On the dimension of the Fock type spaces
aa r X i v : . [ m a t h . C V ] F e b ON THE DIMENSION OF THE FOCK TYPE SPACES
ALEXANDER BORICHEV, VAN AN LE, AND EL HASSAN YOUSSFI
Abstract.
We study the weighted Fock spaces in one and sev-eral complex variables. We evaluate the dimension of these spacesin terms of the weight function extending and completing earlierresults by Rozenblum–Shirokov and Shigekawa. Introduction
Let ψ be a plurisubharmonic function on C n , n ≥
1. The weightedFock space F ψ is the space of entire functions f such that k f k ψ = Z C n | f ( z ) | e − ψ ( z ) dv ( z ) < ∞ , where dv is the volume measure on C n . Note that F ψ is a closedsubspace of L ( C n , e − ψ dv ) and hence is a Hilbert space endowed withthe inner product h f, g i ψ = Z C n f ( z ) g ( z ) e − ψ ( z ) dv ( z ) , f, g ∈ F ψ . In this paper we study when the space F ψ is of finite dimensiondepending on the weight ψ . This problem (at least for the case n = 1)is motivated by some quantum mechanics questions, especially by thestudy of zero modes, eigenfunctions with zero eigenvalues.In [8, Theorem 3.2], Rozenblum and Shirokov proposed a sufficientcondition for the space F ψ to be of infinite dimension, when ψ is asubharmonic function.More precisely, they claimed that if ψ is a finite subharmonic functionon the complex plane such that the measure µ = ∆ ψ is of infinite mass:(1.1) µ ( C ) = Z C dµ ( z ) = ∞ , Key words and phrases.
Fock space, subharmonic function, plurisubharmonicfunction.The results of Section 2 were obtained in the framework of the project 20-61-46016 by the Russian Science Foundation.A. Borichev and H. Youssfi were partially supported by the project ANR-18-CE40-0035. then the space F ψ has infinite dimension.(For the fact that if µ = ∆ ψ a non-trivial doubling measure, then F ψ has infinite dimension see [4, Theorem 11.45]).We improve and extend somewhat the statement of Rozenblum–Shirokov in our paper, give a necessary and sufficient condition on ψ for the space F ψ to be of finite dimension, and calculate this dimension.The situation is much more complicated in C n , n ≥
2. Shigekawaestablished in [10] (see also [4, Theorem 11.20] in a book by Haslinger),the following interesting result.
Theorem A.
Let ψ : C n → R be a C ∞ smooth function and let λ ( z ) be the smallest eigenvalue of the Levi matrix L ψ ( z ) = i∂ ¯ ∂ψ ( z ) = (cid:18) ∂ ψ ( z ) ∂z j ∂z k (cid:19) nj,k =1 . Suppose that (1.2) lim | z |→∞ | z | λ ( z ) = ∞ . Then dim F ψ = ∞ . Note that the condition (1.2) is not necessary. A corresponding ex-ample is given in [4, Section 11.5] ( ψ ( z, w ) = | z | | w | + | w | ). In thispaper, we improve Theorem A by presenting a weaker condition forthe dimension of the Fock space F ψ to be infinite. Furthermore, wegive several examples that show how far is our condition from beingnecessary. Finally, we consider several examples (classes of examples)of weight functions ψ of special form and evaluate the dimension of F ψ .The rest of the paper is organised as follows. The case of dimensionone is considered in Section 2, and the case of higher dimension isconsidered in Section 3. Acknowledgments.
We thank Friedrich Haslinger and Grigori Rozen-blum for helpful remarks.2.
The case of C Given a subharmonic function ψ : C → [ −∞ , ∞ ) denote by µ ψ thecorresponding Riesz measure, µ ψ = ∆ ψ . Next, consider the class M d of the positive σ -finite atomic measures with masses which are integermultiples of 4 π . Given a σ -finite measure µ , consider the correspondingatomic measure µ d , µ d = max n µ ∈ M d : µ ≤ µ o . N THE DIMENSION OF THE FOCK TYPE SPACES 3
In fact, for every atom aδ x of µ , µ d has at the point x an atom of size 4 π times the integer part of a/ (4 π ). Denote µ c = µ − µ d , µ d = P k πδ x k,µ .Denote by M c the class of the positive σ -finite measures µ suchthat µ d = 0. Note that if ψ is finite on the complex plane, then µ ψ has no point masses and µ ψ ∈ M c . Furthermore, if µ ψ ∈ M c , then e − ψ ∈ L loc ( v ). Lemma 2.1.
Let ψ, ψ be two subharmonic functions such that ( µ ψ ) c =( µ ψ ) c . Then dim F ψ = dim F ψ .Proof. Let
F, F be two entire functions with the zero sets, corre-spondingly, { x k,µ ψ } and { x k,µ ψ } (taking into account the multiplici-ties). Then ∆ log | F | = ( µ ψ ) d , ∆ log | F | = ( µ ψ ) d , and the functions h = ψ − log | F | − ψ c , h = ψ − log | F | − ψ c are harmonic. Let h = ℜ H , h = ℜ H for some entire functions H, H .Given an entire function f we have f ∈ F ψ ⇐⇒ Z C | f ( z ) | e − ψ ( z ) dv ( z ) < ∞ ⇐⇒ Z C | f ( z ) | e − ψ c ( z ) − h ( z ) − log | F ( z ) | dv ( z ) < ∞ ⇐⇒ Z C | f ( z ) e − H ( z ) / /F ( z ) | e − ψ c ( z ) dv ( z ) < ∞ ⇐⇒ Z C | f ( z ) e − H ( z ) / /F ( z ) | e − ψ c ( z ) dv ( z ) < ∞ ⇐⇒ Z C | f ( z ) e − H ( z ) / /F ( z ) | e − ψ ( z )+ h ( z )+log | F ( z ) | dv ( z ) < ∞ ⇐⇒ Z C | f ( z ) e − H ( z ) / H ( z ) / | F ( z ) /F ( z ) | e − ψ ( z ) dv ( z ) < ∞ ⇐⇒ f · F F e − H/ H / ∈ F ψ . Thus, dim F ψ = dim F ψ . (cid:3) Lemma 2.2.
Let ψ be a subharmonic function such that µ ψ ∈ M c . If dim F ψ < ∞ , then µ ψ ( C ) < ∞ . See the proof of [8, Theorem 3.2].
Lemma 2.3.
Let ψ be a subharmonic function. Then dim F ψ ≤ l µ ψ ( C )4 π m . Here and later on, given a real number x , ⌈ x ⌉ is the maximal integersmaller than x . ALEXANDER BORICHEV, VAN AN LE, AND EL HASSAN YOUSSFI
Proof.
Set µ = µ ψ and consider a modified logarithmic potential G ofthe measure µ : G ( z ) = 12 π Z D (0 , log | z − w | dµ ( w ) + 12 π Z C \ D (0 , log (cid:12)(cid:12)(cid:12) z − ww (cid:12)(cid:12)(cid:12) dµ ( w )= G ( z ) + G ( z ) . Here and later on, D ( z, r ) = { w ∈ C : | w − z | < r } . Since ∆ G = µ =∆ ψ , by Lemma 2.1 we have dim F ψ = dim F G .Next,(2.1) (cid:12)(cid:12)(cid:12) G ( z ) − µ ( D (0 , π log | z | (cid:12)(cid:12)(cid:12) ≤ π Z D (0 , log (cid:12)(cid:12)(cid:12) − wz (cid:12)(cid:12)(cid:12) dµ ( w ) ≤ C | z | , | z | ≥ , and G ( z ) − µ ( C \ D (0 , π log | z | = 12 π Z C \ D (0 , log (cid:12)(cid:12)(cid:12) z − w (cid:12)(cid:12)(cid:12) dµ ( w ) ≤ , | z | ≥ . Thus, G ( z ) ≤ µ ( C )2 π log(1 + | z | ) + C | z | , z ∈ C . Now, given an entire function f , we have f ∈ F ψ = ⇒ Z C | f ( z ) | (1 + | z | ) − µ ( C ) / (2 π ) dv ( z ) < ∞ . By a Liouville type theorem, f is a polynomial of degree N such that Z ∞ r N r − µ ( C ) / (2 π ) rdr < ∞ . Therefore,
N < − µ ( C ) / (4 π ). Thus, dim F ψ ≤ l µ ( C )4 π m . (cid:3) Lemma 2.4.
Let ψ be a subharmonic function and suppose that µ ψ ∈M c . Then dim F ψ ≥ l µ ψ ( C )4 π m . Proof.
Set µ = µ ψ and choose ε > R > µ ( D (0 , R ))4 π > l µ ( C )4 π m + ε . N THE DIMENSION OF THE FOCK TYPE SPACES 5
Next, increasing R , we can guarantee that µ ( D (0 , R )) > µ ( C ) − . Consider a modified logarithmic potential U of measure µ : U ( z ) = 12 π Z D (0 ,R ) log | z − w | dµ ( w ) + 12 π Z C \ D (0 ,R ) log (cid:12)(cid:12)(cid:12) z − ww (cid:12)(cid:12)(cid:12) dµ ( w )= U ( z ) + U ( z ) . Since ∆ U = µ = ∆ ψ , by Lemma 2.1 we have dim F ψ = dim F U .Arguing as in (2.1), we get U ( z ) ≥ µ ( D (0 , R ))2 π log | z | − C | z | , | z | ≥ R. Next, let | z | ≥ R . Then U ( z ) = 12 π Z C \ ( D (0 ,R ) ∪ D ( z, | z | / log (cid:12)(cid:12)(cid:12) z − ww (cid:12)(cid:12)(cid:12) dµ ( w )+ 12 π Z D ( z, | z | / log (cid:12)(cid:12)(cid:12) z − ww (cid:12)(cid:12)(cid:12) dµ ( w ) ≥ C − π Z D ( z, | z | / log (cid:12)(cid:12)(cid:12) z/ z − w (cid:12)(cid:12)(cid:12) dµ ( w ) = C − U ( z ) . Now, we apply a result by Hayman [5, Lemma 4]. The followingnotation is used there. Let ν be a finite positive measure. Given z ∈ C , h >
0, set n ( z, h ) = ν ( D ( z, h )), N ( z, h ) = R D ( z,h ) log (cid:12)(cid:12)(cid:12) hw − z (cid:12)(cid:12)(cid:12) dν ( w ). Lemma 2.5.
Let z ∈ C , < d < h/ . There exists a set S of area atmost πd such that N ( z, h/ ≤ n ( z , h ) log 16 hd , z ∈ D ( z , h/ \ S. Given m ≥
1, denote A m = { z ∈ C : 2 m R ≤ | z | < m +1 R } . Fix m ≥ k ≥ ν = C \ D (0 ,R ) µ , 2 m R ≤| z | < m +1 R , h = 2 m − R , n ( z , h ) ≤ /
2, and d = 2 m − k − R to get forsome C, C > δ ∈ (0 , m (cid:8) z ∈ A m : U ( z ) > C + δk (cid:9) ≤ C · m R − k , k ≥ . ALEXANDER BORICHEV, VAN AN LE, AND EL HASSAN YOUSSFI
Hence, Z C (1 + | z | ) − − ε e U ( z ) dv ( z ) ≤ C + C X m ≥ X k ≥ − (2+ ε ) m e δk × m (cid:8) z ∈ A m : C + δk ≤ U ( z ) < C + δ ( k + 1) (cid:9) ≤ C + C X m ≥ X k ≥ − (2+ ε ) m e δk m R − k < ∞ . Next, for every 0 ≤ N ≤ l µ ( C )4 π m − Z C | z | N e − U ( z ) dv ( z ) ≤ C Z C | z | N (1 + | z | ) − µ ( D (0 ,R )) / (2 π ) e U ( z ) dv ( z ) ≤ C Z C (1 + | z | ) − − ε e U ( z ) dv ( z ) < ∞ Here we use that µ ψ ∈ M c and, hence, e − U is locally integrable.Finally, we have dim F ψ ≥ l µ ( C )4 π m . (cid:3) Summing up Lemmata 2.1, 2.2, 2.3, and 2.4, we obtain the followingresult, extending and slightly correcting [8, Theorem 3.2].
Theorem 2.6.
Let ψ be a subharmonic function on the complex plane.Then the Fock space F ψ is finite-dimensional if and only if (2.2) ( µ ψ ) c ( C ) < ∞ . If ψ is finite on C , then we can write condition (2.2) as µ ψ ( C ) < ∞ .Finally, if ( µ ψ ) c ( C ) < ∞ , then dim F ψ = l ( µ ψ ) c ( C )4 π m . Remark 2.7.
It is an interesting open question to characterize nonsubharmonic functions ψ such that the space F ψ is of finite dimension.For some results in this direction and some physical interpretations see[9]. N THE DIMENSION OF THE FOCK TYPE SPACES 7 The case of C n , n > C n denote the n -dimensional complex Euclidean space. Given z = ( z , z , . . . , z n ) ∈ C n , we set | z | = p | z | + · · · + | z n | . Denote B n ( z, r ) = { w ∈ C n : | w − z | < r } . Then B n = B n (0 ,
1) isthe unit ball and S n = ∂ B n is the unit sphere in C n . Let dσ be thenormalized surface measure on S n . Theorem 3.1.
Let ψ : C n → R be a C smooth function. Given M > , consider ψ M ( z ) = M log( | z | ) . Suppose that for every M > ,the function ψ − ψ M is plurisubharmonic outside a compact subset of C n . Then dim F ψ = ∞ .Proof. We use the fundamental result of Bedford–Taylor [1] on the so-lutions of the Dirichlet problem for the complex Monge–Amp`ere equa-tion. Given
M >
0, choose r M > ψ − ψ M is plurisubhar-monic on C n \ B n (0 , r M ). Solving the Dirichlet problem for the complexMonge–Amp`ere equation on B n (0 , r M ) with the boundary conditions( ψ − ψ M ) | ∂ B n (0 ,r M ) , we obtain a function u . Set e ψ M ( z ) = ( ( ψ − ψ M )( z ) , z ∈ C n \ B n (0 , r M ) ,u ( z ) , z ∈ B n (0 , r M ) . Then e ψ M is a continuous plurisubharmonic function on C n (see also [3,Section 7]).Now, by the H¨ormander theorem ([6, Theorem 4.4.4], see also [2,Section IV]), there exists an entire function f Z C n | f ( z ) | (1 + | z | ) − n e − e ψ M ( z ) dv ( z ) < ∞ . Hence, for every 0 ≤ k ≤ M − n , we have Z C n | f ( z ) | | z | k e − ψ ( z ) dv ( z ) ≤ C + Z C n \ B n (0 ,r M ) | f ( z ) | | z | k e − ψ ( z ) dv ( z )= C + Z C n \ B n (0 ,r M ) | f ( z ) | | z | k e − ψ M ( z ) e − ( ψ ( z ) − ψ M ( z )) dv ( z ) ≤ C + Z C n \ B n (0 ,r M ) | f ( z ) | | z | − n e − e ψ M ( z ) dv ( z ) < ∞ . Since M is arbitrary, we have dim F ψ = ∞ . (cid:3) ALEXANDER BORICHEV, VAN AN LE, AND EL HASSAN YOUSSFI
Remark 3.2.
Theorem A is an immediate corollary of Theorem 3.1.Indeed, an easy computation shows that if ψ ( z ) = ϕ ( | z | ), ϕ ∈ C ((0 , + ∞ )), then ∂ ψ∂z j ∂ ¯ z k ( z ) = ϕ ′′ ( | z | ) ¯ z j z k + ϕ ′ ( | z | ) δ jk , where δ jk is the Kronecker delta symbol. This implies that i∂ ¯ ∂ψ ( z ) = ϕ ′ ( | z | ) I + ϕ ′′ ( | z | ) z ∗ z, where z ∗ = ¯ z . . . ¯ z n , z ∗ z = (cid:2) ¯ z j z k (cid:3) nj,k =1 . Note also that the spectrum ofthe matrix i∂ ¯ ∂ψ ( z ) is(3.1) σ ( i∂ ¯ ∂ψ ( z )) = (cid:8) ϕ ′ ( | z | ) , ϕ ′ ( | z | ) + | z | ϕ ′′ ( | z | ) (cid:9) . The first eigenvalue has multiplicity n − L ψ ( z ) = i∂ ¯ ∂ψ ( z ) = i∂ ¯ ∂ ( ψ − ψ M )( z ) + M | z | I − M | z | z ∗ z = L ψ − ψ M ( z ) + M | z | I − M | z | z ∗ z. Let z ∈ C n and let V = V . . .V n be a normalized eigenvector correspond-ing to an eigenvalue ν of L ψ − ψ M ( z ). By the hypothesis of Theorem A,for | z | > r M we have λ ( z ) | z | ≥ M , where λ ( z ) is the smallest eigen-value of L ψ ( z ). Thus, ν = h L ψ − ψ M ( z ) V, V i = h L ψ ( z ) V, V i − M | z | + M | z | h z ∗ zV, V i≥ λ ( z ) − M | z | + M | z | | zV | ≥ . Therefore, ψ − ψ M is plurisubharmonic on C n \ B n (0 , r M ), and we arein the conditions of Theorem 3.1. (cid:3) Now we give an easy example when Theorem 3.1 applies while The-orem A does not work.
Example 3.3.
Set ψ ( z ) = ϕ ( | z | ) = (cid:0) log(1 + | z | ) (cid:1) / , z ∈ C n . N THE DIMENSION OF THE FOCK TYPE SPACES 9
Then ϕ ( t ) = (cid:0) log(1 + t ) (cid:1) / , t > F ψ = ∞ . We will show that condition (1.2) fails for ψ while the conditions of Theorem 3.1 are satisfied.We have ϕ ′ ( t ) = 32 11 + t (cid:0) log(1 + t ) (cid:1) / , and ϕ ′′ ( t ) = − (cid:0) log(1 + t ) (cid:1) / (1 + t ) + 34(1 + t ) (cid:0) log(1 + t ) (cid:1) / . By (3.1), the eigenvalues of the matrix L ψ ( z ) are λ ( z ) = 3 (cid:0) log(1 + | z | ) (cid:1) / | z | ) , and λ ( z ) = 3 (cid:0) log(1 + | z | ) (cid:1) / | z | ) + 3 | z | | z | ) (cid:0) log(1 + | z | ) (cid:1) / = 34 2 log(1 + | z | ) + | z | (1 + | z | ) (cid:0) log(1 + | z | ) (cid:1) / . For | z | ≥
2, the smallest eigenvalue of the matrix L ψ ( z ) is λ ( z ) andlim | z |→∞ | z | λ ( z ) = 0 . Thus, condition (1.2) does not hold.On the other hand, for
M >
0, the eigenvalues of matrix L ψ − ψ M ( z )are α ( z ) = λ ( z ) − M | z | , and α ( z ) = λ ( z ) . Since lim | z |→∞ | z | λ ( z ) = ∞ and α ( z ) > z = 0, the conditions ofTheorem 3.1 are satisfied. (cid:3) In the rest of the paper we show that in different situations thesufficient condition of Theorem 3.1 is not necessary for dim F ψ = ∞ . Example 3.4.
Set ψ ( z, w ) = | z | + 2 log(1 + | w | ) , w, z ∈ C . It is clear that dim F ψ = ∞ . Let us verify that for M > ψ − ψ M is not plurisubharmonic at the points (1 , w ), w ∈ C . We start with some easy computations: ∂ψ∂z = z, ∂ ψ∂z∂z = 1 , ∂ ψ∂z∂w = 0 ,∂ψ∂w = 2 w | w | , ∂ ψ∂w∂z = 0 , ∂ ψ∂w∂w = 2(1 + | w | ) . Now, given
M >
0, we have L ψ − ψ M ( z, w )= (cid:18) | w | ) (cid:19) + M ( | z | + | w | ) (cid:18) | z | zwzw | w | (cid:19) − M | z | + | w | I = − M | w | ( | z | + | w | ) Mzw ( | z | + | w | ) Mzw ( | z | + | w | ) | w | ) − M | z | ( | z | + | w | ) ! , and, hence,det( L ψ − ψ M ( z, w ))= 2(1 + | w | ) − M | z | ( | z | + | w | ) − M | w | (1 + | w | ) ( | z | + | w | ) = 2( | z | + | w | ) − M (2 | w | + | z | (1 + | w | ) )(1 + | w | ) ( | z | + | w | ) < M > z = 1 and arbitrary w . Therefore, the conditions of Theo-rem 3.1 do not hold. (cid:3) Weight functions ψ of special form. In this subsection weevaluate the dimension of F ψ and the applicability of our criterion inTheorem 3.1, for some concrete weight functions ψ and for ψ in somespecial classes. Example 3.5.
Let k ≥
3. Set ψ ( z ) = | z k + z k | , z = ( z , z ) ∈ C .Given M >
0, we have L ψ − ψ M ( z ) = k | z | k − − M | z | | z | k ( z z ) k − + M | z | z z k ( z z ) k − + M | z | z z k | z | k − − M | z | | z | ! , N THE DIMENSION OF THE FOCK TYPE SPACES 11 and, hence,det( L ψ − ψ M ( z ))= (cid:18) k | z | k − − M | z | | z | (cid:19) (cid:18) k | z | k − − M | z | | z | (cid:19) − (cid:18) k ( z z ) k − + M | z | z z (cid:19) (cid:18) k ( z z ) k − + M | z | z z (cid:19) = − k M | z | (cid:0) | z | k + | z | k + ( z z ) k + ( z z ) k (cid:1) = − k M | z | | z k + z k | < z k + z k = 0. Thus, for M >
0, the function ψ − ψ M is notplurisubharmonic outside a compact subset of C .Next we are going to verify that dim F ψ = ∞ .We have X := Z C e −| z k + z k | dv ( z ) ≍ Z ∞ Z S r e − r k | ζ k + ζ k | dσ ( ζ , ζ ) dr ≍ Z S | ζ k + ζ k | − /k dσ ( ζ , ζ ) . Given ε >
0, we consider the set T ε = (cid:8) ( ζ , ζ ) ∈ S : | ζ k + ζ k | < ε (cid:9) . Given ( ζ , ζ ) ∈ S such that | ζ | ≥ | ζ | , set ζ = q + r · e iθ and ζ = q − r · e iϕ , r ≥
0. If ( ζ , ζ ) ∈ T ε , then | ζ | − | ζ | < Cε forsome constant C = C ( k ) >
0. Hence, r . ε . Next, since | ζ k + ζ k | < ε ,we obtain that | e ikθ − e ikϕ | . ε . As a result, we obtain that σ ( T ε ) . ε . Set U s = (cid:8) ( ζ , ζ ) ∈ S : 2 − s < | ζ k + ζ k | ≤ − s +1 (cid:9) . Then X ≍ ∞ X s =0 Z U s | ζ k + ζ k | − /k dσ ( ζ , ζ ) . ∞ X s =0 − s s/k = ∞ X s =0 − s (1 − (2 /k )) < ∞ , since k ≥
3. Thus, 1 ∈ F ψ . In the same way, for every α > Z C e − α | z k + z k | dv ( z ) < ∞ . Consider the entire functions f ( z ) = e β ( z k + z k ) , 0 < β < . Since Z C (cid:12)(cid:12) e β ( z k + z k ) (cid:12)(cid:12) e −| z k + z k | dv ( z ) = Z C e β Re(( z k + z k ) ) −| z k + z k | dv ( z ) ≤ Z C e − (1 − β ) | z k + z k | dv ( z ) < ∞ , we conclude that dim F ψ = ∞ . (cid:3) Interestingly, F ψ = 0 if k = 2. Indeed, let ψ (( z , z )) = | z + z | , f ∈ F ψ , f ( z , z ) = ( z + z ) s g ( z , z ) for some s ≥
0, where g ( z , z )is not a multiple of z + z . By the mean value property, for every z ∈ C \ D (0 ,
10) we have | g ( z , iz ) | . (1 + | z | ) Z D ( iz , / (1+ | z | ) \ D ( iz , / (1+ | z | )) | g ( z , z ) | e −| z + z | dv ( z ) . (1 + | z | ) Z D ( iz , / (1+ | z | ) \ D ( iz , / (1+ | z | )) | f ( z , z ) | e −| z + z | dv ( z ) . Hence, Z C | g ( z , iz ) | (1 + | z | ) − dv ( z ) . k f k ψ , and by a Liouville type theorem, g ( z, iz ) ≡
0. Analogously, g ( z, − iz ) ≡
0. Set h ( z, w ) = g ( z − iw, z + iw ). Then h is an entire function and h (0 , w ) = h ( w, ≡
0. Hence, h ( z, w ) = zwh ( z, w ) for another entirefunction h and g ( z , z ) = ( z + z ) g ( z , z ) for some entire function g . This contradiction shows that F ψ = 0.Extending the previous example to C n with n ≥ Example 3.6.
Let n ≥ k ≥ n + 1. Set ψ ( z ) = | z k + · · · + z kn | , z = ( z , . . . , z n ) ∈ C n . Let us verify that for
M >
0, the function ψ − ψ M is not plurisub-harmonic outside a compact subset of C n . N THE DIMENSION OF THE FOCK TYPE SPACES 13
We have L ψ ( z ) = k | z | k − ( z z ) k − . . . ( z z n ) k − ( z z ) k − | z | k − . . . ( z z n ) k − ... ... . . . ...( z z n ) k − ( z z n ) k − . . . | z n | k − = k z k − z k − ... z k − n (cid:0) z k − z k − . . . z nk − (cid:1) . Set A ( z ) = M | z | z z ... z n (cid:0) z z . . . z n (cid:1) . Then L ψ − ψ M ( z ) = L ψ ( z ) + A ( z ) − M | z | I. The spectra of the matrices L ψ ( z ) and A ( z ) are σ L ψ ( z ) = (cid:8) k (cid:0) | z | k − + | z | k − + · · · + | z n | k − (cid:1) , (cid:9) ,σ A ( z ) = (cid:26) M | z | , (cid:27) . Let V be the a unit vector in C n orthogonal to z k − z k − ... z k − n and to z z ... z n .Then h L ψ − ψ M ( z ) V, V i = h L ψ ( z ) V + A ( z ) V − M | z | V, V i = − M | z | < . Thus, for
M >
0, the function ψ − ψ M is plurisubharmonic at no pointsof C n \ { } . Finally, let us verify that dim F ψ = ∞ . Set X := Z C n e −| z k + ... + z kn | dv ( z ) ≍ Z ∞ Z S n r n − e − r k | ζ k + ··· + ζ kn | dσ ( ζ , . . . , ζ n ) dr ≍ Z S n | ζ k + . . . + ζ kn | − n/k dσ ( ζ , . . . , ζ n ) . Given ε >
0, we consider the set T ε = (cid:8) ( ζ , . . . , ζ n ) ∈ S n : | ζ k + . . . + ζ kn | < ε (cid:9) . Set P ( z ) = n X j =1 z kj , z = ( z , . . . , z n ) ∈ C n . Then the function f = log | P | is plurisubharmonic. Following [7], weconsider the Lelong number of f at a ∈ C n , ν f ( a ) = lim r → sup | z |≤ r f ( a + z )log r ∈ [0 , ∞ ] . If f ( a ) = 0, then ν f ( a ) = 0. Otherwise, let a = ( a , . . . , a n ) = 0 and f ( a ) = 0. Without loss of generality, we can assume that a = 0. If0 < r < | a | , then f (cid:0) a + ( r, , . . . , (cid:1) = log | ( a + r ) k − a k | = log | ka k − r + O ( r ) | , r → , and hence, ν f ( a ) = 1. By Theorem 3.1 in [7], applied to Ω = 2 B n , K = B n \ B n , 1 < α <
2, we obtain v (cid:0) { z ∈ K : | P ( z ) | ≤ e − u } (cid:1) = v (cid:0) { z ∈ K : f ( z ) ≤ − u } (cid:1) ≤ C α e − αu , u ≥ . By homogeneity of P , σ ( T ε ) ≤ Cε α , ε > , for some constant C > ∈ F ψ and thenthat dim F ψ = ∞ for k ≥ n + 1. (cid:3) At the end of the paper, we consider two special classes of weightfunctions ψ : radial weight functions and the functions of the form ψ ( z , . . . , z n ) = P nj =1 ψ j ( z j ). N THE DIMENSION OF THE FOCK TYPE SPACES 15
Suppose that ψ ( z ) = ϕ ( | z | ) is a radial plurisubharmonic function ofclass C . By the computations in Remark 3.2,(3.2) ∂ ψ∂z j ∂ ¯ z k ( z ) = ϕ ′′ ( | z | ) ¯ z j z k + ϕ ′ ( | z | ) δ jk . The action of the Monge–Amp`ere operator on ψ is( dd c ψ ) n = 4 n ! det (cid:16) ∂ ψ∂z j ∂ ¯ z k (cid:17) dv = 4 n !( ϕ ′ ( | z | )) n − ( ϕ ′ ( | z | ) + | z | ϕ ′′ ( | z | )) dv. Proposition 3.7.
Suppose that ψ ( z ) = ϕ ( | z | ) is a radial plurisubhar-monic function of class C . Then dim F ψ = ∞ if and only if (3.3) Z C n ( dd c ψ ) n = ∞ . Proof.
Since the spectrum of the matrix (3.2) consists of the eigenvalues ϕ ′ ( | z | ) and ϕ ′ ( | z | ) + | z | ϕ ′′ ( | z | ), the first eigenvalue has multiplicity n − ϕ ′ ≥
0, ( rϕ ′ ( r )) ′ ≥ R + . Furthermore, we have Z C n ( dd c ψ ) n = C Z ∞ ( ϕ ′ ( r )) n − ( ϕ ′ ( r ) + r ϕ ′′ ( r )) dr n = C Z ∞ d (cid:0) ( rϕ ′ ( r )) n (cid:1) . Thus, (3.3) is equivalent to the relation lim r →∞ rϕ ′ ( r ) = ∞ . Now,if rϕ ′ ( r ) is bounded on R + , then ψ ( z ) = O (log | z | ), | z | → ∞ , and aversion of the Liouville theorem shows that dim F ψ < ∞ . On the otherhand, if lim r →∞ rϕ ′ ( r ) = ∞ , then log | z | = o ( ψ ( z )), | z | → ∞ , and thepolynomials belong to F ψ . Hence, dim F ψ = ∞ . (cid:3) For general C plurisubharmonic functions, the radial case suggeststhe following question. Is it true that dim F ψ = ∞ if and only if (3.3)holds? Our last example gives a negative answer to this question. Example 3.8.
Given subharmonic functions ψ j on the complex plane,1 ≤ j ≤ n , set(3.4) ψ ( z , . . . , z n ) = n X j =1 ψ j ( z j ) . Claim: dim F ψ < ∞ if and only if either max j dim F ψ j < ∞ ormin j dim F ψ j = 0. In one direction, by the Fubini theorem, if dim F ψ < ∞ , thenmax j dim F ψ j < ∞ or min j dim F ψ j = 0. In the opposite direction,it is clear that if min j dim F ψ j = 0, then F ψ = 0. It remains to verifythat if max j dim F ψ j < ∞ , then dim F ψ < ∞ .First, suppose that n = 2, dim F ψ < ∞ , N = dim F ψ < ∞ . Fix abasis ( g k ), 1 ≤ k ≤ N , in the space F ψ and choose a family of points( w m ), 1 ≤ m ≤ N , such that det Q = 0, where Q = (cid:0) g k ( w m ) (cid:1) Nk,m =1 .Next, choose f ∈ F ψ . By the mean value property, | f ( z, w ) | ≤ π Z D ( z, | f ( ζ , w | dv ( ζ ) , z, w ∈ C . Therefore, for every z ∈ C , the function f ( z, · ) belongs to F ψ , and,hence, we have f ( z, · ) = N X k =1 a k ( z ) g k . In the same way, the functions f ( · , w j ), 1 ≤ j ≤ N , belong to F ψ .Next, Q − f ( z, w )... f ( z, w N ) = a ( z )... a N ( z ) . Hence, every a j belongs to F ψ . Since dim F ψ < ∞ , we concludethat the space F ψ has finite dimension. For n ≥ ψ satisfying (3.4). We have Z C n ( dd c ψ ) n = C Z C n n Y j =1 ∆ ψ j ( z j ) dv ( z ) = C n Y j =1 Z C ∆ ψ j ( z j ) dv ( z j ) . Now, if n = 2, ψ ( z ) = | z | , ∆ ψ ( z ) = max(1 − | z | , Z C n ( dd c ψ ) n = ∞ , but F ψ = 0. Thus, Proposition 3.7 does not extend to general C -smooth plurisubharmonic functions. References [1] E. Bedford, B. A. Taylor,
The Dirichlet problem for a complex Monge–Amp`ereequation , Invent. Math. (1976), 1–44.[2] E. Bombieri, Algebraic values of meromorphic maps , Invent. Math. (1970),267–287. N THE DIMENSION OF THE FOCK TYPE SPACES 17 [3] J.-P. Demailly,
Potential Theory in Several Complex Variables , Coursdonn´e dans le cadre de l’Ecole d’´et´e d’Analyse Complexe orga-nis´ee par le CIMPA, Nice, Juillet 1989, Manuscript available at ∼ demailly/manuscripts/nice cimpa.pdf [4] F. Haslinger, Complex analysis. A functional analytic approach , De GruyterGraduate, Berlin, 2018.[5] W. Hayman,
The minimum modulus of large integral functions , Proc. LondonMath. Soc. (3) (1952), 469–512.[6] L. H¨ormander, An introduction to complex analysis in several variables , Thirdedition. North-Holland Mathematical Library, . North-Holland Publishing Co.,Amsterdam, 1990.[7] Ch. Kiselman, Ensembles de sous-niveau et images inverses des fonctionsplurisousharmoniques , Bull. Sci. Math. (2000), 75–92.[8] G. Rozenblum, N. Shirokov,
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Entire functions in weighted L and zero modesof the Pauli operator with non-sign definite magnetic field , Cubo (2010),115–132.[10] I. Shigekawa, Spectral properties of Schr¨odinger operators with magnetic fieldsfor a spin / particle , J. Func. Anal. (1991), 255–285. Alexander Borichev: Aix–Marseille University, CNRS, CentraleMarseille, I2M, Marseille, France,St. Petersburg University, Saint Petersburg, Russia
Email address : [email protected] Van An Le: Aix–Marseille University, CNRS, Centrale Marseille,I2M, Marseille, France,University of Quynhon, Department of Mathematics and Statistics,170 An Duong Vuong, Quy Nhon, Vietnam
Email address : [email protected] El Hassan Youssfi: Aix–Marseille University, CNRS, CentraleMarseille, I2M, Marseille, France
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