In this note, we will solve Sarason’s conjecture on the Fock-Sobolev type spaces and give a well solution that if Toeplitz product TuTv¯, with entire symbols u and v, is bounded if and only if u=eq, v=Ce-q, where q is a linear complex polynomial and C is a nonzero constant.
National Natural Science Foundation of China1147108411301101Young Innovative Talent Project of Department of Education of Guangdong Province2017KQNCX220Natural Research Project of Zhaoqing University2017322216221. Introduction
Let Cn denote the complex n-space and dv be the ordinary volume measure on Cn that is normalized so that ∫Cne-z2dv(z)=1. If given any two points z=(z1,z2,…,zn) and w=(w1,w2,…,wn) in Cn, we denote (1)z·w¯=∑j=1nzjwj¯,z=z·z¯.For every 0<p<∞, α∈R, we denote by Lαp(Cn) the space of measurable functions f such that (2)fLαp=∫Cnfze-1/2z2pdvz1+zα1/p<∞.Let H(Cn) be the set of entire functions on Cn. Then for a given 0<p<∞, the Fock-Sobolev type space Fαp with the norm ·Fαp=·Lαp is defined as (3)Fαp=f∈HCn∣fLαp<∞.
Obviously, the Fock-Sobolev type space Fα2 equipped with the natural inner product defined by (4)f,gLα2=∫Cnfzgz¯e-z2dvz1+zαis a reproducing kernel Hilbert space for every real α. As stated in [1], with respect to the above inner product, it is difficult to compute the reproducing kernel of Fα2 explicitly. So we use the equivalent norm with respect to a new measure z-αdv(z). In more detail, for α≤0, we will let (5)f,gα=∫Cnfzgz¯e-z2dvzzα,and for α>0 we let (6)f,gα=∫Cnfα/2-zgα/2-z¯e-z2dvz+∫Cnfα/2+zgα/2+z¯e-z2dvzzα,where fα/2- is the Taylor expansion of f up to order α/2 and fα/2+=f-fα/2-. Now we can bravely make sure that the inner product ·,·α generates a new Hilbert space norm on Fα2 that is equivalent to the Fα2 norm ·Fα2. In particular, if we define the norm ·Fα2~ on Fα2 by, when α≤0, (7)fFα2~=∫Cnfz2e-z2dvzzα1/2and when α>0, (8)fFα2~=∫Cnfα/2-z2e-z2dvz1/2+∫Cnfα/2+z2e-z2dvzzα1/2,and then we have that both ·Fα2~ and ·Fα2 are equivalent norms.
As is well known, Fα2 is indeed a reproducing kernel Hilbert space (see Lemma 2.1 of [1] for more details). Therefore its reproducing kernel is (9)Kzαw=∑βϕβwϕβz¯,where {ϕβ} is any orthonormal basis for Fα2 with respect to ·,·α. Note that polynomials form a dense subset of Fα2 (see Proposition 2.3 in [2]). Also note that monomials are mutually orthogonal, which means that {zβ/zβ,zβα} is an orthonormal basis for Fα2. The arguments that are identical to the ones in the proof of Theorem 4.5 in [2] then give us that (10)Kzαw=I-α/2Kzw,if α≤0;I-α/2Kzw+Kzα/2-w,if α>0.Here Is is the fractional integration operator defined as (11)Isfz=∑k=0∞Γn+kΓn+s+kfkz,if s≥0;∑k>s∞Γn+kΓn+s+kfkz,if s<0,where each fk is a polynomial of degree k. Moreover, for s>0, fs+ is the tail part of the Taylor expansion of f of degree higher than s given by (12)fs+z=∑k>sfkzand we let fs-=f-fs+ (see [2] for more information on fractional differentiation and integration).
Now it is easy to see that if α≤0, (Fα2,·Fα2~) is a closed subspace of Lα2 with respect to ·,·α. In this case, let Pα denote the orthogonal projection, so that (13)Pαfz=f,Kzααfor any f∈Lα2. Unfortunately, the inner product ·,·α does not make sense on Lα2 when α>0. That means we can not define the Toeplitz operator on Fα2 in the usual way in terms of this inner product. However according to the ideas of [1], it makes sense to define the Toeplitz operator with the symbols in Fα2 by the following formula: (14)Tφαfz=∫CnfwφwKzαw¯e-w2dvwwα,if α≤0 and (15)Tφαfz=∫Cnφwfα/2-wKzαα/2-w¯e-w2dvw+∫Cnφwfα/2+wKzαα/2+w¯e-w2dvwwαif α>0, for any φ,f∈Fα2. In the sequel, we can reasonably define the Berezin transform of Toeplitz operator on Fα2 by (16)Tφα~z=Tφαkzα,kzαα=∫Cnφwkzαw2e-w2dvwwαif α≤0, and (17)Tφα~z=Tφαkzα,kzαα=∫Cnφwkzαα/2-w2e-w2dvw+∫Cnφwkzαα/2+w2e-w2dvwwαif α>0 for any φ∈Fα2, where kzα(w) is the normalization of the kernel Kzα(w), that is, kzα(w)=Kzα(w)/Kzα(z).
The original product problem, owed to Sarason firstly in [3], describes the pairs of outer functions g and h in the Hardy space such that the operator TgTh¯ is bounded on the Hardy space. Sequentially, this problem was partially researched for the Hardy space in [4] and for the Bergman space in [5–8]. Unluckily it turns out that the Sarason’s conjecture is not true for both Hardy space and Bergman space of unit disk. See [9, 10] for counterexamples.
We will, in this note, give the equivalent conditions about the Sarason’s conjecture of Toeplitz product on Fock-Sobolev type spaces Fα2. Our main result will be the following.
Main Theorem. Suppose that u and v are two nonzero functions in Fock-Sobolev type spaces Fα2. Then the following conditions are equivalent:
The Toeplitz product TuαTv¯α is bounded on Fock-Sobolev type spaces Fα2.
There exists a complex linear polynomial q(z) on Cn such that u=eq and v=Ce-q, where C is a nonzero complex constant.
The product u2~v2~ is a bounded function on the complex space Cn.
In 2014, Cho et al. studied the products of Toeplitz operators on the classical Fock space (see [11]). In the case of the Fock-Sobolev space, Chen et al. (see [12]) had already proven the same topics and obtained the similar results. What is more, they claimed that if f and g are two nonzero functions in the Fock(-Sobolev) space, then the Toeplitz product TfTg¯ is bounded if and only if f=eq and g=Ce-q, where C is a nonzero constant and q is a linear polynomial. More properties about Toeplitz operators on Fock-Sobolev spaces are referred to in [13]. Sequentially, Bommier-Hato et al. in [14] continued to research Cho’s results on the general Fock-type space with the weight functions exp(-·2m). They took full advantage of the exact form of the reproducing kernel of the general Fock-type space and concluded that if u and v are two nonzero functions, then the Toeplitz product TuTv¯ is bounded if and only if u=eg and v=Ce-g, where C is a nonzero constant and g is a polynomial of degree at most m. The similar techniques are founded in [15, 16]. However, the translations appearing to the classical Fock spaces are not suitable to the generalized Fock space. To tackle the main theorem, we have to use the main ideas of [14], that is, making good use of the explicit properties of the reproducing kernel Kzα in Fock-Sobolev type spaces Fα2 instead of the Weyl operators defined by translations on the complex plane.
At last, it is remarked that, as stated in [1], the Fock-Sobolev type spaces Fα2 are in fact very natural generalization of the Fock-Sobolev spaces and the Fock-Sobolev spaces of fractional order. For example, when α=0, F02 is the classical Fock space F2. Thus in this paper, we always omit discussing the case of α=0 and the similar result of this case is obtained in [11, 14].
Throughout this paper we write X≲Y or Y≳X for nonnegative quantities X and Y whenever there is a constant C>0 independent of X and Y such that X≤CY. Similarly we write X≃Y if X≲Y and Y≲X.
2. Proof of the Main Result
We begin with some properties of the Fock-Sobolev spaces Fα2. See [1] for more information.
Lemma 1.
Suppose that f belongs to the Fock-Sobolev type space Fαp for any real α. Then for any z,w∈Cn, we have (18)fzpe-p/2z211+zα≲fFαpp,and when α<0,(19)Kzαw≲1+zwα/2exp12z2+12w2-18z-w2,when α>0,(20)Kzαw≲1+w·z¯α/2exp12z2+12w2-18z-w2.More specifically, (21)Kzαz≃1+zαez2for any z∈Cn and there is a r>0 such that (22)Kzαw≳1+zαexp12z2+12w2,for any z∈B(w,r).
A consequence of the first estimate in Lemma 1 is that, for any function u∈Fα2, the Toeplitz operators Tu and Tu¯ are both densely defined on Fα2.
Lemma 2.
If the function u belongs to the Fock-Sobolev type space Fα2, we then have (Tuα)∗=Tu¯α.
Proof.
In views of the Lemma 3.4 in [1], we can calculate that, for any polynomial f and g∈F∞ (see [1] for the definition of F∞), (23)Tuα∗f,gα=∫Cnu¯zfzgz¯e-z2zαdvz=Tu¯αf,gα, if α≤0, and if α>0, (24)Tuα∗f,gα=∫Cnu¯zfα/2-zgα/2-z¯e-z2dvz+∫Cnu¯zfα/2+zgα/2+z¯e-z2zαdv. Lastly the fact that the set of all holomorphic polynomials is dense in Fα2 completes the proof.
Lemma 3.
For given u,v∈Fα2, if TuαTv¯α is bounded on Fα2, then TuαTv¯αKzα(w)=u(w)v¯(z)Kzα(w) for any z,w∈Cn.
Proof.
When α≤0, in view of reproducing properties of Kzα(w), Lemma 3.4, the claim (d)⇒(a) of Lemma 3.10 in [1], and Lemma 2, we see that (25)TuαTv¯αKzαw=TuαTv¯αKzα,Kwαα=v¯zKzα,u¯wKwαα=uwv¯zKzαw.
On the other side, when α>0, we have to use the Lemma 3.4, the claim (d)⇒(a) of Lemma 3.10 in [1] to achieve that, if u,v∈Fα2, TuαTv¯α is bounded, (26)TuαTv¯αKzα,Kwαα=∫CnuλTv¯αKzαα/2-λKwαα/2-λ¯e-λ2dvλ+∫CnuλTv¯αKzαα/2+λKwαα/2+λ¯e-λ2λαdvλ.Together with (3.5) in [1], Fubini’s theorem, and the reproducing property, we can see that (27)∫CnuλTv¯αKzαα/2+λKwαα/2+λ¯e-λ2λαdvλ=∫Cn∫Cnuλv¯ξKzαα/2+ξKξαα/2+λKwαα/2+λ¯e-λ2λαe-ξ2ξαdvξdvλ=∫CnuλKξαα/2+λKwαα/2+λ¯e-λ2λαdvλ∫Cnv¯ξKzαα/2+ξe-ξ2ξαdvξ=uw∫Cnv¯ξKξαα/2+wKzαα/2+ξe-ξ2ξαdvξ=uwv¯zKzαα/2+w.Similarly, we can achieve that (28)∫CnuλTv¯αKzαα/2-λKwαα/2-λ¯e-λ2dvλ=uwv¯zKzαα/2-w.Therefore, (29)TuαTv¯αKzαw=TuαTv¯αKzα,Kwαα=uwv¯zKzαw.
Theorem 4 ((1) ⇒ (2)).
If we give that u and v are two nonzero functions in the Fock-Sobolev type space Fα2 such that Toeplitz product TuαTv¯α is bounded on Fα2; then there is a complex linear polynomial q(z) on Cn such that u=eq and v=Ce-q, where C is a nonzero complex constant.
Proof.
This proof is similar to Theorem 2.4 in [12] and here we only give its brief illustration.
If the condition holds that the Toeplitz product TuαTv¯α is bounded on Fα2, by Lemmas 2 and 3 and the Cauchy-Schwarz inequality, we can see that (30)uzvz=TuαTv¯α~≤TuαTv¯α<∞. Sequentially, the local property of reproducing kernel Kz shows us that the module of function (31)Tz,w=TuαTv¯αKwα,KzααKzαzKwαw is equivalent to eq(z)-q(w)¯ when z-w<ϵ0. It implies that Tz,w is bounded in that situation.
On the other side, we give the representation of quadratic polynomial in the case of real inner product as follows: q2(z)=Az,z, where q=q1+q2, q1 is linear, q2 is a homogeneous polynomial of degree 2, and A=An×n is a complex matrix symmetric in the real sense. After we choose w=rξ and z=rξ+(ϵ0/2)η, where r is any real positive number, we achieve that (32)eqz-qw¯=Mexprϵ0Aξ,η is not bounded as r→∞. This contradiction finishes the proof.
Theorem 5 ((2) ⇒ (1)).
If u=eq and v=e-q where q is a complex linear polynomial on Cn, then TuαTv¯α is bounded on the Fock-Sobolev type space Fα2.
Proof.
To prove the boundedness of TuαTv¯α, we will sufficiently obtain that TuαTv¯αfFα2~ is bounded by means of the idea of [1]. In fact we only discuss the case of α>0 because the other case is the same as the proof of Theorem 2.5 in [12]. Using the similar ways, our goal is to obtain that (33)TuαTv¯αfFα2~=∫CnTuαTv¯αfα/2-ze-1/2z22dvz1/2+∫CnTuαTv¯αfα/2+ze-1/2z2z-α/22dvz1/2 is bounded for any f∈Fα2 in view of the definition of the norm ·Fα2~. To the end, we focus our attention on the integrands in it. By formulae (3.4)(3.5) in [1] and the definition of Toeplitz operator, the integrands in the norm are (34)TuαTv¯αfα/2-z=∫Cnv¯ηfα/2-η∫CnuwKwαα/2-η¯Kzαα/2-w¯e-w2dvwe-η2dvη,TuαTv¯αfα/2+z=∫Cnv¯ηfα/2+η∫CnuwKwαα/2+η¯Kzαα/2+w¯e-w2wαdvwe-η2ηαdvη.
By the reproducing property, the estimations of their module are, respectively, coming from (35)TuαTv¯αfα/2-z≲∫Cnuzv¯ηfα/2-ηKzαα/2-ηe-η2dvη, and then, similarly, (36)TuαTv¯αfα/2+z≲∫Cnuzv¯ηfα/2+ηKzαα/2+ηe-η2ηαdvη.Therefore, using the Cauchy-Schwarz inequality, we have the estimation of the first term of the norm as follows: (37)∫CnTuαTv¯αfα/2-ze-1/2z22dvz≲∫Cn∫CnHα1w,zdvw∫CnHα1w,zfα/2-w2e-w2dvwdvz,where Hα1(w,z)=e-1/2z2Kzαα/2-we-(1/2)w2exp(Re(q(z)− q(w)¯)). Similarly the estimation of the second norm has been achieved that (38)∫CnTuαTv¯αfα/2+ze-1/2z2z-α/22dvz≲∫Cn∫CnHα2w,zdvw∫CnHα2w,zfα/2+w2e-w2wαdvwdvz,where Hα2w,z=e-1/2z2z-α/2Kzαα/2+we-1/2w2·w-α/2exp(Re(q(z)-q(w)¯)).
If we can affirm both (39)supz∈Cn∫CnHα1z,wdvw<∞,supz∈Cn∫CnHα2z,wdvw<∞, we would finish the proof because (40)TuαTv¯αfFα2~2≲∫Cnfα/2-w2e-w2dvw+∫Cnfα/2+w2e-w2wαdvw. To the finish, in terms of Lemma 1 and transformation, we can assert that (41)supz∈Cn∫CnHα1z,wdvw≲supz∈Cn∫Cn1+w·z¯α/2e-1/8z-w2eqz-wdvw<∞,supz∈Cn∫CnHα2z,wdvw≲supz∈Cn∫Cnz-α/2w-α/2e-1/8z-w21+z·w-α/2eqz-wdvw<∞.This implies that TuTv¯fFα2~ is bounded and completes the proof.
Theorem 6 ((1) ⇒ (3)).
If u and v are two functions in the Fock-Sobolev type space Fα2, not identically zero, such that the operator TuαTv¯α is bounded on Fα2, then u2~(z)v2~(z) is a bounded function on the complex space.
Proof.
We omit the proof here for it is analogous to Theorem 2.6 in [12].
Theorem 7 ((3) ⇒ (2)).
Suppose that u and v are two functions in the Fock-Sobolev type space Fα2, not identically zero, such that u2~(z)v2~(z) is bounded on Cn. Then there is a complex linear polynomial q(z) on Cn satisfying u=eq and v=Ce-q, where C is a nonzero complex constant.
Proof.
Now we only consider the case of α>0 while the other case would be referred to in Theorem 2.7 in [12].
It is easy to see that, for any u∈Fα2, u~(z)=Tuαkzα,kzαα=u(z). When α>0, we use the triangle inequality and Hölder’s inequality to calculate (42)uz2≲∫Cnuwkzαα/2-w2e-w2dvw∫Cnkzαα/2-w2e-w2dvw+∫Cnuwkzαα/2+w2e-w2wαdvw∫Cnkzαα/2+w2e-w2wαdvw.Because kzα is a unit element, that is, (43)kzαFα2~=∫Cnkzαα/2-w2e-w2dvw1/2+∫Cnkzαα/2+w2e-w2wαdvw1/2,we can see that (44)∫Cnkzαα/2-w2e-w2dvw≲1,∫Cnkzαα/2+w2e-w2wαdvw≲1. From the above inequations, the estimate of uz2 turns into (45)uz2≲∫Cnuw2kzαα/2-w2e-w2dvw+∫Cnuw2kzαα/2+w2e-w2wαdvw=u2~z. If u2~(z)v2~(z) is a bounded function on Cn, v2~(z)u2(z) and u2(z)v2(z) are both bounded on Cn. By Liouville’s theorem, the boundedness of u2(z)v2(z) implies that there exists a constant C such that uv=C. Since neither u nor v is identically zero, we have C≠0. That is, both u and v are nonvanishing. By Lemma 1, there exists a complex polynomial q(z) on Cn with deg(q)≤2 such that u=eq and v=Ce-q.
On the other side, by the definition of Berezin transformation in this case, (46)u2~zv2z=∫Cnuwv¯zkzαα/2-w2e-w2dvw+∫Cnuwv¯zkzαα/2+w2e-w2wαdvw. Now giving a sufficiently small δ>0, we further obtain that (47)u2~zv2z≳∫w>δuwv¯zkzαα/2-w2e-w2dvw+∫w>δuwv¯zkzαα/2+w2e-w2wαdvw≳∫w>δuwv¯zkzαw2e-w2wαdvw≳∫w>δ1+z-α/2e-1/2z2Kzαwe-1/2w22eqw-qzwα/22dvw. When choosing a constant ε>0 satisfying Lemma 1, we can see that (48)u2~zv2z≳∫z-w<εw>δ1+z-α/2e-1/2z2Kzαwe-1/2w22eqw-qzwα/22dvw+∫z-w≥εw>δ1+z-α/2e-1/2z2Kzαwe-1/2w22eqw-qzwα/22dvw.Using the similar method like the case α<0, we can get the desired results and the proof is finished at this moment.
3. Conclusions
In this content, we deal with the Sarason’s problem on the Fock-Sobolev type spaces and have a complete solution that u=eq,v=Ce-q, where q is a linear complex polynomial and C is a nonzero constant. As stated in [1], we know that the Fock-Sobolev type space Fα2 clearly does not fall under the class of weighted Fock spaces Fφ2. Therefore the Sarason’s problem of weighted Fock spaces Fφ2 is still open. We will focus on this open problem in the future study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally. All authors read and approved the final manuscript.
Acknowledgments
This paper is supported by National Natural Science Foundation of China (Grants nos. 11471084 and 11301101), Young Innovative Talent Project of Department of Education of Guangdong Province (no. 2017KQNCX220), and the Natural Research Project of Zhaoqing University (nos. 201732 and 221622).
ChoH. R.IsralowitzJ.JooJ.-C.Toeplitz Operators on Fock–Sobolev Type Spaces20158212-s2.0-8493997591510.1007/s00020-015-2223-8Zbl1326.47029ChoH. R.ChoeB. R.KooH.Fock-Sobolev spaces of fractional order201543219924010.1007/s11118-015-9468-3SarasonD.KhavinV. P.NikolskiN. K.Products of Toeplitz operators19941573Berlin, GermanySpringer318319Lecture Notes in MathZhengD.The distribution function inequality and products of Toeplitz operators and Hankel operators199613824775012-s2.0-003058564710.1006/jfan.1996.0073ParkJ.-D.Bounded toeplitz products on the bergman space of the unit ball in Cn200654457158410.1007/s00020-005-1405-12-s2.0-33645509317StroethoffK.ZhengD.Products of Hankel and Toeplitz Operators on the Bergman Space199916912893132-s2.0-000184014710.1006/jfan.1999.3489StroethoffK.ZhengD.Bounded Toeplitz products on the Bergman space of the polydisk200327811251352-s2.0-003730904810.1016/S0022-247X(02)00578-4Zbl1051.47025StroethoffK.ZhengD.Bounded Toeplitz products on Bergman spaces of the unit ball200732511141292-s2.0-3375036023110.1016/j.jmaa.2006.01.009AlemanA.PottS.RegueraM. C.Sarason’s conjecture on the Bergman space2016130NazarovF.A counterexample to Sarason's conjecture, preprint, available at: http://users.math.msu.edu/users/fedja/prepr.html, 1997ChoH. R.ParkJ.-D.ZhuK.Products of Toeplitz operators on the fock space20141427248324892-s2.0-8492476899310.1090/S0002-9939-2014-12110-1Zbl1303.47035ChenJ. J.WangX. F.XiaJ.CaoG. F.Sarason’s Toeplitz product problem on the Fock–Sobolev space201719https://doi.org/10.1007/s10114-017-5780-810.1007/s10114-017-5780-82-s2.0-85033712783WangX.CaoG.XiaJ.Toeplitz operators on Fock-Sobolev spaces with positive measure symbols20145771443146210.1007/s11425-014-4813-3MR3213881Zbl1302.30072Bommier-HatoH.YoussfiE. H.ZhuK.Sarason's Toeplitz product problem for a class of Fock spaces2017141540844210.1016/j.bulsci.2017.03.0022-s2.0-85020247089SeipK.YoussfiE. H.Hankel operators on Fock spaces and related Bergman kernel estimates20132311702012-s2.0-8487259842310.1007/s12220-011-9241-9Zbl1275.47063WangX.CaoG.ZhuK.BMO and Hankel Operators on Fock-Type Spaces2015253165016652-s2.0-8493156790010.1007/s12220-014-9488-zZbl1337.47044