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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.13249 |
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| _version_ | 1866914377153642496 |
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| author | Kania, Tomasz Maślany, Natalia |
| author_facet | Kania, Tomasz Maślany, Natalia |
| contents | We study Raja's covering index $Θ_X(n)$ for classical $L_p$-spaces and their non-commutative counterparts. For infinite-dimensional Hilbert spaces we compute the covering index exactly, proving \[ Θ_H(n)=n^{-1/2}\qquad(n\in\mathbb N); \] in particular $Θ_H(2)=1/\sqrt2$, thus answering a question of Raja about the precise two-piece covering index of $\elltwo$. For scalar-valued Lebesgue spaces $L_p(μ)$, $1\le p<\infty$, we construct an explicit block decomposition of the unit ball yielding the upper bound $Θ_{L_p(μ)}(n)\le n^{-1/p}$ for all $n\in\mathbb{N}$; in particular $Θ_{\ell_p}(n)\le n^{-1/p}$. For $1<p<\infty$, under the corresponding $p$-AUS renormability hypothesis, this combines with Raja's general lower bound to give the sharp asymptotic estimate $Θ_{L_p(μ)}(n)\asymp n^{-1/p}$.
We also obtain uniform upper bounds $Θ_{L_p(μ;E)}(n)\le n^{-1/p}$ for Bochner spaces $L_p(μ;E)$ over non-atomic $σ$-finite measure spaces, with constants independent of the Banach space $E$; this shows that, at the level of power-type upper estimates, the covering index decays at the same rate regardless of the asymptotic geometry of~$E$ and provides a partial negative answer to a problem of Raja. Finally, using non-commutative Clarkson inequalities, we derive power-type lower bounds $Θ_{L_p(M,τ)}(n)\gtrsim n^{-1/r}$ for non-commutative $L_p(M,τ)$ spaces associated with semifinite von Neumann algebras, where $r=\min\{p,2\}$. We do not attempt to optimise the exponent or constants in the non-commutative setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13249 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Raja's covering index of $L_p$ spaces Kania, Tomasz Maślany, Natalia Functional Analysis Primary 46B20, Secondary 46B03, 46B25 We study Raja's covering index $Θ_X(n)$ for classical $L_p$-spaces and their non-commutative counterparts. For infinite-dimensional Hilbert spaces we compute the covering index exactly, proving \[ Θ_H(n)=n^{-1/2}\qquad(n\in\mathbb N); \] in particular $Θ_H(2)=1/\sqrt2$, thus answering a question of Raja about the precise two-piece covering index of $\elltwo$. For scalar-valued Lebesgue spaces $L_p(μ)$, $1\le p<\infty$, we construct an explicit block decomposition of the unit ball yielding the upper bound $Θ_{L_p(μ)}(n)\le n^{-1/p}$ for all $n\in\mathbb{N}$; in particular $Θ_{\ell_p}(n)\le n^{-1/p}$. For $1<p<\infty$, under the corresponding $p$-AUS renormability hypothesis, this combines with Raja's general lower bound to give the sharp asymptotic estimate $Θ_{L_p(μ)}(n)\asymp n^{-1/p}$. We also obtain uniform upper bounds $Θ_{L_p(μ;E)}(n)\le n^{-1/p}$ for Bochner spaces $L_p(μ;E)$ over non-atomic $σ$-finite measure spaces, with constants independent of the Banach space $E$; this shows that, at the level of power-type upper estimates, the covering index decays at the same rate regardless of the asymptotic geometry of~$E$ and provides a partial negative answer to a problem of Raja. Finally, using non-commutative Clarkson inequalities, we derive power-type lower bounds $Θ_{L_p(M,τ)}(n)\gtrsim n^{-1/r}$ for non-commutative $L_p(M,τ)$ spaces associated with semifinite von Neumann algebras, where $r=\min\{p,2\}$. We do not attempt to optimise the exponent or constants in the non-commutative setting. |
| title | Raja's covering index of $L_p$ spaces |
| topic | Functional Analysis Primary 46B20, Secondary 46B03, 46B25 |
| url | https://arxiv.org/abs/2512.13249 |