Chronological Lasota--Yorke bounds for nonstationary compositions

Definitions and mathematical statement

The source asks for Lasota--Yorke-type inequalities for compositions of operators, but does not specify the operator class, spaces, norms or desired uniformity. The supplied research aid gives this abstract strong/weak formulation. Let $\mathcal B$ be continuously embedded in a weaker normed space, with strong norm $\|\cdot\|_s$ and weak norm $|\cdot|_w$. Suppose a sequence of linear operators $L_j$ acts on these spaces and, for all $v\in\mathcal B$, satisfies \[ \|L_jv\|_s\leq a_j\|v\|_s+b_j|v|_w, \qquad |L_jv|_w\leq c_j|v|_w, \qquad a_j,b_j,c_j\geq0. \] Put $L^{(n)}=L_n\cdots L_1$, so time runs from right to left. The reported chronological composition inequality is \[ \|L^{(n)}v\|_s\leq \left(\prod_{i=1}^n a_i\right)\|v\|_s +\left(\sum_{j=1}^n b_j \prod_{i=j+1}^n a_i\prod_{i=1}^{j-1}c_i\right)|v|_w, \] with empty products equal to $1$. Thus the defect introduced at step $j$ is multiplied by earlier weak factors and later strong factors; interchanging those chronological roles would give a different assertion.

The aid's uniform sufficient condition is: $c_j\leq1$, $b_j\leq B$, $a_j\leq A$ for fixed nonnegative $A,B$, and there are $r\geq1$ and $0<\rho<1$ such that \[ \prod_{i=k+1}^{k+r}a_i\leq\rho\qquad\text{for every }k\geq0. \] Then the strong initial coefficient decays exponentially; explicitly, writing $n=qr+s$ with $0\leq s<r$, the reported bound has the form \[ \|L^{(n)}v\|_s\leq A^s\rho^q\|v\|_s +\frac{B(1+A+\cdots+A^{r-1})}{1-\rho}|v|_w. \] The block condition is sliding and uniform in the starting time, and permits individual $a_j$ to exceed one. The summary also reports a scalar counterexample showing that the weaker assertion $a_j<1$ at every step alone need not give decaying products. No specific dynamical transfer operators or application-dependent strong/weak spaces are supplied, so this abstract criterion is retained as partial progress rather than a universal answer for arbitrary operator compositions.

Short English statement

The question asks how a strong/weak regularity estimate behaves under successive operators. The supplied partial result keeps track of the order in which defects are introduced and gives a uniform bound when each fixed-length sliding block contracts. Individual contraction factors below one do not by themselves guarantee decay without uniform control.

Sources

• Local UnsolvedMath registry, permanent ID 20001391, exact source code catalogue entry; canonical aim-dynamical-systems-notes.json, notes[48], source item 6, tag "problem", workshop Stochastic methods for non-equilibrium dynamical systems; https://aimath.org/pastworkshops/noneqdynsysproblems.pdf.
• The original spells the term "Lasota-York" and contains a line-break hyphen in "operators"; the conventional Lasota--Yorke spelling is used here without changing the question. No proposer or separate authoritative bibliographic reference is supplied. Original remarks and literature field are empty; the verification only introduces the canonical quotation.
• Supplied metadata: PARTIAL-PROGRESS, registry status "partially_solved", statement status "reconstructed_unverified"; attempt 1, result "partial_result", problem status at run "partially_solved", recorded rigor "proved", novelty confidence "low", candidate type "explicit criterion". The machine aid is not peer reviewed and explicitly requires independent verification before citation. No external source was visited.