Sarason ’ s Conjecture of Toeplitz Operators on Fock-Sobolev Type Spaces

For every 0 < p < ∞, α ∈ R, we denote by Lpα(Cn) the space of measurable functions f such that 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩Lpα = (∫Cn 󵄨󵄨󵄨󵄨󵄨󵄨f (z) e−(1/2)|z| 󵄨󵄨󵄨󵄨󵄨󵄨 p dV (z) (1 + |z|)α) 1/p < ∞. (2) Let H(Cn) be the set of entire functions on C. Then for a given 0 < p < ∞, the Fock-Sobolev type space Fp α with the norm ‖ ⋅ ‖Fp α = ‖ ⋅ ‖Lpα is defined as Fp α = {f ∈ H (Cn) | 󵄩󵄩󵄩󵄩f󵄩󵄩󵄩󵄩Lpα < ∞} . (3) Obviously, the Fock-Sobolev type spaceF2 α equippedwith the natural inner product defined by ⟨f, g⟩L2α = ∫Cn f (z) g (z)e−|z| 2 dV (z) (1 + |z|)α (4) is 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 F2 α explicitly. So we use the equivalent normwith respect to a newmeasure |z|−αdV(z). In more detail, for α ≤ 0, we will let ⟨f, g⟩α = ∫ C f (z) g (z)e−|z| dV (z) |z|α , (5) and for α > 0 we let ⟨f, g⟩α = ∫ C f− α/2 (z) g− α/2 (z)e−|z|dV (z) + ∫ C f+ α/2 (z) g+ α/2 (z)e−|z| dV (z) |z|α , (6)


Introduction
Let C  denote the complex -space and V be the ordinary volume measure on C  that is normalized so that ∫    −|| 2 V() = 1.If given any two points  = ( 1 ,  2 , . . .,   ) and  = ( 1 ,  2 , . . .,   ) in C  , we denote For every 0 <  < ∞,  ∈ R, we denote by    (C  ) the space of measurable functions  such that            = (∫ Let (C  ) be the set of entire functions on C  .Then for a given 0 <  < ∞, the Fock-Sobolev type space    with the norm ‖ ⋅ ‖    = ‖ ⋅ ‖    is defined as Obviously, the Fock-Sobolev type space  2  equipped with the natural inner product defined by is a reproducing kernel Hilbert space for every real .As stated in [1], with respect to the above inner product, it is difficult to compute the reproducing kernel of  2  explicitly.So we use the equivalent norm with respect to a new measure || − V().In more detail, for  ≤ 0, we will let and for  > 0 we let where  − /2 is the Taylor expansion of  up to order /2 and  + /2 =  −  − /2 .Now we can bravely make sure that the inner product ⟨⋅, ⋅⟩  generates a new Hilbert space norm on  2   that is equivalent to the  As is well known,  2  is indeed a reproducing kernel Hilbert space (see Lemma 2.1 of [1] for more details).Therefore its reproducing kernel is where {  } is any orthonormal basis for  2  with respect to ⟨⋅, ⋅⟩  .Note that polynomials form a dense subset of  2  (see Proposition 2.3 in [2]).Also note that monomials are mutually orthogonal, which means that {  /√⟨  ,   ⟩  } is an orthonormal basis for  2  .The arguments that are identical to the ones in the proof of Theorem 4.5 in [2] then give us that Here I  is the fractional integration operator defined as where each   is a polynomial of degree .Moreover, for  > 0,  +  is the tail part of the Taylor expansion of  of degree higher than || given by and we let  −  =  −  +  (see [2] for more information on fractional differentiation and integration).Now it is easy to see that if  ≤ 0, ( 2  , ‖ ⋅ ‖ F2

𝛼
) is a closed subspace of  2  with respect to ⟨⋅, ⋅⟩  .In this case, let   denote the orthogonal projection, so that for any  ∈  2  .Unfortunately, the inner product ⟨⋅, ⋅⟩  does not make sense on  2  when  > 0. That means we can not define the Toeplitz operator on  2  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  2   by the following formula:  [3], describes the pairs of outer functions  and ℎ in the Hardy space such that the operator    ℎ 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][6][7][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  2  .Our main result will be the following.
Main Theorem.Suppose that  and V are two nonzero functions in Fock-Sobolev type spaces  2  .Then the following conditions are equivalent: (2) There exists a complex linear polynomial () on C  such that  =   and V =  − , where  is a nonzero complex constant.
(3) The product || 2 |V| 2 is a bounded function on the complex space C  .
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  and  are two nonzero functions in the Fock(-Sobolev) space, then the Toeplitz product     is bounded if and only if  =   and  =  − , where  is a nonzero constant and  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(−| ⋅ | 2 ).They took full advantage of the exact form of the reproducing kernel of the general Fock-type space and concluded that if  and V are two nonzero functions, then the Toeplitz product    V is bounded if and only if  =   and V =  − , where  is a nonzero constant and  is a polynomial of degree at most .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    in Fock-Sobolev type spaces  2  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  2  are in fact very natural generalization of the Fock-Sobolev spaces and the Fock-Sobolev spaces of fractional order.For example, when  = 0,  2 0 is the classical Fock space  2 .Thus in this paper, we always omit discussing the case of  = 0 and the similar result of this case is obtained in [11,14].
Throughout this paper we write  ≲  or  ≳  for nonnegative quantities  and  whenever there is a constant  > 0 independent of  and  such that  ≤ .Similarly we write  ≃  if  ≲  and  ≲ .

Proof of the Main Result
We begin with some properties of the Fock-Sobolev spaces  2  .See [1] for more information.
Lemma 1. Suppose that  belongs to the Fock-Sobolev type space    for any real .Then for any ,  ∈ C  , we have and when  < 0, Proof.In views of the Lemma 3.4 in [1], we can calculate that, for any polynomial  and  ∈  ∞ (see [1] for the definition of  ∞ ), if  ≤ 0, and if  > 0, Lastly the fact that the set of all holomorphic polynomials is dense in  2  completes the proof.
Proof.When  ≤ 0, in view of reproducing properties of    (), Lemma 3.4, the claim () ⇒ () of Lemma 3.10 in [1], and Lemma 2, we see that On the other side, when  > 0, we have to use the Lemma 3.4, the claim () ⇒ () of Lemma 3.10 in [1] to achieve that, if , V ∈  2  ,      V is bounded, Together with (3.5) in [1], Fubini's theorem, and the reproducing property, we can see that Similarly, we can achieve that Therefore, Theorem 4 ((1) ⇒ (2)).If we give that  and V are two nonzero functions in the Fock-Sobolev type space  2  such that Toeplitz product      V is bounded on  2  ; then there is a complex linear polynomial () on C  such that  =   and V =  − , where  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      V is bounded on  2  , by Lemmas 2 and 3 and the Cauchy-Schwarz inequality, we can see that Sequentially, the local property of reproducing kernel   shows us that the module of function is equivalent to | ()−() | when | − | <  0 .It implies that |(, )| 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:  2 () = ⟨, ⟩, where  =  1 +  2 ,  1 is linear,  2 is a homogeneous polynomial of degree 2, and  =  × is a complex matrix symmetric in the real sense.After we choose  =  and  =  + ( 0 /2), where  is any real positive number, we achieve that        ()−()       =  exp ( 0 ⟨, ⟩) is not bounded as  → ∞.This contradiction finishes the proof.
Proof.To prove the boundedness of      V , we will sufficiently obtain that ‖     V ‖ F2  is bounded by means of the idea of [1].In fact we only discuss the case of  > 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 is bounded for any  ∈  2  in view of the definition of the norm ‖ ⋅ ‖ F2  .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 By the reproducing property, the estimations of their module are, respectively, coming from      (     V ) Therefore, using the Cauchy-Schwarz inequality, we have the estimation of the first term of the norm as follows: where Similarly the estimation of the second norm has been achieved that where If we can affirm both sup we would finish the proof because Proof.We omit the proof here for it is analogous to Theorem 2.6 in [12].
Theorem 7 ((3) ⇒ (2)).Suppose that  and V are two functions in the Fock-Sobolev type space  2  , not identically zero, such that || 2 () |V| 2 () is bounded on C  .Then there is a complex linear polynomial () on C  satisfying  =   and V =  − , where  is a nonzero complex constant.
Proof.Now we only consider the case of  > 0 while the other case would be referred to in Theorem 2.7 in [12].
It is easy to see that, for any  ∈ Using the similar method like the case  < 0, we can get the desired results and the proof is finished at this moment.

Conclusions
In this content, we deal with the Sarason's problem on the Fock-Sobolev type spaces and have a complete solution that  =   , V =  − , where  is a linear complex polynomial and  is a nonzero constant.As stated in [1], we know that the Fock-Sobolev type space  2  clearly does not fall under the class of weighted Fock spaces  2  .Therefore the Sarason's problem of weighted Fock spaces  2   is still open.We will focus on this open problem in the future study.
If the function  belongs to the Fock-Sobolev type space  2  , we then have (   ) * =    .