Sharp Gaussian Mixed-Norm Theory for Fock Spaces
Abstract
We develop a structural theory of the Gaussian mixed-norm Fock spaces ℱαp,q, where α>0 and 0<p,q≤∞, which separate angular Lp-means from radial Gaussian Lq-summability. We first prove the sharp circle-mean estimate Mp(f,r)≤ Cα,q(1+r)−1/q eα/2r2lVert frVertℱαp,q, r≥0, where Mp(f,r) denotes the angular Lp-mean of f on the circle lvert zrvert=r with the convention 1/∞=0, and show that the exponent −1/q is optimal. The proof is based on a local mass principle for convex Laplace-type exponents, which also yields an annular discretization of the mixed Gaussian norm. Using this discretization, we characterize the continuous embeddings between mixed-norm Fock spaces and obtain an atomic decomposition in terms of normalized Fock kernels, with coefficients belonging to a corresponding annular mixed sequence space. We further characterize Carleson and vanishing Carleson measures and identify the continuous duals in the Banach, quasi-Banach, and endpoint regimes. For p<1, the atomic decomposition relies on a quantitative re-centering theorem that controls expansions of normalized Fock kernels under perturbations of their centers. This theorem is of independent interest even for the Hilbert Fock space ℱα2.
Related
- Positive Toeplitz Operators Between Gaussian Mixed-Norm Fock Spaces
- Fock Projections between Gaussian Mixed-Norm Spaces: Full-Range Boundedness and Compactness
- Sampling and Interpolation in Gaussian Mixed-Norm Fock Spaces
- Sharp Injective Norm Bounds for Inhomogeneous Gaussian Tensors and Sparse Extensions
- Mixed-norm Herz-type Triebel-Lizorkin spaces
BibTeX
@misc{fang2026sharp,
title = {{Sharp Gaussian Mixed-Norm Theory for Fock Spaces}},
author = {Xiang Fang and Pham Trong Tien},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26821},
eprint = {2609.26821},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26821},
}
RIS
TY - UNPB
TI - Sharp Gaussian Mixed-Norm Theory for Fock Spaces
AU - Xiang Fang
AU - Pham Trong Tien
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.26821
UR - https://arxiv.org/abs/2609.26821
AB - We develop a structural theory of the Gaussian mixed-norm Fock spaces ℱ_α^p,q, where α>0 and 0<p,q≤∞, which separate angular L^p-means from radial Gaussian L^q-summability. We first prove the sharp circle-mean estimate M_p(f,r)≤ C_α,q(1+r)^−1/q e^α/2r^2lVert frVert_ℱ_α^p,q, r≥0, where M_p(f,r) denotes the angular L^p-mean of f on the circle lvert zrvert=r with the convention 1/∞=0, and show that the exponent −1/q is optimal. The proof is based on a local mass principle for convex Laplace-type exponents, which also yields an annular discretization of the mixed Gaussian norm. Using this discretization, we characterize the continuous embeddings between mixed-norm Fock spaces and obtain an atomic decomposition in terms of normalized Fock kernels, with coefficients belonging to a corresponding annular mixed sequence space. We further characterize Carleson and vanishing Carleson measures and identify the continuous duals in the Banach, quasi-Banach, and endpoint regimes. For p<1, the atomic decomposition relies on a quantitative re-centering theorem that controls expansions of normalized Fock kernels under perturbations of their centers. This theorem is of independent interest even for the Hilbert Fock space ℱ_α^2.
ER -