Compactness of Hankel Operators with Symbols Continuous on the Closure of Pseudoconvex Domains

Abstract

Let Ω be a bounded pseudoconvex domain in C 2 with Lipschitz boundary or a bounded convex domain in C n and φ ∈ C(Ω) such that the Hankel operator H φ is compact on the Bergman space A 2 (Ω). Then φ • f is holomorphic for any holomorphic f : D → bΩ.Let Ω be a domain in C n and A 2 (Ω) denote the Bergman space of Ω, the space of square integrable holomorphic functions on Ω. Since A 2 (Ω) is a closed subspace of L 2 (Ω), the space of square integrable functions on Ω, there exists an orthogonal projection P :Hankel operators have been well studied on the Bergman space of the unit disc. Sheldon Axler in [Axl86] proved the following interesting theorem.The space of holomorphic functions satisfying the condition in the theorem is called little Bloch space. One can check that φ(z) = exp((z + 1)/(z − 1)) is bounded on D but it does not belong to the little Bloch space. Hence not every bounded symbol that is smooth on the domain produces compact Hankel operator on the disc. However, Hankel operators with symbols continuous on the closure are compact for bounded domains in C (see, for instance, [Şah12, Proposition 1]). We refer the reader to [Zhu07] for more information on the theory of Hankel operators (as well as Toeplitz operators) on the Bergman space of the unit disc. We note that Sheldon Axler's result has been extended to a small class of domains in C n , such as strongly pseudoconvex domains, by Marco Peloso [Pel94] and Huiping Li [Li94].The situation in C n for n ≥ 2 is radically different. For instance, H z 1 is not compact when Ω is the bidisc (see, for instance, [Le10, Clo17b, CŞ18, Clo17a]). Hence in higher dimensions compactness of Hankel operators is not guaranteed even if the symbol is smooth up to the boundary. We refer the reader to [Str10, Has14] for more information about Hankel operators in higher dimensions and their relations to ∂-Neumann problem.

Convex domains, Hankel operators, and maximal estimates

“…Our first result gives a necessary condition for compactness of Hankel operators; the case q = 1 and n = 2 is in [6] (for symbols in C(Ω)), the case q = 1 but general n is in [7] (for symbols in C ∞ (Ω)).…”

“…Since Ω is convex, such varieties are necessarily contained in affine varieties, see [11], Theorem 1.1 and section 2, and [7], Lemma 2. The proof of Theorem 1, given in section 2, combines ideas from [11] and [6] in a fairly straightforward way.…”

“…Proof of Theorem 1: The proof combines ideas from [11] and [7,6]. (In turn, these ideas can be traced back at least to [3,10].)…”

“…There exists a bounded sequence {F j } ∞ j=1 ⊂ A 2 (Ω) such that the sequence { f j } ∞ j=1 of restrictions to Ω 2 , given by f j (z ′′ ) := F j (0, z ′′ ), belongs to A 2 (Ω 2 ), but does not admit a convergent subsequence. 3 For the rest of the argument, we follow [7,6], with appropriate modifications. Choose a radially symmetric non-negative function χ ∈ C ∞ 0 (D) such that χ(ξ) = 1 for |ξ| ≤ 1/2 and χ(ξ) = 0 for |ξ| ≥ 3/4.…”

“…∂(g k ) instead of −∂(g k )) results from the definition of (·) k , which corresponds to inserting L k into the last slot rather than the first. This affects (∂g) k by a factor (−1) q , and ∂(g k ) by a factor of (−1) q−1 6. Denote by S the increasing reordering of (m, k 1 , · · · , k q−1 ).…”