ANALYSIS OF FRACTAL DIMENSION OF MIXED RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL

Subhash Chandra, Syed Abbas

correctmedium confidence

Category: math.DS
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves that if f ≥ 0 is continuous on [0,1]^2 and dim_B Gr(f)=2, then dim_B Gr(I_γ f)=2 (Theorem 5.1), via an estimate relating N_δ(Gr(I_γ f)) to N_δ(Gr(f)) and a standard covering bound (Lemma 5.1), yielding an upper bound ≤2 and the trivial lower bound ≥2 by projection . The model gives a different—and strictly stronger—argument: using positivity of the Riemann–Liouville kernels and f≥0, I_γ f is coordinatewise nondecreasing; a telescoping sum of mixed second differences plus one-sided increments shows Σ osc_Q(I_γ f)=O(δ^{-1}), and Lemma 5.1 then implies N_δ(Gr(I_γ f))=O(δ^{-2}), hence dim_B=2 without assuming dim_B Gr(f)=2. Thus the paper’s statement is correct but not optimal; the model’s proof is correct and strictly strengthens it (still under f≥0) .

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript offers a coherent study of the fractal (box and Hausdorff) dimensions of graphs of mixed Riemann–Liouville fractional integrals across several function classes and collects useful bounds. The main two-dimensional result (Theorem 5.1) is correct under its hypotheses. However, the nonnegativity requirement appears unused in the proof, and, moreover, a direct positivity/monotonicity argument yields a stronger conclusion when f ≥ 0 without assuming dim\_B Gr(f)=2. Clarifying these aspects and explicitly invoking the Lipschitz projection for the lower bound would sharpen the contribution.