Symmetric Bases and ℓp Embeddings in Musielak--Orlicz Sequence Spaces under δ2
Abstract
We study isomorphic embeddings into Musielak-Orlicz sequence spaces ℓΦ, where the normalized generating sequence Φ={ϕn} satisfies the δ2 condition. Using admissible normalized blocks, we construct an associated limiting family 𝒞Φ,τ of Orlicz functions. If a Banach space X with a symmetric basis embeds into ℓΦ, then there is a function ψ∈𝒞Φ,τ such that the basis of X is equivalent to the canonical basis of ℓψ. An Orlicz sequence space ℓψ embeds if and only if ψ is equivalent near zero to a member of 𝒞Φ,τ; in particular, ℓp embeds exactly when such a member is equivalent to tp. Assuming differentiability of the generators on a common interval near zero, we also derive the necessary bound p≤ bΦ,(cn),δ, where the upper growth index incorporates the summable errors in the δ2 condition. The same index yields cotype q for every q≥2 with q>bΦ,(cn),δ. Examples with nonuniform growth and lacunary weights illustrate this bound. For alternating Nakano exponents, the explicit limiting family gives a sharper restriction than the index alone. For general Nakano exponents, we recover the known exact cotype criterion, including attainment of the optimal exponent. For weighted Orlicz spaces, we express the cotype criterion through summable scalar errors and recover the formula using the global upper growth index when w∈𝒜.
Related
- The best Musielak-Orlicz approximation by linear subspaces
- Isometric embeddings of noncommutative Lp-spaces into noncommutative symmetric spaces
- Embeddings of Derived Categories of Moduli Spaces into Symmetric Powers
- Continuous and compact embeddings for a class of modular sequence spaces
- Rigidity of symmetric holomorphic functions on infinite-dimensional sequence spaces
BibTeX
@misc{georgiev2026symmetric,
title = {{Symmetric Bases and $\ell^p$ Embeddings in Musielak--Orlicz Sequence Spaces under $δ_2$}},
author = {Valentin Georgiev and Boyan Zlatanov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15504},
eprint = {2610.15504},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15504},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Symmetric Bases and ℓ^p Embeddings in Musielak--Orlicz Sequence Spaces under δ_2
AU - Valentin Georgiev
AU - Boyan Zlatanov
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15504
UR - https://arxiv.org/abs/2610.15504
AB - We study isomorphic embeddings into Musielak-Orlicz sequence spaces ℓ_Φ, where the normalized generating sequence Φ={ϕ_n} satisfies the δ_2 condition. Using admissible normalized blocks, we construct an associated limiting family 𝒞_Φ,τ of Orlicz functions. If a Banach space X with a symmetric basis embeds into ℓ_Φ, then there is a function ψ∈𝒞_Φ,τ such that the basis of X is equivalent to the canonical basis of ℓ_ψ. An Orlicz sequence space ℓ_ψ embeds if and only if ψ is equivalent near zero to a member of 𝒞_Φ,τ; in particular, ℓ^p embeds exactly when such a member is equivalent to t^p. Assuming differentiability of the generators on a common interval near zero, we also derive the necessary bound p≤ b_Φ,(c_n),δ, where the upper growth index incorporates the summable errors in the δ_2 condition. The same index yields cotype q for every q≥2 with q>b_Φ,(c_n),δ. Examples with nonuniform growth and lacunary weights illustrate this bound. For alternating Nakano exponents, the explicit limiting family gives a sharper restriction than the index alone. For general Nakano exponents, we recover the known exact cotype criterion, including attainment of the optimal exponent. For weighted Orlicz spaces, we express the cotype criterion through summable scalar errors and recover the formula using the global upper growth index when w∈𝒜.
ER -