IDA and Hankel operators on Fock spaces

We introduce a new space IDA of locally integrable functions whose integral distance to holomorphic functions is finite, and use it to completely characterize boundedness and compactness of Hankel operators on weighted Fock spaces. As an application, for bounded symbols, we show that the Hankel operator $H_f$ is compact if and only if $H_{\bar f}$ is compact, which complements the classical compactness result of Berger and Coburn. Motivated by recent work of Bauer, Coburn, and Hagger, we also apply our results to the Berezin-Toeplitz quantization.


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ZHANGJIAN HU AND JANI A. VIRTANEN where 0 < p < ∞ and ϕ is a suitable weight function (see Section 2 for further details).Obviously, with p = 2 and ϕ(z) = (α/2)|z| 2 , we obtain the weighted Fock space F 2 α .The study of L p -type Fock spaces was initiated in [Janson et al. 1987] and has since grown considerably, as seen in [Zhu 2012].
We also revisit and complement a surprising result due to [Berger and Coburn 1987], which states that for bounded symbols In particular, we give a new proof and show that this phenomenon remains true for Hankel operators from F p (ϕ) to L q (ϕ) for general weights.What also makes this result striking is that it is not true for Hankel operators acting on other important function spaces, such as Hardy or Bergman spaces.
As an application, we will apply our results to the Berezin-Toeplitz quantization, which complements the results in [Bauer et al. 2018].
1A. Main results.We introduce the following new function spaces to characterize bounded and compact Hankel operators.Let 0 < s ≤ ∞ and 0 < q < ∞.For f ∈ L q loc , set where H (B(z, r )) stands for the set of holomorphic functions in the ball B(z, r ).We say that f ∈ L q loc is in IDA s,q if ∥ f ∥ IDA s,q = ∥G q,1 ( f )∥ L s < ∞.
We further write BDA q for IDA ∞,q and say that f ∈ VDA q if lim z→∞ G q,1 ( f )(z) = 0.
The properties of these spaces will be studied in Section 3.
We denote by S the set of all measurable functions f that satisfy the condition in (2-7), which ensures that the Hankel operator H f is densely defined on F p (ϕ) provided that 0 < p < ∞ and ϕ is a suitable weight.Notice that the symbol class S contains all bounded functions.Further, we write Hess ‫ޒ‬ ϕ for the Hessian of ϕ and E for the 2n × 2n identity matrix -these concepts will be discussed in more detail in Section 2. It is important to notice that the condition Hess ‫ޒ‬ ϕ ≃ E in the following theorems is satisfied by the classical Fock space F 2 , the Fock spaces F 2 α generated by standard weights ϕ(z) = (α/2)|z| 2 , α > 0, Fock-Sobolev spaces, and a large class of nonradial weights.
(a) For 0 < p ≤ q < ∞ and q ≥ 1, H f : F p (ϕ) → L q (ϕ) is bounded if and only if f ∈ BDA q , and H f is compact if and only if f ∈ VDA q .For the operator norm of H f , we have the estimate (1-1) (b) For 1 ≤ q < p < ∞, H f : F p (ϕ) → L q (ϕ) is bounded if and only if it is compact, which is equivalent to f ∈ IDA s,q , where s = pq/( p − q), and (1-2) (c) For 0 < p ≤ q ≤ 1 and f ∈ L ∞ , H f : F p (ϕ) → L q (ϕ) is bounded with and compact if and only if f ∈ VDA q .
We first note that Theorem 1.1 is new even for Hankel operators acting from F 2 to L 2 .Previously only characterizations for H f and H f to be simultaneously bounded (or simultaneously compact) were known.These were given in terms of the bounded (or vanishing) mean oscillation of f in [Bauer 2005] for F 2 and in [Hu and Wang 2018] for Hankel operators from F p α to L q α .In Theorem 7.1 of Section 7, we obtain these results as a simple consequence of Theorem 1.1.We also mention our recent work [Hu and Virtanen 2022], which gives a complete characterization of Schatten class Hankel operators.
Theorem 1.1 should also be compared with the results for Hankel operators on Bergman spaces A p .Indeed, characterizations for boundedness and compactness can be found in [Axler 1986] for antianalytic symbols, in [Hagger and Virtanen 2021] for bounded symbols, and in [Hu and Lu 2019;Li 1994;Luecking 1992;Pau et al. 2016] for unbounded symbols.These two cases are different to study because of properties such as F p ⊂ F q for p ≤ q (as opposed to A q ⊂ A p ) and certain nice geometry on the boundary of these bounded domains, which in turn helps with the treatment of the ∂-problem.
What is very different about the results on Hankel operators acting on these two types of spaces is that our next result is only true in Fock spaces (see [Hagger and Virtanen 2021] for an interesting counterexample for the Bergman space).
For Hankel operators on the Fock space F 2 , Theorem 1.2 was proved in [Berger and Coburn 1987] using C * -algebra and Hilbert space techniques and in [Stroethoff 1992] using elementary methods.More recently in [Hagger and Virtanen 2021], limit operator techniques were used to treat the reflexive Fock spaces F p α .However, our result is new even in the Hilbert space case because of the more general weights that we consider.As a natural continuation of our present work, in [Hu and Virtanen 2022], we prove that, for f ∈ L ∞ , the Hankel operator H f is in the Schatten class S p if and only if H f is in the Schatten class S p provided that 1 < p < ∞.
As an application and further generalization of our results, in Section 6, we provide a complete characterization of those f ∈ L ∞ for which lim t→0 ∥T (t)  f T (t) g − T (t) f g ∥ t = 0 (1-4) for all g ∈ L ∞ , where T (t) f = P (t) M f : F 2 t (ϕ) → F 2 t (ϕ) and P (t) is the orthogonal projection of L 2 t (ϕ) onto F 2 t (ϕ).Here L 2 t = L 2 ‫ރ(‬ n , dµ t ) and The importance of the semiclassical limit in (1-4) stems from the fact that it is one of the essential ingredients of the deformation quantization of [Rieffel 1989;1990] in mathematical physics.Our conclusion related to (1-4) extends and complements the main result in [Bauer et al. 2018].
1B. Approach.A careful inspection shows that the methods and techniques used in [Berger and Coburn 1986;1987;Hagger and Virtanen 2021;Perälä et al. 2014;Stroethoff 1992] depend heavily upon the following three aspects.First, the explicit representation of the Bergman kernel K (z, w) for standard weights ϕ(z) = (α/2)|z| 2 has the property that (1-5) However, for the class of weights we consider, this quadratic decay is known not to hold (even in dimension n = 1) and is expected to be very rare [Christ 1991].The second aspect involves the Weyl unitary operator W a defined as where τ a is the translation by a and k a is the normalized reproducing kernel.As a unitary operator on F p α (or on L p α ), W a plays a very important role in the theory of the Fock spaces F p α (see [Zhu 2012]).Unfortunately, no analogue of Weyl operators is currently available for F p (ϕ) when ϕ ̸ = (α/2)|w| 2 .The third aspect we mention is Banach (or Hilbert) space techniques, such as the adjoint (for example, H * f ) and the duality.However, when 0 < p < 1, F p (ϕ) is only an F-space (in the sense of [Rudin 1973]) and the usual Banach space techniques can no longer be applied.
To overcome the three difficulties mentioned above, we introduce function spaces IDA, BDA and VDA, and develop their theory, which we use to characterize those symbols f such that H f are bounded (or compact) from F p (ϕ) to L q (ϕ).Our characterization of the boundedness of H f extends the main results of [Bauer 2005;Hu and Wang 2018;Perälä et al. 2014].It is also worth noting that as a natural generalization of BMO, the space IDA will have its own interest and will likely be useful to study other (related) operators (such as Toeplitz operators).
In our analysis, we appeal to the ∂-techniques several times.As the canonical solution to ∂u = g∂ f, H f g is naturally connected with the ∂-theory.Hörmander's theory provides us with the L 2 -estimate, but less is known about L p -estimates on ‫ރ‬ n when p ̸ = 2.With the help of a certain auxiliary integral operator, we obtain L p -estimates of the Berndtsson-Anderson solution [1982] to the ∂-equation.Our approach to handling weights whose curvature is uniformly comparable to the Euclidean metric form is similar to the treatment in [Schuster and Varolin 2012] which was initiated in [Berndtsson and Ortega Cerdà 1995], and a number of the techniques we use here were inspired by this approach.Although the work in [Berndtsson and Ortega Cerdà 1995] is restricted to n = 1, some of the results were extended to higher dimensions in [Lindholm 2001], and the others are easy to modify.
The outline of the paper is as follows.In Section 2 we study preliminary results on the Bergman kernel which are needed throughout the paper, and we also establish estimates for the ∂-solution developed in [Berndtsson and Andersson 1982].In Section 3, a notion of function spaces IDA s,q is introduced.We obtain a useful decomposition for functions in IDA s,q (compare with the decompositions of BMO and VMO).Using this decomposition, we obtain the completeness of IDA s,q /H ‫ރ(‬ n ) in ∥ • ∥ IDA s,q .In Sections 4 and 5 we prove Theorems 1.1 and 1.2, respectively.For the latter theorem, we also appeal to the Calderón-Zygmund theory of singular integrals, and in particular employ the Ahlfors-Beurling operator to obtain certain estimates on ∂and ∂-derivatives.In Section 6, we present an application of our results to quantization.In the last section, we give further remarks together with two conjectures.
Throughout the paper, C stands for positive constants which may change from line to line, but does not depend on functions being considered.Two quantities A and B are called equivalent, denoted by A ≃ B, if there exists some C such that C −1 A ≤ B ≤ C A.

Preliminaries
Let ‫ރ‬ n = ‫ޒ‬ 2n be the n-dimensional complex Euclidean space and denote by v the Lebesgue measure on ‫ރ‬ n .For z = (z 1 , . . ., z n ) and w = (w 1 , . . ., w n ) in ‫ރ‬ n , we write z ) be the family of all holomorphic functions on ‫ރ‬ n .Given a domain in ‫ރ‬ n and a positive Borel measure µ on , we denote by L p ( , dµ) the space of all Lebesgue measurable functions f on for which For ease of notation, we simply write L p for the space L p ‫ރ(‬ n , dv).
For z ∈ ‫ރ‬ n and r > 0, let B(z, r ) = {w ∈ ‫ރ‬ n : |w − z| < r } be the ball with center at z with radius r .For the proof of the following weighted Bergman inequality, we refer to Proposition 2.3 of [Schuster and Varolin 2012].
Lemma 2.1.Suppose 0 < p ≤ ∞.For each r > 0 there is some C > 0 such that if f ∈ F p (ϕ) then It follows from the preceding lemma that ∥ f ∥ q,ϕ ≤ C∥ f ∥ p,ϕ and This inclusion is completely different from that of the Bergman spaces.
Lemma 2.2.There exist positive constants θ and C 1 , depending only on n, m and M, such that and there exist positive constants C 2 and r 0 such that (2-4) for z ∈ ‫ރ‬ n and w ∈ B(z, r 0 ).
2B.The Bergman projection.For Fock spaces, we denote by P the orthogonal projection of L 2 (ϕ) onto F 2 (ϕ), and refer to it as the Bergman projection.It is well known that P can be represented as an integral operator As a consequence of Lemma 2.2, it follows that the Bergman projection P is bounded on L p (ϕ) for 1 ≤ p ≤ ∞, and P| F p (ϕ) = I for 0 < p ≤ ∞; for further details, see Proposition 3.4 and Corollary 3.7 of [Schuster and Varolin 2012].
2C. Hankel operators.To define Hankel operators with unbounded symbols, consider and the symbol class (2-7) Given f ∈ S, the Hankel operator H f = (I − P)M f with symbol f is well-defined on .According to Proposition 2.5 of [Hu and Virtanen 2020], for 0 < p < ∞, the set is dense in F p (ϕ), and hence the Hankel operator H f is densely defined on F p (ϕ).
Notice that there exists an integer N depending only on the dimension of ‫ރ‬ n such that, for any  2014], which is indispensable to the study of operators between distinct Banach spaces and will be applied to analyze Hankel operators acting from F p (ϕ) to L q (ϕ) in our work.We recall the basic theory of these measures.Let 0 < p, q < ∞ and let µ ≥ 0 be a positive Borel measure on ‫ރ‬ n .We call µ a ( p, q)-Fock Carleson measure if the embedding I : F p (ϕ) → L q ‫ރ(‬ n , e −qϕ dµ) is bounded.Further, the measure µ is referred to as a vanishing ( p, q)-Fock Carleson measure if in addition and converges to 0 uniformly on any compact subset of ‫ރ‬ n as j → ∞.Fock Carleson measures were completely characterized in [Hu and Lv 2014] and we only add the following simple result, which is trivial for Banach spaces and can be easily proved in the other cases.
Proposition 2.3.Let 0 < p, q < ∞ and µ be a positive Borel measure on ‫ރ‬ n .Then µ is a vanishing ( p, q)-Fock Carleson measure if and only if the inclusion map I is compact from F p (ϕ) → L q ‫ރ(‬ n , dµ).
Proof.It is not difficult to show that the image of the unit ball of F p (ϕ) under the inclusion is relatively compact in L q ‫ރ(‬ n , e qϕ dµ).We leave out the details.□ 2F.Differential forms and an auxiliary integral operator.(2-11) Given a weight function ϕ satisfying (2-1), we define an integral operator A ϕ as for ω ∈ L 0,1 , where as denoted on page 92 in [Berndtsson and Andersson 1982].For an (s 1 , t 1 )-form ω A and an (s Therefore, for the (n, n)-form inside the integral of the right-hand side of (2-12), we obtain Recall that Proof.Let z ∈ ‫ރ‬ n .By (2-1), using Taylor expansion of ϕ at ξ , we get Then (2-12) gives (2-13) For l < 2n fixed, define another integral operator A l as It is easy to verify, by interpolation, that A l is bounded on L p for 1 ≤ p ≤ ∞.Therefore, which completes the proof of part (A).

ZHANGJIAN HU AND JANI A. VIRTANEN
Hence, for g ∈ and z ∈ ‫ރ‬ n , it holds that From Proposition 10 of [Berndtsson and Andersson 1982], we get (B) (pay attention to the mistake in the last line of that result where f is left out on the right-hand side).
This shows that H f (g) = u − P(u).□

The space IDA
We now introduce a new space to characterize boundedness and compactness of Hankel operators.The space IDA is related to the space of bounded mean oscillation BMO (see, e.g., [John and Nirenberg 1961;Zhu 2012]), which has played an important role in many branches of analysis and their applications for decades.We find that IDA is also broad in scope and should have more applications in operator theory and related areas.
3A. Definitions and preliminary lemmas.Let 0 < q < ∞ and r > 0. For f ∈ L q loc (the collection of q-th locally Lebesgue integrable functions on ‫ރ‬ n ), following [Luecking 1992], we define G q,r ( f ) as The space IDA ∞,q is also denoted by BDA q .The space VDA q consists of all f ∈ BDA q such that lim z→∞ G q,1 ( f )(z) = 0.
We will see in Section 6 that IDA s,q is an extension of the space IMO s,q introduced in [Hu and Wang 2018].
Notice that the space BDA 2 was first introduced in the context of the Bergman spaces of the unit disk in [Luecking 1992], where it is called the space of functions with bounded distance to analytic functions (BDA).
Remark 3.2.As is the case with the classical BMO q and VMO q spaces, we have BDA q 2 ⊂ BDA q 1 and VDA q 2 ⊂ VDA q 1 properly for 0 < q 1 < q 2 < ∞.
and sup where the constant C is independent of f and r .
Proof.Let f ∈ L q loc , z ∈ ‫ރ‬ n and r > 0. Taking h = 0 in the integrand of (3-1), we get Then for j = 1, 2, . . ., we can pick h j ∈ H (B(z, r )) such that as j → ∞.Hence, for j sufficiently large, This shows that {h j } ∞ j=1 is a normal family.Thus, we can find a subsequence {h j k } ∞ k=1 and a function h ∈ H (B(z, r )) so that lim k→∞ h j k (w) → h(w) for w ∈ B(z, r ).By (3-4), applying Fatou's lemma, we have which proves (3-2).It remains to note that, with the plurisubharmonicity of |h| q , for w ∈ B(z, r/2), we have where h is as in Lemma 3.3.
Proof.For 0 < s < r and w ∈ B(z, r − s), we have B(w, s) ⊂ B(z, r ).Then, the first estimate in (3-6) comes from the definition of G q,s ( f ), while (3-2) yields which completes the proof.□ For z ∈ ‫ރ‬ n and r > 0, let be the q-th Bergman space over B(z, r ).Denote by P z,r the corresponding Bergman projection induced by the Bergman kernel for A 2 (B(z, r ), dv).It is well known that P z,r ( f ) is well-defined for f ∈ L 1 (B(z, r ), dv).
Choose h as in Lemma 3.3.Then h ∈ A q (B(z, r ), dv) because f ∈ L q loc .Thus, P z,r (h) = h.Now for w ∈ B(z, (r − s)/2) and 1 ≤ q < ∞, From this and Lemma 3.3, (3-7) follows.□ Notice that f 1 (z) is a finite sum for every z ∈ ‫ރ‬ n and hence well-defined because we have supp Inspired by a similar treatment on pages 254-255 of [Luecking 1992], using the partition of unity, we can prove the following estimate.
Lemma 3.6.Suppose 0 < q < ∞.For f ∈ L q loc and t > 0, for z ∈ ‫ރ‬ n , where the constant C is independent of f.
Proof.Observe first that f 1 ∈ C 2 ‫ރ(‬ n ) follows directly from the properties of the functions h j and ψ j .For z ∈ ‫ރ‬ n , we may assume z ∈ B(a 1 , t/2) without loss of generality.Then for those j that satisfy ∂ψ j (z) ̸ = 0, G q,t ( f )(a j ).
Thus, using Corollary 3.4 again, we get Similarly, we have , and so Therefore, Combining this and the other two estimates above gives (3-13).□ Given {ψ j } as in (3-11), we have another decomposition f = F 1 + F 2 , where When q = 2, the two decompositions coincide.
Corollary 3.7.Suppose 1 ≤ q < ∞.For f ∈ L q loc and t > 0, we have for z ∈ ‫ރ‬ n , where the constant C is independent of f.
Corollary 3.9.Suppose 1 ≤ q < ∞, and f ∈ L q loc .Then f ∈ BDA q (or VDA q ) if and only if for some (or any) r > 0. Furthermore, , where the infimum is taken over all possible decompositions f = f 1 + f 2 , with f 1 and f 2 satisfying the conditions in (3-26) and (3-27).
Corollary 3.10.Suppose 1 ≤ q < ∞.Different values of r give equivalent seminorms ∥G q,r ( • )∥ L s on IDA s,q when 0 < s < ∞ and on both BDA q and VDA q when s = ∞.
Remark 3.11.Recall that each f in BMO q can be decomposed as f = f 1 + f 2 , where f 1 is of bounded oscillation BO and f 2 has a bounded average BA q (see [Zhu 2012] for the one-dimensional case and [Lv 2019] for the general case).Furthermore, we may choose f 1 to be a Lipschitz function in . ., 2n and f 2 ∈ BA q , or in the language of complex analysis both ∂ f 1 and ∂ f1 are bounded.Therefore, f ∈ BMO q if and only if f, f ∈ BDA q .For a similar relationship between IMO q and the IDA spaces, see Lemma 6.1 of [Hu and Virtanen 2022] and Theorem 7.1 below.
3C. IDA as a Banach space.We next prove that IDA s,q /H ‫ރ(‬ n ) with 1 ≤ s, q < ∞ is a Banach space when equipped with the induced norm (3-28) for f ∈ IDA s,q .
This means that the norm in (3-28) is well-defined on IDA s,q /H ‫ރ(‬ n ).
Let f 1 , f 2 ∈ IDA s,q and z ∈ ‫ރ‬ n .According to Lemma 3.3, there are functions h j holomorphic in B(z, r ) such that M q,r ( f j − h j )(z) = G q,r ( f j )(z) for j = 1, 2.
It remains to prove that the norm is complete.Suppose that { f m } ∞ m=1 is a Cauchy sequence in According to Corollary 3.10, we may assume that { f m } ∞ m=1 is a Cauchy sequence in ∥G q,r ( • )∥ L s with r > 0 fixed.We now embark on proving that, for some f ∈ IDA s,q , lim m→∞ ∥G q,r/2 ( f m − f )∥ L s = 0, which implies { f m } ∞ m=1 converges to some f ∈ IDA s,q in the ∥ • ∥ IDA s,q -topology.For this purpose, let {a j } ∞ j=1 be some t = (r/4)-lattice.We decompose each f m similarly to (3-14) as where {ψ j } ∞ j=1 is the partition of unity subordinate to {B(a j , r/4)} ∞ j=1 as in (3-11).It follows from Corollary 3.7 that This implies that { f m,2 } ∞ j=1 converges to some function f 2 in the L q loc -topology.In addition, by Lemma 3.5, we have Letting k → ∞ and applying Fatou's lemma, we get Integrate both sides over ‫ރ‬ n and apply Fatou's lemma again to obtain the estimate . We may assume ∂ f m,1 → S = n j=1 S j d zj under the L s 0,1 -norm.Since ∂2 = 0, ∂ f m,1 is trivially ∂-closed, and so, as the L s 0,1 limit of { ∂ f m,1 } ∞ m=1 , S is also ∂-closed weakly.Let φ(z) = 1 2 |z| 2 and g = 1 ∈ , and define Then, by Lemma 2.4, and family of all C ∞ functions with compact support) and j = 1, 2, . . ., n, it holds that Hence, ∂ f 1 = S weakly.Then for H B(z,r ) ( ∂ f m,1 − S), the Henkin solution to the equation ∂u = ∂ f m,1 − S on B(z, r ), (3-16) gives (3-31) In addition, according to (3-24), it holds that on B(z, r ).Therefore, by (3-8), (3-9), and (3-31) we have ≤ C∥ ∂ f m,1 − S∥ q L q (B(z,r ),dv) . (3-32) Since S = lim k→∞ ∂ f k,1 in L s 0,1 , by Fatou's lemma, where the last inequality follows from (3-30).We combine (3-32) and (3-33) to get Integrating both sides over ‫ރ‬ n with respect to dv and applying Fatou's lemma once more gives the estimates which completes the proof of the completeness and of the theorem.□ Corollary 3.13.Let 1 ≤ q < ∞.With the norm induced by ∥ • ∥ BDA q , the quotient space BDA q /H ‫ރ(‬ n ) is a Banach space and VDA q is a closed subspace of BDA q .
Proof.The proof of Theorem 3.12 works for s = ∞, so BDA q /H ‫ރ(‬ n ) is a Banach space in ∥ • ∥ BDA q .That VDA q is a closed subspace of BDA q can be proved in a standard way.□ 4. Proof of Theorem 1.1 Given two F-spaces X and Y, we write B(X) for the unit ball of X.A linear operator T from X to Y is bounded (or compact) if T(B(X)) is bounded (or relatively compact) in Y.The collection of all bounded (and compact) operators from X to Y is denoted by B(X, Y) (and by K(X, Y) respectively).We use ∥T∥ X→Y to denote the corresponding operator norm.In particular, we recall that when 0 < p < 1, the Fock space F p (ϕ) with the metric given by d( f, g) = ∥ f − g∥ p p,ϕ is an F-space.To deal with the boundedness and compactness of Hankel operators, we need an additional result involving positive measures and their averages.More precisely, given a positive Borel measure µ on ‫ރ‬ n and r > 0, we write μr (z) = µ(B(z, r )).Notice, in particular, μr is a constant multiple of the averaging function induced by the measure µ.
which completes the proof.□ Remark 4.2.To prove compactness of Hankel operators on spaces that are not necessarily Banach spaces, we use the following result.For 0 < p, q < ∞, H f : This and Fatou's lemma imply that g 0 ∈ B(F p (ϕ)), and hence by the hypothesis, we get 4A.The case 0 < p ≤ q < ∞ and q ≥ 1.
Proof of Theorem 1.1(b).Suppose that H f ∈ B(F p (ϕ), L q (ϕ)).Because the proof of sufficiency is similar to the implication (A) ⇒ (C) of Theorem 4.4 in [Hu and Lu 2019], we only give the sketch here.Indeed, take r 0 as in (4-1), and set t = r 0 /4.Let {a j } ∞ j=1 be a (t/2)-lattice.By Lemma 2.4 of [Hu and Lv 2014], ∞ j=1 λ j k a j p,ϕ ≤ C∥{λ j }∥ l p for all {λ j } ∞ j=1 ∈ l p , where the constant C is independent of {λ j } ∞ j=1 .Let {φ j } ∞ j=1 be the sequence of Rademacher functions on the interval [0, 1].Using the boundedness of H f , we get for s ∈ [0, 1].On the other hand, This and Khintchine's inequality yield Combining this with (4-8) gives for all {|λ j | q } ∞ j=1 ∈ l p/q .By duality with the exponentials p/q and its conjugate, Therefore, by (3-7), which means that f ∈ IDA s,q with the estimate ∥ f ∥ IDA s,q ≤ C∥H f ∥.
It should be pointed out that the right-hand side of the estimate (4.24) (the analogue of (4-10) above) in [Hu and Lu 2019] , and not C∥H f ∥ A p ω →L q ω as stated there.Conversely, suppose f ∈ IDA s,q .As before, decompose f = f 1 + f 2 as in (3-12).From Lemma 3.6 we know that ∥M q,r ( f 2 )∥ pq/( p−q) ≤ C∥ f ∥ IDA s,q .Applying Hölder's inequality to the right-hand side integral in (4-5) with exponent pq/( p − q) and its conjugate exponent t, since we have ∥K ( • , z)∥ t,ϕ < ∞, it follows that This implies f 2 ∈ S, and so also f 1 ∈ S. Now for dν = | ∂ f 1 | q dv, applying Hölder's inequality again with p/( p − q) and its conjugate exponent p/q, we get Theorem 2.8 of [Hu and Lv 2014] shows that ν is a vanishing ( p, q)-Fock Carleson measure; that is, the multiplier and we obtain the norm estimate Similarly to (4-11), using Lemma 3.6, for dµ = | f 2 | q dv, we get Hence, dµ = | f 2 | q dv is a vanishing ( p, q)-Fock Carleson measure.It follows from Proposition 2.3 that the identity operator we see that It remains to notice that the norm equivalence in (1-2) follows from combining the estimates in (4-10), (4-12), and (4-13).□ Remark 4.3.In [Stroethoff 1992], it was proved that for bounded symbols f, the Hankel operator as |λ| → ∞, where φ λ (z) = z + λ.This characterization was recently generalized to F p α with 1 < p < ∞ in [Hagger and Virtanen 2021].Here we note that, using a generalization of Lemma 8.2 of [Zhu 2012] to the setting of ‫ރ‬ n , one can prove that Stroethoff's result remains true for Hankel operators acting from F p α to L q α whenever 1 ≤ p, q < ∞ even for unbounded symbols.
4C.The case 0 < p ≤ q ≤ 1 with bounded symbols.We start with the following preliminary lemma whose proof can be completed with a standard ε argument.
Conversely, suppose that H f is compact from F p (ϕ) to L q (ϕ).Then, as in (4-4), we have for r ∈ (0, r 0 ] fixed.We claim that (4-22) is valid for any r > 0. To see this, we consider the Hankel operator H f on the Fock space F p α .From (4-22), using the sufficiency part, it follows that H f is compact from F p α to L q ‫ރ(‬ n , e −(qα/2)|z| 2 dv).Notice that the equality (1-5) yields inf w∈B(z,r ) |K (w, z)| ≥ C > 0 for any r > 0 fixed, where the constant C is independent of z ∈ ‫ރ‬ n .As in (4-2), we have The following Corollary 4.5 is a direct consequence of the proof of Theorem 1.1(c) which we use to complement and extend the classical result of Berger and Coburn in the next section.

Proof of Theorem 1.2
Proof of the case 0 .

Application to Berezin-Toeplitz quantization
As an application and further generalization of our results, we consider deformation quantization in the sense of [Rieffel 1989;1990] and focus on one of its essential ingredients in the noncompact setting of ‫ރ‬ n that involves the limit condition Recently this and related questions were studied in [Bauer and Coburn 2016;Bauer et al. 2018;Fulsche 2020], which also provide further physical background and references for this type of quantization.
Recall that ϕ ∈ C 2 ‫ރ(‬ n ) is real-valued and Hess ‫ޒ‬ ϕ ≃ E, where E is the 2n × 2n-unit matrix.For t > 0, we set and denote by L 2 t (ϕ) the space of all Lebesgue measurable functions f in ‫ރ‬ n such that in terms of the spaces that were considered in the previous sections.Given f ∈ L ∞ , we use the orthogonal projection P (t) from L 2 t (ϕ) onto F 2 t (ϕ) to define the Toeplitz operator T (t) f and the Hankel operator H (t) f , respectively, by Let U t be the dilation acting on measurable functions in ‫ރ‬ n as It is easy to verify that U t is a unitary operator from L 2 t (ϕ) to L 2 (ϕ) (as well as a unitary operator from F 2 t (ϕ) to F 2 (ϕ)).Further, we have U t P (t) U −1 t = P (1) , which implies that Therefore, and where f S = (1/|S|) S f dv for S ⊂ ‫ރ‬ n measurable.
The following definitions of BMO and VMO are analogous to the classical definition introduced in [John and Nirenberg 1961], but they differ from those widely used in the study of Bergman and Fock spaces.Definition 6.1.We denote by BMO the set of all f ∈ L 2 loc such that Definition 6.2.We define BDA * to be the family of all f ∈ L 2 loc such that and VDA * to be the subspace of all f ∈ BDA * such that lim Given a family X of functions on ‫ރ‬ n , we set X = { f : f ∈ X }.Furthermore, we have for f ∈ L 2 loc .Proof.From a careful inspection of the proof of Proposition 2.5 in [Hu and Wang 2018], it follows that there is a constant C > 0 such that, for f ∈ L 2 loc and z ∈ ‫ރ‬ n , r > 0, there is a constant c(z) for which 1 It is easy to verify that On the other hand, by definition, we have Thus, we have C 1 and C 2 , independent of f, r and z, such that The estimate in (6-4) follows from (6-5).□ Theorem 6.4.Suppose f ∈ L ∞ .Then for all g ∈ L ∞ , it holds that if and only if f ∈ VDA * .

Further remarks
For 1 ≤ p, q < ∞, we have characterized those f ∈ S for which H f : F p (ϕ) → L q (ϕ) is bounded (or compact).For small exponents 0 < p < q < 1, we have proved that this characterization remains true for compactness when f ∈ L ∞ .We also note that when p ≤ q and q ≥ 1, boundedness and compactness of Hankel operators H f : F p (ϕ) → L p (ϕ) depend on q (see Remark 3.2 and Theorem 1.1), while for p > q we cannot say the same -we note that we have no statement analogous to Remark 3.2 for IDA s,q .Moreover, for harmonic symbols f ∈ S and 0 < p, q < ∞, using the Hardy-Littlewood theorem on the submean value (see Lemma 2.1 of [Hu et al. 2007], for example), we are able to characterize boundedness of H f : F p (ϕ) → L q (ϕ) with the space IDA s,q .We will return to this topic in a future publication.
We also note that the space F ∞ (ϕ) does not appear in our results because is not dense in it.Instead, it may be possible to consider the space which can be viewed as the closure of in F ∞ (ϕ), and extend our results to this setting.Regarding weights, the Fock spaces studied in this paper are defined with weights ϕ ∈ C(‫ރ‬ n ) satisfying Hess ‫ޒ‬ ϕ ≃ E. As stated in Section 2A, these weights are contained in the class considered in [Schuster and Varolin 2012].Now, we note that for the weights ϕ in that work, i∂ ∂ϕ ≃ ω 0 , and from Hörmander's theorem on the canonical solution to the ∂-equation it follows that and hence we know that the conclusions of Theorem 1.1 remain true when q = 2 (see Theorem 4.3 of [Hu and Virtanen 2022]).Upon these observations, we raise the following conjecture.
In the literature, there are a number of interesting results on the simultaneous boundedness (and compactness) of Hankel operators H f and H f .These types of characterizations often involve the function spaces BMO q and IMO s,q in their conditions; see, e.g., [Hu and Wang 2018;Zhu 2012].For 1 ≤ q < ∞ and 1 ≤ s ≤ ∞, set IDA s,q = { f : f ∈ IDA s,q }.Then Proposition 2.5 of [Hu and Wang 2018] shows that IDA s,q ∩ IDA s,q = IMO s,q and the results of Section 4 provide a description of the simultaneous boundedness (or compactness) of H f and H f as seen in the following theorem, where as before, we set s = pq/( p − q) if p > q and s = ∞ if p ≤ q.
Theorem 7.1.Let ϕ ∈ C 2 ‫ރ(‬ n ) be real-valued, Hess ‫ޒ‬ ϕ ≃ E, and let f ∈ S. For 1 ≤ p, q < ∞, Hankel operators H f and H f are simultaneously bounded from F p (ϕ) to L q (ϕ)) if and only if f ∈ IMO s,q .
We state one more conjecture related to Theorem 1.2, in which we proved that for f ∈ L ∞ and 0 < p < ∞, H f is compact on F p (ϕ) if and only if H f in compact on F p (ϕ). Recall that this phenomenon does not occur for Hankel operators on the Bergman space or on the Hardy space.As predicted in [Zhu 2012], and verified for Hankel operators on the weighted Fock spaces F p (α) with 1 < p < ∞ in [Hagger and Virtanen 2021], a partial explanation for this difference is the lack of bounded holomorphic or harmonic functions on the entire complex plane.From this point of view it is natural to suggest that a similar result should remain true for Hankel operators mapping from F p (ϕ) to L q (ϕ).
Related to our work on quantization and Theorem 6.4 in particular, we conclude this section with the following problem: characterize those f ∈ L ∞ for which it holds that lim t→0 ∥T (t)  f T (t) g − T (t) f g ∥ S 2 = 0 for all g ∈ L ∞ , where ∥ • ∥ S 2 stands for the Hilbert-Schmidt norm.It would also be important to consider this question for other Schatten classes S p .
where χ E is the characteristic function of E ⊂ ‫ރ‬ n .