Duality of Large Fock Spaces in Several Complex Variables and Compact Localization Operators

<jats:p>In this paper, dual spaces of large Fock spaces <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
                        <msubsup>
                           <mrow>
                              <mi mathvariant="script">F</mi>
                           </mrow>
                           <mrow>
                              <mi>ϕ</mi>
                           </mrow>
                           <mrow>
                              <mi>p</mi>
                           </mrow>
                        </msubsup>
                     </math>
                  </jats:inline-formula> with <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
                        <mn>0</mn>
                        <mo><</mo>
                        <mi>p</mi>
                        <mo><</mo>
                        <mo>∞</mo>
                     </math>
                  </jats:inline-formula> are characterized. Also, algebraic properties and equivalent conditions for compactness of weakly localized operators are obtained on <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <msubsup>
                           <mrow>
                              <mi mathvariant="script">F</mi>
                           </mrow>
                           <mrow>
                              <mi>ϕ</mi>
                           </mrow>
                           <mrow>
                              <mi>p</mi>
                           </mrow>
                        </msubsup>
                        <mfenced open="(" close=")">
                           <mrow>
                              <mn>0</mn>
                              <mo><</mo>
                              <mi>p</mi>
                              <mrow>
                                 <mo><</mo>
                              </mrow>
                              <mrow>
                                 <mo>∞</mo>
                              </mrow>
                           </mrow>
                        </mfenced>
                     </math>
                  </jats:inline-formula>.</jats:p>


Introduction
Let ℂ n be the n-dimensional complex Euclidean space. Let dv denote the Lebesgue volume measure on ℂ n . For any two points z = ðz 1 ,⋯,z n Þ and w = ðw 1 ,⋯,w n Þ in ℂ n , we write hz , wi = z 1 w 1 + ⋯ + z n w n and jzj = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi jz 1 j 2 +⋯+jz n j 2 q . For each z ∈ ℂ n and r >0, denotes the Euclidean ball centered at z with radius r. Let Δ denote the Laplacian operator. Suppose ϕ : ℂ n ⟶ ℝ is a C 2 plurisubharmonic function (see [1]). We say that ϕ belongs to the weight class W if ϕ satisfies the following statements: (A) There exists c > 0 such that for z ∈ ℂ n inf z∈ℂ n sup w∈B z,c ð Þ Δϕ w ð Þ > 0 ; ð2Þ (B) For any z ∈ ℂ n and r > 0, Δϕ satisfies the reverse-Hölder inequality for some 0 < C < ∞; (C) The eigenvalues of H ϕ are comparable, i.e., for every z, u ∈ ℂ n , there exists δ 0 > 0 such that where Suppose 0 < p < ∞, ϕ ∈ W. The space L p ϕ consists of all Lebesgue measurable functions f on ℂ n for which L ∞ ϕ is the set of all Lebesgue measurable functions f on ℂ n with f k k ∞,ϕ = sup Let Hðℂ n Þ be the family of all entire functions on ℂ n . The large Fock space is defined as F p ϕ is a Banach space under k·k p,ϕ if p ≥ 1, and F p ϕ is a quasi-Banach space with distance dð f , gÞ = k f − gk p p,ϕ if 0 < p < 1. Assume that ϕðzÞ = jzj 2 /2, then F p ϕ is the classical Fock space which has been studied in [2][3][4] for example. Also, the weight function ϕ on ℂ n with the restriction that dd c ϕ ⋍ d d c jzj 2 belongs to W, where d = ∂ + ∂ and d c = ð ffiffiffiffiffi ffi −1 p /4Þð ∂ − ∂Þ. See [5,6] for more details.
Particularly, F 2 ϕ is a reproducing kernel Hilbert space. That is, for any f ∈ F 2 ϕ , there exists a unique function K z ∈ F 2 ϕ so that f ðzÞ = h f , We say that the function K z ð·Þ is the reproducing kernel of F 2 ϕ . It is well known that the orthogonal projection P : L 2 ϕ ⟶ F 2 ϕ is given by As we know if 1 ≤ p < ∞ and q is the conjugate exponents of p, then the dual space of L p ϕ can be identified with L q ϕ by the integral pairing h,i F 2 ϕ defined by (9). In general, for 1 ≤ p < ∞, no less important than the Hahn-Banach theorem is the Bergman projection to explore the dual spaces of F p ϕ . However, there are some differences for these quasi-Banach spaces F p ϕ ð0 < p < 1Þ. To do this, we will mainly apply Hörmander's solution of the ∂ equation and the Lebesgue dominated convergence theorem to consider the duality of F p ϕ ð0 < p < 1Þ.
The "weakly localized" operators were introduced for the first time in [7], and the authors studied the compactness of these operators on the Bergman space A p and weighted the Bargmann-Fock space F p φ with 1 < p < ∞. In fact, this kind of operators is interesting since these weakly localized operators contain Toeplitz operators which are induced by bounded symbols. Indeed, Toeplitz operators are a kind of significant operators, and these Toeplitz operators induced by diverse functions enjoy abundant properties, see more in [8,9]. As a further research, Hu, Lv, and Wick characterized the compactness of these weakly localized operators on generalized Fock spaces F p φ with 0 < p ≤ 1, see [5]. Besides, in generalized Bergman space setting [10], there are two questions: whether Toeplitz operators induced by bounded symbols are weakly localized operators? Would these weakly localized operators form an algebra? This paper is devoted to consider the compactness of these weakly localized operators on large Fock spaces F p ϕ with 0 < p < ∞. To ensure the validity of these fascinating operators, we show these localization operators contain Toeplitz operators induced by bounded symbols on F p ϕ , see Theorem 16. Meanwhile, we also give affirmative answer about the second question on our Fock spaces, see Theorem 15. Notice that although in the one-dimensional case, the diverse weight function gives another Bergman metric, and the resulting Bergman disk will be changed. Furthermore, there is no inclusion relation between F p ϕ and F q ϕ if p ≠ q. The above properties are much different from [5], so we have to apply more techniques to discuss the compactness of weakly localized operators in case 0 < p ≤ 1. For case 1 < p < ∞, the ideas to study compact weakly localized operators in [7] are not entirely applicable to the situation we are discussing. Hence, we finally combine the skills in [5,7] to consider the compactness of these operators on F p ϕ ð1 < p<∞Þ. Eventually, when p > 1, we bring new consequences even if F p ϕ is the generalized Fock space in [5].
This paper is organized as follows. In Section 2, we give some lemmas which will play key roles in our proofs. In Section 3, we show some properties of projection and dual spaces of large Fock spaces F p ϕ when 0 < p < ∞. In Section 4, we conclude the algebraic properties and boundedness of localization operators. Finally, in Section 5, we consider the compactness of weakly localized operators on our Fock spaces.
Throughout this paper, we write A ≲ B for two quantities A and B if there is a constant C > 0 such that A ≤ CB. Furthermore, A ⋍ B means that both A ≲ B and B ≲ A are satisfied.

Preliminaries and Basic Estimates
In this section, we will give some useful estimates for our proofs. For z ∈ ℂ n , set In the following, we write ρðzÞ instead of ρ ϕ ðzÞ for short. By [11] (see also [12]), we have the following consequences.
(B) The function ρ is Lipschitz, that is (C) For r ∈ ð0, 1Þ and w ∈ Bðz, rρðzÞ), there holds Let r > 0, we write B r ðzÞ = Bðz, rρðzÞÞ and BðzÞ = B 1 ðzÞ. In fact, it is easily obtained from estimate (14) that there is some constant c r such that c −1 r ρðzÞ ≤ ρðwÞ ≤ c r ρðzÞ, where c r = ð1 − rÞ −1 for any r ∈ ð0, 1Þ. That is for every r ∈ ð0, 1Þ, we have ρðwÞ ⋍ ρðzÞ whenever w ∈ B r ðzÞ. Besides, (14) and the triangle inequality give m 1 and m 2 so that Given r > 0, there is a sequence fa k g ∞ k=1 in ℂ n such that fB r ða k Þg k covers ℂ n , and the balls fB r/5 ða k Þg k are pairwise disjoint. We say the sequence fa k g k is an r-lattice. For the r -lattice fa k g k and m > 0, there exists some integer N such that any z in ℂ n belongs to at most N balls of fB mr ða k Þg k . That is, for every z ∈ ℂ n , Now, we are going to state the properties of the reproducing kernel K z . Let ϕ ∈ W, and it follows from [11][12][13] that (A) For z, w ∈ ℂ n , there are constants ϵ, α > 0 such that (B) For z ∈ ℂ n , there exists β ∈ ð0, 1Þ such that (C) For 0 < p ≤ ∞, there holds With the help of Lemmas 1 and 2 in [12], we get the following lemma.
We will write k p,z ðwÞ = K z ðwÞ/kK z k p,ϕ for the normalized reproducing kernel at z ∈ ℂ n , where 0 < p < ∞ and w ∈ ℂ n . Lemma 4. Let 0 < p ≤ 2. Then, for every z ∈ ℂ n , we have as r ⟶ ∞.
Proof. By joining (18) and (20), we have Here, the last step is from the estimate (12). Thus, the assertion follows from Lemma 3 with k = 0 for any fixed z ∈ ℂ n . The next lemma is immediately from ( [12], Lemma 4) (see also ([11], Lemma 2) for any r > 0.

Lemma 5.
For 0<p<∞, there is a constant C>0 such that for each r > 0, f ∈ Hðℂ n Þ and z ∈ ℂ n , we have For r>0 and some domain Ω ⊂ ℂ n , write Ω + r = S z∈Ω B r ðzÞ . Let dð·, · Þ be the Euclidean distance, and we have the following lemma. ð0,1Þ locally in X and ∂ψ = 0, then the equation ∂u = ψ has a solution u ∈ L 2 loc ðXÞ such that For 1 < p < ∞, we let q be the conjugate exponent of p such that 1/p + 1/q = 1.
If 0 < p ≤ 1, then by Lemma 6, we have This together with (18), Lemma 2 and Fubini's theorem give We now let 1 < p < ∞. Notice that estimate (18) and Journal of Function Spaces So, H€ older ' s inequality and Fubini's theorem show And then, for 0 < p < ∞, we get This combined with (25) means that as j ⟶ ∞.
On the other hand, applying Theorem 7 with a = 2 to the solution of Hence, for z ∈ Ω j and let j be sufficiently large, it follows immediately from esitmates (25), (30), and Lemma 2 that By combining the above estimate and (37), we finally obtain as j ⟶ ∞. This ends the proof.
We now proceed to identify the dual space of F p ϕ when 0 < p < ∞. Arguing as in [16], we let The above inqueality shows that L g is a bounded linear functional on F p ϕ and ∥L g ∥≤C∥g∥ ∞,2n−2n/p,ϕ . For w ∈ ℂ n , define gðwÞ = LðKð·, wÞÞ where L is a bounded linear functional on F p ϕ . Pick an r > 0 such that w + Δw ∈ B r ðwÞ. For some m > 0 and every z ∈ ℂ n , using Cauchy's estimates, we have We note that for any fixed w ∈ ℂ n , the function ð Ð B m 2 r ðwÞ jKðu, ·Þj p dvðuÞÞ 1/p is in L p ϕ . Fix w and z, and we get Thus Lebesgue dominated convergence theorem indicates Hence, for any L ∈ ðF p ϕ Þ * , we obtain gðwÞ ∈ H ðℂ n Þ since 5 Journal of Function Spaces and |gðwÞ | ≤kLkkKð·, wÞk p,ϕ ≤ CkLke ϕðwÞ ρðwÞ 2nð1/p−1Þ . The result is To complete the proof, it only remains to show that Let fa n g n be an r-lattice. For 0 < R < ∞, we consider goes to 0 by letting r ⟶ 0, where χ Bð0,RÞ denotes the characteristic function for the ball Bð0, RÞ. This means that It is clear that, for each fixed z ∈ ℂ n , sup Hence, by the following estimate, and the Lebesgue dominated convergence theorem we deduce Furthermore, we claim that as r ⟶ 0. Here, the last assertion follows from fact that kKð·, a n Þ − Kð·, wÞk p,ϕ ⟶ 0 whenever r ⟶ 0. To see this, by letting r ⟶ 0 and w ∈ B r ða n Þ, we then get K w ⟶ K a n . Indeed, (18) gives us a dominating function, and it is from Lemma 2 that the function is in L p ϕ since kðe ϕðwÞ+ϕðzÞ /ρðwÞ n ρðzÞ n Þe −ϵðjz−wj/ρðwÞÞ α k p,ϕ ≲ e ϕðwÞ ρðwÞ 2n/p−2n for any fixed w. Then, the desired assertion holds by Lebesgue dominated convergence theorem again. So, we have Therefore, we have by Theorem 8, (51) and (55) that This finishes the proof.
Proof. If g ∈ F q ϕ , define For any f ∈ F p ϕ , H€ older ' s inequality gives This means that L g is a bounded linear functional on F p ϕ and ∥L g ∥≤∥g∥ q,ϕ .

Journal of Function Spaces
On the other hand, let L : F p ϕ ⟶ ℂ be a bounded linear functional. The Hahn-Banach extension theorem implies that L can be extended to a bounded linear functionalL on L p ϕ . It follows from the duality theory of L p ϕ that there is a function G ∈ L q ϕ such that ∥G∥ q,ϕ ≤ ∥L∥ = ∥L g ∥ and Set g = PG, then ∥g∥ q,ϕ ≤ ∥P∥∥G∥ q,ϕ since P is bounded. Also, note that Theorem 8 shows This completes the proof.
Corollary 11. Suppose 0 < p < ∞. Then, the linear span E of all reproducing kernel functions K z ð·Þ is dense in F p ϕ .
Proof. Let 0 < p ≤ 1. It is immediately from Theorem 8 and the proof of Theorem 9, for any f ∈ F p ϕ , that Next, we assume that p > 1. By Theorem 10 and the Hahn-Banach theorem, it suffices to show that for any g ∈ E, we have f = 0 if f ∈ F q ϕ satisfies h f , gi = 0. This follows from the fact that f ðzÞ = Pf ðzÞ = hf , K z i = 0 for every z ∈ ℂ n .

Localization Operators
In this section, we will explore some properties of weakly localized operators on our Fock spaces. In particular, we will show the algebraic properties of these localization operators.
Before stating weakly localized operators, we consider firstly the following proposition.
Proof. It is from (11) that there exists some r 0 > 0 such that ρðuÞ > r 0 for each u ∈ ℂ n . Fix w ∈ ℂ n , and we have For every w ∈ ℂ n , let r > 0 be sufficiently large and let |u − w | ≥r, and it follows that estimate (18) together with is a dimensionless. For fixed w ∈ ℂ n , we get r ≃ βρðwÞ, where β ∈ ð0, 1Þ. Hence, Theorem 9 together with (19) shows that which is the desired estimate. Now, with the above preparations, we are ready for the definition of weakly localized operators. where Recall that, for 1 < p < ∞, q is the conjugate exponent of p so that 1/p + 1/q = 1.
7 Journal of Function Spaces Next, we are going to answer the questions raised at the beginning of the paper in our Fock spaces. In fact, each set of these weakly localized operators on F p ϕ is an algebra. Proof. Suppose operators T and S are weakly localized. So, it remains to show that TS is a weakly localized operator because the linear combination of two weakly localized operators is also a weakly localized operator.
We let 0 < p ≤ 1. It follows from (68) that there is some where a = ð1 + rÞ 1/2 + 1 and any ε > 0 (when 1 < p < ∞, estimate (71) gives an analogous representation). For z, x ∈ ℂ n , if x ∈ B r/a ðzÞ then ρðxÞ ≤ ð1 + ðr/aÞÞρðzÞ and by the triangle inequality, we have B r/a ðxÞ ⊂ B r ðzÞ. That is, ðB r ðzÞÞ c ⊂ ðB r/a ðxÞÞ c whenever x ∈ B r/a ðzÞ. By joining Lemma 6 and Fubini's theorem, we get Since T and S are weakly localized, hence Therefore, by combining I 1 and I 2 , we get where the constant C does not depend on ε. This means when r ⟶ ∞. Meanwhile, we also get whenever r ⟶ ∞.
On the other hand, let now 1 < p < ∞, by Fubini's theorem, and we have We split again the above integral on ℂ n into the all go to 0 as r ⟶ ∞. This ends the proof since others are obvious.
Let WL ϕ p denote the algebra generated by weakly localized operators for F p ϕ . Let T f be a Toeplitz operator (see [8]) on F Proof. We first suppose 0 < p ≤ 1. Clearly, it suffices to prove that converges to 0 as r ⟶ ∞.
Since T f k p,z = Pð f k p,z Þ for any fixed z, hence Lemma 4 gives that goes to 0 whenever r ⟶ ∞. Now, assume that 1 < p < ∞. It is easily obtained from (18), (20), and Lemma 2 that ð Thus, we only need to show as r ⟶ ∞. In fact, |w − z | ≥rρðzÞ if w ∈ ðB r ðzÞÞ c . This together with (82) indicates Therefore, the desired conclusion follows when r ⟶ ∞. This ends the proof.
Remark. Moreover, Theorem 16 indicates that the identity operator is also in WL ϕ p . Namely, each algebra WL ϕ p possesses an unit.
We next consider the boundedness of operator T ∈ WL ϕ p for 0 < p < ∞.
Proof. First, we see that Let 0 < p ≤ 1 and let Estimate (20) combined with Lemma 6 yields

Journal of Function Spaces
We now assume that p > 1. Set By Fubini's theorem and H€ older ' s inequality, we obtain Since T, S ∈ WL ϕ p , then Theorem 17 says T and S are bounded on F p ϕ . Thus, the above estimate implies kTSk p,z k p,ϕ ≲ kTk p,ϕ kSk p,ϕ . This completes the proof since the supremum of kTSk p,z k p,ϕ is no more than C times kTk p,ϕ kSk p,ϕ .
Theorem 19. If 0 < p ≤ 1, then WL ϕ p is closed under the operator norm on F p ϕ .
Proof. See Lemma 2.6 of [5]. We omit the details.

Equivalent Conditions for Compactness
For this section, we use the ideas in [5,7] to characterize compactness of weakly localized operators on large Fock spaces. Indeed, it is more complex than [5] because Bergman metric works in a different way than in Euclidean metric.
We begin with the following preparations. Recall that, for fixed r > 0, there is an r-lattice fz j g j such that fB r ðz j Þg j covers ℂ n . Let F j = B r ðz j Þ \ S i<j B r ðz i Þ. It follows that fF j g j is also a covering of ℂ n and F j ∩ F k = ∅ðj ≠ kÞ. We write ðF j Þ + r = S x∈F j B r ðxÞ, and it is from estimate (16) that we con- In what follows, we always define F j and G j as above. Also, there is some constant N such that Lemma 20. If 0 < p ≤ 1 and T ∈ WL ϕ p , then for every ε > 0, there exists sufficiently large r > 0 such that for the covering fF j g j (associated to r), we obtain Proof. Since T ∈ WL which ends the proof since sup w∈B r ðzÞ jhTk p,z , k 2n−2n/p,w i F 2 ϕ j ⟶ 0 as z ⟶ ∞.

Data Availability
No data are used.

Conflicts of Interest
The authors declare that they have no competing interests.