Barnsley-Navascués fractal operators on Banach spaces on the Sierpiński gasket

Asheesh Kumar Yadav, Himanshu Kumar, Saurabh Verma, Bilel Selmi

correctmedium confidence

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:57 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper establishes all eight properties of the Barnsley–Navascués fractal operator F_{α,N,T} on the oscillation space C_β(SG) with the sharp scaling constant 2^{Nβ} and the smallness conditions 2^{Nβ}(∥α∥_∞+∥α∥_{C_β})<1 and 2^{Nβ}(∥α∥_∞+∥α∥_{C_β})∥T∥<1. The candidate solution proves the same properties via a compact operator-theoretic identity (Id−S)F=(Id−ST) with an explicit bound ∥S∥≤2^{Nβ}(∥α∥_∞+∥α∥_{C_β}). Constants, hypotheses, and conclusions match the paper’s statements, though the proofs differ in style. Minor typographical inconsistencies in the paper’s constants (missing carets in 2^{Nβ}) do not affect correctness.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript correctly develops and analyzes Barnsley–Navascués fractal operators on the Sierpiński gasket and proves robust norm and invertibility results in oscillation spaces. The arguments are sound and sufficiently detailed. Minor typographical inconsistencies in the constants (e.g., exponents on 2\^{Nβ}) should be fixed, and the presentation could be tightened by highlighting an operator-theoretic identity that underlies several bounds.