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We show that for an entire function φ belonging to the Fock space F2(Cn) on the complex Euclidean space Cn, the integral operator SφF(z)=∫CnF(w)ez⋅w¯φ(z−w¯)dλ(w),z∈Cn, is bounded on F2(Cn) if and only if there exists a function m∈L∞(Rn) such that φ(z)=∫Rnm(x)e−2(x−[Formula presented]z)2 dx,z∈Cn. Here dλ(w)=π−ne−|w|2 dw is the Gaussian measure on Cn. With this characterization we are able to obtain some fundamental results of the operator Sφ, including the normality, the C⁎ algebraic properties, the spectrum and its compactness. Moreover, we obtain the reducing subspaces of Sφ. In particular, in the case n=1, we give a complete solution to an open problem proposed by K. Zhu for the Fock space F2(C) on the complex plane C (Zhu (2015) ).
|Number of pages||33|
|Journal||Advances in Mathematics|
|Publication status||Published - 25 Mar 2020|
- Fock space
- Singular integral operator
- Bargmann transform
- Riesz transform
- Reducing subspace
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Harmonic analysis and dispersive partial differential equations
Li, J., Guo, Z., Kenig, C. & Nakanishi, K.
31/01/17 → …
Harmonic analysis: function spaces and partial differential equations
Duong, X., Hofmann, S., Ouhabaz, E. M. & Wick, B.
11/02/19 → 10/02/22