Composition Operators Between Weighted Fock Spaces
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Composition Operators Between Weighted Fock Spaces Xingxing Liu1 · Jineng Dai1 Received: 10 August 2020 / Revised: 4 October 2020 / Accepted: 20 October 2020 © Iranian Mathematical Society 2020
Abstract In this paper, we characterize completely the boundedness and compactness of composition operators between weighted Fock spaces. Keywords Weighted Fock space · Composition operator · Carleson measure Mathematics Subject Classification 30H20 · 47B33
1 Introduction Let C n be the n-dimensional complex Euclidean space. For any two points z = n (z 1 , · · · , z n ) and w = (w1 , · · · , wn ) in C n , we write z, w = j=1 z j w j and √ p |z| = z, z. For α real, 0 < β < ∞ and 0 < p < ∞, we denote by Fα,β the space of holomorphic functions f on C n such that 1 p p α − β2 |z|2 f (z)(1 + |z|) e dv(z) < ∞, f p,α,β = Cn
where dv is the Lebesgue volume measure on C n . For p = ∞, we use the notation ∞ to denote the space of holomorphic functions f on C n such that Fα,β β 2 f ∞,α,β = sup | f (z)|(1 + |z|)α e− 2 |z| : z ∈ C n < ∞.
Communicated by Ali Abkar. Supported by the National Natural Science Foundation of China (11771441).
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Jineng Dai [email protected] Xingxing Liu [email protected]
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Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430070, China
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Bulletin of the Iranian Mathematical Society p
2 is a Hilbert space. For When 1 ≤ p ≤ ∞, Fα,β is a Banach space. In particular, Fα,β p p 0 < p < 1, Fα,β is a complete metric space with the distance d( f , g) = f −g p,α,β . p When α > 0, the factor (1 + |z|)α in the definition of Fα,β can be replaced by |z|α p p p with equivalent norms. For α = 0, we write Fβ = F0,β , and the space Fβ is often p called Fock space. Naturally we call the space Fα,β weighted Fock space. Cho et al. p [5] studied explicitly the weighted Fock space Fα,1 . One of their main results is that p p α α f ∈ Fα,1 if and only if R f ∈ F1 , where R is the fractional differentiation operator for α ≥ 0 and the fractional integration operator for α < 0, respectively, as introduced in [5]. So, the weighted Fock space sometimes is called Fock–Sobolev space. Since p the property of the space Fα,1 can be extended trivially to the general weighted Fock p p space Fα,β with any 0 < β < ∞, altogether with the property of the Fock space Fβ p q (see [9, Theorem 2.10]) it follows that Fα,β ⊂ Fα,β for 0 < p < q ≤ ∞. n n For a holomorphic mapping ϕ : C → C , the composition operator Cϕ on the p weighted Fock space is defined by Cϕ f = f ◦ ϕ for f ∈ Fα,β . Carswell, MacCluer and Schuster [1] first studied the composition operator on the Fock space Fβ2 with β = 1/2, and proved that a composition operator is bounded or compact on the Fock space if and only if its inducing function is an affine transformation satisfying certain additional conditions. Their result was generalized to the weighted Fock space 2 for some positive integer m in [4]. With the aid of Fock–Carleson measure, Fm,1 Mengestie [7] characterized the bounded and compact weight
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