Carleson Measures and Integral Operators on Fock--Sobolev Spaces
Abstract
For integers n≥1 and N≥0, let g=(g0,…,gN) be a vector of entire functions. We consider the generalized integral operator \ [T_{\mathbf g}^{(n,N)}f =I^n\!\left(\sum_{j=0}^N g_jf^{(j)}\right), \qquad (Ih)(z)=\int_0^z h(w)\dd w. \] We characterize its boundedness and compactness between Fock--Sobolev spaces Fmp and Fℓq, where 0<p,q<∞ and m,ℓ≥0 are integers. The coefficients of critical total degree determine the operator modulo compact operators and yield essential-norm estimates. In the Hilbert case, a Laurent Toeplitz representation gives the exact essential norm, a Fredholm criterion and an index formula. As an application, we obtain conditions for solutions of linear differential equations to belong to Fock--Sobolev spaces, including a necessary and sufficient polynomial zero condition in the Hilbert case. For entire functions u0,…,uN and φ, we also characterize boundedness, compactness and essential norms of the operators Lu,φ(N)f =∑j=0N uj f(j)∘φ. The proofs use Carleson measure characterizations for finite sums of weighted differentiation operators on Fock--Sobolev spaces.
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BibTeX
@misc{dong2026carleson,
title = {{Carleson Measures and Integral Operators on Fock--Sobolev Spaces}},
author = {Jianxiang Dong and Shibo Hu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01959},
eprint = {2611.01959},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01959},
}
RIS
TY - UNPB
TI - Carleson Measures and Integral Operators on Fock--Sobolev Spaces
AU - Jianxiang Dong
AU - Shibo Hu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01959
UR - https://arxiv.org/abs/2611.01959
AB - For integers n≥1 and N≥0, let g=(g_0,…,g_N) be a vector of entire functions. We consider the generalized integral operator \ [T_{\mathbf g}^{(n,N)}f =I^n\!\left(\sum_{j=0}^N g_jf^{(j)}\right), \qquad (Ih)(z)=\int_0^z h(w)\dd w. \] We characterize its boundedness and compactness between Fock--Sobolev spaces F_m^p and F_ℓ^q, where 0<p,q<∞ and m,ℓ≥0 are integers. The coefficients of critical total degree determine the operator modulo compact operators and yield essential-norm estimates. In the Hilbert case, a Laurent Toeplitz representation gives the exact essential norm, a Fredholm criterion and an index formula. As an application, we obtain conditions for solutions of linear differential equations to belong to Fock--Sobolev spaces, including a necessary and sufficient polynomial zero condition in the Hilbert case. For entire functions u_0,…,u_N and φ, we also characterize boundedness, compactness and essential norms of the operators L_u,φ^(N)f =∑_j=0^N u_j f^(j)∘φ. The proofs use Carleson measure characterizations for finite sums of weighted differentiation operators on Fock--Sobolev spaces.
ER -