A negative minimum modulus theorem and surjectivity of ultradifferential operators
aa r X i v : . [ m a t h . C V ] J a n A NEGATIVE MINIMUM MODULUS THEOREM ANDSURJECTIVITY OF ULTRADIFFERENTIAL OPERATORS
L ´ASZL ´O ZSID ´O
Dedicated to the memory of Professor Ciprian Foia¸s
Abstract.
In 1979 I. Cior˘anescu and L. Zsid´o have proved a minimummodulus theorem for entire functions dominated by the restriction to(0 , + ∞ ) of entire functions of the form ω ( z ) = ∞ Y j =1 (cid:16) izt j (cid:17) , z ∈ C ,with 0 < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , ∞ X j =1 t j < + ∞ , and suchthat + ∞ Z ln | ω ( t ) | t ln t ln | ω ( t ) | d t < + ∞ . It implies that for ω as above,every ω -ultradifferential operator with constant coefficients and of con-vergence type maps some D ρ ′ ⊃ D ω ′ onto itself. Here we show thatthe above results are sharp, by proving the negative counterpart of theabove minimum modulus theorem : if + ∞ Z ln | ω ( t ) | t ln t ln | ω ( t ) | d t = + ∞ ,then always there exists an entire function dominated by the restric-tion to (0 , + ∞ ) of ω , which does not satisfy the minimum modulusconclusion in the 1979 paper. It follows that for such ω there exists an ω -ultradifferential operator with constant coefficients and of convergencetype, which does not map any D ρ ′ ⊃ D ω ′ onto itself. Introduction
The main purpose of this paper is to expose (in a slightly completedform) the surjectivity criterion for ultradifferential operators with constantcoefficients, given in [10], Proposition 2.7, and to prove that this criterion issharp.To avoid ambiguity, we notice that we will use Bourbaki’s terminology:”positive” and ”strictly positive” instead of ”non-negative” and ”positive”,as well as ”increasing” and ”strictly increasing” instead of ”non-decreasing”and ”increasing”.
Date : 15 September 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Ultradistributions, ultradifferential operators, entire functions,minimum modulus theorems.Supported by INdAM and EU.
In Section 2 we present, following [9], the current ultradistribution theo-ries on R . Up to equivalence, there are two of them.The first one is parametrized by entire functions of the form ω ( z ) = ∞ Y j =1 (cid:16) izt j (cid:17) , z ∈ C , where 0 < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , ∞ X j =1 t j < + ∞ , whose set is denoted by Ω . D ω is a strict inductive limit of a sequence ofnuclear Fr´echet spaces, whose elemts are infinitely differentiable functions ofcompact support. The strong dual D ω ′ is the space of ω -ultradistributions. D ω can be naturally considered a subspace of D ω ′ . If ω , ρ ∈ Ω are such that | ω ( t ) | ≤ c | ρ ( t ) | for some constant c > t ∈ R , then D ρ ⊂ D ω and D ω ′ ⊂ D ρ ′ .A second ultradistribution theory is obtained by considering the spaces D ω and D ω ′ only for entire functions ω as above with the t j ’s satisfyingadditionally 0 < t ≤ t ≤ t ≤ ... . Ω will denote the set of these entire functions.In Section 3 we discuss ultradifferential operators and formulate the mainresults.We call a linear map T : D ω −→ D ω ω -ultradifferential operator wheneverthe support of T ϕ is contained in the support of ϕ ∈ D ω . It is of constantcoefficients if it commutes with the translation operators. T is an ω -ultradifferential operator of constant coefficients if and only ifthere exists an entire function f of exponential type 0 such that | f ( it ) | ≤ c | ω ( t ) | n , t ∈ R , for some c > n ≥ T ϕ is the product of the Fourier transform of ϕ multiplied by R ∋ t f ( it ) . In order that T be the convergent Taylor series f ( D ) of thederivation operator D , f must satisfy the stronger majorization property | f ( z ) | ≤ c (cid:12)(cid:12) ω ( | z | ) (cid:12)(cid:12) n , z ∈ C , with c > n ≥ T is called of convergence type.Any ω -ultradifferential operator T of constant coefficients can be uniquelyextended to a continuous linear operator D ω ′ −→ D ω ′ , still denoted by T . Acentral issue is the characterization of the situation T D ω ′ = D ω ′ , when theequation f ( D ) X = F has a solution X ∈ D ω ′ for each F ∈ D ω ′ , in terms ofthe entire function f associated to T . Such a criterion was obtained by I.Cior˘anescu in [8], Proposition 2.4 and Theorem 3.4 : T D ω ′ = D ω ′ if and onlyif f satisfies a certain minimum modulus condition.In [10] a minimum modulus theorem was obtained, which implies that if + ∞ Z ln | ω ( t ) | t ln t ln | ω ( t ) | d t < + ∞ INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 3 then, for every ω -ultradifferential operator T of constant coefficients and ofconvergent type, there exists some ρ ∈ Ω , | ω ( t ) | ≤ c | ρ ( t ) | for some constant c > t ∈ R , hence such that D ω ′ ⊂ D ρ ′ , for which the surjectivity T D ρ ′ = D ρ ′ holds true. We complete this result by proving that if ω ∈ Ω ,then we can choose ρ ∈ Ω (Theorem 3.9). To do this, we completed theminimum modulus theorem from [10] correspondingly (Theorem 3.8).On the other hand we prove (Theorem 3.11) that if + ∞ Z ln | ω ( t ) | t ln t ln | ω ( t ) | d t = + ∞ then, there exists an ω -ultradifferential operator T of constant coefficientsand of convergent type, such that the surjectivity T D ρ ′ = D ρ ′ can not holdfor any ρ ∈ Ω , | ω ( t ) | ≤ c | ρ ( t ) | for some constant c > t ∈ R (Theorem 3.11). This is consequence of the negative minimum modulustheorem (Theorem 3.10), claiming that for ω as above there exists an entirefunction f such that | f ( z ) | ≤ (cid:12)(cid:12) ω ( | z | ) (cid:12)(cid:12) , z ∈ C , but for no increasing function β : (0 , + ∞ ) −→ (0 , + ∞ ) with + ∞ Z β ( t ) t d t < + ∞ can hold the minimummodulus condition sup s ∈ R | s − t |≤ β ( t ) ln | f ( s ) | ≥ − β ( t ) , t > . This negative minimum modulus theorem is the hearth of the paper and isproved in the last, 6th section.In Section 4 we investigate the majorization of positive functions definedon (0 , + ∞ ) with functions α : (0 , + ∞ ) −→ (0 , + ∞ ) belonging to differentregularity classes and satisfying the non-quasianalyticity condition + ∞ Z α ( t ) t d t < + ∞ . (like (0 , + ∞ ) ∋ t ln | ω ( t ) | for ω ∈ Ω ). These topics are used in theproof of Theorem 3.8. Lemma 4.2 could be of interest for itself.Section 5 is devoted to increasing functions α : (0 , + ∞ ) −→ (0 , + ∞ )satisfying + ∞ Z α ( t ) t d t < + ∞ and + ∞ Z α ( t ) t ln tα ( t ) d t = + ∞ . Discretization of the above conditions is investigated (Propositions 5.3 and5.4) and the case α ( t ) = ln | ω ( t ) | , ω ∈ Ω , is characterized (Theorem 5.6). L. ZSID ´O
In particular, for 0 < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , α : (0 , + ∞ ) ∋ t ln (cid:12)(cid:12)(cid:12) ∞ Y j =1 (cid:16) itt j (cid:17)(cid:12)(cid:12)(cid:12) satisfies the above two conditions if and only if ∞ X j =1 t j < + ∞ and ∞ X j =1 ln t j jt j = + ∞ , what happens, for example, if t j = j (ln j ) (ln ln j ) p , j ≥ < p ≤ n j = α (2 j ) , j ≥ α ( t ) = t (ln t )(ln ln t ) , t > e , sent to me in [15]. The proof of Lemma6.1 is based on Hayman’s ideas, it is actually an adaptation of Hayman’sdraft to the general case.2. Ultradistribution theories
In order to enlarge the family of L. Schwartz’s distributions, I. M. Gelfandand G. E. Shilov proposed in [12] (see also [13], Chapters II and IV) thefollowing extension of L. Schwartz’s strategy: consider an appropriate locallyconvex topological vector space B of infinitely differentiable functions suchthat • B is a Fr´echet space or a countable inductive limit of Fr´echet spaces, • the topology of B is stronger than the topology of pointwise conver-gence.The elements of B are called basic functions , and the elements of the dual B ′ , generalized functions . If we ”shrink” B , then B ′ becomes larger.The generalized functions B ′ are usually called ultradistributions when,roughly speaking, disjoint compact sets can be separated by functions whichbelong to B . This yields a ”lower bound” for B . Ultradistribuion theoriesare mostly based on non-quasianaliticity.Let us briefly sketch, following [9], Section 7, what we will here understandby an ultradistribution theory on the real line R (a slightly different pictureis given in [24]).Let S be a parameter set and assume that to each σ ∈ S is associated alocally convex topological vector space D σ of infinitely differentiable func-tions R −→ C with compact support such that, for every σ ∈ S ,(i) D σ is an inductive limit of a sequence of Fr´echet spaces;(ii) the topology of D σ is stronger than the topology of pointwise con-vergence; INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 5 (iii) D σ is an algebra under pointwise multiplication;(iv) for K ⊂ D ⊂ R , K compact and D open, there exists ϕ ∈ D σ suchthat 0 ≤ ϕ ≤ , ϕ ( s ) = 1 for s ∈ K , supp( ϕ ) ⊂ D ;(v) denoting by E σ the multiplier algebra of D σ , that is the set of allfunctions ψ : R −→ C satisfying ϕψ ∈ D σ , ϕ ∈ D σ , and endowing itwith the projective limit topology defined by the linear mappings E σ ∋ ψ ϕψ ∈ D σ , ϕ ∈ D σ , the set A of all real analytic complex functions on R is a dense subsetof E σ .We will say that { D σ } σ ∈ S is a theory of ultradistributions and the elementsof the dual D σ ′ will be called σ -ultradistributions. For σ ∈ S and F ∈ D σ ′ , there is a smallest closed set S ⊂ R such that ϕ ∈ D σ , S ∩ supp( ϕ ) = ∅ = ⇒ F ( ϕ ) = 0 . Then S is called the support of F and is denoted by supp( F ) . The dual E σ ′ can be identified with the vector space of all σ -ultradistributions of compactsupport, since the restriction map E σ ′ ∋ G G ⌈ D σ is a linear isomorphismof D σ ′ onto { F ∈ D σ ′ ; supp( F ) compact } .By a σ -ultradifferential operator we mean a linear operator T : D σ −→ D σ which doesn’t enlarge the support:supp( T ϕ ) ⊂ supp( ϕ ) , ϕ ∈ D σ . Let { D σ } σ ∈ S and { D τ } τ ∈ T be two ultradistribution theories. We saythat the ultradistribution theory { D τ } τ ∈ T is larger than { D σ } σ ∈ S if forevery σ ∈ S there exists some τ ∈ T such that D τ ⊂ D σ , or equivalently, E τ ⊂ E σ . When this happens then the inclusion maps D τ ֒ → D σ and E τ ֒ → E σ are continuous and have a dense range.We notice that if { D σ } σ ∈ S and { D τ } τ ∈ T are ultradistribution theoriesand { D τ } τ ∈ T is larger than { D σ } σ ∈ S , then A ⊂ \ τ ∈ T E τ ⊂ \ σ ∈ S E σ . We say that two ultradistribution theories { D σ } σ ∈ S and { D τ } τ ∈ T are equivalent whenever each one of them is larger than the other.Let us recall the usual ultradistribution theories. They are labeled by oneof the following parameter sets S : • M is the set of all sequences ( M p ) p ≥ in (0 , + ∞ ) , M = 1 , satisfying M p ≤ M p − M p +1 , p ≥ , X p ≥ M p − M p < + ∞ (non-quasianalyticity) . L. ZSID ´O • M is the set of all sequences ( M p ) p ∈ M which satisfy the strongerlogarithmic convexity condition (cid:16) M p p ! (cid:17) ≤ M p − ( p − · M p +1 ( p + 1)! , p ≥ . • A is the set of all continuous functions α : R −→ (0 , + ∞ ) satisfying α (0) = 0 , α ( t + s ) ≤ α ( t ) + α ( s ) for t , s ∈ R (subadditivity) , there exist a ∈ R and b > α ( t ) ≥ a + b ln(1 + | t | ) , t ∈ R , + ∞ Z −∞ α ( t )1 + t d t < + ∞ . • Ω is the set of all entire functions ω of the form(2.1) ω ( z ) = ∞ Y j =1 (cid:16) izt j (cid:17) , z ∈ C , where 0 < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , ∞ X j =1 t j < + ∞ . • Ω is the set of all entire functions ω of the form ω ( z ) = ∞ Y j =1 (cid:16) izt j (cid:17) , z ∈ C , where 0 < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , ∞ X j =1 t j < + ∞ . Let ( M p ) p ∈ M be fixed. For K ⊂ R compact and h > D { M p } ,h ( K )denote the vector space of all infinitely differentiable functions ϕ : R −→ C with supp( ϕ ) ⊂ K , satisfying k ϕ k { M p } ,h := sup s ∈ K, p ≥ h p M p | ϕ ( p ) ( s ) | < + ∞ . Then D { M p } ,h ( K ) , endowed with the norm k · k { M p } ,h , becomes a Banachspace.The Roumieu ultradifferentiable functions of class ( M p ) p ∈ M on R ,having compact support, are D { M p } := lim −→ K ⊂ R compact lim −→ Beurling-Komatsu ultradistribution theories . INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 7 Let now α ∈ A be fixed. For K ⊂ R compact we denote by D α ( K ) thevector space of all continuous functions ϕ : R −→ C with supp( ϕ ) ⊂ K , forwhich k ϕ k α,λ := + ∞ Z −∞ | b ϕ ( t ) | e λα ( t ) d t < + ∞ , λ > , where b ϕ stands for the Fourier transform of ϕ : b ϕ ( t ) = 12 π + ∞ Z −∞ ϕ ( s ) e − its d s Then D α ( K ) , endowed with the family of norms k · k α,λ , λ > Beurling-Bj¨orck ultradifferentiable functions of class α ∈ A on R ,having compact support, are D α := lim −→ K ⊂ R compact D α ( K )(see [2] and [4]). { D α } α ∈ A is the Beurling-Bj¨orck ultradistribution theory .Finally, for ω ∈ Ω and K ⊂ R compact, let D ω ( K ) be the vector space ofall continuous functions ϕ : R −→ C with supp( ϕ ) ⊂ K , for which p ω,n ( ϕ ) := sup t ∈ R | b ϕ ( t ) ω ( t ) n | < + ∞ , n ≥ . Then D ω ( K ) , endowed with the family of norms p ω,n , n ≥ ω -ultradifferentiable functions on R , having compact support, are D ω := lim −→ K ⊂ R compact D ω ( K )(see [9], Section 2). { D ω } ω ∈ Ω is the ω -ultradistribution theory .We have to remark that in [9], Definition III, D ω ( K ) is defined by usingthe norms p ω,L,n , L > , n ≥ p ω,L,n ( ϕ ) := sup t ∈ R | b ϕ ( t ) ω ( L t ) n | . However, with the notation of (2.1), we have(2.2) | ω ( L t ) | = ∞ Y j =1 (cid:16) L t t k (cid:17) / ≤ ∞ Y j =1 (cid:16) t t k (cid:17) L / = | ω ( t ) | L , so the two definitions are equivalent.We notice that the Roumieu, the Beurling-Komatsu and the Beurling-Bj¨orck ultradistribution theories were considered also on open subsets of R d (see [23], [17], [4]), while the ω -ultradistribution theory, originally consideredin [9] only on R , was subsequently extended to the multidimensional setting(see [6] and [1]). However, in this paper we will restrict us to the one-dimensional case of R . L. ZSID ´O In [9], 7.4 it was shown that the ultradistribution theories(2.3) { D { M p } } ( M p ) p ∈ M , { D ( M p ) } ( M p ) p ∈ M , { D ω } ω ∈ Ω are equivalent. Thus they are just different labelings of the same global setof ultradistributions. To work with ultradifferential operators, the setting ofthe ω -ultradistribution theory seems to be the most advantageous. Thereforewe will adopt this setting in the sequel.We notice that, according to [11], Theorem 1, also the ultradistributiontheories(2.4) { D { M p } } ( M p ) p ∈ M , { D ( M p ) } ( M p ) p ∈ M , { D α } α ∈ A , { D ω } ω ∈ Ω are equivalent. As was pointed out in [9], Section 7.7, T ω ∈ Ω E ω = T ω ∈ Ω E ω ,so the ultradistribution theories (2.3) are larger than those in (2.4), but notequivalent to them.3. Ultradifferential operators and the main results For ω ∈ Ω , let us consider the ω -ultradifferentiable function spaces D ω , E ω , as defined in Section 2 ( D ω on page 7, and E ω as indicated in (v) onpage 5). D ω is strict inductive limit of a sequence of nuclear Fr´echet spaces and it isstable under a series of elementary operations like pointwise multiplication,convolution, differentiation, translations etc. Moreover, these operations arecontinuous. E ω is a nuclear Fr´echet space and has similar stability properties as D ω .The set A of all real analytic complex functions on R , as well as D ω , aredense subsets of E ω .The space of the ω -ultradistributions is the strong dual D ω ′ of D ω and,associating to each ϕ ∈ E ω the linear functional D ω ∋ ψ + ∞ Z −∞ ϕ ( s ) ψ ( s )d s , we obtain an inclusion map with dense range E ω ֒ → D ω ′ .For all the above facts we send to [9], Section 2.Let now T : D ω −→ D ω be an ω -ultradifferential operator , that is a linearoperator satisfying the conditionsupp( T ϕ ) ⊂ supp( ϕ ) , ϕ ∈ D ω . Then T is continuous and can be (uniquely) extended to a continuous linearoperator E ω −→ E ω , which will be still denoted by T ([9], Theorem 2.16).We say that an ω -ultradifferential operator is with constant coefficients ifit commutes with every translation operator. An immediate consequence of[9], Theorem 2.21 is Proposition 3.1. If f is an entire function of exponential type such that (3.1) | f ( it ) | ≤ d | ω ( t ) n | , t ∈ R INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 9 for some integer n ≥ and real number d > , then the formula \ ( f ( D ) ϕ )( t ) = f ( it ) b ϕ ( t ) , ϕ ∈ D ω , t ∈ R defines an ω -ultradifferential operator f ( D ) with constant coefficients. Con-versely, any ω -ultradifferential operator f ( D ) with constant coefficients is ofthis form. (cid:3) If f is an entire function of exponential type 0 , satisfying (3.1) for some n ≥ d > ω -ultradifferential operator f ( D ) : E ω −→ E ω with constent coefficients can be extended to a continuous linear operator D ω ′ −→ D ω ′ , which we will still denote by f ( D ) (see [9], discussion beforeTheorem 3.5).Denoting by δ s o the Dirac measure concentrated at s ∈ R , consideredan ω -ultradistribution of support { s } , for each ω -ultradistribution F with-support { s } there exists an entire function as above such that T = f ( D ) δ s (see [9], Theorem 3.5).If ω ( z ) = ∞ Y j =1 (cid:16) izt j (cid:17) , z ∈ C , where0 < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , ∞ X j =1 t j < + ∞ , then, for n ≥ k ≥ a ω,nk the square root of thecoefficient of z k in the power series expansion of the entire function C ∋ z (cid:0) ω ( z ) ω ( z ) (cid:1) n = ∞ Y j =1 (cid:16) z t k (cid:17) n ( ω ( z ) stands here, as usual, for ω ( z ) ). We recall (see [9], page 109): a ω,nk ≤ a ω,n +1 k , n ≥ , k ≥ p ≥ a ω,np | t | p ≤ | ω ( t ) n | ≤ √ p ≥ a ω,np (cid:12)(cid:12) √ t (cid:12)(cid:12) p , t ∈ R . We have also, according to [9], Corollary 2.9,(3.3) (cid:0) a ω,nk (cid:1) ≥ a ω,nk − · a ω,nk +1 , n , k ≥ (cid:16) a ω,nk a ω,nk − (cid:17) k sup p ≥ a ω,np (cid:16) a ω,n k − a ω,nk (cid:17) p = a ω,nk , n , k ≥ (cid:16) a ω,nk a ω,nk − (cid:17) k sup p ≥ a ω,np (cid:16) a ω,n k − a ω,nk (cid:17) p = (cid:16) a ω,nk a ω,nk − (cid:17) k max (cid:18) , sup p ≥ p Y q =1 (cid:16) a ω,nq a ω,nq − · a ω,nk − a ω,nk (cid:17)(cid:19) and, by (3.3), a ω,nq a ω,nq − · a ω,nk − a ω,nk ( ≥ q ≤ k ≤ q ≥ k , we deduce: (cid:16) a ω,nk a ω,nk − (cid:17) k sup p ≥ a ω,np (cid:16) a ω,n k − a ω,nk (cid:17) p = (cid:16) a ω,nk a ω,nk − (cid:17) k k Y q =1 (cid:16) a ω,nq a ω,nq − · a ω,nk − a ω,nk (cid:17) = a ω,nk . (3.4) implies immediately:(3.5) min t>o t k sup p ≥ a ω,np t p = a ω,nk , n ≥ , k ≥ . We notice also the inequality(3.6) (cid:12)(cid:12) ω ( z ) (cid:12)(cid:12) = ∞ Y j =1 (cid:12)(cid:12)(cid:12) izt j (cid:12)(cid:12)(cid:12) ≤ ∞ Y j =1 (cid:16) | z | t j (cid:17) = ω ( − i | z | ) , z ∈ C . If P is a polynomial with complex coefficients and P ( z ) = n P k =0 c k z k , then P ( D ) = n P k =0 c k D k where D is the derivation operator. The next proposition,a variant of [9], Theorem 2.25, characterizes those ω -ultradifferential oper-ators with constant coefficients, which can be expanded in power series in D . Proposition 3.2. Let f be an entire function of exponential type suchthat ( ) holds true for some n ≥ and d > , and f ( z ) = ∞ P k =0 c k z k itsexpansion in a power series. Then the following statements are equivalent :(i) There exist an integer n ≥ and a real number d > such that | f ( z ) | ≤ d (cid:12)(cid:12) ω ( | z | ) n (cid:12)(cid:12) , z ∈ C . (ii) There exist an integer n ≥ and real numbers L , d > such that | c k | ≤ d L k a ω,n k , k ≥ . (iii) We have f ( D ) = ∞ X k =0 c k D k , where the series converges in the vectorspace of all continuous linear maps E ω −→ E ω , endowed with thetopology of the uniform convergence on the bounded subsets of E ω . (iv) We have f ( D ) = ∞ X k =0 c k D k , where the series converges in the vectorspace of all continuous linear maps E ω −→ E ω , endowed with thetopology of the pointwise convergence. INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 11 Proof. For (i) ⇒ (ii). Using the Cauchy estimate, (i) and (3.2), we obtain forany integer k ≥ r > | c k | ≤ r k sup | z | = r | f ( z ) | ≤ d r k sup | z | = r (cid:12)(cid:12) ω ( | z | ) n (cid:12)(cid:12) ≤ √ d r k sup p ≥ a ω,n p (cid:0) √ r (cid:1) p . Using now (3.5), we infer: | c k | ≤ √ d inf r> r k sup p ≥ a ω,n p (cid:0) √ r (cid:1) p = √ d inf t> (cid:16) √ t (cid:17) k sup p ≥ a ω,n p t p = √ d (cid:0) √ (cid:1) k a ω,n k , k ≥ . Thus (ii) holds with n = n , L = √ , d = √ d .For (ii) ⇒ (i). Using (ii) and the first inequality in (3.2), we deduce: | f ( z ) | ≤ ∞ X k =0 | c k | | z | k ≤ d ∞ X k =0 a ω,n k (cid:0) L | z | (cid:1) k = d ∞ X k =0 k a ω,n k (cid:0) L | z | (cid:1) k ≤ d (cid:16) ∞ X k =0 k (cid:17) sup k ≥ a ω,n k (cid:0) L | z | (cid:1) k ≤ d (cid:12)(cid:12) ω (cid:0) L | z | (cid:1) n (cid:12)(cid:12) . Choosing some integer m ≥ L and using (2.2), we obtain | f ( z ) | ≤ d (cid:12)(cid:12) ω (cid:0) m | z | (cid:1) n (cid:12)(cid:12) ≤ d (cid:12)(cid:12) ω ( | z | ) m n (cid:12)(cid:12) , hence (i) holds with n = m n and d = 2 d .Implication (ii) ⇒ (iii) follows by [9], Proposition 2.24, and implication(iii) ⇒ (iv) is trivial.Finally, for the proof of (iv) ⇒ (ii) we adapt the proof of (iv) ⇒ (ii) in [9],Theorem 2.25 as follows.(iv) implies that the sequence ( c k D k ϕ ) k ≥ = ( c k ϕ ( k ) ) k ≥ converges in E ω to 0 for every ϕ ∈ E ω . Therefore the sequence E ω ∋ ϕ c k ϕ ( k ) (0) , k ≥ , is pointwise convergent to 0 in E ω ′ , in particular it is pointwise bounded.Since E ω is a Fr´echet space, and hence barrelled, if follows that the abovesequence in E ω ′ is equicontinuous (see e.g. [5], Ch. III, § 4, Section 1).Recalling that the topology of E ω is defined by the semi-norms r Kω,L,n : E ω ∋ ϕ sup p ≥ (cid:0) L p a ω,np sup s ∈ K | ϕ ( p ) ( s ) | (cid:1) , where K ⊂ R is compact, L > n ≥ K , L , n and of a constant d > | c k ϕ ( k ) (0) | ≤ d · r Kω,L,n ( ϕ ) , k ≥ , ϕ ∈ E ω . Applying this inequality to ϕ = e iα · , α > | c k | · α k ≤ d · sup p ≥ (cid:0) L p a ω,np α p (cid:1) , k ≥ , α > . Therefore, using (3.5), we infer: | c k | ≤ d inf α> α k sup p ≥ a ω,np ( L α ) p = d inf t> (cid:16) Lt (cid:17) k sup p ≥ a ω,np t p = d L k a ω,nk , k ≥ . In other words, (ii) holds with n = n , L = L , d = d . (cid:3) If f is an entire function satisfying the equivalent conditions in Proposition3.2, then we will say that the ω -ultradifferential operator with constantcoefficients f ( D ) is of convergence type .Proposition 3.2 enables to prove a description of those ω ∈ Ω , for whichevery ω -ultradifferential operator with constant coefficients is of convergencetype. This description is essentially [9], Theorem 2.25. Corollary 3.3. The following statements concerning ω ∈ Ω are equivalent :(j) There exist an integer n ≥ and a real number d > such that (cid:12)(cid:12) ω ( − it ) (cid:12)(cid:12) ≤ d (cid:12)(cid:12) ω ( t ) n (cid:12)(cid:12) , t ∈ R . (jj) There exist an integer n ≥ and a real number d > such that (cid:12)(cid:12) ω ( z ) (cid:12)(cid:12) ≤ d (cid:12)(cid:12) ω ( | z | ) n (cid:12)(cid:12) , z ∈ C . (jjj) The ω -ultradifferential operator with constant coefficients ω ( − iD ) is of convergence type. (jw) Every ω -ultradifferential operator with constant coefficients is ofconvergence type.Proof. (j) ⇒ (jj) follows easily by using (3.6): (cid:12)(cid:12) ω ( z ) (cid:12)(cid:12) ( . ) ≤ ω ( − i | z | ) (j) ≤ d (cid:12)(cid:12) ω ( | z | ) n , z ∈ C . On the other hand, implication (jj) ⇒ (j) is trivial: (cid:12)(cid:12) ω ( − it ) (cid:12)(cid:12) (jj) ≤ d (cid:12)(cid:12) ω ( | − it | ) n (cid:12)(cid:12) = d (cid:12)(cid:12) ω ( | t | ) n (cid:12)(cid:12) = d (cid:12)(cid:12) ω ( t ) n (cid:12)(cid:12) , t ∈ R . Thus (j) ⇔ (jj).Next, equivalence (jj) ⇔ (jjj) is an immediate consequence of the definitionof the convergence type by using condition (i) in Proposition 3.2, whileimplication (jw) ⇒ (jjj) is trivial. Thus it remains only to prove, for example,(jj) ⇒ (jw).For let us assume that (jj) is satisfied and f is an arbitrary entire functionof exponential type 0 , satisfying (3.1).Denoting, for convenience, ρ ( z ) := d · ω ( z ) n , z ∈ C ,ρ is an entire function of exponential type 0 , which has no zeros in theopen lower half-plane. Therefore, using the terminology of [18], ChapterVII, § ρ is an entire function of class P . Since f ( i · ) is an entire functionof exponential type 0 and, by (3.1), INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 13 | f ( it ) | ≤ | ρ ( t ) | , t ∈ R , applying [18], Chapter IX, § 4, Lemma 1, we obtain: (cid:12)(cid:12) f ( iz ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) ρ ( z ) (cid:12)(cid:12) and (cid:12)(cid:12) f ( i z ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) ρ ( z ) (cid:12)(cid:12) for all z ∈ C with Im z ≤ , that is(3.7) (cid:12)(cid:12) f ( iz ) (cid:12)(cid:12) ≤ ( d (cid:12)(cid:12) ω ( z ) n (cid:12)(cid:12) for z ∈ C , Im z ≤ d (cid:12)(cid:12) ω ( z ) n (cid:12)(cid:12) for z ∈ C , Im z ≥ z ∈ C (3.8) (cid:12)(cid:12) ω ( z ) (cid:12)(cid:12) ≤ d (cid:12)(cid:12) ω ( | z | ) n (cid:12)(cid:12) and (cid:12)(cid:12) ω ( z ) (cid:12)(cid:12) ≤ d (cid:12)(cid:12) ω ( | z | ) n (cid:12)(cid:12) = d (cid:12)(cid:12) ω ( | z | ) n (cid:12)(cid:12) . Now, by (3.7) and (3.8) we deduce: (cid:12)(cid:12) f ( iz ) (cid:12)(cid:12) ≤ d ( d ) n (cid:12)(cid:12) ω ( | z | ) n · n (cid:12)(cid:12) , z ∈ C . Consequently condition (i) in Proposition 3.2 holds true with n = n · n and d = d ( d ) n , and we conclude that the ω -ultradifferential operatorwith constant coefficients f ( D ) is of convergence type. (cid:3) Following [9], Definition XI, we will say that ω satisfies the strong non-quasianalyticity condition whenever it fulfills the equivalent conditions inCorollary 3.3. Remark 3.4. If ω ( z ) = ∞ Y j =1 (cid:16) izt j (cid:17) , z ∈ C , where < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , ∞ X j =1 t j < + ∞ , then, in order that ω satisfy the strong non-quasianalyticity condition, anecessary condition is ∞ X j =1 ln jt j < + ∞ ( see [9] , Corollary 1.9 ), while a sufficient condition is the existence of aconstant c > such that ∞ X j = k t j ≤ c kt k , k ≥ see [17] , Proposition 4.6 or [9] , comments after Proposition 5.15 ). If weassume also < t ≤ t ≤ t ≤ ... , then ω satisfies the strong non-quasianalyticity condition if and only if thereexists a constant c > such that ∞ X j = k t j ≤ c kt k (cid:16) t k ( t ... t k ) /k (cid:17) , k ≥ see [9] , Proposition 5.15 ).Central issue in the theory of ω -ultradifferential operators with constantcoefficients f ( D ) : D ω ′ −→ D ω ′ is the characterization of its surjectivity, thatis of the existence of a solution X ∈ D ω ′ of the equation f ( D ) X = F for each F ∈ D ω ′ , in terms of f . A surjectivity criterion was proved by I. Cior˘anescuin [8], Proposition 2.4 and Theorem 3.4 : Proposition 3.5. For f an entire function of exponential type such that ( ) holds true for some n ≥ and d > ., the following statements areequivalent :(i) There exists some E ∈ D ω ′ such that f ( D ) E = δ . (ii) f ( D ) : D ω ′ −→ D ω ′ is surjective, that is f ( D ) D ω ′ = D ω ′ . (iii) there are constants c , c ′ > such that sup s ∈ R | s − t |≤ c ln | ω ( t ) | + c ′ ln | f ( s ) | ≥ − c ln | ω ( t ) | − c ′ , t ∈ R . (cid:3) If f satisfies the equivalent conditions of Proposition 3.5, then, following[7], D´efinition III.1-4, and by abuse of language, we will say that f ( D ) is invertible in D ω ′ .If ρ ∈ Ω and | ω ( t ) | ≤ c | ρ ( t ) | , t ∈ R for some c > D ρ ⊂ D ω , where the inclusion is continuous and withdense range, Consequently also D ω ′ ⊂ D ρ ′ , where the inclusion is continu-ous and with dense range. Any ω -ultradifferential operator with constantcoefficients f ( D ) is clearly also a ρ -ultradifferential operator with constantcoefficients, hence we can consider the problem of the invertibility of f ( D )in D ρ ′ . We notice that is f ( D ) is of convergence type as ω -ultradifferentialoperator, then it is of convergence type also as ρ -ultradifferential operator.The main goal of this paper is to give an exact answer to the question: forwhich ω ∈ Ω is every ω -ultradifferential operator with constant coefficientsand of convergence type, ρ -invertible for some ρ ∈ Ω with ω ≤ c ρ , where c > + ∞ Z ln | ω ( t ) | t ln t ln | ω ( t ) | d t < + ∞ . INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 15 Let us call condition (3.9) the mild strong non-quasianalyticity condition .This denomination is justified by the fact that (3.9) is implied by the strongnon-quasianalyticity property. More precisely, we have : Proposition 3.6. For ω ( z ) = ∞ Y j =1 (cid:16) izt j (cid:17) , z ∈ C , where < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , ∞ X j =1 t j < + ∞ , we have ω satisfies the strong non-quasianalyticity condition = ⇒ ∞ X j =1 ln jt j < + ∞ ⇐⇒ ∞ X j =1 ln t j t j < + ∞ = ⇒ ∞ X j =1 ln t j jt j < + ∞ ⇐⇒ ω satisfies the mild strongnon-quasianalyticity condition.Proof. The first implication was already pointed out in Remark 3.4.A proof of equivalence ∞ X j =1 ln jt j < + ∞ ⇐⇒ ∞ X j =1 ln t j t j < + ∞ was given inthe comments after [9], Corollary 1.9 (Page 92).The second implication is trivial, while the last equivalence is (i) ⇔ (iv) in[10], Lemma 2.1. (cid:3) We will need the next calculus lemma : Lemma 3.7. Let α , γ : (0 , + ∞ ) −→ (0 , + ∞ ) be two functions such that γ ( t ) α ( t ) ≥ e , t > . (i) If α and γ are increasing, then also the function (3.10) (0 , + ∞ ) ∋ t α ( t ) ln γ ( t ) α ( t ) ∈ (0 , + ∞ ) is increasing. (ii) If α and γ are twice differentiable and concave, then also the function (3.10) is twice differentiable and concave.Proof. For (i). Assume that α , γ are increasing and let 0 < t < t bearbitrary. Then α ( t ) ln γ ( t ) α ( t ) ≤ α ( t ) ln γ ( t ) α ( t ) = γ ( t ) (cid:16) α ( t ) γ ( t ) ln γ ( t ) α ( t ) (cid:17) . Since γ ( t ) α ( t ) ≥ γ ( t ) α ( t ) ≥ e and x x ln x is decreasing on [ e , + ∞ ) , we get α ( t ) ln γ ( t ) α ( t ) ≤ γ ( t ) (cid:16) α ( t ) γ ( t ) ln γ ( t ) α ( t ) (cid:17) = α ( t ) ln γ ( t ) α ( t ) . For (ii). Assume that α , γ are twice differentiable and concave, hence α ′′ , γ ′′ ≤ t > α ′ ( t ) ln γ ( t ) α ( t ) + γ ′ ( t ) α ( t ) − α ′ ( t ) γ ( t ) γ ( t ) , while its second derivative at t > α ′′ ( t ) (cid:16) ln γ ( t ) α ( t ) − (cid:17) + γ ′′ ( t ) α ( t ) γ ( t ) − (cid:0) γ ′ ( t ) α ( t ) − α ′ ( t ) γ ( t ) (cid:1) α ( t ) γ ( t ) . Since α ′′ ( t ) γ ′′ ( t ) ≤ ln γ ( t ) α ( t ) − ≥ ln e − ≤ t > (cid:3) The next theorem is a slightly extended version of [10], Theorem 2.2. Forits proof we adapted the proof of [10], Theorem 2.2. Theorem 3.8. Let us assume that ω ∈ Ω satisfies the mild strong non-quasianalyticity condition. Then there exists some ρ ∈ Ω satisfying (3.11) | ω ( t ) | ≤ c | ρ ( t ) | , t ∈ R with c > a constant, such that : If f is an entire function and (3.12) | f ( z ) | ≤ d (cid:12)(cid:12) ω ( | z | ) n (cid:12)(cid:12) , z ∈ C for some integer n ≥ and d > , then there are constants c , c ′ > suchthat (3.13) sup s ∈ R | s − t |≤ c ln | ρ ( t ) | + c ′ ln | f ( s ) | ≥ − c ln | ρ ( t ) | − c ′ , t ∈ R . Moreover, if ω ∈ Ω , then we can choose ρ ∈ Ω .Proof. In the case of a general ω ∈ Ω , let α denote the function(0 , + ∞ ) ∋ t ln | ω ( t ) | ∈ (0 , + ∞ ) . In the case of ω ∈ Ω we need for α an infinitely differentiable, increasing,concave function satisfying(3.14) + ∞ Z α ( t ) t d t < + ∞ , + ∞ Z α ( t ) t ln tα ( t ) d t < + ∞ and ln | ω ( t ) | ≤ α ( t ) , t > . To obtain it, let INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 17 < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , ∞ X j =1 t j < + ∞ be such that ω ( z ) = ∞ Y j =1 (cid:16) izt j (cid:17) , z ∈ C , and set α ( t ) := ln 3 + 2 ln (cid:16) ∞ X k =1 (4 t ) k t ... t k (cid:17) , t > . Then (0 , + ∞ ) ∋ t α ( t ) ∈ (0 , + ∞ ) is infinitely differentiable, increasingand, according to Lemma 4.2, concave. On the other hand, by [9], Lemma1.7, we have ln | ω ( t ) | ≤ α ( t ) , t > . Finally, since by [9], Lemma 1.7, α ( t ) ≤ ln 3 + 2 (cid:0) ln 4 + 2 ln | ω (8 t ) | (cid:1) = ln(48) + 4 ln | ω (8 t ) | , t > , and ω satisfies the mild strong non-quasianalyticity condition, Lemma 5.1yields (3.14).An inspection of the proof of [10], Corollary 1.2 shows that there exists aconstant λ > tλ α (2 et ) > e , t > , and the function β : (0 , + ∞ ) ∋ t α (2 et ) ln 1 + tλ α (2 et ) + 8 ∞ X j =1 α (2 j et )4 j ∈ (0 , + ∞ ) , which is, according to Lemma 3.7, increasing and, in the case of ω ∈ Ω ,also concave, satisfies + ∞ Z β ( t ) t d t < + ∞ and has the property :If f is any entire function satisfyingln | f ( z ) | ≤ d α ( | z | ) + d ′ , z ∈ C for some d , d ′ > c , c ′ > t ≤ r ≤ t + c α ( t ) inf | z | = r ln | f ( z ) | ≥ − c β ( t ) − c ′ , t > . We notice that β ( t ) ≥ α (2 et ) ln(8 e ) > α ( t ) for all t > ρ ∈ Ω and a constant d > β ( t ) ≤ ln | ρ ( t ) | + d , t > . Moreover, in the case of ω ∈ Ω , when β is increasing and concave, Theorem4.4 ensures that ρ can be chosen belonging to Ω .Since | ω ( t ) | ≤ e α ( t ) < e β ( t ) ≤ e ln | ρ ( t ) | + d = e d | ρ ( t ) | , t < , (3.11) holds true with c = e d . Let f be an entire function satisfying (3.12). Thenln | f ( z ) | ≤ n ln (cid:12)(cid:12) ω ( | z | ) (cid:12)(cid:12) + ln d ≤ n α ( | z | ) + ln d , z ∈ C . By the choice of β there exist then constants c , c ′ > t ≤ r ≤ t + c α ( t ) inf | z | = r ln | f ( z ) | ≥ − c β ( t ) − c ′ , t > . It follows for every t ∈ R sup s ∈ R | s − t |≤ c ln | ρ ( t ) | + c ′ + c d ln | f ( s ) | ≥ sup s ∈ R | s − t |≤ c β ( t ) | + c ′ ln | f ( s ) |≥ sup | t |≤ r ≤| t | + c α ( t ) inf | z | = r ln | f ( z ) |≥ − c β ( | t | ) − c ′ ≥ − c (cid:12)(cid:12) ρ ( | t | ) (cid:12)(cid:12) − c ′ − c d = − c | ρ ( t ) | − c ′ − c d . Consequently (3.13) holds true with c = c and c ′ = c ′ + c d . (cid:3) Theorem 3.8 implies that mild strong non-quasianalyticity of ω ∈ Ω isa sufficient condition in order that every ω -ultradifferential operator withconstant coefficients and of convergence type be invertible in D ρ ′ for some ρ ∈ Ω satisfying | ω ( t ) | ≤ c | ρ ( t ) | , t ∈ R , with c > Theorem 3.9. If ω ∈ Ω is satisfying the mild strong non-quasianalyticitycondition, then there exist ρ ∈ Ω and a constant c > with | ω ( t ) | ≤ c | ρ ( t ) | , t ∈ R , such that every ω -ultradifferential operator with constant coefficients and ofconvergence type is invertible in D ρ ′ .Moreover, if ω ∈ Ω , then we can choose ρ ∈ Ω .Proof. Choose ρ and c as in Theorem 3.8.According to Propositions 3.1 and 3.2, every ω -ultradifferential operatorwith constant coefficients and of convergence type is of the form f ( D ) with f an entire function satisfying condition (i) in Proposition 3.2. By the choiceof ρ and c , there exist constants c , c ′ > f ( D ) is invertible in D ρ ′ . (cid:3) The main result of this paper is the following theorem, which shows thatTheorem 3.8 is sharp. It will be proved in Section 6. INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 19 Theorem 3.10. Let us assume that ω ∈ Ω does not satisfy the mild strongnon-quasianalyticity condition, that is such that + ∞ Z ln | ω ( t ) | t ln t ln | ω ( t ) | d t = + ∞ . Then there exists an entire function f such that (3.15) | f ( z ) | ≤ (cid:12)(cid:12) ω ( | z | ) (cid:12)(cid:12) , z ∈ C but for no increasing β : (0 , + ∞ ) −→ (0 , + ∞ ) with + ∞ Z β ( t ) t d t < + ∞ can hold the condition (3.16) sup s ∈ R | s − t |≤ β ( t ) ln | f ( s ) | ≥ − β ( t ) , t > . Using Theorem 3.10, we infer that also Theorem 3.9 is sharp : Theorem 3.11. Let us assume that ω ∈ Ω does not satisfy the mild strongnon-quasianalyticity condition. Then there exists some ω -ultradifferentialoperator with constant coefficients and of convergence type, which is notinvertible in D ρ ′ for any ρ ∈ Ω satisfying | ω ( t ) | ≤ c | ρ ( t ) | , t ∈ R for some constant c > .Proof. Let f be an entire function f as in Theorem 3.10. Then, accordingto Propositions 3.1 and 3.2, we can consider the ω -ultradifferential operatorwith constant coefficients f ( D ) , and it is of convergence tyoe.If it would exist some ρ ∈ Ω satisfying | ω ( t ) | ≤ c | ρ ( t ) | , t ∈ R with c > f ( D ) is invertible in D ρ ′ , then Proposition3.5 would imply the existence of constants c , c ′ > s ∈ R | s − t |≤ c ln | ρ ( t ) | + c ′ ln | f ( s ) | ≥ − c ln | ρ ( t ) | − c ′ , t ∈ R . But this is not possible because β : (0 , + ∞ ) ∋ t c ln | ρ ( t ) | + c ′ ∈ (0 , + ∞ )would be a function with + ∞ Z β ( t ) t d t < + ∞ (see e.g. [16], Theorem 1 or [9],Theorem 1.6) such thatsup s ∈ R | s − t |≤ β ( t ) ln | f ( s ) | ≥ − β ( t ) , t ∈ R , in contradiction with the choice of f . (cid:3) On the non-quasianalyticity condition For sake of convenience, we will say that a Lebesgue measurable function α : (0 , + ∞ ) −→ (0 , + ∞ ) satisfies the non-quasianalyticity condition if + ∞ Z α ( t ) t d t < + ∞ . This denomination is suggested by the classical Denjoy-Carleman Theorem(se e.g. [19], 4.1.III) in which non-quasianalyticity is characterized by thiscondition.Examples of functions satisfying the non-quasianalyticity condition : Remark 4.1. If < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , ∞ X j =1 t j < + ∞ , then the increasing functions (1) (0 , + ∞ ) ∋ t n ( t ) with n ( t ) the number of the elements of the set { k ≥ t k ≤ t } ( the distribution function of the sequence ( t j ) j ≥ ) , (2) (0 , + ∞ ) ∋ t α ( t ) := ln (cid:16) ∞ X k =1 t k t ... t k (cid:17) ∈ (0 , + ∞ ) , (3) (0 , + ∞ ) ∋ t N ( t ) := ln max (cid:16) , sup k ≥ t k t t ... t k (cid:17) ∈ (0 , + ∞ ) , (4) (0 , + ∞ ) ∋ t ln | ω ( t ) | ∈ (0 , + ∞ ) , where ω ∈ Ω is defined by ω ( z ) = ∞ Y j =1 (cid:16) izt j (cid:17) , z ∈ C , satisfy the non-quasianalyticity condition. Since t k +1 Z t n ( t ) t d t = k X j =1 t j +1 Z t j n ( t ) t d t = k X j =1 j (cid:16) t j − t j +1 (cid:17) = (cid:16) k X j =1 f j (cid:17) − kt k +1 ,we have ∞ Z t n ( t ) t d t ≤ ∞ X j =1 f j < + ∞ .A proof of the non-quasianalyticity of ln | ω ( · ) | can be found, for example,in the proof of implication (iii) ⇒ (i) in [9], Theorem 1.6.The non-quasianalyticity of N ( · ) follows from the non-quasianalyticityof ln | ω ( · ) | and the clear inequality N ( t ) ≤ ln | ω ( t ) | , t > α is consequence of the inequality1 + ∞ X k =1 t k t ... t k = 1 + ∞ X k =1 k t k ( t / ... ( t k / INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 21 ≤ (cid:16) ∞ X k =1 k (cid:17) max (cid:16) , sup k ≥ t k ( t / ... ( t k / (cid:17) and of the non-quasianalyticity of N ( · ) with t k replaced by t k / • Increasing functions (0 , + ∞ ) −→ (0 , + ∞ ) . • ”Concave like functions” α : (0 , + ∞ ) −→ (0 , + ∞ ) , which can be1) concave: α (cid:0) (1 − λ ) t + λt (cid:1) ≥ (1 − λ ) α ( t )+ λα ( t ) for 0 ≤ λ ≤ t t > , + ∞ ) ∋ t α ( t ) t is decreasing;3) subadditive: α ( t + t ) ≤ α ( t ) + α ( t ) for t , t > ⇒ ⇒ 3) . Indeed, if α is concave and 0 < t < t are arbitrary, then we have for any 0 < ε < t α ( t ) = α (cid:16) t − t t − ε ε + t − εt − ε t (cid:17) ≥ t − t t − ε α ( ε ) + t − εt − ε α ( t ) > t − εt − ε α ( t ) . Letting ε → α ( t ) ≥ t t α ( t ) ⇐⇒ α ( t ) t ≥ α ( t ) t .On the other hand, if (0 , + ∞ ) ∋ t α ( t ) t is decreasing, then wehave for all t , t > α ( t + t ) = t α ( t + t ) t + t + t α ( t + t ) t + t ≤ t α ( t ) t + t α ( t ) t = α ( t ) + α ( t ) . • (0 , + ∞ ) ∋ t ln | ω ( t ) | with ω an entire function belonging to Ω or Ω .The next lemma extends [9], Lemma 1.7 : Lemma 4.2. If < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , then the function α : (0 , + ∞ ) ∋ t ln (cid:16) ∞ X k =1 t k t ... t k (cid:17) ∈ (0 , + ∞ ) is strictly increasing and concave. Assuming additionally that t k k = t k +1 k + 1 for at least one k ≥ , α turns out to be even strictly concave.Proof. α is clearly strictly increasing.For the proof of the concavity it is convenient to denote c k = t k k , k ≥ . Then c ≥ c ≥ c ≥ ... ≥ , c > α ( t ) = ln (cid:18) ∞ X k =0 (cid:16) k Y j =1 c j (cid:17) t k k ! (cid:19) t > , where we agree that k Y j =1 c j = 1 for k = 0 .If c k = t k k = t k +1 k + 1 = c k +1 for all k ≥ α ( t ) = ln e c t = c t , so α is linear, hence concave. We will show that, assuming c k > c k +1 for some k ≥ α is strictly concave, by proving that α ′′ ( t ) < t > k denote the least integer k ≥ c k > c k +1 .Denoting f ( t ) = ∞ X k =0 (cid:16) k Y j =1 c j (cid:17) t k k ! , we have α ′′ ( t ) = (cid:0) ln f ( t ) (cid:1) ′′ = (cid:16) f ′ ( t ) f ( t ) (cid:17) ′ = f ′′ ( t ) f ( t ) − f ′ ( t ) f ( t ) . Therefore out task is to prove that f ( t ) − f ′′ ( t ) f ( t ) > t > f ′ ( t ) = ∞ X k =0 (cid:16) k +1 Y j =1 c j (cid:17) t k k ! , f ′′ ( t ) = ∞ X k =0 (cid:16) k +2 Y j =1 c j (cid:17) t k k ! and f ( t ) − f ′′ ( t ) f ( t )= ∞ X k =0 (cid:18) X p,q ≥ p + q = k p ! q ! (cid:16) p +1 Y j =1 c j (cid:17)(cid:16) q +1 Y j =1 c j (cid:17) − X p,q ≥ p + q = k p ! q ! (cid:16) p +2 Y j =1 c j (cid:17)(cid:16) q Y j =1 c j (cid:17)(cid:19) t k . Hence the proof will be done once we show that C k := X p,q ≥ p + q = k p ! q ! (cid:16) p +1 Y j =1 c j (cid:17)(cid:16) q +1 Y j =1 c j (cid:17) − X p,q ≥ p + q = k p ! q ! (cid:16) p +2 Y j =1 c j (cid:17)(cid:16) q Y j =1 c j (cid:17) ≥ k ≥ C k − > C = c c − c c = c ( c − c ) ≥ k = 1 , it remains that we prove that C k ≥ k ≥ C k − > k ≥ k ≥ INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 23 X p,q ≥ p + q = k p ! q ! (cid:16) p +1 Y j =1 c j (cid:17)(cid:16) q +1 Y j =1 c j (cid:17) = 1 k ! c k +1 Y j =1 c j + X p ≥ ,q ≥ p + q = k p ! q ! (cid:16) p +1 Y j =1 c j (cid:17)(cid:16) q +1 Y j =1 c j (cid:17) and X p,q ≥ p + q = k p ! q ! (cid:16) p +2 Y j =1 c j (cid:17)(cid:16) q Y j =1 c j (cid:17) = 1 k ! k +2 Y j =1 c j + X p ≥ ,q ≥ p + q = k p ! q ! (cid:16) p +2 Y j =1 c j (cid:17)(cid:16) q Y j =1 c j (cid:17) = 1 k ! k +2 Y j =1 c j + X p ′ ≥ ,q ′ ≥ p ′ + q ′ = k p ′ − q ′ + 1)! (cid:16) p ′ +1 Y j =1 c j (cid:17)(cid:16) q ′ +1 Y j =1 c j (cid:17) , we obtain(4.1) C k = 1 k ! (cid:0) c − c k +2 (cid:1) k +1 Y j =1 c j + S k , where(4.2) S k = X p ≥ ,q ≥ p + q = k (cid:16) p ! q ! − p − q + 1)! (cid:17)(cid:16) p +1 Y j =1 c j (cid:17)(cid:16) q +1 Y j =1 c j (cid:17) . Therefore it is enough to show that S k ≥ k ≥ C k ≥ k ≥ k ≥ k − ≥ c − c k +1 ≥ c k − c k +1 > C k − = 1( k − (cid:0) c − c k +1 (cid:1) k Y j =1 c j + S k − > S k − ≥ . Direct computation shows that S k ≥ ≤ k ≤ S = 0 , S = 12 c c ( c − c ) ≥ , S = 23 c c c ( c − c ) ≥ ,S = 18 c c c c ( c − c ) + 112 c c c ( c − c ) ≥ ,S = 130 c c c c c ( c − c ) + 124 c c c c ( c − c ) ≥ . It remains to show that S k ≥ k ≥ k ≥ p = k + 12 (what canhappen only for odd k ) we have1 p ! q ! − p − q + 1)! = 0 , hence S k = X ≤ p ≤ k/ p + q = k (cid:16) p ! q ! − p − q + 1)! (cid:17)(cid:16) p +1 Y j =1 c j (cid:17)(cid:16) q +1 Y j =1 c j (cid:17) + X k/ ≤ p ≤ kp + q = k (cid:16) p ! q ! − p − q + 1)! (cid:17)(cid:16) p +1 Y j =1 c j (cid:17)(cid:16) q +1 Y j =1 c j (cid:17) = X ≤ p ≤ k/ p + q = k (cid:16) p ! q ! − p − q + 1)! (cid:17)(cid:16) p +1 Y j =1 c j (cid:17)(cid:16) q +1 Y j =1 c j (cid:17) + X ≤ q ≤ k/ − p + q = k (cid:16) p ! q ! − p − q + 1)! (cid:17)(cid:16) p +1 Y j =1 c j (cid:17)(cid:16) q +1 Y j =1 c j (cid:17) = X ≤ p ≤ k/ p + q = k (cid:16) p ! q ! − p − q + 1)! (cid:17)(cid:16) p +1 Y j =1 c j (cid:17)(cid:16) q +1 Y j =1 c j (cid:17) + X ≤ p ≤ k/ − p + q = k (cid:16) q ! p ! − q − p + 1)! (cid:17)(cid:16) q +1 Y j =1 c j (cid:17)(cid:16) p +1 Y j =1 c j (cid:17) ; . Denoting by p the unique integer for which k − ≤ p ≤ k S k = X ≤ p ≤ p − p + q = k (cid:16) p ! q ! − p − q + 1)! (cid:17)(cid:16) p +1 Y j =1 c j (cid:17)(cid:16) q +1 Y j =1 c j (cid:17) + (cid:16) p !( k − p )! − p − k − p + 1)! (cid:17)(cid:16) p +1 Y j =1 c j (cid:17)(cid:16) k − p +1 Y j =1 c j (cid:17) + X ≤ p ≤ p − p + q = k (cid:16) p ! q ! − p + 1)!( q − (cid:17)(cid:16) p +1 Y j =1 c j (cid:17)(cid:16) q +1 Y j =1 c j (cid:17) + (cid:16) k ! − k − (cid:17) c (cid:16) k +1 Y j =1 c j (cid:17) = (cid:16) p !( k − p )! − p − k − p + 1)! (cid:17)(cid:16) p +1 Y j =1 c j (cid:17)(cid:16) k − p +1 Y j =1 c j (cid:17) + (cid:16) k ! − k − (cid:17) c (cid:16) k +1 Y j =1 c j (cid:17) + X ≤ p ≤ p − p + q = k (cid:16) p ! q ! − p − q + 1)! − p + 1)!( q − (cid:17)(cid:16) p +1 Y j =1 c j (cid:17)(cid:16) q +1 Y j =1 c j (cid:17) . Set d k,p := 1 p !( k − p )! − p − k − p + 1)! = k − p + 1 p !( k − p + 1)! > INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 25 and, for 1 ≤ p ≤ p − d k,p := 2 p !( k − p )! − p − k − p + 1)! − p + 1)!( k − p − . Then S k = d k,p (cid:16) p +1 Y j =1 c j (cid:17)(cid:16) k − p +1 Y j =1 c j (cid:17) − (cid:16) k − − k ! (cid:17) c (cid:16) k +1 Y j =1 c j (cid:17) + X ≤ p ≤ p − d k,p (cid:16) p +1 Y j =1 c j (cid:17)(cid:16) k − p +1 Y j =1 c j (cid:17) . (4.3)It is easy to see that(4.4) d k,p ≥ p ≥ k − √ k + 22 , d k,p < p < k − √ k + 22 . Let p denote the unique integer for which k − √ k + 22 ≤ p < k − √ k + 22 + 1 . If k = 6 , then p = 3 and p = 2 , while if k ≥ ≤ k − √ k + 22 ≤ k − ≤ p < k − √ k + 22 ≤ k − 12 + 1 ≤ p . Thus we always have 2 ≤ p ≤ p − { , , , ... , p } ∋ p (cid:16) p +1 Y j =1 c j (cid:17)(cid:16) k − p +1 Y j =1 c j (cid:17) = (cid:16) p +1 Y j =1 c j (cid:17) (cid:16) k − p +1 Y j = p +2 c j (cid:17) is increasing and, according to (4.4), d k,p ≥ p ≥ p , d k,p < p ≤ p − , we deduce d k,p (cid:16) p +1 Y j =1 c j (cid:17)(cid:16) k − p +1 Y j =1 c j (cid:17) ≥ d k,p (cid:16) p +1 Y j =1 c j (cid:17)(cid:16) k − p +1 Y j =1 c j (cid:17) , ≤ p ≤ p − . We have also (cid:16) p +1 Y j =1 c j (cid:17)(cid:16) k − p +1 Y j =1 c j (cid:17) ≥ (cid:16) p +1 Y j =1 c j (cid:17)(cid:16) k − p +1 Y j =1 c j (cid:17) ,c (cid:16) k +1 Y j =1 c j (cid:17) ≤ (cid:16) p +1 Y j =1 c j (cid:17)(cid:16) k − p +1 Y j =1 c j (cid:17) , so (4.3) yields(4.5) S k ≥ (cid:18) d k,p − (cid:16) k − − k ! (cid:17) + X ≤ p ≤ p − d k,p (cid:19)(cid:16) p +1 Y j =1 c j (cid:17)(cid:16) k − p +1 Y j =1 c j (cid:17) In order to compute the sum s k := d k,p − (cid:16) k − − k ! (cid:17) + X ≤ p ≤ p − d k,p , we notice that, according to (4.3), s k is equal to S k with c = c = ... .Computing S k in this case by using the formula (4.2) instead of (4.3), weobtain s k = k X p =1 (cid:16) p !( k − p )! − p − k − p + 1)! (cid:17) . But this is a telescoping sum, hence it is equal to 1 k !0! − k ! = 0 . Using now (4.5), we deduce the desired result: S k ≥ (cid:3) The next majorization theorem is essentially [16], Theorem 1 and [9],Theorem 1.6, claiming that any increasing function, which satisfies the non-quasianalyticity condition, can be majorized by some function c + ln | ω ( · ) | with ω ∈ Ω and c ≥ Theorem 4.3. For f : (0 , + ∞ ) −→ (0 , + ∞ ) the following conditons areequivalent :(i) f ( t ) ≤ α ( t ) , t > , for α : (0 , + ∞ ) −→ (0 , + ∞ ) some increasingfunction satisfying the non-quasianalyticity condition. (ii) There exist < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , ∞ X j =1 t j < + ∞ and a constant c ≥ , such that f ( t ) ≤ c + ln max (cid:16) , sup k ≥ t k t t ... t k (cid:17) , t > . (iii) f ( t ) ≤ c + ln | ω ( t ) | , t > , for some ω ∈ Ω and constant c ≥ .A necessary condition that f satisfies the above equivalent conditions is (4.6) lim t → + ∞ f ( t ) t = 0 . Proof. The equivalences (i) ⇔ (ii) ⇔ (iii) are immediate consequences ofthe corresponding equivalences in [9], Theorem 1.6.Also the necessary condition (4.6) is well-known. Here is a short proof ofit :Let α be as in (i). Then0 ≤ f ( t ) t ≤ α ( t ) t = + ∞ Z t α ( t ) s d s ≤ + ∞ Z t α ( s ) s d s t → + ∞ −−−−−→ . (cid:3) INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 27 The second majorization theorem is an extended version of [16], Theorem2 and [9], Theorem 1.8. It claims essentially that Lebesgue measurablepositive subadditive functions on (0 , + ∞ ) , which are bounded on (0 , 1] andsatisfy the non-quasianalyticity condition, can be majorized by a continuous,increasing, concave function satisfying the non-quasianalyticity condition, orby a function of the form c + ln | ω ( · ) | with ω ∈ Ω and c ≥ Theorem 4.4. For f : (0 , + ∞ ) −→ (0 , + ∞ ) the following conditons areequivalent :(i) f ( t ) ≤ α ( t ) , t > , for α : (0 , + ∞ ) −→ (0 , + ∞ ) some Lebesguemeasurable, subadditive function, bounded on (0 , and satisfyingthe non-quasianalyticity condition. (ii) f ( t ) ≤ α ( t ) , t > , for α : (0 , + ∞ ) −→ (0 , + ∞ ) some continuousfunction, bounded on (0 , and satisfying the non-quasianalyticitycondition, such that (0 , + ∞ ) ∋ t α ( t ) t is decreasing. (iii) f ( t ) ≤ α ( t ) , t > , with α : (0 , + ∞ ) −→ (0 , + ∞ ) some increasing,concave function satisfying the non-quasianalyticity condition. (iv) f ( t ) ≤ α ( t ) , t > , with α : (0 , + ∞ ) −→ (0 , + ∞ ) some infinitelydifferentiable, increasing, concave function satisfying the non-quasianalyticity condition. (v) There exist < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , ∞ X j =1 t j < + ∞ and a constant c ≥ , such that f ( t ) ≤ c + ln (cid:16) ∞ X k =1 t k t ... t k (cid:17) , t > . (vi) There exist < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , ∞ X j =1 t j < + ∞ and a constant c ≥ , such that f ( t ) ≤ c + ln max (cid:16) , sup k ≥ t k t t ... t k (cid:17) , t > . (vii) f ( t ) ≤ c + ln | ω ( t ) | , t > , for some ω ∈ Ω and constant c ≥ .A necessary condition that f satisfies the above equivalent conditions is (4.7) lim t → + ∞ f ( t ) ln tt = 0 . Proof. The equivalences (i) ⇔ (vi) ⇔ (vii) are immediate consequences ofthe equivalences (iii) ⇔ (ii) ⇔ (iv) in [9], Theorem 1.8.Implications (vi) ⇒ (v) and (iv) ⇒ (iii) are trivial, while implication(v) ⇒ (iv) follows by Lemma 4.2 and Remark 4.1. Finally, the implications (iii) ⇒ (ii) ⇒ (i) where proved in the discussionbefore Lemma 4.2.The necessary condition (4.7) is, like (4.6) in Theorem 4.3, well-known.We provide a short proof of it, essentially reproducing the proof of [4],Corollary 1.2.8 :Let α be as in (ii). We have for every t > + ∞ Z √ t α ( s ) s d s ≥ t Z √ t α ( s ) s d s ≥ t Z √ t α ( t ) t s d s = α ( t ) t ln t √ t = 12 α ( t ) ln tt , so 0 ≤ α ( t ) ln tt ≤ + ∞ Z √ t α ( s ) s d s t → + ∞ −−−−−→ . (cid:3) We notice that (i) ⇔ (iii) in Theorem 4.4 was originally proved by A.Beurling (see [2], lemma 1, [4], Theorem 1.2.7, [3], Lemma V), [14], Lemma3.3). A new feature of Theorem 4.4 consists in the exhibition (thanks toLemma 4.2) of a rather explicite α in (iii), obtaining thus the equivalentconditions (iv) and (v).Clearly, every f , which satisfies the equivalent conditions in Theorem 4.4,satisfies also the equivalent conditions in Theorem 4.3. It is an intriguingquestion: does it exist f satisfying the conditions in Theorem 4.3, but notthose in Theorem 4.4 ? The answer is yes : Corollary 4.5. There exists an increasing function (0 , + ∞ ) −→ (0 , + ∞ ) satisfying the non-quasianalyticity condition, which can not be majorized byany Lebesgue measurable, subadditive function on (0 , + ∞ ) , which is boundedon (0 , and satisfies the non-quasianalyticity condition.Consequently there exists ω ∈ Ω such that | ω ( · ) | can not be majorized bya scalar multiple of some | ρ ( · ) | with ρ ∈ Ω .Proof. Let e = t < t < t < ... be a sequence such that ∞ X k =1 t k < + ∞ (for example, t k = e k ). Defining the function f : (0 , + ∞ ) −→ (0 , + ∞ ) by f ( t ) := 0 for 0 < t < t ,f ( t ) := t k ln t k for t k ≤ t < t k +1 , k ≥ ,f will be increasing and satisfying the non-quasianalyticity condition : + ∞ Z f ( t ) t d t = ∞ X k =1 t k +1 Z t k f ( t ) t d t = ∞ X k =1 t k ln t k (cid:16) t k − t k +1 (cid:17) ≤ ∞ X k =1 t k < + ∞ . INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 29 The above defined f can not be majorized by any Lebesgue measurable,subadditive function on (0 , + ∞ ) , which is bounded on (0 , 1] and satisfiesthe non-quasianalyticity condition. Indeed, otherwise (4.7) would hold truebyTheorem 4.4, contradicting f ( t k ) ln t k t k = 1 , k ≥ (cid:3) The mild strong non-quasianalyticity condition First at all we notice that if α , β : (0 , + ∞ ) −→ (0 , + ∞ ) are increasing, + ∞ Z α ( t ) t d t < + ∞ and lim t → + ∞ β ( t ) t = 0 , then + ∞ Z α ( t ) t ln tβ ( t ) d t > −∞ is awell defined improper integral. Indeed, if t ≥ β ( t ) t < t ≥ t , then [ t , + ∞ ) ∋ t α ( t ) t ln tβ ( t ) is a positive Lebesgue measurablefunction.In particular, if α : (0 , + ∞ ) −→ (0 , + ∞ ) is an increasing function and + ∞ Z α ( t ) t d t < + ∞ , then + ∞ Z α ( t ) t ln tα ( t ) d t > −∞ is a well defined improperintegral. Indeed, we have lim t → + ∞ α ( t ) t = 0 by Theorem 4.3.Let us say that an increasing function α : (0 , + ∞ ) −→ (0 , + ∞ ) satisfiesthe mild strong non-quasianalyticity condition if + ∞ Z α ( t ) t d t < + ∞ and + ∞ Z α ( t ) t ln tα ( t ) d t < + ∞ . We notice that, for ω ∈ Ω , (0 , + ∞ ) ∋ t ω ( t ) | ∈ (0 , + ∞ ) satisfies themild non-quasianalyticity condition exactly when condition (3.9) is satisfied,that is when ω satisfies the mild non-quasianalyticity condition as definedin Section 2. Proposition 5.1. Let α : (0 , + ∞ ) −→ (0 , + ∞ ) be an increasing functionsatisfying the mild strong non-quasianalyticity condition. Then (i) c · α and α ( L · ) satisfy the mild strong non-quasianalyticity conditionfor each c > and L > α + β satisfies the mild strong non-quasianalyticity condition foreach increasing β : (0 , + ∞ ) −→ (0 , + ∞ ) satisfying the mild strongnon-quasianalyticity condition ;(iii) any increasing β : (0 , + ∞ ) −→ (0 , + ∞ ) , β ≤ α , satisfies the mildstrong non-quasianalyticity condition.Proof. The proof of (i) is immediate. Also (ii) is easily seen by using that α ( t ) + β ( t ) t ln tα ( t ) + β ( t ) d t ≤ α ( t ) t ln tα ( t ) d t + β ( t ) t ln tβ ( t ) d t . For the proof of (iii) we notice that, according to Theorem 4.3, there existsa t ≥ α ( t ) t < e ⇔ α ( t ) < te for all t ≥ t . Then β ( t ) ln tβ ( t ) ≤ α ( t ) ln tα ( t ) , t ≥ t . Indeed, (cid:16) , te (cid:17) ∋ x x ln tx is increasing and 0 < β ( t ) ≤ α ( t ) < te . (cid:3) At first view, the next characterization of mild strong non-quasianalyticity(more precisely, of its negation) can appear surprising : Proposition 5.2. Let α : (0 , + ∞ ) −→ (0 , + ∞ ) be an increasing functionsuch that + ∞ Z α ( t ) t d t < + ∞ . Then the following conditions are equivalent :(i) α does not satisfy the mild strong non-quasianalyticity condition,that is + ∞ Z α ( t ) t ln tα ( t ) d t = + ∞ . (ii) For any increasing function β : (0 , + ∞ ) −→ (0 , + ∞ ) such that + ∞ Z β ( t ) t d t < + ∞ , we have + ∞ Z α ( t ) t ln tβ ( t ) d t = + ∞ . The above two conditions imply the condition (iii) + ∞ Z e α ( t ) t ln ln( t ) d t = + ∞ and, if α is also subadditive or α = ln | ω ( · ) | with ω ∈ Ω , then all the abovethree conditions are equivalent.Proof. Implication (ii) ⇒ (i) is trivial. For (i) ⇒ (ii) : since α ( t ) t ln tβ ( t ) ≥ α ( t ) t ln tα ( t ) + β ( t ) = α ( t ) t ln tα ( t ) − α ( t ) t ln α ( t ) + β ( t ) α ( t ) ≥ α ( t ) t ln tα ( t ) − α ( t ) t α ( t ) + β ( t ) α ( t ) = α ( t ) t ln tα ( t ) − α ( t ) + β ( t ) t , we have + ∞ Z α ( t ) t ln tβ ( t ) d t ≥ + ∞ Z α ( t ) t ln tα ( t ) d t − + ∞ Z α ( t ) + β ( t ) t d t = + ∞ . INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 31 (ii) ⇒ (iii) follows by applying (ii) to β ( t ) = t (ln t ) for t ≥ e ,t < t < e . Finally we prove that, if α is also subadditive or α = ln | ω ( · ) | with ω ∈ Ω , then (iii) ⇒ (i). For we recall that, according to Theorem 4.4,lim t → + ∞ α ( t ) ln tt = 0 . Consequently there exists some t ≥ e such that α ( t ) ln tt < ⇔ tα ( t ) > ln t for t ≥ t . We deduce : + ∞ Z t α ( t ) t ln tα ( t ) d t ≥ + ∞ Z t α ( t ) t ln ln( t ) d t = + ∞ . (cid:3) The non-quasianalyticity and mild strong non-quasianalyticity conditionsfor increasing functions can be rewritten in discretized form : Proposition 5.3. Let α : (0 , + ∞ ) −→ (0 , + ∞ ) be an increasing function. (i) + ∞ Z α ( t ) t d t < + ∞ if and only if ∞ X j =1 α (2 j )2 j < + ∞ . (ii) Assuming that + ∞ Z α ( t ) t d t < + ∞ , we have + ∞ Z α ( t ) t ln tα ( t ) d t = + ∞ if and only if ∞ X j =1 α (2 j )2 j ln 2 j β (2 j ) = + ∞ for any increasing function β : (0 , + ∞ ) −→ (0 , + ∞ ) satisfying + ∞ Z β ( t ) t d t < + ∞ . (iii) + ∞ Z e α ( t ) t ln ln( t ) d t < + ∞ if and only if ∞ X j =1 α (2 j )2 j ln j < + ∞ .Proof. (i) follows by noticing that j +1 Z j α ( t ) t d t ≤ α (2 j +1 ) j +1 Z j t d t = α (2 j +1 )2 j +1 and j +1 Z j α ( t ) t d t ≥ α (2 j ) j +1 Z j t d t = α (2 j )2 j +1 . Let us now assume that + ∞ Z α ( t ) t d t < + ∞ and + ∞ Z α ( t ) t ln tα ( t ) d t = + ∞ .Let further β : (0 , + ∞ ) −→ (0 , + ∞ ) be an arbitrary increasing function.such that + ∞ Z β ( t ) t d t < + ∞ . Then also β (2 · ) is increasing and such that + ∞ Z β (2 t ) t d t < + ∞ , so Proposition 5.2 yields + ∞ Z α ( t ) t ln tβ (2 t ) d t = + ∞ .Since j +1 Z j α ( t ) t ln tβ (2 t ) d t ≤ α (2 j +1 ) ln 2 j +1 β (2 · j ) j +1 Z j t d t = α (2 j +1 )2 j +1 ln 2 j +1 β (2 j +1 ) , we obtain ∞ X j =1 α (2 j )2 j ln 2 j β (2 j ) ≥ + ∞ Z α ( t ) t ln tβ (2 t ) d t = + ∞ . Assume now that + ∞ Z α ( t ) t d t < + ∞ and ∞ X j =1 α (2 j )2 j ln 2 j β (2 j ) = + ∞ for any increasing function β : (0 , + ∞ ) −→ (0 , + ∞ ) with + ∞ Z β ( t ) t d t < + ∞ .Since α (2 · ) is such a function, we have ∞ X j =1 α (2 j )2 j ln 2 j α (2 j +1 ) = + ∞ . Since j +1 Z j α ( t ) t ln tα ( t ) d t ≥ α (2 j ) ln 2 j α (2 j +1 ) j +1 Z j t d t = α (2 j )2 j +1 ln 2 j α (2 j +1 ) , we deduce + ∞ Z α ( t ) t ln tα ( t ) d t ≥ ∞ X j =1 α (2 j )2 j ln 2 j α (2 j +1 ) = + ∞ .Finally, (iii) follows by using the estimations j +1 Z j α ( t ) t ln ln( t ) d t ≤ α (2 j +1 ) ln ln(2 j +1 ) j +1 Z j t d t ≤ α (2 j +1 )2 j +1 ln( j + 1) , j +1 Z j α ( t ) t ln ln( t ) d t ≥ α (2 j ) ln ln(2 j ) j +1 Z j t d t ≥ α (2 j )2 j +1 ln j α (2 j )2 j ln j , INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 33 the second one, in which ln ln(2 j ) ≥ ln j j ≥ (cid:3) The condition for the sequence (cid:0) α (2 j ) (cid:1) j ≥ , formulated in Proposition5.3 to characterize the negation of the mild strong non-quasianalyticity foran increasing α : (0 , + ∞ ) −→ (0 , + ∞ ) satisfying the non-quasianalyticitycondition, has an important permanence property which will be used in thenext section to prove Theorem 3.10 : Proposition 5.4. Let (cid:0) a j (cid:1) j ≥ be a sequence in [ 0 , + ∞ ) such that ∞ X j =1 a j j < + ∞ and ∞ X j =1 a j j ln 2 j β (2 j ) = + ∞ for any increasing function β : (0 , + ∞ ) −→ (0 , + ∞ ) with + ∞ Z β ( t ) t d t < + ∞ .Then the sequence (cid:0) ( a j +1 − a j ) + (cid:1) j ≥ , where λ + = ( λ for λ ≥ , for λ < , has the same two properties.Proof. First of all, ∞ X j =1 ( a j +1 − a j ) + j ≤ ∞ X j =1 a j +1 + a j j ≤ ∞ X j =1 a j j < + ∞ . Now let β : (0 , + ∞ ) −→ (0 , + ∞ ) be any increasing function satisfying + ∞ Z β ( t ) t d t < + ∞ . By Theorem 4.3 lim t → + ∞ β ( t ) t = 0 , so there is an integer n ≥ β (2 n ) ≤ n for n ≥ n .For each n > n , n X j = n ( a j +1 − a j ) + j ln 2 j β (2 j ) ≥ n X j = n a j +1 − a j j ln 2 j β (2 j )= − a n n ln 2 n β (2 n ) + n X j = n +1 a j j (cid:16) j − β (2 j − ) − ln 2 j β (2 j ) (cid:17) + a n +1 n ln 2 n β (2 n ) ≥ − a n n ln 2 n β (2 n ) + n X j = n +1 a j j ln (cid:16) j − β (2 j − ) · β (2 j )2 j (cid:17) ≥ − a n n ln 2 n β (2 n ) + n X j = n +1 a j j ln 2 j − β (2 j ) , so n X j = n ( a j +1 − a j ) + j ln 2 j β (2 j ) ≥ − a n n ln 2 n β (2 n ) − n X j = n +1 a j j ln 4 + n X j = n +1 a j j ln 2 j β (2 j ) = + ∞ . (cid:3) We denote ln + t := ( ln t for t > 00 for t ≤ . The next lemma completes Proposition 3.6 : Lemma 5.5. For < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , are equivalent :(i) ∞ X j =2 ln ln jt j < + ∞ ; (ii) lim j →∞ t j = + ∞ and ∞ X j =1 ln + ln t j t j < + ∞ .Moreover, (i) and (ii) imply (iii) lim j →∞ t j j = + ∞ and ∞ X j =1 ln + t j jt j < + ∞ ,and, if < t ≤ t ≤ t ≤ ... , then (i) , (ii) and (iii) are all equivalent.Proof. First we prove that (i) implies (ii) and (iii).Clearly, t j −→ + ∞ . For each j ≥ t j ≤ j thenln + ln t j t j ≤ ln ln( j ) t j = 2 ln ln jt j , while if t j > j , thenln + ln t j t j = ln ln t j t j < ln ln( j ) j = 2 ln ln jj . Therefore ∞ X j =3 ln + ln t j t j ≤ ∞ X j =3 ln ln jt j + 2 ∞ X j =3 ln ln jj < + ∞ . INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 35 On the other hand, since ln ln j > j ≥ 16 , we have ∞ X j =1 t j < + ∞ .Furthermore, for each j ≥ t j j ≤ (ln j ) thenln + t j jt j ≤ ln(ln j ) t j = 2 ln ln jt j , while if t j j > (ln j ) > e , thenln + t j jt j j = ln t j jt j j < ln(ln j ) (ln j ) = 2 ln ln j (ln j ) , hence ln + t j jt j < jj (ln j ) . We conclude that ∞ X j =6 ln + t j jt j j ≤ ∞ X j =6 ln ln jt j + 2 ∞ X j =6 ln ln jj (ln j ) < + ∞ . Next we prove implication (ii) ⇒ (i).Since t j −→ + ∞ , we have eventually ln + ln t j ≥ ∞ X j =1 t j < + ∞ .Consequently, (see e.g. [9], Lemma 1.5 (ii)), lim j →∞ t j j = + ∞ . In particular,we have eventually j ≤ t j and the convergence of ∞ X j =2 ln ln jt j follows.Finally we show that if 0 < t ≤ t ≤ t ≤ ... , then (iii) ⇒ (i).Since t j j −→ + ∞ , we have eventually ln + t j j ≥ ∞ X j =1 t j < + ∞ .By [9], Lemma 1.5 (iii) it follows that t j j ln j −→ + ∞ . In particular, wehave eventually ln j ≤ t j j and the convergence of ∞ X j =2 ln ln jt j follows. (cid:3) We end this section with a summary of several characterizations the mildstrong non-quasianalyticity condition for functions of the form | ω ( · ) | with ω ∈ Ω . Theorem 5.6. For < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , ∞ X j =1 t j < + ∞ ,let us denote n ( t ) = { k ≥ t k ≤ t } , t > ,N ( t ) = ln max (cid:16) , sup k ≥ t k t t ... t k (cid:17) , t > ,ω ( z ) = ∞ Y j =1 (cid:16) izt j (cid:17) , z ∈ C . Then the following conditions are equivalent :(i) ∞ X j =1 ln t j jt j < + ∞ ;(ii) + ∞ Z n ( t ) | t ln tn ( t ) d t < + ∞ ; (iii) + ∞ Z N ( t ) | t ln tN ( t ) d t < + ∞ ; (iv) + ∞ Z ln | ω ( t ) | t ln t ln | ω ( t ) | d t < + ∞ . ( In the above conditions we take when it occurs. ) The above conditions are implied by the next equivalent conditions :(v) ∞ X j =2 ln ln jt j < + ∞ ;(vi) ∞ X j =2 ln + ln t j t j < + ∞ . Finally, if < t ≤ t ≤ t ≤ ... , then all the above six conditions areequivalent.Proof. Statement (i) ⇔ (ii) ⇔ (iii) ⇔ (iv) is [10], Lemma 2.1, while (v) ⇔ (vi)and the relationship between the above two groups of equivalent conditionsis Lemma 5.5. (cid:3) Proof of the negative minimum modulus theorem In this section we provide a proof for Theorem 3.10, a negative minimummodulus theorem. The idea of the proof, located in the proof of the next INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 37 Lemma 6.1, is due to W. K. Hayman ([15]), while the technical execution isbased upon the topics of Section 5. Lemma 6.1. Let n , n , ... ≥ be integers such that (6.1) ∞ X j =1 n j j < + ∞ and (6.2) ∞ X j =1 n j j ln 2 j β (2 j ) = + ∞ for any increasing β : (0 , + ∞ ) −→ (0 , + ∞ ) such that + ∞ Z β ( t ) t d t < + ∞ .The the formulas ω ( z ) = ∞ Y j =1 (cid:0) iz j (cid:17) n j , f ( z ) = ∞ Y j =1 (cid:18) − (cid:16) z j (cid:17) (cid:19) n j , z ∈ C define a function ω ∈ Ω and an entire function f with (6.3) | f ( z ) | ≤ (cid:12)(cid:12) ω ( | z | ) (cid:12)(cid:12) , z ∈ C , such that there exists no increasing function β : (0 , + ∞ ) −→ (0 , + ∞ ) with + ∞ Z β ( t ) t d t < + ∞ satisfying (6.4) sup s ∈ R | s − t |≤ β ( t ) ln | f ( s ) | ≥ − β ( t ) , t > . Proof. (6.1) yields ω ∈ Ω and, since (cid:12)(cid:12)(cid:12) − (cid:16) z j (cid:17) (cid:12)(cid:12)(cid:12) ≤ (cid:16) | z | j (cid:17) = (cid:12)(cid:12)(cid:12) i | z | j (cid:12)(cid:12)(cid:12) ,(6.3) holds true.For the remaining part of the proof, we need a particular upper estimateof | f ( z ) | for z in the disk of radius 2 j , centered at 2 j .Let j ≥ j is a zero of multiplicity n j of f , we canapply the general Schwarz’ lemma (see e.g. [20], Chapter XII, § 3, Section2, page 359, or [21], Chapter 9, § 2, Exercise 1, page 274), obtaining for any z ∈ C , | z − j | ≤ j , | f ( z ) | ≤ (cid:16) sup | z ′ − j | =2 j | f ( z ′ ) | (cid:17)(cid:16) | z − j | j (cid:17) n j ( . ) ≤ (cid:16) sup | z ′ − j | =2 j (cid:12)(cid:12) ω ( | z ′ | ) (cid:12)(cid:12) (cid:17)(cid:16) | z − j | j (cid:17) n j = (cid:12)(cid:12) ω (2 j +1 ) (cid:12)(cid:12) (cid:16) | z − j | j (cid:17) n j . Thus, for each 0 < δ ≤ | f ( z ) | ≤ (cid:12)(cid:12) ω (2 j +1 ) (cid:12)(cid:12) + n j ln δ , z ∈ C , | z − j | ≤ j δ . Now we assume that for some increasing β : (0 , + ∞ ) −→ (0 , + ∞ ) with + ∞ Z β ( t ) t d t < + ∞ we have (6.4) and show that this assumption leads to acontradiction.By (6.4) we have(6.6) sup s ∈ R | s − j |≤ β (2 j ) ln | f ( s ) | ≥ − β (2 j ) , j ≥ . On the other hand, using Theorem 4.3, we deduce lim t →∞ β ( t ) t = 0 , so thereexists j ≥ β (2 j )2 j ≤ , j ≥ j . Applying (6.5) with δ = β (2 j )2 j , we obtain(6.7) sup z ∈ C | z − j |≤ β (2 j ) ln | f ( z ) | ≤ (cid:12)(cid:12) ω (2 j +1 ) (cid:12)(cid:12) + n j ln β (2 j )2 j , j ≥ j . (6.6) and (6.7) imply successively for every j ≥ j − β (2 j ) ≤ (cid:12)(cid:12) ω (2 j +1 ) (cid:12)(cid:12) + n j ln β (2 j )2 j ,n j ln 2 j β (2 j ) ≤ (cid:12)(cid:12) ω (2 j +1 ) (cid:12)(cid:12) + β (2 j ) . Consequently ∞ X j = j n j j ln 2 j β (2 j ) ≤ ∞ X j = j ln (cid:12)(cid:12) ω (2 j +1 ) (cid:12)(cid:12) j +1 + ∞ X j = j β (2 j )2 j . But this is not possible, because the left-hand side of the above inequalityis + ∞ according to the assumption (6.2), while the right-hand side is finitebecause of Remark 4.1 (4) and Proposition 5.3 (i). (cid:3) Now we are ready to prove Theorem 3.10 , the main goal of this section : Proof (of Theorem 3.10 ). Let 0 < t ≤ t ≤ t ≤ ... ≤ + ∞ , t < + ∞ , ∞ X j =1 t j < + ∞ , be such that ω ( z ) = ∞ Y j =1 (cid:16) izt j (cid:17) , z ∈ C . Let us denote, for every t > n ( t ) the number of the elements of INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 39 the set { k ≥ t k ≤ t } . By Remark 4.1 (1) we have ∞ Z n ( t ) t d t < + ∞ and,according to Proposition 5.3 (i), it follows ∞ X j =1 n (2 j )2 j < + ∞ .On the other hand, Theorem 5.6 yields + ∞ Z n ( t ) | t ln tn ( t ) d t = + ∞ . Applying Proposition 5.3 (ii) to (0 , + ∞ ) ∋ t n ( t ) , we deduce that ∞ X j =1 n (2 j )2 j ln 2 j β (2 j ) = + ∞ for any increasing β : (0 , + ∞ ) −→ (0 , + ∞ ) satisfying + ∞ Z β ( t ) t d t < + ∞ .Set n := n (2) , n j := n (2 j ) − n (2 j − ) for j ≥ . By Proposition 5.4 we infer that ∞ X j =1 n j j < + ∞ and ∞ X j =1 n j j ln 2 j β (2 j ) = + ∞ for any increasing β : (0 , + ∞ ) −→ (0 , + ∞ ) satisfying + ∞ Z β ( t ) t d t < + ∞ .In other words, the sequence ( n j ) j ≥ satisfies conditions (6.1) and (6.2), soLemma 6.1 implies that the formula f ( z ) = ∞ Y j =1 (cid:18) − (cid:16) z j (cid:17) (cid:19) n j , z ∈ C defines an entire function f such that there exists no increasing function β : (0 , + ∞ ) −→ (0 , + ∞ ) with + ∞ Z β ( t ) t d t < + ∞ satisfying (6.4) = (3.16).It remains only to verify (3.15) : we have for every z ∈ C | f ( z ) | ≤ ∞ Y j =1 (cid:18) (cid:16) | z | j (cid:17) (cid:19) n j = (cid:18) (cid:16) | z | (cid:17) (cid:19) n (2) ∞ Y j =2 (cid:18) (cid:16) | z | j (cid:17) (cid:19) n (2 j ) − n (2 j − )0 L. ZSID ´O ≤ (cid:20) n (2) Y k =1 (cid:18) (cid:16) | z | t k (cid:17) (cid:19) n j (cid:21) ∞ Y j =2 (cid:20) n (2 j ) Y k = n (2 j − )+1 (cid:18) (cid:16) | z | t k (cid:17) (cid:19)(cid:21) = ∞ Y k =1 (cid:18) (cid:16) | z | t k (cid:17) (cid:19) = (cid:12)(cid:12) ω ( | z | ) (cid:12)(cid:12) . (cid:3) References [1] A. V. Abanin, Ω -ultradistributions (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. (2008), 3–38.[2] A. Beurling, Quasi-analiticity and general distributions , Lectures 4 and 5 (mimeo-graphed), A.M.S. Summer Institute, Stanford, 1961.[3] A. Beurling, Analytic continuation across a linear boundary , Acta Math. (1972),153–182.[4] G. Bj¨orck, Linear partial differential operators and generalized distributions , Ark.Mat. (1966), 351–407.[5] N. Bourbaki, Topological Vector Spaces , Chapters 1-5, Springer-Verlag, 1987.[6] R. W. Braun, R. Meise and B. A. Taylor, Ultradifferentiable functions and Fourieranalysis , Result. Math. (1990), 206–237.[7] Ch.-Ch. Chou, La transformation de Fourier complexe et l’´equation de convolution ,Lecture Notes in Math., Vol. 325, Springer-Verlag, 1973.[8] I. Cior˘anescu, Convolution equations in ω -ultradistribution spaces , rev. Roum. Math.Pures et Appl. (1980), 719–737.[9] I. Cior˘anescu and L. Zsid´o, ω -Ultradistributions and their application to operatortheory , in Banach center Publications, Vol. 8, Warsaw, 1982, 77–220.[10] I. Cior˘anescu and L. Zsid´o, A minimum modulus theorem and applications to ultra-differential operators , Ark. Mat. (1979), 151–166.[11] I. Cior˘anescu and L. Zsid´o, Ultradistributions and the Levinson condition , in´OGeneralized functions, operator theory, and dynamical systems ´O (Editors: I. An-toniou and G. Lumer), CRC Research Notes in Mathematics, Vol. 399, Chapman &Hall, 1999, 96-106.[12] I. M. Gelfand and G. E. Shilov, Quelques applications de la th´eorie des fonctionsg´en´eralis´ees , J. Math. Pures Appl. (9) (1956), 383–413.[13] I. M. Gelfand and G. E. Shilov, Generalized Functions , Vol. 2 - Spaces of Fundamentaland generalized Functions, Original Russian Edition: Fizmatgiz, Moscow, 1958 ;English Translation: Academic Press, New York and London, 1968.[14] V. P. Gurari˘i, Harmonic analysis in spaces with a weight (in Russian), Trudy Mosk.Mat. Obs. (1976), 21–76.[15] W. K. Hayman, Letter of 8th August, 1983.[16] I. O. Inozemcev and V. A. Marchenko, On majorants of genus zero (in Russian),Uspekhi Mat. Nauk (1956) Issue 2(68), 173–178.[17] H. Komatsu, Ultradistributions I. Structure theorems and a characterization , J. Fac.Sci. Tokyo Sect. IA Math. (1973), 25–105.[18] B. Ya. Levin, Distribution of zeros of entire functions (in Russian), G.I.T-T.L.,Moskow, 1956 ; English translation (2nd ed.): Transl. Math. Monographs, Vol. 5,Amer. Math. Soc., Providence, Rhode Island, 1980.[19] S. Mandelbrojt, S´eries adh´erentes, r´egularisation des suites, applications , Gauthier-Villars, Paris, 1952.[20] I. I. Privalov, Introduction to the theory of functions of a complex variable (in Rus-sian), 13-th edition, Nauka, Moscow, 1984. INIMUM MODULUS THEOREM AND ULTRADIFFERENTIAL OPERATORS 41 [21] R. Remmert, Theory of complex functions , Springer-Verlag, New York, 1991.[22] C. Roumieu, Sur quelques extensions de la notion de distributions , Ann. Sci. ´EcoleNorm. Sup. 3 e s´erie (1960) No. 1, 41–121.[23] C. Roumieu, Ultra-distributions d´efinies sur R n et sur certaines classes de vari´et´esdif´erentiables , J. Anal. Math. (1962), 153-192.[24] P. Schapira, Sur les ultre-distributions , Ann. Sci. ´Ecole Norm. Sup. 4 e s´erie (1968)No. 3, 395–415.(L. Zsid´o) Dipartimento di Matematica, Universit`a di Roma ”Tor Vergata”,Via della Ricerca Scientifica 1, 00133 Roma, ITALIA Email address ::