Fock Projections between Gaussian Mixed-Norm Spaces: Full-Range Boundedness and Compactness
Abstract
The Fock--Bargmann projection Pt is initially an orthogonal projection on a Gaussian Hilbert space. We characterize the boundedness and compactness of Pt:Lαp1,q1⟶ ℱβp2,q2 and of its associated maximal projection Pt+:Lαp1,q1⟶ Lβp2,q2 for all t,α,β>0 and 0<p1,q1,p2,q2≤∞. Set Δ(t;α,β):=t2−2βt+αβ, d(p,q):=1/q−1/p. For both operators, boundedness holds exactly when p1,q1≥1 and either Δ(t;α,β)<0, or Δ(t;α,β)=0 with q1≤ q2, d(p2,q2)≤ d(p1,q1). Both operators are compact exactly when p1,q1≥1 and either Δ(t;α,β)<0, or Δ(t;α,β)=0 with both of the above boundary inequalities strict. Thus boundedness forces the source exponents into the Banach range, whereas when Δ(t;α,β)<0, every target exponent is admissible, including quasi-Banach ones. Annular discretization reduces the problem to drifted Gaussian sequence operators, whose comparison with diagonal sequence operators yields the endpoint criteria and quantitative tail estimates. A dilation--multiplier transfer gives corresponding results for Gaussian-twisted Fock operators, including Berezin transforms.
Related
- Positive Toeplitz Operators Between Gaussian Mixed-Norm Fock Spaces
- Sharp Gaussian Mixed-Norm Theory for Fock Spaces
- Sampling and Interpolation in Gaussian Mixed-Norm Fock Spaces
- A note on the boundedness of the multidimensional Katugampola operator in Campanato spaces
- Some New Sharp Boundedness Result Concerning Linear functionals on Banach spaces
BibTeX
@misc{fang2026fock,
title = {{Fock Projections between Gaussian Mixed-Norm Spaces: Full-Range Boundedness and Compactness}},
author = {Xiang Fang and Pham Trong Tien},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41461},
eprint = {2610.41461},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41461},
}
RIS
TY - UNPB
TI - Fock Projections between Gaussian Mixed-Norm Spaces: Full-Range Boundedness and Compactness
AU - Xiang Fang
AU - Pham Trong Tien
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41461
UR - https://arxiv.org/abs/2610.41461
AB - The Fock--Bargmann projection P_t is initially an orthogonal projection on a Gaussian Hilbert space. We characterize the boundedness and compactness of P_t:L_α^p_1,q_1⟶ ℱ_β^p_2,q_2 and of its associated maximal projection P_t^+:L_α^p_1,q_1⟶ L_β^p_2,q_2 for all t,α,β>0 and 0<p_1,q_1,p_2,q_2≤∞. Set Δ(t;α,β):=t^2−2βt+αβ, d(p,q):=1/q−1/p. For both operators, boundedness holds exactly when p_1,q_1≥1 and either Δ(t;α,β)<0, or Δ(t;α,β)=0 with q_1≤ q_2, d(p_2,q_2)≤ d(p_1,q_1). Both operators are compact exactly when p_1,q_1≥1 and either Δ(t;α,β)<0, or Δ(t;α,β)=0 with both of the above boundary inequalities strict. Thus boundedness forces the source exponents into the Banach range, whereas when Δ(t;α,β)<0, every target exponent is admissible, including quasi-Banach ones. Annular discretization reduces the problem to drifted Gaussian sequence operators, whose comparison with diagonal sequence operators yields the endpoint criteria and quantitative tail estimates. A dilation--multiplier transfer gives corresponding results for Gaussian-twisted Fock operators, including Berezin transforms.
ER -