Absence of local unconditional structure in spaces of smooth functions on two-dimensional torus
aa r X i v : . [ m a t h . F A ] J a n Absence of local unconditional structure in spaces of smoothfunctions on two-dimensional torus
Anton Tselishchev
Abstract
Consider a finite collection { T , . . . , T J } of differential operators with constant coefficients on T and the space of smooth functions generated by this collection, namely, the space of functions f suchthat T j f ∈ C ( T ). We prove that under a certain natural condition this space is not isomorphic to aquotient of a C ( S )-space and does not have a local unconditional structure. This fact generalizes thepreviously known result that such spaces are not isomorphic to a complemented subspace of C ( S ). It is well known and easy to see that the space C k ( T ) of k times continuously differentiable functionson the unit circle is isomorphic to C ( T ). Also, it has long been known that in higher dimensions thesituation is different — already for two dimensions the space C k ( T ) is not isomorphic to C ( T ).This fact was first announced in [4] and later generalized in many directions (see [5, 7, 8, 14, 18, 16,13, 9, 10, 15]). However, the most general and natural framework was introduced only in the quite recentpaper [11] (see also the preprint [12] for the two-dimensional case).More specifically, suppose we have a collection T = { T , T , . . . , T J } of differential operators withconstant coefficients on the torus T . So, each T j is a linear combination of operators ∂ α ∂ β . We call thenumber α + β the order of such a differential monomial and the order of T j is the maximal order amongall monomials involved in it. We consider the following seminorm on trigonometric polynomials f : k f k T = max ≤ j ≤ J k T j f k C ( T ) . Now define the Banach space C T ( T ) by this seminorm (that is, factorize over the null space andconsider the completion). For example, whet T consists of all differential monomials of order at most k ,we get the space C k ( T ).In the papers [9, 10] the following statement was proved. Suppose that all differential monomialsinvolved in any of T j are of order not exceeding k . Let us drop the junior part of each T j (this means thatwe drop all monomials whose order is strictly smaller than k ). If among the remaining senior parts thereare at least two linearly independent, then C T ( T ) is not isomorphic to a complemented subspace of C ( S ). (We denote by S an arbitrary uncountable compact metric space. According to Milutin theorem,all the resulting C ( S ) spaces are isomorphic.) However, if all senior parts are multiples of one of them,the situation was unclear.So, in the preprint [12] (and in the paper [11] for arbitrary dimensions) a refinement of this statementwas proved. In order to state it, we need the concept of mixed homogeneity.Fix some mixed homogeneity pattern , that is, a line Λ that intersects the positive semiaxes. Theequation of such a line is xa + yb = 1 where a and b are positive numbers. We call such a line admissible if all multiindices ( α, β ) such that ∂ α ∂ β is involved in one of T j lie below Λ or on it. This means thatall such multiindices must satisfy the following inequality: αa + βb ≤ . Now we define the senior part of T j as the sum of all differential monomials involved in T j whosemultiindices lie on the line Λ and the junior part as the sum of all other monomials of T j . The seniorpart is denoted by σ j and the junior by τ j . 1uppose that for some choice of Λ there are at least two linearly independent among all senior parts σ j . Then it was proved in [11, 12] that C T ( T ) is not isomorphic to a complemented subspace of a C ( S )space.However, in a less general setting, this is not the best known statement. For example, in [7] it wasproved that C k ( T ) is not isomorphic to any quotient space of C ( S ). The following theorem generalizesthis statement in the described setting. Theorem 1.
If for the collection T there are at least two linearly independent operators among σ j ( forsome choice of an admissible line Λ) , then C T ( T ) is not isomorphic to any quotient space of C ( S ) . This is the first result of this paper.Also, we note that there is another generalization of the theorem from [11, 12] (again in a less generalsetting). In [13] it was proved that if all operators in the collection T are differential monomials andat least two senior monomials (with respect to some pattern) are linearly independent, then the space C T ( T ) does not have local unconditional structure.Following [3] we give the definition. A Banach space X is said to have local unconditional structureif there exists a constant C > F ⊂ X there exists aBanach space E with 1-unconditional basis and two linear operators R : F → E and S : E → X suchthat SRx = x for all x ∈ F and k S k · k R k ≤ C . A basis { e n } is 1-unconditional if for any numbers ε n with | ε n | ≤ α n ) the following inequality holds: k P ε n α n x n | ≤ k P α n x n k .It is worth noting that X has local unconditional structure if and only if its conjugate X ∗ is a directfactor of a Banach lattice (see [17]). Since the space C ( S ) does have local unconditional structure, thenon-isomorphism of C T ( T ) to a complemented space of C ( S ) would also follow once it is proved that C T ( T ) does not have local unconditional structure. This is exactly the statement of the next theorem. Theorem 2.
If for a collection T there are at least two linearly independent operators among σ j ( forsome choice of an admissible line Λ) , then C T ( T ) does not have local unconditional structure. The main ingredients of our proofs are the same as in [11, 12]. We use the new embedding theoremestablished there together with some facts about p -summing operators.At first, we introduce some definitions. A distribution f on the torus T is called proper if ˆ f ( s, t ) = 0whenever s = 0 or t = 0. Next, we need a notion of Sobolev spaces with nonintegral smoothness: W α,β ( T ) = { f ∈ C ∞ ( T ) ′ : { (1 + m ) α/ (1 + n ) β/ ˆ f ( m, n ) } ∈ ℓ ( Z ) } . Of course, the norm of f in W α,β ( T ) is defined as k{ (1 + m ) α/ (1 + n ) β/ ˆ f ( m, n ) }k ℓ .Now we state the embedding theorem (see Theorem 0.2 and Remark 1.6 in [11]) which we are goingto use. Fact 1.
Suppose that proper distributions φ , . . . , φ N satisfy the following system of equations : − ∂ k ϕ = µ ; ∂ l ϕ j − ∂ k ϕ j +1 = µ j , j = 1 , . . . , N − ∂ l φ N = µ N , (1) where µ , . . . , µ N are functions in L ( T ) ( or measures ) . Then N X j =1 k ϕ j k W k − , l − ( T ) . N X j =0 k µ j k . Here (and everywhere in this paper) the symbol A . B means that there exists some constant C > A ≤ CB .Several remarks are in order. First, Theorems 1 and 2 hold also for the torus of arbitrary dimension, T n . But this fact cannot be derived from 2-dimensional statements (or at least it is unclear how to dothis, see [11] for some explanations). The proofs in higher dimensions are somewhat similar, however,they are much more technically sophisticated (and even require a different embedding theorem, again,see [11] and Theorem 1.1 there). So, in this article we restrict ourselves to the two-dimensional case.In this paper we present the proofs of Theorem 1 and Theorem 2. We start with the first theorembecause its proof is easier and contains less technical details (however, the reader will see that the proofsof both theorems are quite similar and similar to the proof from the preprint [12]).2e note that in the paper [13] it was also proved (again, in case when all operators in T are differentialmonomials and there are at least two linearly independent operators among their senior parts) that if C T ( T ) ∗ is isomorphic to a subspace of a space Y with local unconditional structure, then Y containsthe spaces ℓ k ∞ uniformly (again, for the definition see [13]). The same statement can also be proved inour situation, but we do not present the details here, because our main goal is to show that, using theembedding theorem from [12], we can adapt various techniques to a more general context. And althoughthis statement implies Theorem 1, we choose to sacrifice the generality for the sake of simplicity andtransparency of presentation.Also, a few words should be said about the notation. As it has already been mentioned, we write A . B if A ≤ CB for some constant C >
0. It will always be clear from the context from whichparameters C can depend and from which it cannot. Besides that, the notation A ≍ B means that A . B and B . A .The author is kindly grateful to his scientific advisor, S. V. Kislyakov, for posing these problems, forvery helpful discussions during the process of their solution and for great help with editing this text. C ( S ) -space Like it was done in [12], we start our proof of Theorem 1 with some simple but helpful observations.
We denote the space of proper functions in C T ( T ) by C T ( T ). It is clear that this space is comple-mented in C T ( T ) (a projection is given by convolution with some measure), so we can prove Theorem1 for C T ( T ) instead of C T ( T ).Next, suppose that the admissible line Λ is given by the equation x/a + y/b = 1. Let us show thatwithout loss of generality we may assume that a and b are positive integers. Indeed, according to theconditions of Theorem 1, there are at least two points ( r , r ) and ( ρ , ρ ) with nonnegative integralcoordinates on Λ. We may assume that r > ρ and r < ρ . Then the equation of Λ can be written inthe following form: xr − ρ + yρ − r = ρ r − ρ + ρ ρ − r . Now note that we can shift the line Λ (and the whole construction) by a vector with integral coordi-nates. This means that we can change the collection T by the collection { T ∂ u ∂ v , . . . T J ∂ u ∂ v } . Thecorresponding spaces C { T ,...,T J } ( T ) and C { T ∂ u ∂ v ,...T J ∂ u ∂ v } ( T )are isomorphic — isomorphism is given by the map f ∂ u ∂ v f . So, by doing this shift we may assumethat the equation of Λ is the following: xr − ρ + yρ − r = ρ + ur − ρ + ρ + vρ − r . If we write this equation in the form x/a + y/b = 1, then a and b are the following: ρ + u + ( ρ + v ) r − ρ ρ − r and ρ + v + ( ρ + u ) ρ − r r − ρ . Clearly, we can find positive integers u and v so that these two expressions become integers.So, we assume that the equation of Λ is x/a + y/b = 1 where a and b are positive integers. Wedenote their greatest common divisor by N and then all points on Λ are of the form ( jm, ( N − j ) n ) with0 ≤ j ≤ N (here m = a/N and n = b/N so m and n are coprime). Suppose that C T ( T ) is isomorphic to a quotient space of C ( S ). Denote by P the quotient map, P : C ( S ) → C T ( T ). 3ue to the reductions we have done, the senior part of every operator from T has the following form: σ s = N X j =0 a sj ∂ jm ∂ ( N − j ) n . We note that the space C T ( T ) depends only on the linear span of operators in T so we can change ourcollection if these changes do not affect its linear span.Now we consider the matrix ( a sj ). Suppose j is the smallest index such that a sj = 0 for at leastone s . Without loss of generality we may assume that a j = 0. Then, multiplying T by a constant andsubtracting a multiple of T from other operators, we can ensure that a j = − a sj = 0 for every s >
1. By the assumption of the theorem, there exists j such that a sj = 0 for some s >
1. Again,without loss of generality we assume that a j = 1 and a sj = 0 for all s > T and T , whose senior parts are linearly independent. Then forsimplicity we denote the coefficients of their senior parts by a j and b j respectively, that is σ = N X j =0 a j ∂ jm ∂ ( N − j ) n ; σ = N X j =0 b j ∂ jm ∂ ( N − j ) n . Moreover, T is the only operator in T that involves the differential monomial ∂ j m ∂ ( N − j ) m and T isthe only operator in T besides maybe T that includes the monomial ∂ j m ∂ ( N − j ) m .Consider the embedding of the space C T ( T ) in C T ,T ( T ) (denote it by i ). Next, we can embedthis space into W T ,T ( T ). We denote this embedding by g . Here, clearly, the spaces C T ,T ( T ) and W T ,T ( T ) are defined by the seminorms max {k T f k C ( T ) , k T f k C ( T ) } and max {k T f k L ( T ) , k T f k L ( T ) } respectively and consist only of proper functions. We note that operator g is 1-summing, this followseasily from the Pietsch factorization theorem. A good reference on the theory of p -summing operatorsis the book [19] (see Chapter III.F there).Next, we are going to construct an operator s from W T ,T ( T ) into W m − , n − ( T ). Again, theconstruction will be very similar to that in [12] with certain simplifications.First, we need the following simple fact. Fact 2.
The system (1) with proper measures ( or L functions ) µ j is solvable if and only if the followingrelation holds true : N X j =0 ∂ jk ∂ ( N − j ) l µ j = 0 . (2)The proof is quite easily done by induction and can be found in [12] (see Lemma 2.1 there).Now take any f ∈ W T ,T ( T ) and consider the pair of functions ( f , f ) = ( T f, T f ). Clearly, theysatisfy the equation T f − T f = 0. This is a differential equation and now we rewrite it in a differentform. In order to do this, we note that if α/a + β/b <
1, then we can express the differential monomial ∂ α ∂ β in terms of ∂ a and ∂ b , using Fourier multipliers: ∂ α ∂ β f = I αβ ∂ a f + J αβ ∂ b f, where I αβ and J αβ are Fourier multipliers with the following symbols:( iu ) α + a ( iv ) β ( iu ) a ± ( iv ) b and ± ( iu ) α ( iv ) β + b ( iu ) a ± ( iv ) b , respectively. By this we mean that they act on a function g ∈ L ( T ) by multiplying its Fouriercoefficients ˆ g ( u, v ) by these expressions. The choice of a sign ± is determined by the condition ( − a = ± ( − b , so that the denominators do not vanish when u and v are not equal to zero. In [12] it wasproved that such multipliers are bounded on L ( T ).4 act 3. The Fourier multipliers I αβ and J αβ defined as above are bounded on L ( T ) . Using these multipliers, we can write the junior parts of operators T and T in the following form: X α,β c αβ ( I αβ ∂ a + J αβ ∂ b ) . Therefore, we can regroup the terms in the expression T f − T f and rewrite it as N X j =0 ∂ jm ∂ ( N − j ) n µ j = 0 , where the µ j are precisely the functions b j f − a j f when j = 0 , N , µ is equal to b f − a f plus somelinear combination of the operators J αβ applied to f and f , and µ N equals b N f − a N f plus somelinear combination of the operators I αβ applied to f and f .Now we use Fact 2 and find a solution of the following system of differential equations: − ∂ m ϕ = µ ; ∂ n ϕ j − ∂ m ϕ j +1 = µ j , j = 1 , . . . , N − ∂ n ϕ N = µ N . (3)By Fact 1, all functions ϕ j lie in W m − , n − ( T ). We take the function ϕ j +1 ∈ W m − , n − ( T )(it depends linearly on the initial function f ) and therefore we get a bounded linear operator s from W T ,T ( T ) into W m − , n − ( T ). Summing up, we have the following diagram: C ( S ) P −→ C T ( T ) i −→ C T ,T ( T ) g −→ W T ,T ( T ) s −→ W m − , n − ( T ) . Now we pass to the final part of the proof. We will construct an operator from a finite-dimensionalsubspace of W m − , n − ( T ) to C ( S ) and use some standard facts from Banach space theory (mainly,about absolutely summing operators) to get a contradiction. Now let us pass to the details.Consider the function v pq := z p z q ∈ C T ( T ). We are going to assume that natural numbers p and q satisfy the inequality δ q n ≤ p m ≤ δq n , where δ is a small fixed constant (depending of course on our collection T but not on p and q ) whichwill be chosen later. Also, we will consider only large values of p : p > C for some big constant C . Wealways assume that the numbers p and q satisfy these conditions and do not emphasize this later in thepresent section.First of all, we note that k v pq k C T ( T ) ≍ p mN .Indeed, if we take any differential monomial ∂ α ∂ β involved in a junior pat of any operator from T ,then we have: ∂ α ∂ β z p z q = ( ip ) α ( iq ) β z p z q . Since this monomial is in a junior part of some operator, thefollowing inequality holds: αNm + βNn <
1. Therefore, if α = α m , then β = ( N − α − c ) n for some c >
0. Hence, the norm of ∂ α ∂ β v pq in C ( T ) is equal to p α q β = p α m q ( N − α − c ) n ≍ p m ( N − c ) . Clearly,this quantity can be made arbitrarily smaller than p mN if we make p sufficiently large.On the other hand, if we apply any differential monomial involved in the senior part of one of theoperators (which is of the form ∂ jm ∂ ( N − j ) n ) to v pq , we get a function whose norm is p jm q ( N − j ) n ≍ p mN .Moreover, if j > j , then p jm q ( N − j ) n ≍ δ j q nN and this quantity can be made arbitrarily smaller than p j m q ( N − j ) n ≍ δ j q nN if we make δ small, therefore k T v pq k C ( T ) ≍ p mN . All these facts easily implythat indeed k v pq k C T ( T ) ≍ p mN .Similarly, k v pq k C T ,T ( T ) ≍ p mN and k v pq k W T ,T ( T ) ≍ p mN . Therefore, we consider functions w pq := v pq p mN . By the discussion above, we have k w pq k C T ( T ) ≍
1. Hence, there exist functions f pq ∈ C ( S ) suchthat P ( f pq ) = w pq and k f pq k C ( S ) ≤ C . Besides that, we see that T w pq = c pq v pq and T w pq = d pq v pq where | c pq | , | d pq | ≍
1. 5ext, we need to solve the system of differential equations (3). We recall that µ equals b c pq v pq − a d pq v pq plus some linear combination of the operators I αβ and J αβ applied to T w pq and T w pq . There-fore, we write it in the following form: µ = ξ pq c pq v pq + η pq d pq v pq + ( b c pq v pq − a d pq v pq ) . It is easy to see that ξ pq , η pq = O ( p − ε ) for some small fixed ε >
0. Indeed, we simply recall that thesymbol of any Fourier multiplier I αβ is of the form( ip ) α + a ( iq β )( ip ) a ± ( iq ) b . The absolute value of this expression can be estimated by (cid:12)(cid:12)(cid:12) ( ip ) α + Nm ( iq β )( ip ) Nm ± ( iq ) Nn (cid:12)(cid:12)(cid:12) ≍ p α q β p Nm ≍ p α · p mn β p Nm . Here α/m + β/n < N , therefore this expression indeed equals O ( p − ε ). The same is true for all operators I αβ and J αβ .Now we find a solution of the system of differential equations (3). Specifically, we are interested inthe function ϕ j +1 (that is how we defined the operator s ).If j = 0, then we need only the first differential equation to find ϕ . By construction, j = 0 meansthat a = − b = 0. Therefore, clearly we have ϕ = k pq v pq p m where | k pq | ≍ j >
0, then again by construction a = b = 0 and we use the first equation from system (3) toconclude that ϕ = ξ (1) pq · v pq p m , where ξ (1) pq = O ( p − ε ) . Note that in this case | ∂ n ϕ | = | ξ (1) pq q n p m v pq | ≍ | ξ (1) pq v pq | . Now, if j = 1, then we use the second equationto conclude that ϕ = k pq v pq p m with | k pq | ≍ µ = b c pq v pq − a d pq v pq and since j = −
1, we see that a = − , b = 0). If j >
1, then we conclude from the second equation that ϕ = ξ (2) pq v pq with | ξ (2) pq | = O ( p − ε ), etc.Anyway, we see that the following relation holds for a function ϕ j +1 : ϕ j +1 = k pq v pq p m , where | k pq | ≍ . Now we emphasize the dependence of ϕ j +1 on p and q so we denote ϕ ( p,q ) := ϕ j . We see that { ϕ ( p,q ) } is an orthogonal system in W m − , n − and k ϕ ( p,q ) k W m − , n − ≍ p − m p m − q n − ≍ p − / q − / . Finally, we consider the finite-dimensional operator A : W m − , n − → C ( S ) which takes p / q / ϕ ( p,q ) to α pq f pq where ( α pq ) is an arbitrary sequence of numbers such that P | α pq | = 1. Here we take p and q satisfying the previous conditions and such that p ≤ M for some big number M . To be more precise,the operator A is the composition of the orthogonal projection onto span { ϕ p,q } p Let X be a Banach space having local unconditional structure. Then every -summing operator T from X to an arbitrary Banach space Y can be factored through the space L , i.e., there is a measure µ and operators V : X → L ( µ ) and U : L ( µ ) → Y ∗∗ such that U V = κT , where κ : Y → Y ∗∗ is thecanonical embedding. Using this fact, we get the following commutative diagram:7 H ( T ) C T ( T ) C T ,T ( T ) W T ,T ( T ) W m − , n − ( T ) L ( µ ) j i V g sU Now we can consider the dual diagram: C H ( T ) ∗ C T ( T ) ∗ C T ,T ( T ) ∗ W T ,T ( T ) ∗ W m − , n − ( T ) L ∞ ( µ ) j ∗ i ∗ g ∗ s ∗ U ∗ V ∗ The next step of the proof is to construct a specific operator which would take elements of C H ( T ) ∗ to elements of the space W H / ( T ) (which is quasi-Banach; the definition will be given below). Thisconstruction is the same as in the paper [13] but for the sake of completeness we repeat it here.Consider the space W H ( T ) that is defined by means of the following seminorm (and contains onlyproper functions): k f k W H ( T ) = max T ∈ H k T f k L ( T ) . Clearly, this is a Hilbert space. Recall that H = { ∂ jm ∂ ( N − j ) n } Nj =0 . The space W H ( T ) can beidentified with a subspace of L ( T ) ⊕ . . . ⊕ L ( T ) (there is N +1 copy of L ( T ) here); this identificationis given by the map f ( ∂ jm ∂ ( N − j ) n f ) Nj =0 . Therefore, we can consider the orthogonal projection from the direct sum of N + 1 copies of L ( T )to W H ( T ). We denote this projection by P . We need some properties of these operators, so now westate these properties here. All of them are listed in [13].First, it is easy to see how P acts on a natural basis of L ( T ) ⊕ . . . ⊕ L ( T ). Suppose that k = ( k , k )is a pair of integers and denote by φ lk the following element of the space L ( T ) ⊕ . . . ⊕ L ( T ): φ lk = (0 , , . . . , , z k z k , , . . . , , where z k z k is at the l th position, 0 ≤ l ≤ N . Then the following statement can be proved by simplecalculations. Fact 5. If either k or k equals , then P ( φ lk ) = 0 . Otherwise, P ( φ lk ) = ¯ λ l (cid:16) N X j =0 | λ j | (cid:17) − ( λ z k z k , . . . λ N z k z k ) , where λ j = ( ik ) jm ( ik ) ( N − j ) n . Now we need to understand how P acts on the space C H ( T ) ∗ . The space C H ( T ) can be identifiedwith a subspace of C ( T ) ⊕ . . . ⊕ C ( T ) (in the same way as W H ( T ) is identified with a subspace of L ( T ) ⊕ . . . ⊕ L ( T )). Therefore, we have: C H ( T ) ∗ = ( M ( T ) ⊕ . . . ⊕ M ( T )) / X , where X is the annihilator of C H ( T ) in C ( T ) ⊕ . . . ⊕ C ( T ), that is, X = n ( µ , µ , . . . , µ N ) : N X j =0 Z ∂ jm ∂ ( N − j ) n g d ¯ µ j = 0 ∀ g ∈ C H ( T ) o . At this point, formally speaking, we should consider an operator Φ M that is convolution with the M th Fej´er kernel in both variables, and the operators P M such that P M ( F ) = P (Φ M µ , Φ M µ , . . . , Φ M µ N ) , F ∈ C H ( T ) ∗ , where ( µ , . . . , µ N ) is any representative of a functional F . This formula is meaningful because P is an orthogonal projection and if ( ν , . . . , ν N ) lies in X , then (Φ M ν , . . . , Φ M ν N ) lies in X ∩ ( L ( T ) ⊕ . . . ⊕ L ( T )), that is, in the kernel of the projection P . Now we state the following fact from [13].8 act 6. The operators P M : C H ( T ) → W H / ( T ) are uniformly bounded in M . The definition of the space W H / ( T ) should now be clear from the context.The proof (modulo some technical details) follows from the theory of singular integrals (and Fouriermultipliers) with mixed homogeneity developed in [2] (we see from the formula in Fact 5 that thecomponents of P are Fourier multipliers with a certain homogeneity) and it is even true that theseoperators are uniformly of weak type (1 , R n instead of T n . On the other hand, these differences can beovercome quite easily, again, some details can be found in [13].Since all the estimates are uniform in M , we omit the letter M in our notation. Now we have thefollowing commutative diagram: W H / ( T ) C H ( T ) ∗ C T ( T ) ∗ C T ,T ( T ) ∗ W T ,T ( T ) ∗ W m − , n − ( T ) L ∞ ( µ ) P j ∗ i ∗ g ∗ s ∗ U ∗ V ∗ Now we are going to use some facts from the theory of p -summing operators. A good reference hereis [6]. The space W H / ( T ) is a quasi-Banach space of cotype 2 and L ∞ ( µ ) is a space of type C ( K ).Therefore, P j ∗ V ∗ is a 2-summing operator (this is a generalization of Grothendieck’s theorem; see [6]for details). Hence, P j ∗ i ∗ g ∗ s ∗ is also 2-summing. In the next subsection we are going to show that thisis not the case. As in the proof of Theorem 1, we denote by v pq the function z p z q . Again, we consider only sufficientlylarge values of p and assume that the pair ( p, q ) in question satisfies the following condition: δ q n ≤ p m ≤ δq n . We see that k v pq k W m − , n − ≍ p m − q n − ≍ p m p − / q − / . Now denote by w pq the function v pq k v pq k . This is an orthonormal system in the space W m − , n − andhence it is weakly 2-summable. Therefore, since P j ∗ i ∗ g ∗ s ∗ is a 2-summing operator (which by definitionsmeans that it takes weakly 2-summable sequences to 2-summable sequences), we have: X k P j ∗ i ∗ g ∗ s ∗ w pq k W H / < ∞ . First, let us realize where the operator j ∗ i ∗ g ∗ s ∗ takes the function w pq . Take any function v ˜ p ˜ q = z ˜ p z ˜ q ∈ C ( T ); linear combinations of such functions are dense in C ( T ). We write: h v ˜ p ˜ q , ( j ∗ i ∗ g ∗ s ∗ ) w pq i = h ( sgij ) v ˜ p ˜ q , w pq i = h sv ˜ p ˜ q , w pq i W m − , n − . (4)Now we need to recall how s acts on the function v ˜ p ˜ q . We need to solve the system of equations (3) andby definition all functions µ j are multiples of z ˜ p z ˜ q . Therefore, the solution is also a multiple of z ˜ p z ˜ q andso we see that h sv ˜ p ˜ q , w pq i W m − , n − = 0 only if p = ˜ p and q = ˜ q .So, now we need to determine the function s ( v pq ). Recall that in the previous section (where weproved Theorem 1) we showed that s takes v pq p mN to k pq v pq p m where | k pq | ≍ 1. Therefore, we have: s ( v pq ) = k pq p mN p − m v pq . Hence, we have the following identity: h sv pq , w pq i = k pq p mN p − m k v pq k W m − , n − ≍ k pq p mN p − m p − / q − / = k pq p mN p − / q − / . h sv ˜ p ˜ q , w pq i W m − , n − = ( , ( p, q ) = (˜ p, ˜ q ) ,k pq p mN p − / q − / , ( p, q ) = (˜ p, ˜ q ) . Recall that the following element of C ( T ) ⊕ . . . ⊕ C ( T ) corresponds to v pq ∈ C H ( T ):( ∂ jm ∂ ( N − j ) n v pq ) Nj =0 = (( ip ) jm ( iq ) ( N − j ) n v pq ) Nj =0 . So, since p mN ≍ q nN , we can take the following representative from the equivalence class correspondingto ( j ∗ i ∗ g ∗ s ∗ ) w pq : ( l pq p − / q − / v pq , , , . . . , , where | l pq | ≍ . Finally, we apply the projection P (using the formula from Fact 5; in our case, | λ j | = p jm q ( N − j ) n ≍ p mN and hence ¯ λ l λ k ( P | λ j | ) − ≍ X k l pq p − / q − / v pq k L / ≍ X p − q − , and it was already established that this sum is divergent. Therefore, we get a contradiction and thetheorem is proved. References [1] O. V. Besov, V. P. Il’in, S. M. 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