Isometries of weak-$L_p$ spaces under natural quasi-norms

Characterize the positive linear isometries of commutative and noncommutative weak-$L_p$ spaces equipped with their natural quasi-norms.

Background

The paper identifies the natural weak-LpL_p quasi-norms as a setting in which logarithmic-submajorisation monotonicity fails. It notes that existing isometry results primarily concern symmetrically quasi-normed spaces whose quasi-norms do satisfy this monotonicity, leaving the corresponding classification problem for natural weak-LpL_p quasi-norms unresolved.

References

This fact leads to several open problems, e.g., whether the H"older type inequalities given in Proposition 5.8, 5.12 hold without the assumption of monotonicity with respect to $\prec\prec_{\log}$, and how to describe (positive) linear isometries on (commutative and noncommutative) weak-$L_p$ spaces under the natural quasi-norms (see and references therein for results concerning isometries on symmetrically (quasi-)normed spaces whose (quasi-)norms are monotone with respect to $\prec\prec_{\log}$).

— Norms of multiplication operators: answering Fialkow--Loebl question  (2608.18449 - Huang et al., 19 Aug 2026) in Example (Example \ref{Lp}), final paragraph

We do not know the description of surjective isometries on $L_{p,\infty}\left(\mathcal{M},\tau\right), ~0<p\le1$, when $\mathcal{M}$ is non-atomic.

— Extreme points of the unit ball and isometries of noncommutative quasi-Banach Marcinkiewicz spaces  (2610.01043 - Fang et al., 1 Oct 2026) in Chapter 1, Section “Isometries on noncommutative weak Lp-spaces, 1<p<∞ and C_{p,∞}, 0<p<∞”, Remark following Theorem σ-finite

However, the noncommutative version of Theorem 2 is not established yet.

— Extreme points of the unit ball and isometries of noncommutative quasi-Banach Marcinkiewicz spaces  (2610.01043 - Fang et al., 1 Oct 2026) in Chapter 4, Section “Descriptions of positive isometries”, paragraph immediately preceding the proof of Theorem Marcin6