Schatten p class commutators on the weighted Bergman space L²_a (\mathbbB_n, dv_γ) for (2n)/(n + 1 + γ) < p < ∞

2011/09/05 by Joshua Isralowitz, Isralowitz, Joshua
Mathematics · #47B35 #47G10 #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1109.0864

openalex publication_date 2011/09/05 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

Let P_γ be the orthogonal projection from the space L ² (\mathbbB_n, dv_γ) to the standard weighted Bergman space L_a ² (\mathbbB_n, dv_γ). In this paper, we characterize the Schatten p class membership of the commutator [M_f, P_γ] when (2n)/(n + 1 + γ) &lt; p &lt; ∞. In particular, if (2n)/(n + 1 + γ) &lt; p &lt; ∞, then we show that [M_f, P_γ] is in the Schatten p class if and only if the mean oscillation MO_γ(f) is in L^p(\mathbbB_n, dζ) where dζ is the Möbius invariant measure on \mathbbB_n. This answers a question recently raised by K. Zhu.

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