ′ ν , ν ′ ( ε ) is well defined and C in ε ∈ E \ I { ν , ν ′ } ( γ ) and M σ,σ ′ ν , ν ′ ( ε ) = 0 for ε ∈ I { ν , ν ′ } ( γ ). By definition { L E σ ,σ ν , ν ( ε /N , ε, { M σ,σ ′ ν , ν ′ ( ε ) } ) } iswell defined as a continuously differentiable application from B in itself, since, for each tree θ ∈ R ( k ) σ ,σ R,h, ν , ν ,the value Val E ( θ ) by definition smoothes out to zero the value of each line propagator g ℓ in the corre-sponding intervals I { ν ℓ , ν ′ ℓ } (2 γ ) \I { ν ℓ , ν ′ ℓ } ( γ ). Again by definition L E (0 , , 0) = 0 and | ∂ M L (0 , , | op = 0,so that we can apply the implicit function theorem.Now we pass to the proof of item 2 in Proposition 2. We need some preliminary results. Lemma 16. Let A = A ( ε ) a self-adjoint matrix piecewise differentiable in the parameter ε . Then,if λ ( a ) ( A ) and φ ( a ) ( A ) denote the eigenvalues and the (normalised) eigenvectors of A , respectively, thefollowing holds.1. One has | λ ( a ) ( A ( ε )) | ≤ k A ( ε ) k . . The eigenvalues λ ( a ) ( A ( ε )) are piecewise differentiable in ε .3. One has | ∂ ε λ ( a ) ( A ( ε )) | ≤ k ∂ ε A ( ε ) k .Proof. See [19] for items 1 and 2. Moreover, for each interval in which A is differentiable, let A n be ananalytic approximation of A in such an interval, with A n → A as n → ∞ : then the eigenvalues φ ( a ) ( A n )are piecewise differentiable [19], and one has ∂ ε λ ( a ) ( A n ) = ∂ ε (cid:16) φ ( a ) , A n φ ( a ) (cid:17) = λ ( a ) ( A n ) ∂ ε (cid:16) φ ( a ) , φ ( a ) (cid:17) + (cid:16) φ ( a ) , ∂ ε A n φ ( a ) (cid:17) = (cid:16) φ ( a ) , ∂ ε A n φ ( a ) (cid:17) , which yields item 3 when the limit n → ∞ is taken.For M ∈ B κ we can write c M = L j M j , where M j are block matrices, so that we can define k c M k =sup j k M j k , with k M j k given as in Definition 7. Lemma 17. For M ∈ B κ one has k c M k ≤ Cε for some constant C depending on κ and ρ .Proof. If M ∈ B κ then c M = L j M j , with M j a block matrix with dimension d j depending on j ,and M j ( a, b ) = M σ,σ ′ ν , ν ′ , for suitable ν , ν ′ , σ, σ ′ such that | M σ,σ ′ ν , ν ′ | ≤ Dε e − κ | ν − ν ′ | ρ for some constant D .Therefore k M j k = max | x | ≤ | M j x | ≤ max | x | ≤ d j X a,b,c =1 | M j ( a, b ) | | x ( b ) | | M j ( a, c ) | | x ( c ) |≤ 12 max | x | ≤ d j X a,b,c =1 | M j ( a, b ) | | M j ( a, c ) | (cid:16) | x ( b ) | + | x ( c ) | (cid:17) ≤ max | x | ≤ d j X a =1 | M j ( a, b ) | d j X c =1 | M j ( a, c ) | d j X b =1 | x ( b ) | ≤ d j X a =1 | M j ( a, b ) | , which yields the assertion. Lemma 18. Let A, B be two self-adjoint d × d matrices. Then (cid:12)(cid:12)(cid:12) λ ( a ) ( A + B ) − λ ( a ) ( A ) (cid:12)(cid:12)(cid:12) ≤ d X b =1 (cid:12)(cid:12)(cid:12) λ ( b ) ( B ) (cid:12)(cid:12)(cid:12) for all a = 1 , . . . , d .Proof. The result follows from Lidskii’s lemma; cf. [19].Define E = { ε ∈ E : x ν ( ε ) ≥ γ/p τ ν ( ε ) ∀ ν ∈ O } and E = { ε ∈ E : || δ ν ( ε ) |− ¯ γ | ≥ γ/ | ν | τ ∀ ν ∈ O } ,and set E = E ∩ E .We can denote by λ σ ν ( A ), with ν ∈ O and σ = ± , the eigenvalues of the block matrix A = D + c M . If | δ ν ( ε ) | ≥ ¯ γ then λ σ ν ( ε ) = δ ν ( ε ). Moreover for each ε ∈ E and each ν ∈ O such that | δ ν ( ε ) | < ¯ γ , thereexists a block A ν ( ε ) of the matrix A , of size d ν ( ε ) ≤ C p α ν ( ε ) such that λ ± ν ( A ) depends only on theentries of such a block. This follows from Remarks 6 and 12.Therefore we have to discard from E only values of ε such that | δ ν ( ε ) | < ¯ γ for some ν ∈ O : for allsuch ν the matrix A ν ( ε ) is well defined, and one has λ σ ν ( A ) = λ σ ν ( A ν ( ε )).One has, by item 3 in Lemma 4, x ν ( ε ) ≥ p ξ ν ( ε ) min a =1 ,...,d ν ( ε ) (cid:12)(cid:12)(cid:12) λ ( a ) ( A ν ( ε )) (cid:12)(cid:12)(cid:12) ≥ p ξ ν ( ε ) min ν ′ ∈C ν ( ε ) min σ = ± | λ σ ν ′ ( A ν ( ε )) | , (7.1)30o that, by using that λ σ ν ′ ( A ν ( ε )) = λ σ ν ′ ( A ν ′ ( ε )) = λ σ ν ′ ( A ) for all ν ′ ∈ C ν ( ε ), we shall impose theconditions | λ σ ν ( A ν ( ε )) | ≥ γ | ν | τ , ν ∈ O , σ = ± , (7.2)for suitable γ > γ . Thus, the conditions (7.2), together with the bound | ν | ≤ p ν ( ε ) (cf. Remark 18),will imply through (7.1) the bounds (5.3) for x ν ( ε ).Define K σ ν = (cid:26) ε ∈ E : | λ σ ν ( A ) | ≤ γ | ν | τ (cid:27) , ν ∈ O , σ = ± , (7.3)with τ = τ − ξ , so that we can estimatemeas( E \ E ) ≤ X ν ∈ O X σ = ± meas( K σ ν ) . (7.4)Moreover, by defining H ν ,σ = (cid:26) ε ∈ E : | δ ν ( ε ) − σ ¯ γ | ≤ γ | ν | τ (cid:27) , ν ∈ C j , j ∈ N , σ = ± , (7.5)with τ to be determined, one hasmeas( E \ E ) ≤ X ν ∈ Z D +1 X σ = ± meas( H ν ,σ ) . (7.6) Lemma 19. There exists constants w and w such that K ± ν = ∅ for all ν such that | ν | ≤ w /ε w . Thereexists constants y and y such that H ν , ± = ∅ for all ν such that | ν | ≤ y /ε y .Proof. We start by considering the sets K σ ν for ν ∈ O and σ = ± . If | δ ν ( ε ) | < ¯ γ one can write A ν ( ε ) = diag { δ ν ′ (0) , δ ν ′ (0) } ν ′ ∈C ν ( ε ) + B ν ( ε ), which defines the matrix B ν ( ε ) as B ν ( ε ) = diag { δ ν ′ ( ε ) − δ ν ′ (0) } σ = ± ν ′ ∈C ν ( ε ) + M ν ( ε ) , where M ν ( ε ) is the block of M ( ε ) with entries M σ ,σ ν , ν ( ε ) such that ν , ν ∈ C ν ( ε ). By Lemma 18, onehas | λ σ ν ( A ) − δ ν (0) | ≤ d ν ( ε ) X a =1 (cid:12)(cid:12)(cid:12) λ ( a ) ( B ν ( ε )) (cid:12)(cid:12)(cid:12) ≤ C p α ν ( ε ) k B ν ( ε ) k , ν ∈ O , σ = ± , (7.7)where we have used Remark 10 to bound d ν ( ε ).One has | δ ν (0) | ≥ γ / | ν | τ ≥ γ / (2 p ν ( ε )) τ by item 2 in Hypothesis 1, whereas k B ν ( ε ) k ≤ c (2 p ν ( ε )) c ε + k M ν ( ε ) k , by items 1 and 2 in Hypothesis 1, and k M ν ( ε ) k ≤ k M ( ε ) k ≤ C ε byLemma 17. Therefore (7.7) implies | λ σ ν ( A ) | ≥ γ (2 p ν ( ε )) τ − Cp c +1 ν ( ε ) ε , for a suitable constant C , so that, by setting w = c + 1 + τ and choosing suitably the constants γ , τ and w , one has | λ ± ν ( A ) | ≥ γ / p ν ( ε )) τ ≥ γ /p τ ν ( ε ) for all ν such that | ν | ≤ w /ε w .For the sets H ν ,σ , one can reason in the same way, by using that ¯ γ ∈ G (cf. Definition 4). Lemma 20. Let ξ > ξ and ε = η N be fixed as in Lemma 14. There exist constants γ , τ and τ suchthat meas( E \ E ) = o ( ε ) . roof. First of all we have to discard from E the sets H ν ,σ . It is easy to see that one hasmeas( H ν ,σ ) ≤ γ | ν | τ c | ν | c , for some positive constant C , so that, by using the second assertion in Lemma 19, we find X ν ∈ O X σ = ± meas( H ν ,σ ) ≤ X ν ∈ O | ν |≥ y /ε y X σ = ± meas( H ν ,σ ) ≤ Cε y ( τ + c − D − , for some constant C , provided τ + c − D > 1, so that we shall require for τ to be such that τ + c − D > y ( τ + c − D − > K ± ν . For all ν ∈ O consider A ν ( ε ) and write A ν ( ε ) = δ ν ( ε ) I + B ν ( ε ), whichdefines the matrix B ν ( ε ) as B ν ( ε ) = diag { δ ν ′ ( ε ) − δ ν ( ε ) } σ = ± ν ∈C ν ( ε ) + M ν ( ε ) , with M ν ( ε ) defined as in the proof of Lemma 19.Then the eigenvalues of A ν ( ε ) are of the form λ ( a ) ( A ν ( ε )) = δ ν ( ε ) + λ ( a ) ( B ν ( ε )), so that for all ε ∈ E \ I ν ( γ ) one has (cid:12)(cid:12)(cid:12) ∂ ε λ ( a ) ( A ν ) (cid:12)(cid:12)(cid:12) ≥ | ∂ ε δ ν ( ε ) | − k ∂ ε B ν ( ε ) k , where item 3 in Lemma 16 has been used. One has | ∂ ε δ ν ( ε ) | ≥ c | ν | c , by item 2 in Hypothesis 1,and k ∂ ε B ν ( ε ) k ≤ max ν ′ ∈C ν ( ε ) | ∂ ε ( δ ν ′ ( ε ) − δ ν ( ε )) | + k ∂ ε M ν ( ε ) k ≤ ζc p ν ( ε ) p c − ν ( ε ) + ε Cp c ν ( ε ), for asuitable constant C , as follows from item 4 in Hypothesis 1, from Hypothesis 3 (see Remark 6 for thedefinition of ζ ), from Lemma 14, and from Lemma 15. Hence we can bound | ∂ ε λ ( a ) ( A ν ) | ≥ c | ν | c / ε small enough.Therefore one has meas( K σ ν ) ≤ γ | ν | τ ( ε ) 2 c | ν | c (cid:16) C | ν | ( α + β )( D +1) (cid:17) , (7.8)for some constant C , where the last factor C | ν | ( α + β )( D +1) arises for the following reason. The eigenvalues λ σ ν ( A ) are differentiable in ε except for those values ε such that for some ν ′ ∈ C ν one has | δ ν ′ ( ε ) | = ¯ γ and | δ ν ( ε ) | < ¯ γ . Because of item 3 in Hypothesis 1 all functions δ ν ′ ( ε ) are monotone in ε as far as | δ ν ′ ( ε ) | < / 2, so that for each ν ′ ∈ C ν the condition | δ ν ′ ( ε ) | = ¯ γ can occur at most twice. The number of ν ′ ∈ C ν such that the conditions | δ ν ′ ( ε ) | = ¯ γ and | δ ν ( ε ) | < ¯ γ can occur for some ε ∈ E is bounded by thevolume of a sphere of centre ν and radius proportional to | ν | α + β (cf. Remark 6). Hence C | ν | ( α + β )( D +1) counts the number of intervals in E \ I ν ( γ ).Thus, (7.8) yields, by making use of the first assertion of Lemma 19, X ν ∈ O X σ = ± meas( K σ ν ) ≤ X ν ∈ Z D +1 | ν |≥ w /ε w γc | ν | − τ − c (cid:0) C | ν | α (cid:1) ≤ Cε w ( τ + c − α − D − , for some positive constant C , provided τ + c − α − D = τ + c − α − D − ξ > 1, so that (7.4) impliesthat meas( E \ E ) ≤ Cε w ( τ + c − D − .Therefore, the assertion follows provided min { τ , τ − α } > D − c + 1, y ( τ + c − D − > w ( τ + c − α − D − > 1. 32 Proof of Lemma 1 Lemma 1 is a consequence of the following elementary proposition in Galois theory. Proposition 3. If p , . . . , p k are distinct primes then the field F := Q [ √ p , √ p , . . . , √ p k ] obtained from the rational numbers Q by adding the k square roots √ p i has dimension k over Q withbasis the elements Q i ∈ I √ p i as I varies on the k subsets of { , , . . . , k } .The group of automorphisms of F which fix Q (i.e. the Galois group of F/ Q ) is an Abelian groupgenerated by the automorphisms τ i defined by τ i ( √ p j ) = ( − δ ( i,j ) √ p j .Proof. We prove by induction both statements. Let us assume the statements valid for p , . . . , p k − andlet F ′ := Q [ √ p , √ p , . . . , √ p k − ] so that F = F ′ [ √ p k ]. We first prove that √ p k / ∈ F ′ . Assume it to befalse. Since ( √ p k ) is integer, each element – say τ – of the Galois group of F ′ / Q must either fix √ p k ortransform it into −√ p k (by definition τ ( p k ) = τ ( √ p k ) = p k ).Now any element b ∈ F ′ is by induction uniquely expressed as b = X I ⊂{ , ,...,k − } a I Y i ∈ I √ p i , a I ∈ Q . 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